Charged Quark Stars and Extreme Compact Objects in Regularized 4D Einstein-Gauss-Bonnet Gravity

Michael Gammon mgammon@uwaterloo.ca Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1    Robert B. Mann rbmann@uwaterloo.ca Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1    Sarah Rourke sarah.a.rourke@mail.mcgill.ca Department of Physics, McGill University, Montreal, Quebec, Canada
Abstract

Since the derivation of a well-defined D4𝐷4D\rightarrow 4italic_D → 4 limit for 4 dimensional Einstein Gauss-Bonnet (4DEGB) gravity coupled to a scalar field, there has been interest in testing it as an alternative to Einstein’s general theory of relativity. Using the Tolman-Oppenheimer-Volkoff (TOV) equations modified for charge and 4DEGB gravity, we model the stellar structure of charged, non-interacting quark stars. We find that increasing the Gauss-Bonnet coupling constant α𝛼\alphaitalic_α or the charge Q𝑄Qitalic_Q both tend to increase the mass-radius profiles of quark stars described by this theory, allowing a given central pressure to support larger quark stars in general. We also derive a generalization of the Buchdahl bound for charged stars in 4DEGB gravity. As in the uncharged case, we find that quark stars can exist below the general relativistic Buchdahl bound (BB) and Schwarzschild radius R=2M𝑅2𝑀R=2Mitalic_R = 2 italic_M, due to the lack of a mass gap between black holes and compact stars in the 4DEGB theory. Even for α𝛼\alphaitalic_α well within current observational constraints, we find that quark star solutions in this theory can describe Extreme Compact Charged Objects (ECCOs), objects whose radii are smaller than what is allowed by general relativity.

I Introduction

Modified theories of gravity continue to attract attention despite the empirical success of general relativity (GR). These theories are motivated by a variety of problems, including addressing issues in modern cosmology bueno2016 ; sotiriou_2010 ; nojiribook ; clifton_2012 , quantizing gravity ahmed2017 ; stelle1977 , eliminating singularities brandenberger1992nonsingular ; BRANDENBERGER_1993 ; brandenberger1995implementing , and, perhaps most importantly, finding viable phenomenological competitors against which GR can be tested in the most stringent manner possible.

Higher curvature theories (or HCTs) are amongst the most popular modifications. An HCT modifies the assumed linear relationship in GR between the curvature and the stress-energy, replacing the former with an arbitrary sum of powers of the curvature tensor (appropriately contracted to two indices). Such modifications provide us with a foil to further challenge the empirical success of GR, while also making new testable predictions.

Lovelock theories lovelock1971 have long been at the forefront of this search, since they possess the distinctive feature of having 2nd order differential equations of motion. The physical significance of such theories has been unclear, however, since their higher order terms yield non-trivial contributions to the equations of motion only in more than four spacetime dimensions (D>4𝐷4D>4italic_D > 4).

Recently this restriction was circumvented hennigar_2020_on ; Fernandes:2020nbq in the quadratic case, or what is better known as “Einstein-Gauss-Bonnet” gravity . The Gauss-Bonnet (GB) contribution to the gravitational action is

SDGB=αdDxg[RμνρτRμνρτ4RμνRμν+R2]αdDxg𝒢superscriptsubscript𝑆𝐷𝐺𝐵𝛼superscript𝑑𝐷𝑥𝑔delimited-[]superscript𝑅𝜇𝜈𝜌𝜏subscript𝑅𝜇𝜈𝜌𝜏4superscript𝑅𝜇𝜈subscript𝑅𝜇𝜈superscript𝑅2𝛼superscript𝑑𝐷𝑥𝑔𝒢S_{D}^{GB}=\alpha\int d^{D}x\sqrt{-g}\left[R^{\mu\nu\rho\tau}R_{\mu\nu\rho\tau% }-4R^{\mu\nu}R_{\mu\nu}+R^{2}\right]\equiv\alpha\int d^{D}x\sqrt{-g}\mathcal{G}italic_S start_POSTSUBSCRIPT italic_D end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_G italic_B end_POSTSUPERSCRIPT = italic_α ∫ italic_d start_POSTSUPERSCRIPT italic_D end_POSTSUPERSCRIPT italic_x square-root start_ARG - italic_g end_ARG [ italic_R start_POSTSUPERSCRIPT italic_μ italic_ν italic_ρ italic_τ end_POSTSUPERSCRIPT italic_R start_POSTSUBSCRIPT italic_μ italic_ν italic_ρ italic_τ end_POSTSUBSCRIPT - 4 italic_R start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT italic_R start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT + italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] ≡ italic_α ∫ italic_d start_POSTSUPERSCRIPT italic_D end_POSTSUPERSCRIPT italic_x square-root start_ARG - italic_g end_ARG caligraphic_G (1)

(where Rμνρτsubscript𝑅𝜇𝜈𝜌𝜏R_{\mu\nu\rho\tau}italic_R start_POSTSUBSCRIPT italic_μ italic_ν italic_ρ italic_τ end_POSTSUBSCRIPT is the Riemann curvature tensor), which becomes the integral of a total derivative in D=4𝐷4D=4italic_D = 4, and thus cannot contribute to a system’s gravitational dynamics in less than five dimensions. For this reason it is often referred to as a “topological term” having no relevance to physical problems. Indeed, the Lovelock theorem lovelock1971 ensures that a D=4𝐷4D=4italic_D = 4 dimensional metric theory of gravity must incorporate additional fields in order to have second order equations of motion and diffeomorphism invariance.

A few years ago it was noted glavan2020 that several exact solutions to D𝐷Ditalic_D-dimensional Einstein-Gauss Bonnet gravity have a sensible limit under the rescaling

limD4(D4)αα,subscript𝐷4𝐷4𝛼𝛼\lim_{D\to 4}(D-4)\alpha\rightarrow\alpha,roman_lim start_POSTSUBSCRIPT italic_D → 4 end_POSTSUBSCRIPT ( italic_D - 4 ) italic_α → italic_α , (2)

of the Gauss-Bonnet coupling constant. Using this approach a variety of 4-dimensional metrics can be obtained, including cosmological glavan2020 ; li2020 ; kobayashi2020 , spherical black hole glavan2020 ; kumar2022 ; fernandes2020 ; kumar2020 ; kumar2022_2 , collapsing malafarina2020 , star-like doneva2021 ; charmousis2022 , and radiating ghosh2020 metrics, each carrying imprints of the quadratic curvature effects of their D>4𝐷4D>4italic_D > 4 counterparts. However a number of objections to this approach were subsequently raised gurses2020 ; ai2020 ; shu2020 , based on the fact that the existence of a limiting solution does not imply the existence of a well-defined 4D theory whose field equations have that solution. This shortcoming was quickly addressed when it was shown that the D4𝐷4D\to 4italic_D → 4 limit in (2) can be taken in the gravitational action hennigar_2020_on ; Fernandes:2020nbq , generalizing an earlier procedure employed in obtaining the D2𝐷2D\to 2italic_D → 2 limit of GR Mann:1992ar . One can also compactify D𝐷Ditalic_D-dimensional Gauss-Bonnet gravity on a (D4)𝐷4(D-4)( italic_D - 4 )-dimensional maximally symmetric space and then use (2) to obtain a D=4𝐷4D=4italic_D = 4 HCT Lu:2020iav . This approach yields the same result (up to trivial field redefinitions), in addition to terms depending on the curvature of the maximally symmetric (D4)𝐷4(D-4)( italic_D - 4 )-dimensional space. Taking this to vanish yields

S4GB=superscriptsubscript𝑆4𝐺𝐵absent\displaystyle S_{4}^{GB}=italic_S start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_G italic_B end_POSTSUPERSCRIPT = αd4xg[ϕ𝒢+4Gμνμϕνϕ4(ϕ)2ϕ+2(ϕ)4]𝛼superscript𝑑4𝑥𝑔delimited-[]italic-ϕ𝒢4subscript𝐺𝜇𝜈superscript𝜇italic-ϕsuperscript𝜈italic-ϕ4superscriptitalic-ϕ2italic-ϕ2superscriptitalic-ϕ4\displaystyle\alpha\int d^{4}x\sqrt{-g}\left[\phi\mathcal{G}+4G_{\mu\nu}\nabla% ^{\mu}\phi\nabla^{\nu}\phi-4(\nabla\phi)^{2}\square\phi+2(\nabla\phi)^{4}\right]italic_α ∫ italic_d start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_x square-root start_ARG - italic_g end_ARG [ italic_ϕ caligraphic_G + 4 italic_G start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ∇ start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT italic_ϕ ∇ start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT italic_ϕ - 4 ( ∇ italic_ϕ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT □ italic_ϕ + 2 ( ∇ italic_ϕ ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT ] (3)

where we see that an additional scalar field ϕitalic-ϕ\phiitalic_ϕ appears. Surprisingly, the spherically symmetric black hole solutions to the field equations match those from the naïve D4𝐷4D\rightarrow 4italic_D → 4 limit of D>4𝐷4D>4italic_D > 4 solutions glavan2020 . The resultant 4D scalar-tensor theory is a particular type of Horndeski theory horndeski1974 , and solutions to its equations of motion can be obtained without ever referencing a higher dimensional spacetime hennigar_2020_on .

We are interested here in what is called 4D Einstein-Gauss-Bonnet gravity (4DEGB), whose action is given by (3) plus the Einstein-Hilbert term:

S𝑆\displaystyle Sitalic_S =SGR+S4GBabsentsuperscript𝑆𝐺𝑅superscriptsubscript𝑆4𝐺𝐵\displaystyle=S^{GR}+S_{4}^{GB}= italic_S start_POSTSUPERSCRIPT italic_G italic_R end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_G italic_B end_POSTSUPERSCRIPT
=18πGd4xg[R+α{ϕ𝒢+4Gμνμϕνϕ4(ϕ)2ϕ+2(ϕ)4}]absent18𝜋𝐺superscript𝑑4𝑥𝑔delimited-[]𝑅𝛼italic-ϕ𝒢4subscript𝐺𝜇𝜈superscript𝜇italic-ϕsuperscript𝜈italic-ϕ4superscriptitalic-ϕ2italic-ϕ2superscriptitalic-ϕ4\displaystyle=\frac{1}{8\pi G}\int d^{4}x\sqrt{-g}\left[R+\alpha\left\{\phi% \mathcal{G}+4G_{\mu\nu}\nabla^{\mu}\phi\nabla^{\nu}\phi-4(\nabla\phi)^{2}% \square\phi+2(\nabla\phi)^{4}\right\}\right]= divide start_ARG 1 end_ARG start_ARG 8 italic_π italic_G end_ARG ∫ italic_d start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_x square-root start_ARG - italic_g end_ARG [ italic_R + italic_α { italic_ϕ caligraphic_G + 4 italic_G start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ∇ start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT italic_ϕ ∇ start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT italic_ϕ - 4 ( ∇ italic_ϕ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT □ italic_ϕ + 2 ( ∇ italic_ϕ ) start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT } ] (4)

which has been shown to be an interesting phenomenological competitor to GR Clifton:2020xhc ; charmousis2022 ; Zanoletti:2023ori . Despite much exploration Fernandes:2022zrq , the role played by these higher curvature terms in real gravitational dynamics is still not fully understood. One important arena for testing such theories against standard general relativity is via observations of compact astrophysical objects like neutron stars. The correct theory should be able to accurately describe recent gravitational wave observations of astrophysical objects existing in the mass gap between the heaviest compact stars and the lightest black holes.

Modern observational astrophysics is rich in findings of compact objects and as such our understanding of highly dense gravitational objects is rapidly advancing. However, there is as of yet no strong consensus on their underlying physics. A number of such objects have been recently observed that are inconsistent with standard GR and a simple neutron star equation of state. It was recently shown zhang_2021_unified ; zhang_2021_stellar that in standard general relativity, the secondary component of the merger GW190814 could feasibly be a quark star with an interacting equation of state governed by a single parameter λ𝜆\lambdaitalic_λ. This parameterization of strong interaction effects was inspired by another recent theoretical study showing that non-strange quark matter (QM) could feasibly be the ground state of baryonic matter at sufficient density and temperature holdom_2018_quark . Similar analyses with a different equation of state and/or QM phase miao_2021_bayesian ; lopes_2022_onthe ; oikonomou_2023_colourflavour found similarly promising results. This same object was subsequently shown to be well described as a slowly-rotating neutron star in the 4DEGB theory without resorting to exotic quark matter EOSs, while also demonstrating that the equilibrium sequence of neutron stars asymptotically matches the black hole limit, thus closing the mass gap between NS/black holes of the same radius charmousis2022 . More recently, some groups have also been interested in modelling ECOs (extreme compact objects) zhang_2023_rescaling ; gammon_2024 ; Mann:2021mnc ; Conklin:2021cbc as well as unusually light compact stars horvath_2023_alight ; oikonomou_2023_colourflavour (like that in the gamma ray remnant J1731-347, which is inconsistent with minimum mass calculations of neutron stars generated by iron cores) as quark stars to explain their unusual properties.

To further illuminate the range of possibilities, we consider in this paper charged quark star solutions to the 4DEGB theory, as the confining nature of the strong interaction make quark stars one of the few remaining candidates for charged stellar objects. Although charged quark star solutions have previously been considered in the context of 4DEGB pretel_2022 , the upper limit of the coupling α𝛼\alphaitalic_α was taken to be much smaller than that allowed by current observational constraints Fernandes:2022zrq ; Clifton:2020xhc , and the relation with the charged 4DEGB Buchdahl bound was not discussed (as the bound was not derived). In considering values of α𝛼\alphaitalic_α closer to presently allowed bounds, we obtain a number of interesting novel results.

Our most intriguing result (extending similar results found in gammon_2024 ) is that charged quark stars in 4DEGB can be Extreme Compact Charged Objects (ECCOs), objects whose radii are smaller than that allowed by the Buchdahl bound and Schwarzschild radius in GR. Indeed, there exist uncharged quark stars in 4DEGB whose radii are smaller than that of a corresponding black hole of the same mass in GR111This phenomenon was also shown to be present for neutron stars charmousis2022 , though the relationship with the Buchdahl bound was not noted.. Observations of these latter objects, apart from indicating a new class of astrophysical phenomena Mann:2021mnc , would provide strong evidence for 4DEGB as a physical theory. These ECOs respect a generalization of the Buchdahl bound, whose small-radius limit is that of the horizon radius of the corresponding minimum mass black hole. ECCOs likewise respect a generalization of the charged Buchdahl bound as we shall demonstrate.

We also find in general that for a given central pressure, charged quark stars in 4DEGB have a larger mass and radius than their GR counterparts with the same pressure. This can be attributed to the ‘less attractive’ nature of gravity in 4DEGB: for α>0𝛼0\alpha>0italic_α > 0 the gradient of the effective potential yields a weaker force than pure GR. We consequently find a larger maximal mass for a given value of Q𝑄Qitalic_Q in 4DEGB than in general relativity. Similarly, increasing Q𝑄Qitalic_Q also weakens the overall force gradient compared to the uncharged case, and leads to larger quark stars relative to their uncharged counterparts.

The outline of our paper is as follows: In section II we introduce the basic theory underlying 4DEGB gravity, as well as the non-interacting quark matter equation of state that we make use of. Following this, a electromagnetic perfect fluid stress-energy tensor is employed to derive the charged 4DEGB TOV equations, and current observational constraints on the coupling constant are briefly discussed. We use section III to derive a generalization of Buchdahl’s bound for charged stars in 4DEGB gravity to be plotted alongside the solutions to the modified TOV equations. Section IV outlines the results of our numerical calculations. We conclude our results with a brief analysis of the stability of charged 4DEGB quark stars. Section VI summarizes our key findings and suggests topics for future study.

II Theory

II.1 4D Einstein-Gauss-Bonnet Gravity

To model the structure of a quark star in 4DEGB gravity, we add to the action (4) a perfect-fluid term corresponding to the energy of the quark matter and

SMax=18πGg(14FμνFμν+AμJμ)subscript𝑆Max18𝜋𝐺𝑔14subscript𝐹𝜇𝜈superscript𝐹𝜇𝜈subscript𝐴𝜇superscript𝐽𝜇S_{\mathrm{Max}}=-\frac{1}{8\pi G}\int\sqrt{-g}\left({\frac{1}{4}}F_{\mu\nu}F^% {\mu\nu}+A_{\mu}J^{\mu}\right)italic_S start_POSTSUBSCRIPT roman_Max end_POSTSUBSCRIPT = - divide start_ARG 1 end_ARG start_ARG 8 italic_π italic_G end_ARG ∫ square-root start_ARG - italic_g end_ARG ( divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_F start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT italic_F start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT + italic_A start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_J start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT ) (5)

where Jμsuperscript𝐽𝜇J^{\mu}italic_J start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT is the electromagnetic current of the charged quark matter. The field equations of 4DEGB are obtained from a straightforward variational principle applied to this action.

Variation with respect to the scalar ϕitalic-ϕ\phiitalic_ϕ yields

ϕ=subscriptitalic-ϕabsent\displaystyle\mathcal{E}_{\phi}=caligraphic_E start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT = 𝒢+8Gμννμϕ+8Rμνμϕνϕ8(ϕ)2+8(ϕ)2ϕ+16aϕνϕνμϕ𝒢8superscript𝐺𝜇𝜈subscript𝜈subscript𝜇italic-ϕ8superscript𝑅𝜇𝜈subscript𝜇italic-ϕsubscript𝜈italic-ϕ8superscriptitalic-ϕ28superscriptitalic-ϕ2italic-ϕ16superscript𝑎italic-ϕsuperscript𝜈italic-ϕsubscript𝜈subscript𝜇italic-ϕ\displaystyle-\mathcal{G}+8G^{\mu\nu}\nabla_{\nu}\nabla_{\mu}\phi+8R^{\mu\nu}% \nabla_{\mu}\phi\nabla_{\nu}\phi-8(\square\phi)^{2}+8(\nabla\phi)^{2}\square% \phi+16\nabla^{a}\phi\nabla^{\nu}\phi\nabla_{\nu}\nabla_{\mu}\phi- caligraphic_G + 8 italic_G start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT ∇ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ∇ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_ϕ + 8 italic_R start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT ∇ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_ϕ ∇ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT italic_ϕ - 8 ( □ italic_ϕ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 8 ( ∇ italic_ϕ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT □ italic_ϕ + 16 ∇ start_POSTSUPERSCRIPT italic_a end_POSTSUPERSCRIPT italic_ϕ ∇ start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT italic_ϕ ∇ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ∇ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_ϕ (6)
+8νμϕνμϕ8subscript𝜈subscript𝜇italic-ϕsuperscript𝜈superscript𝜇italic-ϕ\displaystyle\qquad+8\nabla_{\nu}\nabla_{\mu}\phi\nabla^{\nu}\nabla^{\mu}\phi+ 8 ∇ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ∇ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_ϕ ∇ start_POSTSUPERSCRIPT italic_ν end_POSTSUPERSCRIPT ∇ start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT italic_ϕ
=\displaystyle==  0 0\displaystyle\;0

while the variation with respect to the metric gives

μνsubscript𝜇𝜈\displaystyle\mathcal{E}_{\mu\nu}caligraphic_E start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT =Λgμν+Gμν+α[ϕHμν2R[(μϕ)(νϕ)+νμϕ]+8R(μσν)σϕ+8R(μσ(ν)ϕ)(σϕ)\displaystyle=\Lambda g_{\mu\nu}+G_{\mu\nu}+\alpha\left[\phi H_{\mu\nu}-2R% \left[\left(\nabla_{\mu}\phi\right)\left(\nabla_{\nu}\phi\right)+\nabla_{\nu}% \nabla_{\mu}\phi\right]+8R_{(\mu}^{\sigma}\nabla_{\nu)}\nabla_{\sigma}\phi+8R_% {(\mu}^{\sigma}\left(\nabla_{\nu)}\phi\right)\left(\nabla_{\sigma}\phi\right)\right.= roman_Λ italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT + italic_G start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT + italic_α [ italic_ϕ italic_H start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT - 2 italic_R [ ( ∇ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_ϕ ) ( ∇ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT italic_ϕ ) + ∇ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ∇ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_ϕ ] + 8 italic_R start_POSTSUBSCRIPT ( italic_μ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT ∇ start_POSTSUBSCRIPT italic_ν ) end_POSTSUBSCRIPT ∇ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT italic_ϕ + 8 italic_R start_POSTSUBSCRIPT ( italic_μ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT ( ∇ start_POSTSUBSCRIPT italic_ν ) end_POSTSUBSCRIPT italic_ϕ ) ( ∇ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT italic_ϕ ) (7)
2Gμν[(ϕ)2+2ϕ]4[(μϕ)(νϕ)+νμϕ]ϕ[gμν(ϕ)24(μϕ)(νϕ)](ϕ)22subscript𝐺𝜇𝜈delimited-[]superscriptitalic-ϕ22italic-ϕ4delimited-[]subscript𝜇italic-ϕsubscript𝜈italic-ϕsubscript𝜈subscript𝜇italic-ϕitalic-ϕdelimited-[]subscript𝑔𝜇𝜈superscriptitalic-ϕ24subscript𝜇italic-ϕsubscript𝜈italic-ϕsuperscriptitalic-ϕ2\displaystyle-2G_{\mu\nu}\left[(\nabla\phi)^{2}+2\square\phi\right]-4\left[% \left(\nabla_{\mu}\phi\right)\left(\nabla_{\nu}\phi\right)+\nabla_{\nu}\nabla_% {\mu}\phi\right]\square\phi-\left[g_{\mu\nu}(\nabla\phi)^{2}-4\left(\nabla_{% \mu}\phi\right)\left(\nabla_{\nu}\phi\right)\right](\nabla\phi)^{2}- 2 italic_G start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT [ ( ∇ italic_ϕ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 □ italic_ϕ ] - 4 [ ( ∇ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_ϕ ) ( ∇ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT italic_ϕ ) + ∇ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT ∇ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_ϕ ] □ italic_ϕ - [ italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ( ∇ italic_ϕ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 4 ( ∇ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_ϕ ) ( ∇ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT italic_ϕ ) ] ( ∇ italic_ϕ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT
+8((μϕ)(ν)σϕ)σϕ4gμνRσρ[σρϕ+(σϕ)(ρϕ)]+2gμν(ϕ)2\displaystyle+8\left(\nabla_{(\mu}\phi\right)\left(\nabla_{\nu)}\nabla_{\sigma% }\phi\right)\nabla^{\sigma}\phi-4g_{\mu\nu}R^{\sigma\rho}\left[\nabla_{\sigma}% \nabla_{\rho}\phi+\left(\nabla_{\sigma}\phi\right)\left(\nabla_{\rho}\phi% \right)\right]+2g_{\mu\nu}(\square\phi)^{2}+ 8 ( ∇ start_POSTSUBSCRIPT ( italic_μ end_POSTSUBSCRIPT italic_ϕ ) ( ∇ start_POSTSUBSCRIPT italic_ν ) end_POSTSUBSCRIPT ∇ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT italic_ϕ ) ∇ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT italic_ϕ - 4 italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT italic_R start_POSTSUPERSCRIPT italic_σ italic_ρ end_POSTSUPERSCRIPT [ ∇ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ∇ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT italic_ϕ + ( ∇ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT italic_ϕ ) ( ∇ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT italic_ϕ ) ] + 2 italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ( □ italic_ϕ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT
4gμν(σϕ)(ρϕ)(σρϕ)+4(σνϕ)(σμϕ)4subscript𝑔𝜇𝜈superscript𝜎italic-ϕsuperscript𝜌italic-ϕsubscript𝜎subscript𝜌italic-ϕ4subscript𝜎subscript𝜈italic-ϕsuperscript𝜎subscript𝜇italic-ϕ\displaystyle-4g_{\mu\nu}\left(\nabla^{\sigma}\phi\right)\left(\nabla^{\rho}% \phi\right)\left(\nabla_{\sigma}\nabla_{\rho}\phi\right)+4\left(\nabla_{\sigma% }\nabla_{\nu}\phi\right)\left(\nabla^{\sigma}\nabla_{\mu}\phi\right)- 4 italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ( ∇ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT italic_ϕ ) ( ∇ start_POSTSUPERSCRIPT italic_ρ end_POSTSUPERSCRIPT italic_ϕ ) ( ∇ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ∇ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT italic_ϕ ) + 4 ( ∇ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ∇ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT italic_ϕ ) ( ∇ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT ∇ start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT italic_ϕ )
2gμν(σρϕ)(σρϕ)+4Rμνσρ[(σϕ)(ρϕ)+ρσϕ]]\displaystyle\left.-2g_{\mu\nu}\left(\nabla_{\sigma}\nabla_{\rho}\phi\right)% \left(\nabla^{\sigma}\nabla^{\rho}\phi\right)+4R_{\mu\nu\sigma\rho}\left[\left% (\nabla^{\sigma}\phi\right)\left(\nabla^{\rho}\phi\right)+\nabla^{\rho}\nabla^% {\sigma}\phi\right]\right]- 2 italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT ( ∇ start_POSTSUBSCRIPT italic_σ end_POSTSUBSCRIPT ∇ start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT italic_ϕ ) ( ∇ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT ∇ start_POSTSUPERSCRIPT italic_ρ end_POSTSUPERSCRIPT italic_ϕ ) + 4 italic_R start_POSTSUBSCRIPT italic_μ italic_ν italic_σ italic_ρ end_POSTSUBSCRIPT [ ( ∇ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT italic_ϕ ) ( ∇ start_POSTSUPERSCRIPT italic_ρ end_POSTSUPERSCRIPT italic_ϕ ) + ∇ start_POSTSUPERSCRIPT italic_ρ end_POSTSUPERSCRIPT ∇ start_POSTSUPERSCRIPT italic_σ end_POSTSUPERSCRIPT italic_ϕ ] ]
=Tμνabsentsubscript𝑇𝜇𝜈\displaystyle=\;T_{\mu\nu}= italic_T start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT

where Tμνsubscript𝑇𝜇𝜈T_{\mu\nu}italic_T start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT is the stress-energy tensor of the charged quark matter and the electromagnetic field, and

Hμν=2[RRμν2RμανβRαβ+RμαβσRναβσ2RμαRνα14gμν𝒢]subscript𝐻𝜇𝜈2delimited-[]𝑅subscript𝑅𝜇𝜈2subscript𝑅𝜇𝛼𝜈𝛽superscript𝑅𝛼𝛽subscript𝑅𝜇𝛼𝛽𝜎superscriptsubscript𝑅𝜈𝛼𝛽𝜎2subscript𝑅𝜇𝛼superscriptsubscript𝑅𝜈𝛼14subscript𝑔𝜇𝜈𝒢\displaystyle H_{\mu\nu}=2\Big{[}RR_{\mu\nu}-2R_{\mu\alpha\nu\beta}R^{\alpha% \beta}+R_{\mu\alpha\beta\sigma}R_{\nu}^{\alpha\beta\sigma}-2R_{\mu\alpha}R_{% \nu}^{\alpha}-\frac{1}{4}g_{\mu\nu}\mathcal{G}\Big{]}italic_H start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT = 2 [ italic_R italic_R start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT - 2 italic_R start_POSTSUBSCRIPT italic_μ italic_α italic_ν italic_β end_POSTSUBSCRIPT italic_R start_POSTSUPERSCRIPT italic_α italic_β end_POSTSUPERSCRIPT + italic_R start_POSTSUBSCRIPT italic_μ italic_α italic_β italic_σ end_POSTSUBSCRIPT italic_R start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_α italic_β italic_σ end_POSTSUPERSCRIPT - 2 italic_R start_POSTSUBSCRIPT italic_μ italic_α end_POSTSUBSCRIPT italic_R start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_g start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT caligraphic_G ] (8)

is the Gauss-Bonnet tensor. These field equations satisfy the following relationship

gμνTμν=gμνμν+α2ϕ=4ΛRα2𝒢superscript𝑔𝜇𝜈subscript𝑇𝜇𝜈superscript𝑔𝜇𝜈subscript𝜇𝜈𝛼2subscriptitalic-ϕ4Λ𝑅𝛼2𝒢g^{\mu\nu}T_{\mu\nu}=g^{\mu\nu}\mathcal{E}_{\mu\nu}+\frac{\alpha}{2}\mathcal{E% }_{\phi}=4\Lambda-R-\frac{\alpha}{2}\mathcal{G}italic_g start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT italic_T start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT = italic_g start_POSTSUPERSCRIPT italic_μ italic_ν end_POSTSUPERSCRIPT caligraphic_E start_POSTSUBSCRIPT italic_μ italic_ν end_POSTSUBSCRIPT + divide start_ARG italic_α end_ARG start_ARG 2 end_ARG caligraphic_E start_POSTSUBSCRIPT italic_ϕ end_POSTSUBSCRIPT = 4 roman_Λ - italic_R - divide start_ARG italic_α end_ARG start_ARG 2 end_ARG caligraphic_G (9)

which can act as a useful consistency check to see whether prior solutions generated via the solution-limit method glavan2020 are even possible solutions to the theory. For example, using (9) it is easy to verify that the rotating metrics generated from a Newman-Janis algorithm kumar2020 ; wei2020 are not solutions to the field equations of the scalar-tensor 4DEGB theory.

II.2 Charged 4DEGB TOV Equations

Throughout the following we assume a non-interacting quark equation of state

P(r)=13(ρ4Beff)𝑃𝑟13𝜌4subscript𝐵effP(r)=\frac{1}{3}(\rho-4B_{\mathrm{eff}})italic_P ( italic_r ) = divide start_ARG 1 end_ARG start_ARG 3 end_ARG ( italic_ρ - 4 italic_B start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT ) (10)

where Beffsubscript𝐵effB_{\mathrm{eff}}italic_B start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT is the MIT bag constant (for which we use a benchmark value of 60MeV/fm360MeVsuperscriptfm360\;\mathrm{MeV}/\mathrm{fm}^{3}60 roman_MeV / roman_fm start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT zhang_2021_stellar ).

The standard Tolman-Oppenheimer-Volkoff (TOV) equations for stellar structure are well-known in GR. Variation of the action with respect to the gauge potential Aμsubscript𝐴𝜇A_{\mu}italic_A start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT, together with (6) and (7) yield the TOV equations for the charged 4DEGB quark star. To obtain these we begin with a static, spherically symmetric metric ansatz in natural units (G=c=1𝐺𝑐1G=c=1italic_G = italic_c = 1):

ds2=eΦ(r)c2dt2+eΛ(r)dr2+r2dΩ2.𝑑superscript𝑠2superscript𝑒Φ𝑟superscript𝑐2𝑑superscript𝑡2superscript𝑒Λ𝑟𝑑superscript𝑟2superscript𝑟2𝑑superscriptΩ2ds^{2}=-e^{\Phi(r)}c^{2}dt^{2}+e^{\Lambda(r)}dr^{2}+r^{2}d\Omega^{2}.italic_d italic_s start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = - italic_e start_POSTSUPERSCRIPT roman_Φ ( italic_r ) end_POSTSUPERSCRIPT italic_c start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d italic_t start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_e start_POSTSUPERSCRIPT roman_Λ ( italic_r ) end_POSTSUPERSCRIPT italic_d italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_d roman_Ω start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT . (11)

As usual hennigar_2020_on ; gammon_2024 , so long as eΦ(r)=eΛ(r)superscript𝑒Φ𝑟superscript𝑒Λ𝑟e^{\Phi(r)}=e^{-\Lambda(r)}italic_e start_POSTSUPERSCRIPT roman_Φ ( italic_r ) end_POSTSUPERSCRIPT = italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT outside the star, the combination 0011superscriptsubscript00superscriptsubscript11\mathcal{E}_{0}^{0}-\mathcal{E}_{1}^{1}caligraphic_E start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT - caligraphic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT of the field equations can be used to derive the following equation for the scalar field:

(ϕ2+ϕ′′)(eΛ(r)(rϕ1)2)=0.superscriptitalic-ϕ2superscriptitalic-ϕ′′superscript𝑒Λ𝑟superscript𝑟superscriptitalic-ϕ120\left(\phi^{\prime 2}+\phi^{\prime\prime}\right)\left(e^{\Lambda(r)}-\left(r% \phi^{\prime}-1\right)^{2}\right)=0.( italic_ϕ start_POSTSUPERSCRIPT ′ 2 end_POSTSUPERSCRIPT + italic_ϕ start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) ( italic_e start_POSTSUPERSCRIPT roman_Λ ( italic_r ) end_POSTSUPERSCRIPT - ( italic_r italic_ϕ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT - 1 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) = 0 . (12)

which, apart from the irrelevant ϕ=ln(rr0l)italic-ϕ𝑟subscript𝑟0𝑙\phi=\ln\left(\frac{r-r_{0}}{l}\right)italic_ϕ = roman_ln ( divide start_ARG italic_r - italic_r start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG italic_l end_ARG ) (with r0subscript𝑟0r_{0}italic_r start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and l𝑙litalic_l being constants of integration), has the solution

ϕ±=1±eΛ(r)/2r𝑑rsubscriptitalic-ϕplus-or-minusplus-or-minus1superscript𝑒Λ𝑟2𝑟differential-d𝑟\phi_{\pm}=\int\frac{1\pm e^{\Lambda(r)/2}}{r}dritalic_ϕ start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT = ∫ divide start_ARG 1 ± italic_e start_POSTSUPERSCRIPT roman_Λ ( italic_r ) / 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r end_ARG italic_d italic_r (13)

where ϕsubscriptitalic-ϕ\phi_{-}italic_ϕ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT falls off as as 1r1𝑟\frac{1}{r}divide start_ARG 1 end_ARG start_ARG italic_r end_ARG in asymptotically flat spacetimes. Choosing ϕ=ϕitalic-ϕsubscriptitalic-ϕ\phi=\phi_{-}italic_ϕ = italic_ϕ start_POSTSUBSCRIPT - end_POSTSUBSCRIPT ensures that (6) is automatically satisfied.

Modelling the charged quark matter by a perfect fluid electromagnetic matter source, the stress-energy tensor is (setting G=c=1𝐺𝑐1G=c=1italic_G = italic_c = 1)

Tνμ=(P+ρ)uμuν+Pδνμ+14π(FμαFαν14δνμFαβFαβ),superscriptsubscript𝑇𝜈𝜇𝑃𝜌superscript𝑢𝜇subscript𝑢𝜈𝑃superscriptsubscript𝛿𝜈𝜇14𝜋superscript𝐹𝜇𝛼subscript𝐹𝛼𝜈14superscriptsubscript𝛿𝜈𝜇subscript𝐹𝛼𝛽superscript𝐹𝛼𝛽T_{\nu}^{\mu}=(P+\rho)u^{\mu}u_{\nu}+P\delta_{\nu}^{\mu}+\frac{1}{4\pi}\left(F% ^{\mu\alpha}F_{\alpha\nu}-\frac{1}{4}\delta_{\nu}^{\mu}F_{\alpha\beta}F^{% \alpha\beta}\right),italic_T start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT = ( italic_P + italic_ρ ) italic_u start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT italic_u start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT + italic_P italic_δ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT + divide start_ARG 1 end_ARG start_ARG 4 italic_π end_ARG ( italic_F start_POSTSUPERSCRIPT italic_μ italic_α end_POSTSUPERSCRIPT italic_F start_POSTSUBSCRIPT italic_α italic_ν end_POSTSUBSCRIPT - divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_δ start_POSTSUBSCRIPT italic_ν end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT italic_F start_POSTSUBSCRIPT italic_α italic_β end_POSTSUBSCRIPT italic_F start_POSTSUPERSCRIPT italic_α italic_β end_POSTSUPERSCRIPT ) , (14)

from which we obtain the equations

e2Λ(r)(r2eΛ(r)(1eΛ(r))r3eΛ(r)Λ(r))r4superscript𝑒2Λ𝑟superscript𝑟2superscript𝑒Λ𝑟1superscript𝑒Λ𝑟superscript𝑟3superscript𝑒Λ𝑟superscriptΛ𝑟superscript𝑟4\displaystyle\frac{e^{-2\Lambda(r)}\left(r^{2}e^{\Lambda(r)}\left(1-e^{\Lambda% (r)}\right)-r^{3}e^{\Lambda(r)}\Lambda^{\prime}(r)\right)}{r^{4}}divide start_ARG italic_e start_POSTSUPERSCRIPT - 2 roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ( italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ( 1 - italic_e start_POSTSUPERSCRIPT roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ) - italic_r start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT roman_Λ ( italic_r ) end_POSTSUPERSCRIPT roman_Λ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) ) end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG (15)
+αe2Λ(r)(2reΛ(r)Λ(r)+2rΛ(r)eΛ(r)(1eΛ(r))eΛ(r)+1)r4=ρ(r)E(r)28π𝛼superscript𝑒2Λ𝑟2𝑟superscript𝑒Λ𝑟superscriptΛ𝑟2𝑟superscriptΛ𝑟superscript𝑒Λ𝑟1superscript𝑒Λ𝑟superscript𝑒Λ𝑟1superscript𝑟4𝜌𝑟𝐸superscript𝑟28𝜋\displaystyle+\alpha\frac{e^{-2\Lambda(r)}\left(-2re^{\Lambda(r)}\Lambda^{% \prime}(r)+2r\Lambda^{\prime}(r)-e^{\Lambda(r)}\left(1-e^{\Lambda(r)}\right)-e% ^{\Lambda(r)}+1\right)}{r^{4}}=-\rho(r)-\frac{E(r)^{2}}{8\pi}+ italic_α divide start_ARG italic_e start_POSTSUPERSCRIPT - 2 roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ( - 2 italic_r italic_e start_POSTSUPERSCRIPT roman_Λ ( italic_r ) end_POSTSUPERSCRIPT roman_Λ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) + 2 italic_r roman_Λ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) - italic_e start_POSTSUPERSCRIPT roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ( 1 - italic_e start_POSTSUPERSCRIPT roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ) - italic_e start_POSTSUPERSCRIPT roman_Λ ( italic_r ) end_POSTSUPERSCRIPT + 1 ) end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG = - italic_ρ ( italic_r ) - divide start_ARG italic_E ( italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 8 italic_π end_ARG
e2Λ(r)(r3eΛ(r)Φ(r)+r2eΛ(r)(1eΛ(r)))r4superscript𝑒2Λ𝑟superscript𝑟3superscript𝑒Λ𝑟superscriptΦ𝑟superscript𝑟2superscript𝑒Λ𝑟1superscript𝑒Λ𝑟superscript𝑟4\displaystyle\frac{e^{-2\Lambda(r)}\left(r^{3}e^{\Lambda(r)}\Phi^{\prime}(r)+r% ^{2}e^{\Lambda(r)}\left(1-e^{\Lambda(r)}\right)\right)}{r^{4}}divide start_ARG italic_e start_POSTSUPERSCRIPT - 2 roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ( italic_r start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT roman_Λ ( italic_r ) end_POSTSUPERSCRIPT roman_Φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) + italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ( 1 - italic_e start_POSTSUPERSCRIPT roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ) ) end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG
+αe2Λ(r)(2reΛ(r)Φ(r)2rΦ(r)eΛ(r)(1eΛ(r))eΛ(r)+1)r4=P(r)E(r)28π𝛼superscript𝑒2Λ𝑟2𝑟superscript𝑒Λ𝑟superscriptΦ𝑟2𝑟superscriptΦ𝑟superscript𝑒Λ𝑟1superscript𝑒Λ𝑟superscript𝑒Λ𝑟1superscript𝑟4𝑃𝑟𝐸superscript𝑟28𝜋\displaystyle+\alpha\frac{e^{-2\Lambda(r)}\left(2re^{\Lambda(r)}\Phi^{\prime}(% r)-2r\Phi^{\prime}(r)-e^{\Lambda(r)}\left(1-e^{\Lambda(r)}\right)-e^{\Lambda(r% )}+1\right)}{r^{4}}=P(r)-\frac{E(r)^{2}}{8\pi}+ italic_α divide start_ARG italic_e start_POSTSUPERSCRIPT - 2 roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ( 2 italic_r italic_e start_POSTSUPERSCRIPT roman_Λ ( italic_r ) end_POSTSUPERSCRIPT roman_Φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) - 2 italic_r roman_Φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) - italic_e start_POSTSUPERSCRIPT roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ( 1 - italic_e start_POSTSUPERSCRIPT roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ) - italic_e start_POSTSUPERSCRIPT roman_Λ ( italic_r ) end_POSTSUPERSCRIPT + 1 ) end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG = italic_P ( italic_r ) - divide start_ARG italic_E ( italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 8 italic_π end_ARG (16)
(ρ(r)+P(r))=2Φ(r)(dPdrq4πr4dqdr)𝜌𝑟𝑃𝑟2superscriptΦ𝑟d𝑃d𝑟𝑞4𝜋superscript𝑟4d𝑞d𝑟\displaystyle(\rho(r)+P(r))=-\frac{2}{\Phi^{\prime}(r)}\left(\frac{\mathrm{d}P% }{\mathrm{~{}d}r}-\frac{q}{4\pi r^{4}}\frac{\mathrm{d}q}{\mathrm{~{}d}r}\right)( italic_ρ ( italic_r ) + italic_P ( italic_r ) ) = - divide start_ARG 2 end_ARG start_ARG roman_Φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) end_ARG ( divide start_ARG roman_d italic_P end_ARG start_ARG roman_d italic_r end_ARG - divide start_ARG italic_q end_ARG start_ARG 4 italic_π italic_r start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG divide start_ARG roman_d italic_q end_ARG start_ARG roman_d italic_r end_ARG ) (17)

in 4DEGB.

Imposing asymptotic flatness implies that Φ()=Λ()=0ΦΛ0\Phi(\infty)=\Lambda(\infty)=0roman_Φ ( ∞ ) = roman_Λ ( ∞ ) = 0, and regularity at the center of the star implies Λ(0)=0Λ00\Lambda(0)=0roman_Λ ( 0 ) = 0. Using the tt𝑡𝑡ttitalic_t italic_t field equation it is easy to show that

eΛ(r)=1+r22α[11+4α(2m(r)r3q(r)2r4)],superscript𝑒Λ𝑟1superscript𝑟22𝛼delimited-[]114𝛼2𝑚𝑟superscript𝑟3𝑞superscript𝑟2superscript𝑟4e^{-\Lambda(r)}=1+\frac{r^{2}}{2\alpha}\left[1-\sqrt{1+4\alpha\left(\frac{2m(r% )}{r^{3}}-\frac{q(r)^{2}}{r^{4}}\right)}\right],italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT = 1 + divide start_ARG italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_α end_ARG [ 1 - square-root start_ARG 1 + 4 italic_α ( divide start_ARG 2 italic_m ( italic_r ) end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG - divide start_ARG italic_q ( italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG ) end_ARG ] , (18)

in agreement with what was found in pretel_2022 , recalling that E(r)2=q(r)2r4𝐸superscript𝑟2𝑞superscript𝑟2superscript𝑟4E(r)^{2}=\frac{q(r)^{2}}{r^{4}}italic_E ( italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = divide start_ARG italic_q ( italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG. With this we arrive at the 4DEGB modified TOV equations, namely

dqdr𝑑𝑞𝑑𝑟\displaystyle\frac{dq}{dr}divide start_ARG italic_d italic_q end_ARG start_ARG italic_d italic_r end_ARG =4πr2ρeeΛ(r)2absent4𝜋superscript𝑟2subscript𝜌𝑒superscript𝑒Λ𝑟2\displaystyle=4\pi r^{2}\rho_{e}e^{\frac{\Lambda(r)}{2}}= 4 italic_π italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ρ start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT divide start_ARG roman_Λ ( italic_r ) end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT (19)
dmdr𝑑𝑚𝑑𝑟\displaystyle\frac{dm}{dr}divide start_ARG italic_d italic_m end_ARG start_ARG italic_d italic_r end_ARG =4πr2ρ(r)+q(r)rdqdrabsent4𝜋superscript𝑟2𝜌𝑟𝑞𝑟𝑟𝑑𝑞𝑑𝑟\displaystyle=4\pi r^{2}\rho(r)+\frac{q(r)}{r}\frac{dq}{dr}= 4 italic_π italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ρ ( italic_r ) + divide start_ARG italic_q ( italic_r ) end_ARG start_ARG italic_r end_ARG divide start_ARG italic_d italic_q end_ARG start_ARG italic_d italic_r end_ARG (20)
dPdr𝑑𝑃𝑑𝑟\displaystyle\frac{dP}{dr}divide start_ARG italic_d italic_P end_ARG start_ARG italic_d italic_r end_ARG =(P(r)+ρ(r))(r3(Γ+8παP(r)1)2αm(r))Γr2((Γ1)r22α)+q(r)4πr4dqdrabsent𝑃𝑟𝜌𝑟superscript𝑟3Γ8𝜋𝛼𝑃𝑟12𝛼𝑚𝑟Γsuperscript𝑟2Γ1superscript𝑟22𝛼𝑞𝑟4𝜋superscript𝑟4𝑑𝑞𝑑𝑟\displaystyle=(P(r)+\rho(r))\frac{\left(r^{3}(\Gamma+8\pi\alpha P(r)-1)-2% \alpha m(r)\right)}{\Gamma r^{2}\left((\Gamma-1)r^{2}-2\alpha\right)}+\frac{q(% r)}{4\pi r^{4}}\frac{dq}{dr}= ( italic_P ( italic_r ) + italic_ρ ( italic_r ) ) divide start_ARG ( italic_r start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ( roman_Γ + 8 italic_π italic_α italic_P ( italic_r ) - 1 ) - 2 italic_α italic_m ( italic_r ) ) end_ARG start_ARG roman_Γ italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( ( roman_Γ - 1 ) italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 2 italic_α ) end_ARG + divide start_ARG italic_q ( italic_r ) end_ARG start_ARG 4 italic_π italic_r start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG divide start_ARG italic_d italic_q end_ARG start_ARG italic_d italic_r end_ARG (21)

where Γ=1+4α(2m(r)r3q(r)2r4)Γ14𝛼2𝑚𝑟superscript𝑟3𝑞superscript𝑟2superscript𝑟4\Gamma=\sqrt{1+4\alpha\left(\frac{2m(r)}{r^{3}}-\frac{q(r)^{2}}{r^{4}}\right)}roman_Γ = square-root start_ARG 1 + 4 italic_α ( divide start_ARG 2 italic_m ( italic_r ) end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG - divide start_ARG italic_q ( italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG ) end_ARG (matching those found in pretel_2022 ). The vacuum solution is given by m(r)=M𝑚𝑟𝑀m(r)=Mitalic_m ( italic_r ) = italic_M and q(r)=Q𝑞𝑟𝑄q(r)=Qitalic_q ( italic_r ) = italic_Q where M𝑀Mitalic_M and Q𝑄Qitalic_Q are constants (the total mass and charge, respectively), implying that Φ=ΛΦΛ\Phi=-\Lambdaroman_Φ = - roman_Λ. Writing eΛ(r)=1+2φ(r)superscript𝑒Λ𝑟12𝜑𝑟e^{-\Lambda(r)}=1+2\varphi(r)italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT = 1 + 2 italic_φ ( italic_r ), we can compute the gravitational force in 4DEGB due to a spherical body

F=dφdrr^=r2α(12αM+r3r8αrM4αQ2+r4)r^,𝐹𝑑𝜑𝑑𝑟^𝑟𝑟2𝛼12𝛼𝑀superscript𝑟3𝑟8𝛼𝑟𝑀4𝛼superscript𝑄2superscript𝑟4^𝑟\vec{F}=-\frac{d\varphi}{dr}\hat{r}=-\frac{r}{2\alpha}\left(1-\frac{2\alpha M+% r^{3}}{r\sqrt{8\alpha rM-4\alpha Q^{2}+r^{4}}}\right)\hat{r}\,,over→ start_ARG italic_F end_ARG = - divide start_ARG italic_d italic_φ end_ARG start_ARG italic_d italic_r end_ARG over^ start_ARG italic_r end_ARG = - divide start_ARG italic_r end_ARG start_ARG 2 italic_α end_ARG ( 1 - divide start_ARG 2 italic_α italic_M + italic_r start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r square-root start_ARG 8 italic_α italic_r italic_M - 4 italic_α italic_Q start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_r start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG end_ARG ) over^ start_ARG italic_r end_ARG , (22)

which is smaller in magnitude than its Newtonian α=0𝛼0\alpha=0italic_α = 0 counterpart (FN=Mr2r^subscript𝐹𝑁𝑀superscript𝑟2^𝑟\vec{F}_{N}=-\frac{M}{r^{2}}\hat{r}over→ start_ARG italic_F end_ARG start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT = - divide start_ARG italic_M end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG over^ start_ARG italic_r end_ARG) for α>0𝛼0\alpha>0italic_α > 0. The force in (22) vanishes at r=(αM)1/3𝑟superscript𝛼𝑀13r=(\alpha M)^{1/3}italic_r = ( italic_α italic_M ) start_POSTSUPERSCRIPT 1 / 3 end_POSTSUPERSCRIPT if Q=0𝑄0Q=0italic_Q = 0, but this is always at a smaller value of r𝑟ritalic_r than the outer horizon Rh=M+M2Q2αsubscript𝑅𝑀superscript𝑀2superscript𝑄2𝛼R_{h}=M+\sqrt{M^{2}-Q^{2}-\alpha}italic_R start_POSTSUBSCRIPT italic_h end_POSTSUBSCRIPT = italic_M + square-root start_ARG italic_M start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_Q start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_α end_ARG of the corresponding black hole. Hence the gravitational force outside of any spherical body, while weaker than that in GR, is always attractive provided α>0𝛼0\alpha>0italic_α > 0. If α<0𝛼0\alpha<0italic_α < 0 then the corresponding gravitational force is more attractive than in GR. Similarly for nonzero charge, if all other parameters are held constant, increasing the charge weakens the gravitational attraction, which only vanishes in regions disallowed by the black hole horizon.

Rescaling the various quantities using

ρ¯=ρ4Beffp¯=p4Beffρ¯e=ρe4Beffformulae-sequence¯𝜌𝜌4subscript𝐵effformulae-sequence¯𝑝𝑝4subscript𝐵effsubscript¯𝜌𝑒subscript𝜌𝑒4subscript𝐵eff\bar{\rho}=\frac{\rho}{4B_{\mathrm{eff}}}\qquad\bar{p}=\frac{p}{4B_{\mathrm{% eff}}}\qquad\bar{\rho}_{e}=\frac{\rho_{e}}{4B_{\mathrm{eff}}}over¯ start_ARG italic_ρ end_ARG = divide start_ARG italic_ρ end_ARG start_ARG 4 italic_B start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT end_ARG over¯ start_ARG italic_p end_ARG = divide start_ARG italic_p end_ARG start_ARG 4 italic_B start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT end_ARG over¯ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = divide start_ARG italic_ρ start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT end_ARG start_ARG 4 italic_B start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT end_ARG (23)

and

m¯=m4Beffr¯=r4Beffq¯=q4Beffα¯=α4Beff,formulae-sequence¯𝑚𝑚4subscript𝐵effformulae-sequence¯𝑟𝑟4subscript𝐵effformulae-sequence¯𝑞𝑞4subscript𝐵eff¯𝛼𝛼4subscript𝐵eff\bar{m}=m\sqrt{4B_{\mathrm{eff}}}\qquad\bar{r}=r\sqrt{4B_{\mathrm{eff}}}\qquad% \bar{q}=q\sqrt{4B_{\mathrm{eff}}}\qquad\bar{\alpha}=\alpha\cdot 4B_{\mathrm{% eff}},over¯ start_ARG italic_m end_ARG = italic_m square-root start_ARG 4 italic_B start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT end_ARG over¯ start_ARG italic_r end_ARG = italic_r square-root start_ARG 4 italic_B start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT end_ARG over¯ start_ARG italic_q end_ARG = italic_q square-root start_ARG 4 italic_B start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT end_ARG over¯ start_ARG italic_α end_ARG = italic_α ⋅ 4 italic_B start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT , (24)

we obtain the dimensionless equations

dq¯dr¯𝑑¯𝑞𝑑¯𝑟\displaystyle\frac{d\bar{q}}{d\bar{r}}divide start_ARG italic_d over¯ start_ARG italic_q end_ARG end_ARG start_ARG italic_d over¯ start_ARG italic_r end_ARG end_ARG =4πr¯2ρ¯eeΛ(r¯)2absent4𝜋superscript¯𝑟2subscript¯𝜌𝑒superscript𝑒Λ¯𝑟2\displaystyle=4\pi\bar{r}^{2}\bar{\rho}_{e}e^{\frac{\Lambda(\bar{r})}{2}}= 4 italic_π over¯ start_ARG italic_r end_ARG start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT over¯ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT divide start_ARG roman_Λ ( over¯ start_ARG italic_r end_ARG ) end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT (25)
dm¯dr¯𝑑¯𝑚𝑑¯𝑟\displaystyle\frac{d\bar{m}}{d\bar{r}}divide start_ARG italic_d over¯ start_ARG italic_m end_ARG end_ARG start_ARG italic_d over¯ start_ARG italic_r end_ARG end_ARG =4πr¯2ρ¯(r¯)+q¯(r¯)r¯dq¯dr¯absent4𝜋superscript¯𝑟2¯𝜌¯𝑟¯𝑞¯𝑟¯𝑟𝑑¯𝑞𝑑¯𝑟\displaystyle=4\pi\bar{r}^{2}\bar{\rho}(\bar{r})+\frac{\bar{q}(\bar{r})}{\bar{% r}}\frac{d\bar{q}}{d\bar{r}}= 4 italic_π over¯ start_ARG italic_r end_ARG start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT over¯ start_ARG italic_ρ end_ARG ( over¯ start_ARG italic_r end_ARG ) + divide start_ARG over¯ start_ARG italic_q end_ARG ( over¯ start_ARG italic_r end_ARG ) end_ARG start_ARG over¯ start_ARG italic_r end_ARG end_ARG divide start_ARG italic_d over¯ start_ARG italic_q end_ARG end_ARG start_ARG italic_d over¯ start_ARG italic_r end_ARG end_ARG (26)
dP¯dr¯𝑑¯𝑃𝑑¯𝑟\displaystyle\frac{d\bar{P}}{d\bar{r}}divide start_ARG italic_d over¯ start_ARG italic_P end_ARG end_ARG start_ARG italic_d over¯ start_ARG italic_r end_ARG end_ARG =(P¯(r¯)+ρ¯(r¯))(r¯3(Γ+8πα¯P¯(r¯)1)2α¯m¯(r¯))Γr¯2((Γ1)r¯22α¯)+q¯(r¯)4πr¯4dq¯dr¯absent¯𝑃¯𝑟¯𝜌¯𝑟superscript¯𝑟3Γ8𝜋¯𝛼¯𝑃¯𝑟12¯𝛼¯𝑚¯𝑟Γsuperscript¯𝑟2Γ1superscript¯𝑟22¯𝛼¯𝑞¯𝑟4𝜋superscript¯𝑟4𝑑¯𝑞𝑑¯𝑟\displaystyle=(\bar{P}(\bar{r})+\bar{\rho}(\bar{r}))\frac{\left(\bar{r}^{3}(% \Gamma+8\pi\bar{\alpha}\bar{P}(\bar{r})-1)-2\bar{\alpha}\bar{m}(\bar{r})\right% )}{\Gamma\bar{r}^{2}\left((\Gamma-1)\bar{r}^{2}-2\bar{\alpha}\right)}+\frac{% \bar{q}(\bar{r})}{4\pi\bar{r}^{4}}\frac{d\bar{q}}{d\bar{r}}= ( over¯ start_ARG italic_P end_ARG ( over¯ start_ARG italic_r end_ARG ) + over¯ start_ARG italic_ρ end_ARG ( over¯ start_ARG italic_r end_ARG ) ) divide start_ARG ( over¯ start_ARG italic_r end_ARG start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT ( roman_Γ + 8 italic_π over¯ start_ARG italic_α end_ARG over¯ start_ARG italic_P end_ARG ( over¯ start_ARG italic_r end_ARG ) - 1 ) - 2 over¯ start_ARG italic_α end_ARG over¯ start_ARG italic_m end_ARG ( over¯ start_ARG italic_r end_ARG ) ) end_ARG start_ARG roman_Γ over¯ start_ARG italic_r end_ARG start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( ( roman_Γ - 1 ) over¯ start_ARG italic_r end_ARG start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 2 over¯ start_ARG italic_α end_ARG ) end_ARG + divide start_ARG over¯ start_ARG italic_q end_ARG ( over¯ start_ARG italic_r end_ARG ) end_ARG start_ARG 4 italic_π over¯ start_ARG italic_r end_ARG start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG divide start_ARG italic_d over¯ start_ARG italic_q end_ARG end_ARG start_ARG italic_d over¯ start_ARG italic_r end_ARG end_ARG (27)

which may be solved numerically. In the limit α0𝛼0\alpha\to 0italic_α → 0, the above equations reduce back to the well-known TOV equations for a charged, static, spherically symmetric gravitating body in GR.

To solve (25), (26) and (27) numerically we impose the boundary conditions

m(0)=0,q(0)=0,ρ(0)=ρc,formulae-sequence𝑚00formulae-sequence𝑞00𝜌0subscript𝜌cm(0)=0,\quad q(0)=0,\quad\rho(0)=\rho_{\mathrm{c}},italic_m ( 0 ) = 0 , italic_q ( 0 ) = 0 , italic_ρ ( 0 ) = italic_ρ start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT , (28)

where the star’s surface radius R𝑅Ritalic_R is defined via p¯(R¯)=0¯𝑝¯𝑅0\bar{p}(\bar{R})=0over¯ start_ARG italic_p end_ARG ( over¯ start_ARG italic_R end_ARG ) = 0, namely the radius at which the pressure goes to 0 (i.e. p(R)=0𝑝𝑅0p(R)=0italic_p ( italic_R ) = 0). We similarly define the total mass of the star to be M=m(R)𝑀𝑚𝑅M=m(R)italic_M = italic_m ( italic_R ).

For the charge density there are several benchmark models used in the literature. We shall restrict ourselves to the model in which the charge density is proportional to energy density (ie. ρe=γρsubscript𝜌𝑒𝛾𝜌\rho_{e}=\gamma\rhoitalic_ρ start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = italic_γ italic_ρ or ρ¯e=γρ¯subscript¯𝜌𝑒𝛾¯𝜌\bar{\rho}_{e}=\gamma\bar{\rho}over¯ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = italic_γ over¯ start_ARG italic_ρ end_ARG where 0γ10𝛾10\leq\gamma\leq 10 ≤ italic_γ ≤ 1) zhang_2021_stellar ; ray2003 ; arbanil2015 . Another popular charge model sets charge proportional to spatial volume. However, this leads to exotic pressure profiles that do not decrease monotonically zhang_2021_stellar , and thus are not consistent with arguments we shall present in section III. As we are primarily interested in how charged 4DEGB quark stars affect the modified Buchdahl bound (since many novel results were noticed in the uncharged case gammon_2024 ), we consider only the model where ρ¯e=γρ¯subscript¯𝜌𝑒𝛾¯𝜌\bar{\rho}_{e}=\gamma\bar{\rho}over¯ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT = italic_γ over¯ start_ARG italic_ρ end_ARG. Furthermore, since the interaction with the Buchdahl bound is the main feature of interest, we choose to present results with a fixed Q𝑄Qitalic_Q rather than a fixed γ𝛾\gammaitalic_γ, with this charge parameter taking on standard values from the literature (Q=(0,1,2)×1020C𝑄012superscript1020CQ=(0,1,2)\times 10^{20}\mathrm{\;C}italic_Q = ( 0 , 1 , 2 ) × 10 start_POSTSUPERSCRIPT 20 end_POSTSUPERSCRIPT roman_C, Q¯=(0,1.538,3.076)×102¯𝑄01.5383.076superscript102\bar{Q}=(0,1.538,3.076)\times 10^{-2}over¯ start_ARG italic_Q end_ARG = ( 0 , 1.538 , 3.076 ) × 10 start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT) for comparison arbanil2015 ; zhang_2021_stellar . This is done by solving the equations for a trial value of γ𝛾\gammaitalic_γ and iterating until our desired fixed charge is attained.

With the above, numerical solutions can thus be obtained by scanning through a range of values of ρcsubscript𝜌𝑐\rho_{c}italic_ρ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT and solving for the star’s total mass and radius.

Before proceeding to solve the TOV equations, we consider the behaviour of the scalar field ϕitalic-ϕ\phiitalic_ϕ in the interior. By continuity with the exterior solution, the scalar is still described by (13) inside the star charmousis2022 . In considering the interior behaviour it is instructive to rewrite q(r)rq~(r)𝑞𝑟𝑟~𝑞𝑟q(r)\to r\tilde{q}(r)italic_q ( italic_r ) → italic_r over~ start_ARG italic_q end_ARG ( italic_r ) (which can be done by virtue of equation (19) and the fact that limr0eΛ(r)2const+𝒪(r)similar-tosubscript𝑟0superscript𝑒Λ𝑟2const𝒪𝑟\lim_{r\to 0}e^{\frac{\Lambda(r)}{2}}\sim\mathrm{const}+\mathcal{O}(r)roman_lim start_POSTSUBSCRIPT italic_r → 0 end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT divide start_ARG roman_Λ ( italic_r ) end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT ∼ roman_const + caligraphic_O ( italic_r )). Inserting the interior solution (18) into (12) and making the latter substitution, we find

ϕ(r)superscriptitalic-ϕ𝑟\displaystyle\phi^{\prime}(r)italic_ϕ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) =m(0)2αr3m(0)4α+m(0)r(α(q~(0)22m(0))5m(0)2)42(αm(0))3/2absent𝑚02𝛼𝑟3𝑚04𝛼𝑚0𝑟𝛼~𝑞superscript022superscript𝑚05𝑚superscript0242superscript𝛼𝑚032\displaystyle=-\sqrt{\frac{m(0)}{2\alpha r}}-\frac{3m(0)}{4\alpha}+\frac{m(0)% \sqrt{r}\left(\alpha\left(\tilde{q}(0)^{2}-2m^{\prime}(0)\right)-5m(0)^{2}% \right)}{4\sqrt{2}(\alpha m(0))^{3/2}}= - square-root start_ARG divide start_ARG italic_m ( 0 ) end_ARG start_ARG 2 italic_α italic_r end_ARG end_ARG - divide start_ARG 3 italic_m ( 0 ) end_ARG start_ARG 4 italic_α end_ARG + divide start_ARG italic_m ( 0 ) square-root start_ARG italic_r end_ARG ( italic_α ( over~ start_ARG italic_q end_ARG ( 0 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 2 italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( 0 ) ) - 5 italic_m ( 0 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG start_ARG 4 square-root start_ARG 2 end_ARG ( italic_α italic_m ( 0 ) ) start_POSTSUPERSCRIPT 3 / 2 end_POSTSUPERSCRIPT end_ARG (29)
+r(4α(6m(0)+3q~(0)2+2)35m(0)2)32α2+𝒪(r3/2)𝑟4𝛼6superscript𝑚03~𝑞superscript02235𝑚superscript0232superscript𝛼2𝒪superscript𝑟32\displaystyle+\frac{r\left(4\alpha\left(-6m^{\prime}(0)+3\tilde{q}(0)^{2}+2% \right)-35m(0)^{2}\right)}{32\alpha^{2}}+\mathcal{O}\left(r^{3/2}\right)+ divide start_ARG italic_r ( 4 italic_α ( - 6 italic_m start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( 0 ) + 3 over~ start_ARG italic_q end_ARG ( 0 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + 2 ) - 35 italic_m ( 0 ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG start_ARG 32 italic_α start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG + caligraphic_O ( italic_r start_POSTSUPERSCRIPT 3 / 2 end_POSTSUPERSCRIPT )

Furthermore, provided m(r)𝑚𝑟m(r)italic_m ( italic_r ) and q(r)𝑞𝑟q(r)italic_q ( italic_r ) both vanish at least quadratically in r𝑟ritalic_r for small r𝑟ritalic_r (which is ensured from (19) and (20) for the boundary conditions (28)) we find that near the origin

limr0ϕ(r)subscript𝑟0superscriptitalic-ϕ𝑟\displaystyle\lim_{r\to 0}\phi^{\prime}(r)roman_lim start_POSTSUBSCRIPT italic_r → 0 end_POSTSUBSCRIPT italic_ϕ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) (0)2αr+r4α+𝒪(r3/2)0similar-toabsent02𝛼𝑟𝑟4𝛼𝒪superscript𝑟320\displaystyle\sim-\sqrt{\frac{\mathcal{M}(0)}{2\alpha}}\sqrt{r}+\frac{r}{4% \alpha}+\mathcal{O}\left(r^{3/2}\right)\approx 0∼ - square-root start_ARG divide start_ARG caligraphic_M ( 0 ) end_ARG start_ARG 2 italic_α end_ARG end_ARG square-root start_ARG italic_r end_ARG + divide start_ARG italic_r end_ARG start_ARG 4 italic_α end_ARG + caligraphic_O ( italic_r start_POSTSUPERSCRIPT 3 / 2 end_POSTSUPERSCRIPT ) ≈ 0 (30)
limr0ϕ(r)subscript𝑟0italic-ϕ𝑟\displaystyle\lim_{r\to 0}\phi(r)roman_lim start_POSTSUBSCRIPT italic_r → 0 end_POSTSUBSCRIPT italic_ϕ ( italic_r ) r3/232(0)α+r28α+KKsimilar-toabsentsuperscript𝑟32320𝛼superscript𝑟28𝛼𝐾𝐾\displaystyle\sim-\frac{r^{3/2}}{3}\sqrt{\frac{2\mathcal{M}(0)}{\alpha}}+\frac% {r^{2}}{8\alpha}+K\approx K∼ - divide start_ARG italic_r start_POSTSUPERSCRIPT 3 / 2 end_POSTSUPERSCRIPT end_ARG start_ARG 3 end_ARG square-root start_ARG divide start_ARG 2 caligraphic_M ( 0 ) end_ARG start_ARG italic_α end_ARG end_ARG + divide start_ARG italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 8 italic_α end_ARG + italic_K ≈ italic_K

(where (r)=r2m(r)𝑟superscript𝑟2𝑚𝑟\mathcal{M}(r)=r^{-2}m(r)caligraphic_M ( italic_r ) = italic_r start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT italic_m ( italic_r ) and K𝐾Kitalic_K is a constant) and thus regularity of the scalar at the origin is ensured.

Finally we note that the effective bag constant Beff=60subscript𝐵eff60B_{\mathrm{eff}}=60italic_B start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT = 60 MeV/fm3superscriptfm3\mathrm{fm}^{3}roman_fm start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT can be converted to units of inverse length squared (‘gravitational units’) with the factor G/c4𝐺superscript𝑐4G/c^{4}italic_G / italic_c start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT, yielding

Beff=7.84×105km2.subscript𝐵eff7.84superscript105superscriptkm2B_{\mathrm{eff}}=7.84\times 10^{-5}\;\mathrm{km}^{-2}.italic_B start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT = 7.84 × 10 start_POSTSUPERSCRIPT - 5 end_POSTSUPERSCRIPT roman_km start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT . (31)

II.3 Observational Constraints on the 4DEGB Coupling Constant

An investigation of the observational constraints on the coupling α𝛼\alphaitalic_α yielded Clifton:2020xhc ; Fernandes:2022zrq

1030m2<α<1010m2superscript1030superscriptm2𝛼superscript1010superscriptm2-10^{-30}\;\textrm{m}^{2}<\alpha<10^{10}\;\textrm{m}^{2}- 10 start_POSTSUPERSCRIPT - 30 end_POSTSUPERSCRIPT m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT < italic_α < 10 start_POSTSUPERSCRIPT 10 end_POSTSUPERSCRIPT m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT (32)

where the lower bound comes from “early universe cosmology and atomic nuclei” data charmousis2022 , and the upper bound follows from LAGEOS satellite observations. Regarding the lower bound as negligibly close to zero, the dimensionless version of (32) reads

0<α¯3.2.0¯𝛼less-than-or-similar-to3.20<\bar{\alpha}\lesssim 3.2.0 < over¯ start_ARG italic_α end_ARG ≲ 3.2 . (33)

We note that inclusion of preliminary calculations on recent GW data suggest these constraints could potentially tighten to 0<α107m20𝛼less-than-or-similar-tosuperscript107superscriptm20<\alpha\lesssim 10^{7}\;\textrm{m}^{2}0 < italic_α ≲ 10 start_POSTSUPERSCRIPT 7 end_POSTSUPERSCRIPT m start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, or alternatively 0<α¯0.00320¯𝛼less-than-or-similar-to0.00320<\bar{\alpha}\lesssim 0.00320 < over¯ start_ARG italic_α end_ARG ≲ 0.0032. This would mean that deviations from GR due to 4DEGB would only be detectable in extreme environments such as in the very early universe or near the surface of extremely massive objects. Even tighter bounds were assumed in previous studies of quark stars, where only solutions with α𝛼\alphaitalic_α below 6 km2superscriptkm2\mathrm{km}^{2}roman_km start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT (α¯0.0019¯𝛼0.0019\bar{\alpha}\leq 0.0019over¯ start_ARG italic_α end_ARG ≤ 0.0019) were considered banerjee_2021_strange ; banerjee_2021_quark ; pretel_2022 . Adopting such a tight bound would make compact stars near the upper end of the mass gap an ideal candidate for investigation the effects of 4DEGB theory.

At this point in time such tighter bounds are not warranted. A proper study of the effects of gravitational radiation in 4DEGB has yet to be carried out. In view of this we shall assume the bound (33), which has strong observational support Clifton:2020xhc ; Fernandes:2022zrq .

III Generalization of the Buchdahl Bound for Charged Compact Stars in 4DEGB Gravity

III.1 Derivation of a Maximal Mass for Charged Compact Objects in 4DEGB Gravity

A generalization of the Buchdahl bound has been derived for charged stars in GR bohmer_2007 and for uncharged objects in 4DEGB theory chakraborty_2020 . In the following we extend these calculations for the case of a charged star in 4DEGB gravity.

By virtue of equation (18) it is straightforward to check that

ddr(1eΛ(r))2r2=11+4α(2m(r)r3q(r)2r4)ddr(m(r)r3q(r)22r4)𝑑𝑑𝑟1superscript𝑒Λ𝑟2superscript𝑟2114𝛼2𝑚𝑟superscript𝑟3𝑞superscript𝑟2superscript𝑟4𝑑𝑑𝑟𝑚𝑟superscript𝑟3𝑞superscript𝑟22superscript𝑟4\frac{d}{dr}\frac{(1-e^{-\Lambda(r)})}{2r^{2}}=\frac{1}{\sqrt{1+4\alpha(\frac{% 2m(r)}{r^{3}}-\frac{q(r)^{2}}{r^{4}})}}\frac{d}{dr}\left(\frac{m(r)}{r^{3}}-% \frac{q(r)^{2}}{2r^{4}}\right)divide start_ARG italic_d end_ARG start_ARG italic_d italic_r end_ARG divide start_ARG ( 1 - italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ) end_ARG start_ARG 2 italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 1 + 4 italic_α ( divide start_ARG 2 italic_m ( italic_r ) end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG - divide start_ARG italic_q ( italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG ) end_ARG end_ARG divide start_ARG italic_d end_ARG start_ARG italic_d italic_r end_ARG ( divide start_ARG italic_m ( italic_r ) end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG - divide start_ARG italic_q ( italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_r start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG ) (34)

The next step of the argument hinges on the assumption that the quantity (1eΛ(r))r21superscript𝑒Λ𝑟superscript𝑟2\frac{(1-e^{-\Lambda(r)})}{r^{2}}divide start_ARG ( 1 - italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ) end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG decreases as radial distance increases chakraborty_2020 . In the uncharged case this is equivalent to assuming decreasing mass density as the star’s surface is approached (though we note that some exotic star models do not have monotonically decreasing density profiles zhang_2021_stellar ). This assumption also seems reasonable in the case of a charged sphere chakraborty_2020 and we employ it in what follows:

Utilizing the conservation equation

(ρ(r)+P(r))=2Φ(r)(dPdrq4πr4dqdr)𝜌𝑟𝑃𝑟2superscriptΦ𝑟d𝑃d𝑟𝑞4𝜋superscript𝑟4d𝑞d𝑟(\rho(r)+P(r))=-\frac{2}{\Phi^{\prime}(r)}\left(\frac{\mathrm{d}P}{\mathrm{~{}% d}r}-\frac{q}{4\pi r^{4}}\frac{\mathrm{d}q}{\mathrm{~{}d}r}\right)( italic_ρ ( italic_r ) + italic_P ( italic_r ) ) = - divide start_ARG 2 end_ARG start_ARG roman_Φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) end_ARG ( divide start_ARG roman_d italic_P end_ARG start_ARG roman_d italic_r end_ARG - divide start_ARG italic_q end_ARG start_ARG 4 italic_π italic_r start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG divide start_ARG roman_d italic_q end_ARG start_ARG roman_d italic_r end_ARG ) (35)

we can write

dPdr=q4πr4dqdr12(ρ(r)+P(r))Φ(r).𝑑𝑃𝑑𝑟𝑞4𝜋superscript𝑟4𝑑𝑞𝑑𝑟12𝜌𝑟𝑃𝑟superscriptΦ𝑟\frac{dP}{dr}=\frac{q}{4\pi r^{4}}\frac{dq}{dr}-\frac{1}{2}(\rho(r)+P(r))\Phi^% {\prime}(r).divide start_ARG italic_d italic_P end_ARG start_ARG italic_d italic_r end_ARG = divide start_ARG italic_q end_ARG start_ARG 4 italic_π italic_r start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG divide start_ARG italic_d italic_q end_ARG start_ARG italic_d italic_r end_ARG - divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( italic_ρ ( italic_r ) + italic_P ( italic_r ) ) roman_Φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) . (36)

Recalling the rr𝑟𝑟rritalic_r italic_r component of the field equations, we obtain

1r2[rΦ(r)eΛ(r)(1eΛ(r))]1superscript𝑟2delimited-[]𝑟superscriptΦ𝑟superscript𝑒Λ𝑟1superscript𝑒Λ𝑟\displaystyle\frac{1}{r^{2}}\left[r\Phi^{\prime}(r)e^{-\Lambda(r)}-\left(1-e^{% -\Lambda(r)}\right)\right]divide start_ARG 1 end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG [ italic_r roman_Φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT - ( 1 - italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ) ] (37)
+α(1eΛ(r))r4[2rΦ(r)eΛ(r)+(1eΛ(r))]=κ(P(r)q(r)28πr4)𝛼1superscript𝑒Λ𝑟superscript𝑟4delimited-[]2𝑟superscriptΦ𝑟superscript𝑒Λ𝑟1superscript𝑒Λ𝑟𝜅𝑃𝑟𝑞superscript𝑟28𝜋superscript𝑟4\displaystyle\qquad+\alpha\frac{\left(1-e^{-\Lambda(r)}\right)}{r^{4}}\left[2r% \Phi^{\prime}(r)e^{-\Lambda(r)}+\left(1-e^{-\Lambda(r)}\right)\right]=\kappa% \left(P(r)-\frac{q(r)^{2}}{8\pi r^{4}}\right)+ italic_α divide start_ARG ( 1 - italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ) end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG [ 2 italic_r roman_Φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT + ( 1 - italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ) ] = italic_κ ( italic_P ( italic_r ) - divide start_ARG italic_q ( italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 8 italic_π italic_r start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG )

and subsequently get

κ(dPdr+q(r)22πr5q(r)4πr4dqdr)=κ(q(r)22πr512(ρ(r)+P(r))Φ(r))𝜅𝑑𝑃𝑑𝑟𝑞superscript𝑟22𝜋superscript𝑟5𝑞𝑟4𝜋superscript𝑟4𝑑𝑞𝑑𝑟𝜅𝑞superscript𝑟22𝜋superscript𝑟512𝜌𝑟𝑃𝑟superscriptΦ𝑟\displaystyle\kappa\left(\frac{dP}{dr}+\frac{q(r)^{2}}{2\pi r^{5}}-\frac{q(r)}% {4\pi r^{4}}\frac{dq}{dr}\right)=\kappa\left(\frac{q(r)^{2}}{2\pi r^{5}}-\frac% {1}{2}(\rho(r)+P(r))\Phi^{\prime}(r)\right)italic_κ ( divide start_ARG italic_d italic_P end_ARG start_ARG italic_d italic_r end_ARG + divide start_ARG italic_q ( italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_π italic_r start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT end_ARG - divide start_ARG italic_q ( italic_r ) end_ARG start_ARG 4 italic_π italic_r start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG divide start_ARG italic_d italic_q end_ARG start_ARG italic_d italic_r end_ARG ) = italic_κ ( divide start_ARG italic_q ( italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_π italic_r start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT end_ARG - divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( italic_ρ ( italic_r ) + italic_P ( italic_r ) ) roman_Φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) ) (38)

by differentiating the left hand side of (37) using (36).

Addition of the tt𝑡𝑡ttitalic_t italic_t and rr𝑟𝑟rritalic_r italic_r equations yields

(ρ(r)+P(r))=1κ(1reΛ(r)(Λ(r)+Φ(r))+2αr3eΛ(r)(1eΛ(r))(Λ(r)+Φ(r)))𝜌𝑟𝑃𝑟1𝜅1𝑟superscript𝑒Λ𝑟Λsuperscript𝑟Φsuperscript𝑟2𝛼superscript𝑟3superscript𝑒Λ𝑟1superscript𝑒Λ𝑟Λsuperscript𝑟Φsuperscript𝑟(\rho(r)+P(r))=\frac{1}{\kappa}\left(\frac{1}{r}e^{-\Lambda(r)}\left(\Lambda(r% )^{\prime}+\Phi(r)^{\prime}\right)+\frac{2\alpha}{r^{3}}e^{-\Lambda(r)}\left(1% -e^{-\Lambda(r)}\right)\left(\Lambda(r)^{\prime}+\Phi(r)^{\prime}\right)\right)( italic_ρ ( italic_r ) + italic_P ( italic_r ) ) = divide start_ARG 1 end_ARG start_ARG italic_κ end_ARG ( divide start_ARG 1 end_ARG start_ARG italic_r end_ARG italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ( roman_Λ ( italic_r ) start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT + roman_Φ ( italic_r ) start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) + divide start_ARG 2 italic_α end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ( 1 - italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ) ( roman_Λ ( italic_r ) start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT + roman_Φ ( italic_r ) start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ) (39)

after division by r2superscript𝑟2r^{2}italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. This can be substituted into (38) to obtain

κ(dPdr+q(r)22πr5q(r)4πr4dqdr)𝜅𝑑𝑃𝑑𝑟𝑞superscript𝑟22𝜋superscript𝑟5𝑞𝑟4𝜋superscript𝑟4𝑑𝑞𝑑𝑟\displaystyle\kappa\left(\frac{dP}{dr}+\frac{q(r)^{2}}{2\pi r^{5}}-\frac{q(r)}% {4\pi r^{4}}\frac{dq}{dr}\right)italic_κ ( divide start_ARG italic_d italic_P end_ARG start_ARG italic_d italic_r end_ARG + divide start_ARG italic_q ( italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_π italic_r start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT end_ARG - divide start_ARG italic_q ( italic_r ) end_ARG start_ARG 4 italic_π italic_r start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG divide start_ARG italic_d italic_q end_ARG start_ARG italic_d italic_r end_ARG ) =κq(r)22πr512(1reΛ(r)+2αr3eΛ(r)(1eΛ(r)))(Λ(r)+Φ(r))Φ(r).absent𝜅𝑞superscript𝑟22𝜋superscript𝑟5121𝑟superscript𝑒Λ𝑟2𝛼superscript𝑟3superscript𝑒Λ𝑟1superscript𝑒Λ𝑟Λsuperscript𝑟Φsuperscript𝑟superscriptΦ𝑟\displaystyle=\kappa\frac{q(r)^{2}}{2\pi r^{5}}-\frac{1}{2}\left(\frac{1}{r}e^% {-\Lambda(r)}+\frac{2\alpha}{r^{3}}e^{-\Lambda(r)}\left(1-e^{-\Lambda(r)}% \right)\right)\left(\Lambda(r)^{\prime}+\Phi(r)^{\prime}\right)\Phi^{\prime}(r).= italic_κ divide start_ARG italic_q ( italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_π italic_r start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT end_ARG - divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( divide start_ARG 1 end_ARG start_ARG italic_r end_ARG italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT + divide start_ARG 2 italic_α end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ( 1 - italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ) ) ( roman_Λ ( italic_r ) start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT + roman_Φ ( italic_r ) start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) roman_Φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) . (40)

Making use of equation (37) we find

ddr(1r2[rΦ(r)eΛ(r)(1eΛ(r))]+α(1eΛ(r))r4[2rΦ(r)eΛ(r)+(1eΛ(r))])𝑑𝑑𝑟1superscript𝑟2delimited-[]𝑟superscriptΦ𝑟superscript𝑒Λ𝑟1superscript𝑒Λ𝑟𝛼1superscript𝑒Λ𝑟superscript𝑟4delimited-[]2𝑟superscriptΦ𝑟superscript𝑒Λ𝑟1superscript𝑒Λ𝑟\displaystyle\frac{d}{dr}\left(\frac{1}{r^{2}}\left[r\Phi^{\prime}(r)e^{-% \Lambda(r)}-\left(1-e^{-\Lambda(r)}\right)\right]+\alpha\frac{\left(1-e^{-% \Lambda(r)}\right)}{r^{4}}\left[2r\Phi^{\prime}(r)e^{-\Lambda(r)}+\left(1-e^{-% \Lambda(r)}\right)\right]\right)divide start_ARG italic_d end_ARG start_ARG italic_d italic_r end_ARG ( divide start_ARG 1 end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG [ italic_r roman_Φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT - ( 1 - italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ) ] + italic_α divide start_ARG ( 1 - italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ) end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG [ 2 italic_r roman_Φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT + ( 1 - italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ) ] ) (41)
=κq(r)22πr512(1reΛ(r)+2αr3eΛ(r)(1eΛ(r)))(Λ(r)+Φ(r))Φ(r).absent𝜅𝑞superscript𝑟22𝜋superscript𝑟5121𝑟superscript𝑒Λ𝑟2𝛼superscript𝑟3superscript𝑒Λ𝑟1superscript𝑒Λ𝑟Λsuperscript𝑟Φsuperscript𝑟superscriptΦ𝑟\displaystyle=\kappa\frac{q(r)^{2}}{2\pi r^{5}}-\frac{1}{2}\left(\frac{1}{r}e^% {-\Lambda(r)}+\frac{2\alpha}{r^{3}}e^{-\Lambda(r)}\left(1-e^{-\Lambda(r)}% \right)\right)\left(\Lambda(r)^{\prime}+\Phi(r)^{\prime}\right)\Phi^{\prime}(r).= italic_κ divide start_ARG italic_q ( italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_π italic_r start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT end_ARG - divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( divide start_ARG 1 end_ARG start_ARG italic_r end_ARG italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT + divide start_ARG 2 italic_α end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ( 1 - italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ) ) ( roman_Λ ( italic_r ) start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT + roman_Φ ( italic_r ) start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) roman_Φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) .

If we define β(r)=1eΛ(r)r2𝛽𝑟1superscript𝑒Λ𝑟superscript𝑟2\beta(r)=\frac{1-e^{-\Lambda(r)}}{r^{2}}italic_β ( italic_r ) = divide start_ARG 1 - italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG the above can be rewritten as

ddr(Φ(r)eΛ(r)r)(1+2αβ(r))+Z(r)=κq(r)22πr5(Φ(r)+Λ(r))2Φ(r)eΛ(r)r(1+2αβ(r))𝑑𝑑𝑟superscriptΦ𝑟superscript𝑒Λ𝑟𝑟12𝛼𝛽𝑟𝑍𝑟𝜅𝑞superscript𝑟22𝜋superscript𝑟5superscriptΦ𝑟superscriptΛ𝑟2superscriptΦ𝑟superscript𝑒Λ𝑟𝑟12𝛼𝛽𝑟\displaystyle\frac{d}{dr}\left(\frac{\Phi^{\prime}(r)e^{-\Lambda(r)}}{r}\right% )(1+2\alpha\beta(r))+Z(r)=\kappa\frac{q(r)^{2}}{2\pi r^{5}}-\frac{\left(\Phi^{% \prime}(r)+\Lambda^{\prime}(r)\right)}{2}\frac{\Phi^{\prime}(r)e^{-\Lambda(r)}% }{r}(1+2\alpha\beta(r))divide start_ARG italic_d end_ARG start_ARG italic_d italic_r end_ARG ( divide start_ARG roman_Φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT end_ARG start_ARG italic_r end_ARG ) ( 1 + 2 italic_α italic_β ( italic_r ) ) + italic_Z ( italic_r ) = italic_κ divide start_ARG italic_q ( italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_π italic_r start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT end_ARG - divide start_ARG ( roman_Φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) + roman_Λ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) ) end_ARG start_ARG 2 end_ARG divide start_ARG roman_Φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT end_ARG start_ARG italic_r end_ARG ( 1 + 2 italic_α italic_β ( italic_r ) ) (42)

where Z(r)=(2αΦ(r)eΛ(r)r1+2αβ(r))β(r)𝑍𝑟2𝛼superscriptΦ𝑟superscript𝑒Λ𝑟𝑟12𝛼𝛽𝑟superscript𝛽𝑟Z(r)=\left(\frac{2\alpha\Phi^{\prime}(r)e^{-\Lambda(r)}}{r}-1+2\alpha\beta(r)% \right)\beta^{\prime}(r)italic_Z ( italic_r ) = ( divide start_ARG 2 italic_α roman_Φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT end_ARG start_ARG italic_r end_ARG - 1 + 2 italic_α italic_β ( italic_r ) ) italic_β start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ). Equivalently,

2e(Φ(r)+Λ(r))/2ddr[eΛ(r)/2rdeΦ(r)/2dr](1+2αβ(r))κq(r)22πr52superscript𝑒Φ𝑟Λ𝑟2𝑑𝑑𝑟delimited-[]superscript𝑒Λ𝑟2𝑟𝑑superscript𝑒Φ𝑟2𝑑𝑟12𝛼𝛽𝑟𝜅𝑞superscript𝑟22𝜋superscript𝑟5\displaystyle 2e^{-(\Phi(r)+\Lambda(r))/2}\frac{d}{dr}\left[\frac{e^{-\Lambda(% r)/2}}{r}\frac{de^{\Phi(r)/2}}{dr}\right](1+2\alpha\beta(r))-\kappa\frac{q(r)^% {2}}{2\pi r^{5}}2 italic_e start_POSTSUPERSCRIPT - ( roman_Φ ( italic_r ) + roman_Λ ( italic_r ) ) / 2 end_POSTSUPERSCRIPT divide start_ARG italic_d end_ARG start_ARG italic_d italic_r end_ARG [ divide start_ARG italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) / 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r end_ARG divide start_ARG italic_d italic_e start_POSTSUPERSCRIPT roman_Φ ( italic_r ) / 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_r end_ARG ] ( 1 + 2 italic_α italic_β ( italic_r ) ) - italic_κ divide start_ARG italic_q ( italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_π italic_r start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT end_ARG
=(12αΦ(r)eΛ(r)r2αβ(r))β(r)absent12𝛼superscriptΦ𝑟superscript𝑒Λ𝑟𝑟2𝛼𝛽𝑟superscript𝛽𝑟\displaystyle\qquad\qquad=\left(1-\frac{2\alpha\Phi^{\prime}(r)e^{-\Lambda(r)}% }{r}-2\alpha\beta(r)\right)\beta^{\prime}(r)= ( 1 - divide start_ARG 2 italic_α roman_Φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT end_ARG start_ARG italic_r end_ARG - 2 italic_α italic_β ( italic_r ) ) italic_β start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) (43)

which can be shown using the identity

(ddr(Φ(r)eΛ(r)r)+12(Φ(r)eΛ(r)r)(Φ(r)+Λ(r)))=2e(Φ(r)+Λ(r))/2ddr[eΛ(r)/2rdeΦ(r)/2dr].𝑑𝑑𝑟superscriptΦ𝑟superscript𝑒Λ𝑟𝑟12superscriptΦ𝑟superscript𝑒Λ𝑟𝑟superscriptΦ𝑟superscriptΛ𝑟2superscript𝑒Φ𝑟Λ𝑟2𝑑𝑑𝑟delimited-[]superscript𝑒Λ𝑟2𝑟𝑑superscript𝑒Φ𝑟2𝑑𝑟\left(\frac{d}{dr}\left(\frac{\Phi^{\prime}(r)e^{-\Lambda(r)}}{r}\right)+\frac% {1}{2}(\frac{\Phi^{\prime}(r)e^{-\Lambda(r)}}{r})(\Phi^{\prime}(r)+\Lambda^{% \prime}(r))\right)=2e^{-(\Phi(r)+\Lambda(r))/2}\frac{d}{dr}\left[\frac{e^{-% \Lambda(r)/2}}{r}\frac{de^{\Phi(r)/2}}{dr}\right].( divide start_ARG italic_d end_ARG start_ARG italic_d italic_r end_ARG ( divide start_ARG roman_Φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT end_ARG start_ARG italic_r end_ARG ) + divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( divide start_ARG roman_Φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT end_ARG start_ARG italic_r end_ARG ) ( roman_Φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) + roman_Λ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) ) ) = 2 italic_e start_POSTSUPERSCRIPT - ( roman_Φ ( italic_r ) + roman_Λ ( italic_r ) ) / 2 end_POSTSUPERSCRIPT divide start_ARG italic_d end_ARG start_ARG italic_d italic_r end_ARG [ divide start_ARG italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) / 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r end_ARG divide start_ARG italic_d italic_e start_POSTSUPERSCRIPT roman_Φ ( italic_r ) / 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_r end_ARG ] . (44)

To proceed further we assume the factors (1+2αβ(r))12𝛼𝛽𝑟(1+2\alpha\beta(r))( 1 + 2 italic_α italic_β ( italic_r ) ) and (12αΦ(r)eΛ(r)r2αβ(r))12𝛼superscriptΦ𝑟superscript𝑒Λ𝑟𝑟2𝛼𝛽𝑟\left(1-\frac{2\alpha\Phi^{\prime}(r)e^{-\Lambda(r)}}{r}-2\alpha\beta(r)\right)( 1 - divide start_ARG 2 italic_α roman_Φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT end_ARG start_ARG italic_r end_ARG - 2 italic_α italic_β ( italic_r ) ) are both non-negative, which is valid for sufficiently small, positive α𝛼\alphaitalic_α. The right hand side of (43) is then negative due to the monotonically decreasing nature of β(r)𝛽𝑟\beta(r)italic_β ( italic_r ). We can thus write

2e(Φ(r)+Λ(r))/2ddr[eΛ(r)/2rdeΦ(r)/2dr](1+2αβ(r))κq(r)22πr502superscript𝑒Φ𝑟Λ𝑟2𝑑𝑑𝑟delimited-[]superscript𝑒Λ𝑟2𝑟𝑑superscript𝑒Φ𝑟2𝑑𝑟12𝛼𝛽𝑟𝜅𝑞superscript𝑟22𝜋superscript𝑟502e^{-(\Phi(r)+\Lambda(r))/2}\frac{d}{dr}\left[\frac{e^{-\Lambda(r)/2}}{r}\frac% {de^{\Phi(r)/2}}{dr}\right](1+2\alpha\beta(r))-\kappa\frac{q(r)^{2}}{2\pi r^{5% }}\leq 02 italic_e start_POSTSUPERSCRIPT - ( roman_Φ ( italic_r ) + roman_Λ ( italic_r ) ) / 2 end_POSTSUPERSCRIPT divide start_ARG italic_d end_ARG start_ARG italic_d italic_r end_ARG [ divide start_ARG italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) / 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r end_ARG divide start_ARG italic_d italic_e start_POSTSUPERSCRIPT roman_Φ ( italic_r ) / 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_d italic_r end_ARG ] ( 1 + 2 italic_α italic_β ( italic_r ) ) - italic_κ divide start_ARG italic_q ( italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_π italic_r start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT end_ARG ≤ 0 (45)

which upon integration yields

eΦ(r′′)/2κr′′𝑑rreΛ(r)/2r𝑑re(Φ(r)+Λ(r))/2q(r)24πr5(1+2αβ(r))1.superscript𝑒Φsuperscript𝑟′′2𝜅superscriptsuperscript𝑟′′differential-dsuperscript𝑟superscript𝑟superscript𝑒Λsuperscript𝑟2superscriptsuperscript𝑟differential-d𝑟superscript𝑒Φ𝑟Λ𝑟2𝑞superscript𝑟24𝜋superscript𝑟5superscript12𝛼𝛽𝑟1e^{\Phi(r^{\prime\prime})/2}\leq\kappa\int^{r^{\prime\prime}}dr^{\prime}r^{% \prime}e^{\Lambda(r^{\prime})/2}\int^{r^{\prime}}dre^{(\Phi(r)+\Lambda(r))/2}% \frac{q(r)^{2}}{4\pi r^{5}}(1+2\alpha\beta(r))^{-1}\;.italic_e start_POSTSUPERSCRIPT roman_Φ ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT ≤ italic_κ ∫ start_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT italic_d italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT roman_Λ ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT ∫ start_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT italic_d italic_r italic_e start_POSTSUPERSCRIPT ( roman_Φ ( italic_r ) + roman_Λ ( italic_r ) ) / 2 end_POSTSUPERSCRIPT divide start_ARG italic_q ( italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 4 italic_π italic_r start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT end_ARG ( 1 + 2 italic_α italic_β ( italic_r ) ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT . (46)

Making for convenience the following definitions

𝒬𝒬\displaystyle\mathcal{Q}caligraphic_Q e(Φ(r)+Λ(r))/2q(r)24πr5(1+2αβ(r))1absentsuperscript𝑒Φ𝑟Λ𝑟2𝑞superscript𝑟24𝜋superscript𝑟5superscript12𝛼𝛽𝑟1\displaystyle\equiv e^{(\Phi(r)+\Lambda(r))/2}\frac{q(r)^{2}}{4\pi r^{5}}(1+2% \alpha\beta(r))^{-1}≡ italic_e start_POSTSUPERSCRIPT ( roman_Φ ( italic_r ) + roman_Λ ( italic_r ) ) / 2 end_POSTSUPERSCRIPT divide start_ARG italic_q ( italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 4 italic_π italic_r start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT end_ARG ( 1 + 2 italic_α italic_β ( italic_r ) ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT (47)
η(r′′)𝜂superscript𝑟′′\displaystyle\eta(r^{\prime\prime})italic_η ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) r′′eΛ(r)/2rr𝒬(r)𝑑r𝑑rabsentsuperscriptsuperscript𝑟′′superscript𝑒Λsuperscript𝑟2superscript𝑟superscriptsuperscript𝑟𝒬𝑟differential-d𝑟differential-dsuperscript𝑟\displaystyle\equiv\int^{r^{\prime\prime}}e^{\Lambda(r^{\prime})/2}r^{\prime}% \int^{r^{\prime}}\mathcal{Q}(r)drdr^{\prime}≡ ∫ start_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT roman_Λ ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ∫ start_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT caligraphic_Q ( italic_r ) italic_d italic_r italic_d italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT
ψ𝜓\displaystyle\psiitalic_ψ eΦ(r′′)/2η(r′′)ξr′′reΛ(r)/2𝑑rformulae-sequenceabsentsuperscript𝑒Φsuperscript𝑟′′2𝜂superscript𝑟′′𝜉superscriptsuperscript𝑟′′superscript𝑟superscript𝑒Λsuperscript𝑟2differential-dsuperscript𝑟\displaystyle\equiv e^{\Phi(r^{\prime\prime})/2}-\eta(r^{\prime\prime})\qquad% \xi\equiv\int^{r^{\prime\prime}}r^{\prime}e^{\Lambda(r^{\prime})/2}dr^{\prime}≡ italic_e start_POSTSUPERSCRIPT roman_Φ ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT - italic_η ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) italic_ξ ≡ ∫ start_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT roman_Λ ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT italic_d italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT

it is straightforward to show that

dψdξ=1r′′eΛ(r′′)/2ddr′′eΦ(r′′)/2r′′𝑑r𝒬(r),𝑑𝜓𝑑𝜉1superscript𝑟′′superscript𝑒Λsuperscript𝑟′′2𝑑𝑑superscript𝑟′′superscript𝑒Φsuperscript𝑟′′2superscriptsuperscript𝑟′′differential-d𝑟𝒬superscript𝑟\displaystyle\frac{d\psi}{d\xi}=\frac{1}{r^{\prime\prime}}e^{-\Lambda(r^{% \prime\prime})/2}\frac{d}{dr^{\prime\prime}}e^{\Phi(r^{\prime\prime})/2}-\int^% {r^{\prime\prime}}dr\mathcal{Q}(r^{\prime}),divide start_ARG italic_d italic_ψ end_ARG start_ARG italic_d italic_ξ end_ARG = divide start_ARG 1 end_ARG start_ARG italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_ARG italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT divide start_ARG italic_d end_ARG start_ARG italic_d italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_ARG italic_e start_POSTSUPERSCRIPT roman_Φ ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT - ∫ start_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT italic_d italic_r caligraphic_Q ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) , (48)
d2ψdξ2=1r′′eΛ(r′′)/2ddr[1r′′eΛ(r′′)/2ddr′′eΦ(r′′)/2]1r′′eΛ(r′′)/2𝒬(r′′)0superscript𝑑2𝜓𝑑superscript𝜉21superscript𝑟′′superscript𝑒Λsuperscript𝑟′′2𝑑𝑑𝑟delimited-[]1superscript𝑟′′superscript𝑒Λsuperscript𝑟′′2𝑑𝑑superscript𝑟′′superscript𝑒Φsuperscript𝑟′′21superscript𝑟′′superscript𝑒Λsuperscript𝑟′′2𝒬superscript𝑟′′0\displaystyle\frac{d^{2}\psi}{d\xi^{2}}=\frac{1}{r^{\prime\prime}}e^{-\Lambda(% r^{\prime\prime})/2}\frac{d}{dr}\left[\frac{1}{r^{\prime\prime}}e^{-\Lambda(r^% {\prime\prime})/2}\frac{d}{dr^{\prime\prime}}e^{\Phi(r^{\prime\prime})/2}% \right]-\frac{1}{r^{\prime\prime}}e^{-\Lambda(r^{\prime\prime})/2}\mathcal{Q}(% r^{\prime\prime})\leq 0divide start_ARG italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ψ end_ARG start_ARG italic_d italic_ξ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG = divide start_ARG 1 end_ARG start_ARG italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_ARG italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT divide start_ARG italic_d end_ARG start_ARG italic_d italic_r end_ARG [ divide start_ARG 1 end_ARG start_ARG italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_ARG italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT divide start_ARG italic_d end_ARG start_ARG italic_d italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_ARG italic_e start_POSTSUPERSCRIPT roman_Φ ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT ] - divide start_ARG 1 end_ARG start_ARG italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_ARG italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT caligraphic_Q ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) ≤ 0 (49)

where integration is from the center (r′′>r>rsuperscript𝑟′′superscript𝑟𝑟r^{\prime\prime}>r^{\prime}>ritalic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT > italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT > italic_r), or alternatively

d2ψdξ2reΛ(r)/20superscript𝑑2𝜓𝑑superscript𝜉2𝑟superscript𝑒Λ𝑟20\frac{d^{2}\psi}{d\xi^{2}}re^{\Lambda(r)/2}\leq 0divide start_ARG italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ψ end_ARG start_ARG italic_d italic_ξ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG italic_r italic_e start_POSTSUPERSCRIPT roman_Λ ( italic_r ) / 2 end_POSTSUPERSCRIPT ≤ 0 (50)

which follows from (45). Since eΛ(r)/2>0superscript𝑒Λ𝑟20e^{\Lambda(r)/2}>0italic_e start_POSTSUPERSCRIPT roman_Λ ( italic_r ) / 2 end_POSTSUPERSCRIPT > 0 and r>0𝑟0r>0italic_r > 0 this is equivalent to

d2ψdξ20.superscript𝑑2𝜓𝑑superscript𝜉20\frac{d^{2}\psi}{d\xi^{2}}\leq 0.divide start_ARG italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_ψ end_ARG start_ARG italic_d italic_ξ start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ≤ 0 . (51)

Trivially integrating and applying the mean value theorem bohmer_2007 we see that

dψdξψ(ξ)ψ(0)ξ0ψ(ξ)ξ𝑑𝜓𝑑𝜉𝜓𝜉𝜓0𝜉0𝜓𝜉𝜉\frac{d\psi}{d\xi}\leq\frac{\psi(\xi)-\psi(0)}{\xi-0}\leq\frac{\psi(\xi)}{\xi}divide start_ARG italic_d italic_ψ end_ARG start_ARG italic_d italic_ξ end_ARG ≤ divide start_ARG italic_ψ ( italic_ξ ) - italic_ψ ( 0 ) end_ARG start_ARG italic_ξ - 0 end_ARG ≤ divide start_ARG italic_ψ ( italic_ξ ) end_ARG start_ARG italic_ξ end_ARG (52)

since at the center of the star ξ=0𝜉0\xi=0italic_ξ = 0 and ψ(0)>0𝜓00\psi(0)>0italic_ψ ( 0 ) > 0 since eΦ(r)/2>0superscript𝑒Φ𝑟20e^{\Phi(r)/2}>0italic_e start_POSTSUPERSCRIPT roman_Φ ( italic_r ) / 2 end_POSTSUPERSCRIPT > 0 inside the star.

Putting everything together we find

(1reΛ(r′′)/2ddreΦ(r′′)/2r′′𝑑r𝒬(r))(r′′𝑑rreΛ(r)/2)1𝑟superscript𝑒Λsuperscript𝑟′′2𝑑𝑑𝑟superscript𝑒Φsuperscript𝑟′′2superscriptsuperscript𝑟′′differential-dsuperscript𝑟𝒬superscript𝑟superscriptsuperscript𝑟′′differential-dsuperscript𝑟superscript𝑟superscript𝑒Λsuperscript𝑟2\displaystyle\left(\frac{1}{r}e^{-\Lambda(r^{\prime\prime})/2}\frac{d}{dr}e^{% \Phi(r^{\prime\prime})/2}-\int^{r^{\prime\prime}}dr^{\prime}\mathcal{Q}(r^{% \prime})\right)\left(\int^{r^{\prime\prime}}dr^{\prime}r^{\prime}e^{\Lambda(r^% {\prime})/2}\right)( divide start_ARG 1 end_ARG start_ARG italic_r end_ARG italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT divide start_ARG italic_d end_ARG start_ARG italic_d italic_r end_ARG italic_e start_POSTSUPERSCRIPT roman_Φ ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT - ∫ start_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT italic_d italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT caligraphic_Q ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ) ( ∫ start_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT italic_d italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT roman_Λ ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT )
eΦ(r′′)/2r′′reΛint/2r𝒬(r)𝑑r𝑑r.absentsuperscript𝑒Φsuperscript𝑟′′2superscriptsuperscript𝑟′′superscript𝑟superscript𝑒subscriptΛint2superscriptsuperscript𝑟𝒬𝑟differential-d𝑟differential-dsuperscript𝑟\displaystyle\qquad\qquad\qquad\leq e^{\Phi(r^{\prime\prime})/2}-\int^{r^{% \prime\prime}}r^{\prime}e^{\Lambda_{\mathrm{int}}/2}\int^{r^{\prime}}\mathcal{% Q}(r)drdr^{\prime}.≤ italic_e start_POSTSUPERSCRIPT roman_Φ ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT - ∫ start_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT roman_Λ start_POSTSUBSCRIPT roman_int end_POSTSUBSCRIPT / 2 end_POSTSUPERSCRIPT ∫ start_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT caligraphic_Q ( italic_r ) italic_d italic_r italic_d italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT . (53)

Recalling that β(r)=1eΛ(r)r2=12α[11+κα2π(2m(r)r3q(r)2r4)]𝛽𝑟1superscript𝑒Λ𝑟superscript𝑟212𝛼delimited-[]11𝜅𝛼2𝜋2𝑚𝑟superscript𝑟3𝑞superscript𝑟2superscript𝑟4\beta(r)=\frac{1-e^{-\Lambda(r)}}{r^{2}}=-\frac{1}{2\alpha}\left[1-\sqrt{1+% \frac{\kappa\alpha}{2\pi}\left(\frac{2m(r)}{r^{3}}-\frac{q(r)^{2}}{r^{4}}% \right)}\right]italic_β ( italic_r ) = divide start_ARG 1 - italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG = - divide start_ARG 1 end_ARG start_ARG 2 italic_α end_ARG [ 1 - square-root start_ARG 1 + divide start_ARG italic_κ italic_α end_ARG start_ARG 2 italic_π end_ARG ( divide start_ARG 2 italic_m ( italic_r ) end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG - divide start_ARG italic_q ( italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG ) end_ARG ] such that

eΛ(r)=1r2β(r),superscript𝑒Λ𝑟1superscript𝑟2𝛽𝑟e^{-\Lambda(r)}=1-r^{2}\beta(r),italic_e start_POSTSUPERSCRIPT - roman_Λ ( italic_r ) end_POSTSUPERSCRIPT = 1 - italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_β ( italic_r ) , (54)

and that β(r)β(r)β(r′′)𝛽𝑟𝛽superscript𝑟𝛽superscript𝑟′′\beta(r)\geq\beta(r^{\prime})\geq\beta(r^{\prime\prime})italic_β ( italic_r ) ≥ italic_β ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ≥ italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) (more primes corresponding to further from the center), we obtain

ξr′′𝑑rreΛ(r)/2=r′′𝑑rr11r2β(r)𝜉superscriptsuperscript𝑟′′differential-dsuperscript𝑟superscript𝑟superscript𝑒Λsuperscript𝑟2superscriptsuperscript𝑟′′differential-dsuperscript𝑟superscript𝑟11superscript𝑟2𝛽superscript𝑟\displaystyle\xi\equiv\int^{r^{\prime\prime}}dr^{\prime}r^{\prime}e^{\Lambda(r% ^{\prime})/2}=\int^{r^{\prime\prime}}dr^{\prime}r^{\prime}\frac{1}{\sqrt{1-r^{% \prime 2}\beta(r^{\prime})}}italic_ξ ≡ ∫ start_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT italic_d italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT roman_Λ ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT = ∫ start_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT italic_d italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG square-root start_ARG 1 - italic_r start_POSTSUPERSCRIPT ′ 2 end_POSTSUPERSCRIPT italic_β ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_ARG end_ARG r′′𝑑rr11r2β(r′′)absentsuperscriptsuperscript𝑟′′differential-dsuperscript𝑟superscript𝑟11superscript𝑟2𝛽superscript𝑟′′\displaystyle\geq\int^{r^{\prime\prime}}dr^{\prime}r^{\prime}\frac{1}{\sqrt{1-% r^{\prime 2}\beta(r^{\prime\prime})}}≥ ∫ start_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT italic_d italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG square-root start_ARG 1 - italic_r start_POSTSUPERSCRIPT ′ 2 end_POSTSUPERSCRIPT italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG end_ARG (55)
=1β(r′′)(11r′′2β(r′′))absent1𝛽superscript𝑟′′11superscript𝑟′′2𝛽superscript𝑟′′\displaystyle=\frac{1}{\beta(r^{\prime\prime})}\left(1-\sqrt{1-r^{\prime\prime 2% }\beta(r^{\prime\prime})}\right)= divide start_ARG 1 end_ARG start_ARG italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG ( 1 - square-root start_ARG 1 - italic_r start_POSTSUPERSCRIPT ′ ′ 2 end_POSTSUPERSCRIPT italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG )

as a bound on ξ𝜉\xiitalic_ξ.

Similarly we can consider the term

r′′𝒬(r)𝑑rsuperscriptsuperscript𝑟′′𝒬superscript𝑟differential-d𝑟\displaystyle\int^{r^{\prime\prime}}\mathcal{Q}(r^{\prime})dr∫ start_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT caligraphic_Q ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) italic_d italic_r =κr′′e(Φ(r)+Λ(r))/2q(r)24πr5(1+2αβ(r))1𝑑rabsent𝜅superscriptsuperscript𝑟′′superscript𝑒Φsuperscript𝑟Λsuperscript𝑟2𝑞superscriptsuperscript𝑟24𝜋superscript𝑟5superscript12𝛼𝛽superscript𝑟1differential-d𝑟\displaystyle=\kappa\int^{r^{\prime\prime}}e^{(\Phi(r^{\prime})+\Lambda(r^{% \prime}))/2}\frac{q(r^{\prime})^{2}}{4\pi r^{\prime 5}}(1+2\alpha\beta(r^{% \prime}))^{-1}dr= italic_κ ∫ start_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT ( roman_Φ ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) + roman_Λ ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ) / 2 end_POSTSUPERSCRIPT divide start_ARG italic_q ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 4 italic_π italic_r start_POSTSUPERSCRIPT ′ 5 end_POSTSUPERSCRIPT end_ARG ( 1 + 2 italic_α italic_β ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_d italic_r (56)
=κ4πr′′eΦ(r)/21r2β(r)(1+2αβ(r))q(r)2r5𝑑r.absent𝜅4𝜋superscriptsuperscript𝑟′′superscript𝑒Φsuperscript𝑟21superscript𝑟2𝛽superscript𝑟12𝛼𝛽superscript𝑟𝑞superscriptsuperscript𝑟2superscript𝑟5differential-dsuperscript𝑟\displaystyle=\frac{\kappa}{4\pi}\int^{r^{\prime\prime}}\frac{e^{\Phi(r^{% \prime})/2}}{\sqrt{1-r^{\prime 2}\beta(r^{\prime})}\left(1+2\alpha\beta(r^{% \prime})\right)}\frac{q(r^{\prime})^{2}}{r^{\prime 5}}dr^{\prime}.= divide start_ARG italic_κ end_ARG start_ARG 4 italic_π end_ARG ∫ start_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT divide start_ARG italic_e start_POSTSUPERSCRIPT roman_Φ ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT end_ARG start_ARG square-root start_ARG 1 - italic_r start_POSTSUPERSCRIPT ′ 2 end_POSTSUPERSCRIPT italic_β ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_ARG ( 1 + 2 italic_α italic_β ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ) end_ARG divide start_ARG italic_q ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r start_POSTSUPERSCRIPT ′ 5 end_POSTSUPERSCRIPT end_ARG italic_d italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT .

Combining our previous assumption on β𝛽\betaitalic_β with the following assumption regarding the behaviour of the charge:

eΦint(r)/2q(r)2r5eΦint(r)/2q(r)2r5eΦint(r′′)/2q(r′′)2r′′5superscript𝑒subscriptΦint𝑟2𝑞superscript𝑟2superscript𝑟5superscript𝑒subscriptΦintsuperscript𝑟2𝑞superscriptsuperscript𝑟2superscript𝑟5superscript𝑒subscriptΦintsuperscript𝑟′′2𝑞superscriptsuperscript𝑟′′2superscript𝑟′′5e^{\Phi_{\mathrm{int}}(r)/2}\frac{q(r)^{2}}{r^{5}}\geq e^{\Phi_{\mathrm{int}}(% r^{\prime})/2}\frac{q(r^{\prime})^{2}}{r^{\prime 5}}\geq e^{\Phi_{\mathrm{int}% }(r^{\prime\prime})/2}\frac{q(r^{\prime\prime})^{2}}{r^{\prime\prime 5}}italic_e start_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_int end_POSTSUBSCRIPT ( italic_r ) / 2 end_POSTSUPERSCRIPT divide start_ARG italic_q ( italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT end_ARG ≥ italic_e start_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_int end_POSTSUBSCRIPT ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT divide start_ARG italic_q ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r start_POSTSUPERSCRIPT ′ 5 end_POSTSUPERSCRIPT end_ARG ≥ italic_e start_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_int end_POSTSUBSCRIPT ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT divide start_ARG italic_q ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r start_POSTSUPERSCRIPT ′ ′ 5 end_POSTSUPERSCRIPT end_ARG (57)

the above integral can be bounded as follows:

κ4πr′′eΦ(r)/21r2β(r)(1+2αβ(r))q(r)2r5𝑑r𝜅4𝜋superscriptsuperscript𝑟′′superscript𝑒Φsuperscript𝑟21superscript𝑟2𝛽superscript𝑟12𝛼𝛽superscript𝑟𝑞superscriptsuperscript𝑟2superscript𝑟5differential-dsuperscript𝑟\displaystyle\frac{\kappa}{4\pi}\int^{r^{\prime\prime}}\frac{e^{\Phi(r^{\prime% })/2}}{\sqrt{1-r^{\prime 2}\beta(r^{\prime})}\left(1+2\alpha\beta(r^{\prime})% \right)}\frac{q(r^{\prime})^{2}}{r^{\prime 5}}dr^{\prime}divide start_ARG italic_κ end_ARG start_ARG 4 italic_π end_ARG ∫ start_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT divide start_ARG italic_e start_POSTSUPERSCRIPT roman_Φ ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT end_ARG start_ARG square-root start_ARG 1 - italic_r start_POSTSUPERSCRIPT ′ 2 end_POSTSUPERSCRIPT italic_β ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_ARG ( 1 + 2 italic_α italic_β ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ) end_ARG divide start_ARG italic_q ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r start_POSTSUPERSCRIPT ′ 5 end_POSTSUPERSCRIPT end_ARG italic_d italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT κ4πeΦint(r′′)/2(1+2αβ(0))q(r′′)2r′′5r′′dr1r2β(r′′)absent𝜅4𝜋superscript𝑒subscriptΦintsuperscript𝑟′′212𝛼𝛽0𝑞superscriptsuperscript𝑟′′2superscript𝑟′′5superscriptsuperscript𝑟′′𝑑superscript𝑟1superscript𝑟2𝛽superscript𝑟′′\displaystyle\geq\frac{\kappa}{4\pi}\frac{e^{\Phi_{\mathrm{int}}(r^{\prime% \prime})/2}}{\left(1+2\alpha\beta(0)\right)}\frac{q(r^{\prime\prime})^{2}}{r^{% \prime\prime 5}}\int^{r^{\prime\prime}}\frac{dr^{\prime}}{\sqrt{1-r^{\prime 2}% \beta(r^{\prime\prime})}}≥ divide start_ARG italic_κ end_ARG start_ARG 4 italic_π end_ARG divide start_ARG italic_e start_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_int end_POSTSUBSCRIPT ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( 1 + 2 italic_α italic_β ( 0 ) ) end_ARG divide start_ARG italic_q ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r start_POSTSUPERSCRIPT ′ ′ 5 end_POSTSUPERSCRIPT end_ARG ∫ start_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT divide start_ARG italic_d italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG start_ARG square-root start_ARG 1 - italic_r start_POSTSUPERSCRIPT ′ 2 end_POSTSUPERSCRIPT italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG end_ARG (58)
=κ4πeΦint(r′′)/2(1+2αβ(0))q(r′′)2r′′5arcsin(β(r′′)r′′)β(r′′).absent𝜅4𝜋superscript𝑒subscriptΦintsuperscript𝑟′′212𝛼𝛽0𝑞superscriptsuperscript𝑟′′2superscript𝑟′′5arcsin𝛽superscript𝑟′′superscript𝑟′′𝛽superscript𝑟′′\displaystyle=\frac{\kappa}{4\pi}\frac{e^{\Phi_{\mathrm{int}}(r^{\prime\prime}% )/2}}{\left(1+2\alpha\beta(0)\right)}\frac{q(r^{\prime\prime})^{2}}{r^{\prime% \prime 5}}\frac{\mathrm{arcsin}\left(\sqrt{\beta(r^{\prime\prime})}r^{\prime% \prime}\right)}{\sqrt{\beta(r^{\prime\prime})}}.= divide start_ARG italic_κ end_ARG start_ARG 4 italic_π end_ARG divide start_ARG italic_e start_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_int end_POSTSUBSCRIPT ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( 1 + 2 italic_α italic_β ( 0 ) ) end_ARG divide start_ARG italic_q ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r start_POSTSUPERSCRIPT ′ ′ 5 end_POSTSUPERSCRIPT end_ARG divide start_ARG roman_arcsin ( square-root start_ARG italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG start_ARG square-root start_ARG italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG end_ARG .

Finally we can consider the most complicated integral term:

r′′reΛint/2r𝒬(r)𝑑r𝑑r=κ4πr′′reΛint/2reΦint/21r2β(r)(1+2αβ(r))q(r)2r5𝑑r𝑑rsuperscriptsuperscript𝑟′′superscript𝑟superscript𝑒subscriptΛint2superscriptsuperscript𝑟𝒬𝑟differential-d𝑟differential-dsuperscript𝑟𝜅4𝜋superscriptsuperscript𝑟′′superscript𝑟superscript𝑒subscriptΛint2superscriptsuperscript𝑟superscript𝑒subscriptΦint21superscript𝑟2𝛽𝑟12𝛼𝛽𝑟𝑞superscript𝑟2superscript𝑟5differential-d𝑟differential-dsuperscript𝑟\displaystyle\int^{r^{\prime\prime}}r^{\prime}e^{\Lambda_{\mathrm{int}}/2}\int% ^{r^{\prime}}\mathcal{Q}(r)drdr^{\prime}=\frac{\kappa}{4\pi}\int^{r^{\prime% \prime}}r^{\prime}e^{\Lambda_{\mathrm{int}}/2}\int^{r^{\prime}}\frac{e^{\Phi_{% \mathrm{int}}/2}}{\sqrt{1-r^{2}\beta(r)}\left(1+2\alpha\beta(r)\right)}\frac{q% (r)^{2}}{r^{5}}drdr^{\prime}∫ start_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT roman_Λ start_POSTSUBSCRIPT roman_int end_POSTSUBSCRIPT / 2 end_POSTSUPERSCRIPT ∫ start_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT caligraphic_Q ( italic_r ) italic_d italic_r italic_d italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = divide start_ARG italic_κ end_ARG start_ARG 4 italic_π end_ARG ∫ start_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT roman_Λ start_POSTSUBSCRIPT roman_int end_POSTSUBSCRIPT / 2 end_POSTSUPERSCRIPT ∫ start_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT divide start_ARG italic_e start_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_int end_POSTSUBSCRIPT / 2 end_POSTSUPERSCRIPT end_ARG start_ARG square-root start_ARG 1 - italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_β ( italic_r ) end_ARG ( 1 + 2 italic_α italic_β ( italic_r ) ) end_ARG divide start_ARG italic_q ( italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT end_ARG italic_d italic_r italic_d italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT (59)
κ4π(1+2αβ(0))1r′′re(Φint(r)+Λint(r))/2q(r)2r5arcsin(β(r)r)β(r)𝑑rabsent𝜅4𝜋superscript12𝛼𝛽01superscriptsuperscript𝑟′′superscript𝑟superscript𝑒subscriptΦintsuperscript𝑟subscriptΛintsuperscript𝑟2𝑞superscriptsuperscript𝑟2superscript𝑟5arcsin𝛽superscript𝑟superscript𝑟𝛽superscript𝑟differential-dsuperscript𝑟\displaystyle\geq\frac{\kappa}{4\pi}\left(1+2\alpha\beta(0)\right)^{-1}\int^{r% ^{\prime\prime}}r^{\prime}e^{(\Phi_{\mathrm{int}}(r^{\prime})+\Lambda_{\mathrm% {int}}(r^{\prime}))/2}\frac{q(r^{\prime})^{2}}{r^{\prime 5}}\frac{\mathrm{% arcsin}\left(\sqrt{\beta(r^{\prime})}r^{\prime}\right)}{\sqrt{\beta(r^{\prime}% )}}dr^{\prime}≥ divide start_ARG italic_κ end_ARG start_ARG 4 italic_π end_ARG ( 1 + 2 italic_α italic_β ( 0 ) ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ∫ start_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT ( roman_Φ start_POSTSUBSCRIPT roman_int end_POSTSUBSCRIPT ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) + roman_Λ start_POSTSUBSCRIPT roman_int end_POSTSUBSCRIPT ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) ) / 2 end_POSTSUPERSCRIPT divide start_ARG italic_q ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r start_POSTSUPERSCRIPT ′ 5 end_POSTSUPERSCRIPT end_ARG divide start_ARG roman_arcsin ( square-root start_ARG italic_β ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_ARG italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_ARG start_ARG square-root start_ARG italic_β ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_ARG end_ARG italic_d italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT
κ4π(1+2αβ(0))1eΦint(r′′)/2q(r′′)2r′′5r′′r211r2β(r)arcsin(β(r)r)β(r)r𝑑r.absent𝜅4𝜋superscript12𝛼𝛽01superscript𝑒subscriptΦintsuperscript𝑟′′2𝑞superscriptsuperscript𝑟′′2superscript𝑟′′5superscriptsuperscript𝑟′′superscript𝑟211superscript𝑟2𝛽superscript𝑟arcsin𝛽superscript𝑟superscript𝑟𝛽superscript𝑟superscript𝑟differential-dsuperscript𝑟\displaystyle\geq\frac{\kappa}{4\pi}\left(1+2\alpha\beta(0)\right)^{-1}e^{\Phi% _{\mathrm{int}}(r^{\prime\prime})/2}\frac{q(r^{\prime\prime})^{2}}{r^{\prime% \prime 5}}\int^{r^{\prime\prime}}r^{\prime 2}\frac{1}{\sqrt{1-r^{\prime 2}% \beta(r^{\prime})}}\frac{\mathrm{arcsin}\left(\sqrt{\beta(r^{\prime})}r^{% \prime}\right)}{\sqrt{\beta(r^{\prime})}r^{\prime}}dr^{\prime}.≥ divide start_ARG italic_κ end_ARG start_ARG 4 italic_π end_ARG ( 1 + 2 italic_α italic_β ( 0 ) ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_int end_POSTSUBSCRIPT ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT divide start_ARG italic_q ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r start_POSTSUPERSCRIPT ′ ′ 5 end_POSTSUPERSCRIPT end_ARG ∫ start_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ 2 end_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG square-root start_ARG 1 - italic_r start_POSTSUPERSCRIPT ′ 2 end_POSTSUPERSCRIPT italic_β ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_ARG end_ARG divide start_ARG roman_arcsin ( square-root start_ARG italic_β ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_ARG italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_ARG start_ARG square-root start_ARG italic_β ( italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_ARG italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG italic_d italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT .

Since the function arcsin(x)/xarcsin𝑥𝑥\mathrm{arcsin}(x)/xroman_arcsin ( italic_x ) / italic_x monotonically increases on the interval x[0,1]𝑥01x\in[0,1]italic_x ∈ [ 0 , 1 ], the integral is minimal when the smallest β𝛽\betaitalic_β is chosen. Thus

r′′reΛint/2r𝒬(r)𝑑r𝑑rsuperscriptsuperscript𝑟′′superscript𝑟superscript𝑒subscriptΛint2superscriptsuperscript𝑟𝒬𝑟differential-d𝑟differential-dsuperscript𝑟\displaystyle\int^{r^{\prime\prime}}r^{\prime}e^{\Lambda_{\mathrm{int}}/2}\int% ^{r^{\prime}}\mathcal{Q}(r)drdr^{\prime}∫ start_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT roman_Λ start_POSTSUBSCRIPT roman_int end_POSTSUBSCRIPT / 2 end_POSTSUPERSCRIPT ∫ start_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT caligraphic_Q ( italic_r ) italic_d italic_r italic_d italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT (60)
κ4π(1+2αβ(0))1eΦint(r′′)/2q(r′′)2r′′5r′′r211r2β(r′′)arcsin(β(r′′)r)β(r′′)r𝑑rabsent𝜅4𝜋superscript12𝛼𝛽01superscript𝑒subscriptΦintsuperscript𝑟′′2𝑞superscriptsuperscript𝑟′′2superscript𝑟′′5superscriptsuperscript𝑟′′superscript𝑟211superscript𝑟2𝛽superscript𝑟′′arcsin𝛽superscript𝑟′′superscript𝑟𝛽superscript𝑟′′superscript𝑟differential-dsuperscript𝑟\displaystyle\geq\frac{\kappa}{4\pi}\left(1+2\alpha\beta(0)\right)^{-1}e^{\Phi% _{\mathrm{int}}(r^{\prime\prime})/2}\frac{q(r^{\prime\prime})^{2}}{r^{\prime% \prime 5}}\int^{r^{\prime\prime}}r^{\prime 2}\frac{1}{\sqrt{1-r^{\prime 2}% \beta(r^{\prime\prime})}}\frac{\mathrm{arcsin}\left(\sqrt{\beta(r^{\prime% \prime})}r^{\prime}\right)}{\sqrt{\beta(r^{\prime\prime})}r^{\prime}}dr^{\prime}≥ divide start_ARG italic_κ end_ARG start_ARG 4 italic_π end_ARG ( 1 + 2 italic_α italic_β ( 0 ) ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_int end_POSTSUBSCRIPT ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT divide start_ARG italic_q ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r start_POSTSUPERSCRIPT ′ ′ 5 end_POSTSUPERSCRIPT end_ARG ∫ start_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT italic_r start_POSTSUPERSCRIPT ′ 2 end_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG square-root start_ARG 1 - italic_r start_POSTSUPERSCRIPT ′ 2 end_POSTSUPERSCRIPT italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG end_ARG divide start_ARG roman_arcsin ( square-root start_ARG italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ) end_ARG start_ARG square-root start_ARG italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_ARG italic_d italic_r start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT
=κ4π(1+2αβ(0))1eΦint(r′′)/2q(r′′)2r′′51β(r′′)(r′′1β(r′′)r′′2β(r′′)arcsin(β(r′′)r′′)).absent𝜅4𝜋superscript12𝛼𝛽01superscript𝑒subscriptΦintsuperscript𝑟′′2𝑞superscriptsuperscript𝑟′′2superscript𝑟′′51𝛽superscript𝑟′′superscript𝑟′′1𝛽superscript𝑟′′superscript𝑟′′2𝛽superscript𝑟′′arcsin𝛽superscript𝑟′′superscript𝑟′′\displaystyle=\frac{\kappa}{4\pi}\left(1+2\alpha\beta(0)\right)^{-1}e^{\Phi_{% \mathrm{int}}(r^{\prime\prime})/2}\frac{q(r^{\prime\prime})^{2}}{r^{\prime% \prime 5}}\frac{1}{\beta(r^{\prime\prime})}\left(r^{\prime\prime}-\sqrt{\frac{% 1-\beta(r^{\prime\prime})r^{\prime\prime 2}}{\beta(r^{\prime\prime})}}\mathrm{% arcsin}(\sqrt{\beta(r^{\prime\prime})}r^{\prime\prime})\right).= divide start_ARG italic_κ end_ARG start_ARG 4 italic_π end_ARG ( 1 + 2 italic_α italic_β ( 0 ) ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT roman_Φ start_POSTSUBSCRIPT roman_int end_POSTSUBSCRIPT ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT divide start_ARG italic_q ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r start_POSTSUPERSCRIPT ′ ′ 5 end_POSTSUPERSCRIPT end_ARG divide start_ARG 1 end_ARG start_ARG italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT - square-root start_ARG divide start_ARG 1 - italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) italic_r start_POSTSUPERSCRIPT ′ ′ 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG end_ARG roman_arcsin ( square-root start_ARG italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) ) .

Inserting these results into (III.1) yields

(e(Φ(r′′)Λ(r′′))/2Φ(r′′)2rκ4πeΦ(r′′)/2(1+2αβ(0))q(r′′)2r′′5arcsin(β(r′′)r′′)β(r′′))((11r′′2β(r′′))β(r′′))superscript𝑒Φsuperscript𝑟′′Λsuperscript𝑟′′2superscriptΦsuperscript𝑟′′2𝑟𝜅4𝜋superscript𝑒Φsuperscript𝑟′′212𝛼𝛽0𝑞superscriptsuperscript𝑟′′2superscript𝑟′′5arcsin𝛽superscript𝑟′′superscript𝑟′′𝛽superscript𝑟′′11superscript𝑟′′2𝛽superscript𝑟′′𝛽superscript𝑟′′\displaystyle\left(e^{(\Phi(r^{\prime\prime})-\Lambda(r^{\prime\prime}))/2}% \frac{\Phi^{\prime}(r^{\prime\prime})}{2r}-\frac{\kappa}{4\pi}\frac{e^{\Phi(r^% {\prime\prime})/2}}{\left(1+2\alpha\beta(0)\right)}\frac{q(r^{\prime\prime})^{% 2}}{r^{\prime\prime 5}}\frac{\mathrm{arcsin}\left(\sqrt{\beta(r^{\prime\prime}% )}r^{\prime\prime}\right)}{\sqrt{\beta(r^{\prime\prime})}}\right)\left(\frac{% \left(1-\sqrt{1-r^{\prime\prime 2}\beta(r^{\prime\prime})}\right)}{\beta(r^{% \prime\prime})}\right)( italic_e start_POSTSUPERSCRIPT ( roman_Φ ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) - roman_Λ ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) ) / 2 end_POSTSUPERSCRIPT divide start_ARG roman_Φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG start_ARG 2 italic_r end_ARG - divide start_ARG italic_κ end_ARG start_ARG 4 italic_π end_ARG divide start_ARG italic_e start_POSTSUPERSCRIPT roman_Φ ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( 1 + 2 italic_α italic_β ( 0 ) ) end_ARG divide start_ARG italic_q ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r start_POSTSUPERSCRIPT ′ ′ 5 end_POSTSUPERSCRIPT end_ARG divide start_ARG roman_arcsin ( square-root start_ARG italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG start_ARG square-root start_ARG italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG end_ARG ) ( divide start_ARG ( 1 - square-root start_ARG 1 - italic_r start_POSTSUPERSCRIPT ′ ′ 2 end_POSTSUPERSCRIPT italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG ) end_ARG start_ARG italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG ) (61)
eΦ(r′′)/2κ4π(1+2αβ(0))1eΦ(r′′)/2q(r′′)2r′′51β(r′′)(r′′1β(r′′)r′′2β(r′′)arcsin(β(r′′)r′′))absentsuperscript𝑒Φsuperscript𝑟′′2𝜅4𝜋superscript12𝛼𝛽01superscript𝑒Φsuperscript𝑟′′2𝑞superscriptsuperscript𝑟′′2superscript𝑟′′51𝛽superscript𝑟′′superscript𝑟′′1𝛽superscript𝑟′′superscript𝑟′′2𝛽superscript𝑟′′arcsin𝛽superscript𝑟′′superscript𝑟′′\displaystyle\qquad\leq e^{\Phi(r^{\prime\prime})/2}-\frac{\kappa}{4\pi}\left(% 1+2\alpha\beta(0)\right)^{-1}e^{\Phi(r^{\prime\prime})/2}\frac{q(r^{\prime% \prime})^{2}}{r^{\prime\prime 5}}\frac{1}{\beta(r^{\prime\prime})}\left(r^{% \prime\prime}-\sqrt{\frac{1-\beta(r^{\prime\prime})r^{\prime\prime 2}}{\beta(r% ^{\prime\prime})}}\mathrm{arcsin}(\sqrt{\beta(r^{\prime\prime})}r^{\prime% \prime})\right)≤ italic_e start_POSTSUPERSCRIPT roman_Φ ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT - divide start_ARG italic_κ end_ARG start_ARG 4 italic_π end_ARG ( 1 + 2 italic_α italic_β ( 0 ) ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_e start_POSTSUPERSCRIPT roman_Φ ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT divide start_ARG italic_q ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r start_POSTSUPERSCRIPT ′ ′ 5 end_POSTSUPERSCRIPT end_ARG divide start_ARG 1 end_ARG start_ARG italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT - square-root start_ARG divide start_ARG 1 - italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) italic_r start_POSTSUPERSCRIPT ′ ′ 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG end_ARG roman_arcsin ( square-root start_ARG italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) )

and cancelling a factor of eΦ(r′′)/2superscript𝑒Φsuperscript𝑟′′2e^{\Phi(r^{\prime\prime})/2}italic_e start_POSTSUPERSCRIPT roman_Φ ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) / 2 end_POSTSUPERSCRIPT leaves

(1r′′2β(r′′)r′′Φ(r′′)2κ4π1(1+2αβ(0))q(r′′)2r′′5arcsin(β(r′′)r′′)β(r′′))((11r′′2β(r′′))β(r′′))1superscript𝑟′′2𝛽superscript𝑟′′superscript𝑟′′superscriptΦsuperscript𝑟′′2𝜅4𝜋112𝛼𝛽0𝑞superscriptsuperscript𝑟′′2superscript𝑟′′5arcsin𝛽superscript𝑟′′superscript𝑟′′𝛽superscript𝑟′′11superscript𝑟′′2𝛽superscript𝑟′′𝛽superscript𝑟′′\displaystyle\left(\frac{\sqrt{1-r^{\prime\prime 2}\beta(r^{\prime\prime})}}{r% ^{\prime\prime}}\frac{\Phi^{\prime}(r^{\prime\prime})}{2}-\frac{\kappa}{4\pi}% \frac{1}{\left(1+2\alpha\beta(0)\right)}\frac{q(r^{\prime\prime})^{2}}{r^{% \prime\prime 5}}\frac{\mathrm{arcsin}\left(\sqrt{\beta(r^{\prime\prime})}r^{% \prime\prime}\right)}{\sqrt{\beta(r^{\prime\prime})}}\right)\left(\frac{\left(% 1-\sqrt{1-r^{\prime\prime 2}\beta(r^{\prime\prime})}\right)}{\beta(r^{\prime% \prime})}\right)( divide start_ARG square-root start_ARG 1 - italic_r start_POSTSUPERSCRIPT ′ ′ 2 end_POSTSUPERSCRIPT italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG end_ARG start_ARG italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT end_ARG divide start_ARG roman_Φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG start_ARG 2 end_ARG - divide start_ARG italic_κ end_ARG start_ARG 4 italic_π end_ARG divide start_ARG 1 end_ARG start_ARG ( 1 + 2 italic_α italic_β ( 0 ) ) end_ARG divide start_ARG italic_q ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r start_POSTSUPERSCRIPT ′ ′ 5 end_POSTSUPERSCRIPT end_ARG divide start_ARG roman_arcsin ( square-root start_ARG italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG start_ARG square-root start_ARG italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG end_ARG ) ( divide start_ARG ( 1 - square-root start_ARG 1 - italic_r start_POSTSUPERSCRIPT ′ ′ 2 end_POSTSUPERSCRIPT italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG ) end_ARG start_ARG italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG ) (62)
1κ4π(1+2αβ(0))1q(r′′)2r′′51β(r′′)(r′′1β(r′′)r′′2β(r′′)arcsin(β(r′′)r′′))absent1𝜅4𝜋superscript12𝛼𝛽01𝑞superscriptsuperscript𝑟′′2superscript𝑟′′51𝛽superscript𝑟′′superscript𝑟′′1𝛽superscript𝑟′′superscript𝑟′′2𝛽superscript𝑟′′arcsin𝛽superscript𝑟′′superscript𝑟′′\displaystyle\qquad\qquad\leq 1-\frac{\kappa}{4\pi}\left(1+2\alpha\beta(0)% \right)^{-1}\frac{q(r^{\prime\prime})^{2}}{r^{\prime\prime 5}}\frac{1}{\beta(r% ^{\prime\prime})}\left(r^{\prime\prime}-\sqrt{\frac{1-\beta(r^{\prime\prime})r% ^{\prime\prime 2}}{\beta(r^{\prime\prime})}}\mathrm{arcsin}(\sqrt{\beta(r^{% \prime\prime})}r^{\prime\prime})\right)≤ 1 - divide start_ARG italic_κ end_ARG start_ARG 4 italic_π end_ARG ( 1 + 2 italic_α italic_β ( 0 ) ) start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT divide start_ARG italic_q ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r start_POSTSUPERSCRIPT ′ ′ 5 end_POSTSUPERSCRIPT end_ARG divide start_ARG 1 end_ARG start_ARG italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT - square-root start_ARG divide start_ARG 1 - italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) italic_r start_POSTSUPERSCRIPT ′ ′ 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG end_ARG roman_arcsin ( square-root start_ARG italic_β ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) end_ARG italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ) )

which depends explicitly on Φ(r′′)superscriptΦsuperscript𝑟′′\Phi^{\prime}(r^{\prime\prime})roman_Φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r start_POSTSUPERSCRIPT ′ ′ end_POSTSUPERSCRIPT ). This can be determined from the rr𝑟𝑟rritalic_r italic_r field equation (16) to give

Φ(r)=α+e2Λ(r)(α8πr4P(r)+q(r)2r2)+eΛ(r)(r22α)2αrreΛ(r)(2α+r2).superscriptΦ𝑟𝛼superscript𝑒2Λ𝑟𝛼8𝜋superscript𝑟4𝑃𝑟𝑞superscript𝑟2superscript𝑟2superscript𝑒Λ𝑟superscript𝑟22𝛼2𝛼𝑟𝑟superscript𝑒Λ𝑟2𝛼superscript𝑟2\Phi^{\prime}(r)=\frac{\alpha+e^{2\Lambda(r)}\left(\alpha-8\pi r^{4}P(r)+q(r)^% {2}-r^{2}\right)+e^{\Lambda(r)}\left(r^{2}-2\alpha\right)}{2\alpha r-re^{% \Lambda(r)}\left(2\alpha+r^{2}\right)}.roman_Φ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( italic_r ) = divide start_ARG italic_α + italic_e start_POSTSUPERSCRIPT 2 roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ( italic_α - 8 italic_π italic_r start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_P ( italic_r ) + italic_q ( italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) + italic_e start_POSTSUPERSCRIPT roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ( italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 2 italic_α ) end_ARG start_ARG 2 italic_α italic_r - italic_r italic_e start_POSTSUPERSCRIPT roman_Λ ( italic_r ) end_POSTSUPERSCRIPT ( 2 italic_α + italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG . (63)

Making this substitution and simplifying, we find the desired inequality

(1R2β(R)1)(q(R)2+R4β(R)(1+αβ(R)))2R4β(R)1R2β(R)(1+2αβ(R))1superscript𝑅2𝛽𝑅1𝑞superscript𝑅2superscript𝑅4𝛽𝑅1𝛼𝛽𝑅2superscript𝑅4𝛽𝑅1superscript𝑅2𝛽𝑅12𝛼𝛽𝑅\displaystyle\frac{\left(\sqrt{1-R^{2}\beta(R)}-1\right)\left(q(R)^{2}+R^{4}% \beta(R)(-1+\alpha\beta(R))\right)}{2R^{4}\beta(R)\sqrt{1-R^{2}\beta(R)}(1+2% \alpha\beta(R))}divide start_ARG ( square-root start_ARG 1 - italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_β ( italic_R ) end_ARG - 1 ) ( italic_q ( italic_R ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_R start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_β ( italic_R ) ( - 1 + italic_α italic_β ( italic_R ) ) ) end_ARG start_ARG 2 italic_R start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_β ( italic_R ) square-root start_ARG 1 - italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_β ( italic_R ) end_ARG ( 1 + 2 italic_α italic_β ( italic_R ) ) end_ARG (64)
1+2q(r)2R4β(R)(arcsin(β(R)R)β(R)R1)absent12𝑞superscript𝑟2superscript𝑅4𝛽𝑅arcsin𝛽𝑅𝑅𝛽𝑅𝑅1\displaystyle\qquad\qquad\qquad\leq 1+\frac{2q(r)^{2}}{R^{4}\beta(R)}\left(% \frac{\mathrm{arcsin}(\sqrt{\beta(R)}R)}{\sqrt{\beta(R)}R}-1\right)≤ 1 + divide start_ARG 2 italic_q ( italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_R start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT italic_β ( italic_R ) end_ARG ( divide start_ARG roman_arcsin ( square-root start_ARG italic_β ( italic_R ) end_ARG italic_R ) end_ARG start_ARG square-root start_ARG italic_β ( italic_R ) end_ARG italic_R end_ARG - 1 )

which is the generalized Buchdahl bound for charged spheres in 4DEGB gravity.

In the q0𝑞0q\to 0italic_q → 0 limit we recover the following inequality chakraborty_2020

1β(R)R2(1+αβ(R))>13(1αβ(R))1𝛽𝑅superscript𝑅21𝛼𝛽𝑅131𝛼𝛽𝑅\sqrt{1-\beta(R)R^{2}}\left(1+\alpha\beta(R)\right)>\frac{1}{3}\left(1-\alpha% \beta(R)\right)square-root start_ARG 1 - italic_β ( italic_R ) italic_R start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ( 1 + italic_α italic_β ( italic_R ) ) > divide start_ARG 1 end_ARG start_ARG 3 end_ARG ( 1 - italic_α italic_β ( italic_R ) ) (65)

for uncharged objects in 4DEGB.

Alternatively, in the small α𝛼\alphaitalic_α limit βα0(r)=2m(r)r3q(r)2r4subscript𝛽𝛼0𝑟2𝑚𝑟superscript𝑟3𝑞superscript𝑟2superscript𝑟4\beta_{\mathrm{\alpha\to 0}}(r)=\frac{2m(r)}{r^{3}}-\frac{q(r)^{2}}{r^{4}}italic_β start_POSTSUBSCRIPT italic_α → 0 end_POSTSUBSCRIPT ( italic_r ) = divide start_ARG 2 italic_m ( italic_r ) end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG - divide start_ARG italic_q ( italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG, and consequently

(1r2β(r)1)(q(r)2rm(r))r41r2β(r)β(r)+2q(r)2r4(sin1(r2β(r))r2β(r)1)1superscript𝑟2𝛽𝑟1𝑞superscript𝑟2𝑟𝑚𝑟superscript𝑟41superscript𝑟2𝛽𝑟𝛽𝑟2𝑞superscript𝑟2superscript𝑟4superscript1superscript𝑟2𝛽𝑟superscript𝑟2𝛽𝑟1\frac{\left(\sqrt{1-r^{2}\beta(r)}-1\right)\left(q(r)^{2}-rm(r)\right)}{r^{4}% \sqrt{1-r^{2}\beta(r)}}\leq\beta(r)+\frac{2q(r)^{2}}{r^{4}}\left(\frac{\sin^{-% 1}\left(\sqrt{r^{2}\beta(r)}\right)}{\sqrt{r^{2}\beta(r)}}-1\right)divide start_ARG ( square-root start_ARG 1 - italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_β ( italic_r ) end_ARG - 1 ) ( italic_q ( italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_r italic_m ( italic_r ) ) end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT square-root start_ARG 1 - italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_β ( italic_r ) end_ARG end_ARG ≤ italic_β ( italic_r ) + divide start_ARG 2 italic_q ( italic_r ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_r start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT end_ARG ( divide start_ARG roman_sin start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( square-root start_ARG italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_β ( italic_r ) end_ARG ) end_ARG start_ARG square-root start_ARG italic_r start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_β ( italic_r ) end_ARG end_ARG - 1 ) (66)

which matches222At the time of publication bohmer_2007 has a factor of 2 missing from their final expression which Dr. Harko has kindly confirmed to be a typo. We note that in bohmer_2007 α(r)=βα0(r)r32m(r)𝛼𝑟subscript𝛽𝛼0𝑟superscript𝑟32𝑚𝑟\alpha(r)=\beta_{\mathrm{\alpha\to 0}}(r)\frac{r^{3}}{2m(r)}italic_α ( italic_r ) = italic_β start_POSTSUBSCRIPT italic_α → 0 end_POSTSUBSCRIPT ( italic_r ) divide start_ARG italic_r start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT end_ARG start_ARG 2 italic_m ( italic_r ) end_ARG in our notation. the final inequality from bohmer_2007 .

III.2 Relation to the Black Hole Horizon

For a given α𝛼\alphaitalic_α the radius of a black hole is R=M+M2Q2α𝑅𝑀superscript𝑀2superscript𝑄2𝛼R=M+\sqrt{M^{2}-Q^{2}-\alpha}italic_R = italic_M + square-root start_ARG italic_M start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_Q start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_α end_ARG. Since the minimum mass Mmin=Q2+αsubscript𝑀minsuperscript𝑄2𝛼M_{\mathrm{min}}=\sqrt{Q^{2}+\alpha}italic_M start_POSTSUBSCRIPT roman_min end_POSTSUBSCRIPT = square-root start_ARG italic_Q start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_α end_ARG, we obtain RMmin=Q2+αsubscript𝑅Mminsuperscript𝑄2𝛼R_{\mathrm{Mmin}}=\sqrt{Q^{2}+\alpha}italic_R start_POSTSUBSCRIPT roman_Mmin end_POSTSUBSCRIPT = square-root start_ARG italic_Q start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_α end_ARG. Substituting this into the Buchdahl bound (64) we find that it is automatically satisfied as an equality; in other words the Buchdahl bound intersects the minimum mass point of the black hole horizon. The M/R𝑀𝑅M/Ritalic_M / italic_R curves for nonzero α𝛼\alphaitalic_α also smoothly join this point in the limit of large central pressure, as can be seen in figures 1 and 2.

IV Results

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Figure 1: Mass vs. radius/mass vs. central density curves for charged 4DEGB quark stars for three different fixed charges. The (blue, orange, green) curves correspond to charges Q¯=(0,1.538,3.076)×102¯𝑄01.5383.076superscript102\bar{Q}=(0,1.538,3.076)\times 10^{-2}over¯ start_ARG italic_Q end_ARG = ( 0 , 1.538 , 3.076 ) × 10 start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT respectively, with black dots corresponding to local maximum mass points (when present). Note that the orange and green curves begin at a nonzero value of R¯¯𝑅\overline{R}over¯ start_ARG italic_R end_ARG. The grey solid and dashed lines are the uncharged GR Schwarzschild and Buchdahl bounds, respectively. The coloured dashed lines are the 4DEGB Buchdahl bounds for the three different charges, with the colours corresponding to the associated M/R𝑀𝑅M/Ritalic_M / italic_R curve. The black curves are the 4DEGB black hole horizons, with charges matching the Buchdahl/MR𝑀𝑅MRitalic_M italic_R curves that intersect them.
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Figure 2: Mass vs. radius/mass vs. central density curves for charged 4DEGB quark stars for three different fixed charges. The (blue, orange, green) curves correspond to charges Q¯=(0,1.538,3.076)×102¯𝑄01.5383.076superscript102\bar{Q}=(0,1.538,3.076)\times 10^{-2}over¯ start_ARG italic_Q end_ARG = ( 0 , 1.538 , 3.076 ) × 10 start_POSTSUPERSCRIPT - 2 end_POSTSUPERSCRIPT respectively, with black dots corresponding to local maximum mass points (when present). Note that the orange and green curves begin at a nonzero value of R¯¯𝑅\overline{R}over¯ start_ARG italic_R end_ARG and almost perfectly overlap with the blue curve at larger α𝛼\alphaitalic_α. The grey solid and dashed lines are the uncharged GR Schwarzschild and Buchdahl bounds, respectively. The coloured dashed lines are the 4DEGB Buchdahl bounds for the three different charges, with the colours corresponding to the associated M/R𝑀𝑅M/Ritalic_M / italic_R curve. The black curves are the 4DEGB black hole horizons, with charges matching the Buchdahl/MR𝑀𝑅MRitalic_M italic_R curves which intersect them.

The mass/radius (M/R𝑀𝑅M/Ritalic_M / italic_R) and mass/central density (M/ρc𝑀subscript𝜌𝑐M/\rho_{c}italic_M / italic_ρ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT) curves for charged 4DEGB quark stars are presented in figures 1 and 2. As expected from previous work gammon_2024 ; zhang_2021_stellar , we observe the general trends that for a larger α𝛼\alphaitalic_α and/or Q𝑄Qitalic_Q, the mass/radius profiles of the quark stars increase in size. For a fixed nonzero charge the stars have a minimum size, below which the gravitational attraction cannot overcome the self-repulsion of the charge. This is most evident in the right-hand lower panels of figure 2. As α𝛼\alphaitalic_α gets large the M/R𝑀𝑅M/Ritalic_M / italic_R curves converge for all values of fixed Q𝑄Qitalic_Q (and similarly with the M/ρc𝑀subscript𝜌𝑐M/\rho_{c}italic_M / italic_ρ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT curves) with a slight divergence near ρc¯=1¯subscript𝜌𝑐1\bar{\rho_{c}}=1over¯ start_ARG italic_ρ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT end_ARG = 1 in the M/ρc𝑀subscript𝜌𝑐M/\rho_{c}italic_M / italic_ρ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT plots to account for the structures mentioned above.

As in gammon_2024 , even for small, nonzero α𝛼\alphaitalic_α we find curves for the Buchdahl bound that intersect the black hole horizon at the minimum mass point. A new feature, evidently not previously noted, is that this same interaction between the black hole horizon and Buchdahl bound can be observed in GR when Q0𝑄0Q\neq 0italic_Q ≠ 0. This is evident in the orange and green dashed curves in the upper left panel of figure 1. The difference is that for nonzero α𝛼\alphaitalic_α the M/R𝑀𝑅M/Ritalic_M / italic_R curves approach this intersection point smoothly, whereas in GR the curves turn away from the horizon/BB and thus never meet.

On the solution curves we have indicated maximum mass points (when this point doesn’t occur at the intersection with the horizon) with a black dot - in general relativity these maximum mass points indicate a transition point from stability against radial perturbations to instability for uncharged stars. In 4DEGB it’s not clear whether this coincidence holds, and offers an interesting avenue for future research.

In the uncharged case a criticality condition was previously noted gammon_2024 wherein for α¯34π¯𝛼34𝜋\bar{\alpha}\geq\frac{3}{4\pi}over¯ start_ARG italic_α end_ARG ≥ divide start_ARG 3 end_ARG start_ARG 4 italic_π end_ARG a critical central pressure was present - for central pressures below this critical value, the pressure function diverges with no real roots to define a star’s surface. Above this critical pressure we find numerical solutions which lie at or beyond the black hole horizon, indicating a lack of stable physical stars in this critical regime. Deriving an analogous bound for the charged case is considerably more complicated, however numerically we find that for Q>0𝑄0Q>0italic_Q > 0 (and hence γ>0𝛾0\gamma>0italic_γ > 0), such criticality can take place for α¯<34π¯𝛼34𝜋\bar{\alpha}<\frac{3}{4\pi}over¯ start_ARG italic_α end_ARG < divide start_ARG 3 end_ARG start_ARG 4 italic_π end_ARG. Because of this lack of an analytic bound for the charged case, we do not present numerical solutions in which such behaviour is manifest. With this, we note that we have considered values of α¯0.1¯𝛼0.1\bar{\alpha}\leq 0.1over¯ start_ARG italic_α end_ARG ≤ 0.1. For larger values of α¯¯𝛼\bar{\alpha}over¯ start_ARG italic_α end_ARG ‘critical central pressure’ behaviour emerges soon after for the charged cases. Examination of these cases requires a more detailed stability analysis, which we leave for future investigation.

Most interestingly we note, similar to the uncharged case gammon_2024 , that the stars described by this theory can be ECCOs (Extreme Compact Charged Objects): charged stellar objects that are smaller (more compact) than the uncharged Buchdahl bound (and sometimes even the Schwarzschild radius) of general relativity. For large enough α𝛼\alphaitalic_α the associated M/R𝑀𝑅M/Ritalic_M / italic_R curves do not reach their maximum mass until they intersect the black hole horizon - these are particularly interesting candidates for stable ECCOs since in pure uncharged GR we only see stability against radial perturbations when dM/dρ>0𝑑𝑀𝑑𝜌0dM/d\rho>0italic_d italic_M / italic_d italic_ρ > 0. However, a net charge offsets the stability point from this maximum mass point zhang_2021_stellar - it is likely that the modifications to gravity will have a similar offsetting effect which should be investigated in future studies before declaring these solutions stable against perturbations.

V Stability

We now briefly consider the stability of stars respecting the generalized Buchdahl bound. In general relativity a necessary but insufficient condition for an uncharged compact star to be stable against radial perturbations is dM/dρc>0𝑑𝑀𝑑subscript𝜌𝑐0dM/d\rho_{c}>0italic_d italic_M / italic_d italic_ρ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT > 0 zhang_2021_stellar ; glendenningbook ; arbanil2015 , corresponding to the part of the solution curve before a maximal mass point is reached. In Einstein-Maxwell theory a net charge has been shown to offset the stability point from the maximum mass point in either direction zhang_2021_stellar . In a similar vein, when the coupling to higher curvature gravity is nonzero, it is not obvious whether this coincidence of stability and the maximum mass point will hold for uncharged stars. While we leave a thorough analysis of the fundamental radial oscillation modes for future work (where stability against radial oscillations can be ensured), the speed of sound and effective adiabatic index inside the star can still be discussed.

In the interior of a stable star, the sound speed cssubscript𝑐𝑠c_{s}italic_c start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT must never exceed the speed of light c𝑐citalic_c. The non-interacting quark equation of state (10) has a constant subluminal sound speed of cs=c3subscript𝑐𝑠𝑐3c_{s}=\frac{c}{\sqrt{3}}italic_c start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT = divide start_ARG italic_c end_ARG start_ARG square-root start_ARG 3 end_ARG end_ARG, and thus the causality condition is always satisfied. Similarly, the effective adiabatic index

γeff(1+ρP)(dPdρ)Ssubscript𝛾eff1𝜌𝑃subscript𝑑𝑃𝑑𝜌𝑆\gamma_{\mathrm{eff}}\equiv\left(1+\frac{\rho}{P}\right)\left(\frac{dP}{d\rho}% \right)_{S}italic_γ start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT ≡ ( 1 + divide start_ARG italic_ρ end_ARG start_ARG italic_P end_ARG ) ( divide start_ARG italic_d italic_P end_ARG start_ARG italic_d italic_ρ end_ARG ) start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT (67)

is often used as another indicator of stability as it is seen as a bridge between “the relativistic structure of a spherical static object and the equation of state of the interior fluid” moustakidis2017_stability . The subscript S𝑆Sitalic_S in the above equation indicates that we consider the sound speed at a constant specific entropy. In principle a critical value for γeffdelimited-⟨⟩subscript𝛾eff\langle\gamma_{\mathrm{eff}}\rangle⟨ italic_γ start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT ⟩ exists, below which configurations are unstable against radial perturbations. In standard general relativity this critical value can be written chandrasekhar1964 ; moustakidis2017_stability γcr=43+1942βsubscript𝛾𝑐𝑟431942𝛽\gamma_{cr}=\frac{4}{3}+\frac{19}{42}\betaitalic_γ start_POSTSUBSCRIPT italic_c italic_r end_POSTSUBSCRIPT = divide start_ARG 4 end_ARG start_ARG 3 end_ARG + divide start_ARG 19 end_ARG start_ARG 42 end_ARG italic_β, where β=2M/R=RS/R𝛽2𝑀𝑅subscript𝑅𝑆𝑅\beta=2M/R=R_{S}/Ritalic_β = 2 italic_M / italic_R = italic_R start_POSTSUBSCRIPT italic_S end_POSTSUBSCRIPT / italic_R is the compactness parameter. If β0𝛽0\beta\to 0italic_β → 0 the well-known classical Newtonian limit is recovered as expected (γeff43delimited-⟨⟩subscript𝛾eff43\langle\gamma_{\mathrm{eff}}\rangle\geq\frac{4}{3}⟨ italic_γ start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT ⟩ ≥ divide start_ARG 4 end_ARG start_ARG 3 end_ARG).

An equivalent bound has not yet been derived for the 4DEGB theory. Despite this, it is common practice to plot the adiabatic index of the star relative to the Newtonian critical value banerjee_2021_quark ; banerjee_2021_strange ; hansraj_2020_isotropic ; singh_2022_anisotropic . Since we restrict ourselves to a simple, non-interacting equation of state (10) it is straightforward to show that γeff=13(4+1P(r))subscript𝛾eff1341𝑃𝑟\gamma_{\mathrm{eff}}=\frac{1}{3}(4+\frac{1}{P(r)})italic_γ start_POSTSUBSCRIPT roman_eff end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 3 end_ARG ( 4 + divide start_ARG 1 end_ARG start_ARG italic_P ( italic_r ) end_ARG ) and hence the Newtonian bound of 4/3 is always satisfied for non-negative pressure.

VI Summary

In this paper we have investigated the stellar structure of strongly interacting quark stars in the 4D Einstein Gauss-Bonnet theory of gravity for different combinations of the charge Q𝑄Qitalic_Q and 4DEGB coupling constant α𝛼\alphaitalic_α. In accord with the lack of a mass gap in the 4DEGB theory charmousis2022 , we find that even for small α𝛼\alphaitalic_α the quark star solutions asymptotically approach the 4DEGB black hole horizon radius, and thus have solutions with smaller radii than the GR Buchdahl/Schwarzschild limits. In general, larger Q𝑄Qitalic_Q and α𝛼\alphaitalic_α tend to increase the mass-radius profile of quark stars, with large α𝛼\alphaitalic_α suppressing the differences between different charges. These findings are generally consistent with what was found in the regime of weak coupling to the 4DEGB theory pretel_2022 .

We have found many additional features in the unexplored regions of parameter space, the most striking of which is that 4DEGB charged quark stars can exist with radii not only smaller than the general relativistic Buchdahl bound, but also smaller than the 2M2𝑀2M2 italic_M Schwarzschild radius. Such Extreme Compact Charged Objects represent a possible new state of matter. A full analysis of the stability of such objects would be an interesting topic for future study.

Acknowledgements

This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. We would like to thank Dr. Tiberiu Harko for useful correspondence regarding the existing literature.

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