General relativistic heat flow from first order hydrodynamics

Bhera Ram; Bibhas Ranjan Majhi

REVIEW 3 major objections 5 minor 43 references

General relativistic heat flow from first order hydrodynamics

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A non-viscous fluid in normal flow on static or stationary spacetimes obeys a heat equation for the redshifted temperature, with analytic solutions for four black hole backgrounds.

desk verdict A clean, correct extension of heat-flow hydrodynamics to stationary black holes, with a narrow scope that the authors mostly acknowledge. read the letter →

arxiv 2412.02364 v2 pith:6LT62OYK submitted 2024-12-03 gr-qc hep-th

Bhera Ram , Bibhas Ranjan Majhi This is my paper · ORCID

classification gr-qchep-th

keywords generalrelativistichydrodynamicsfirst-orderheatflowequationTolman-EhrenfestrelationredshiftfactorSchwarzschild-deSitterKerr-dechemicalpotentialprofile

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper derives an equation for heat flow in a gravitational field starting from the causal, stable first-order relativistic hydrodynamics formalism used in the paper. For a non-viscous fluid whose four-velocity is normal to time slices in a static or stationary spacetime, it finds that $\sqrt{N^2}\,q^a$ is conserved, $\nabla_a(\sqrt{N^2}\,q^a)=0$, and that the redshifted temperature $T\sqrt{N^2}$ satisfies a heat equation. Solving that equation for radial heat flow gives explicit temperature and chemical-potential profiles on Schwarzschild, Schwarzschild--de Sitter, Kerr, and Kerr--de Sitter backgrounds. When the heat flux vanishes, the profiles reduce to the Tolman--Ehrenfest relation and Klein's law. This gives analytic benchmarks for numerical studies of dissipative relativistic fluids around black holes.

What carries the argument

The central object is the first-order heat-flux constitutive relation together with the normal-flow condition $u_a=-N\nabla_a t$. Under static or stationary symmetry and radial heat flow, the $\tau_q$ corrections in that relation drop out, leaving $q^a=-\kappa T\nabla^a\ln(T\sqrt{N^2})$. Substituting this into the energy-momentum conservation law $\nabla_a q^a+q^a\dot{u}_a=0$ turns the conservation statement into $\nabla_a(\sqrt{N^2}\,q^a)=0$ and then into the scalar heat equation for the redshifted temperature, $\nabla_a[\kappa\nabla^a(T\sqrt{N^2})]=0$. The factor $N^2$ is exactly the Tolman--Ehrenfest redshift factor.

What would settle it

Reduce the paper's framework to a concrete test on Schwarzschild: solve the full first-order hydrodynamic equations for a non-viscous fluid whose four-velocity has a nonzero radial component. If $\nabla_a(\sqrt{N^2}\,q^a)=0$ and the profile (31) fail under that tilted flow, the normal-flow assumption is load-bearing; if they survive, the result extends beyond the stated restriction.

Watch

Extended reading notes

Core claim

The paper's central claim is that gravity modifies heat conduction through the same redshift factor that governs thermal equilibrium: in the static and stationary cases considered, it is not $q^a$ but $\sqrt{N^2}\,q^a$ that is divergence-free, $\nabla_a(\sqrt{N^2}\,q^a)=0$. Combined with the reduced heat-flux law $q^a=-\kappa\,T\,\nabla^a\ln(T\sqrt{N^2})$, this yields the heat equation $\nabla_a[\kappa\,\nabla^a(T\sqrt{N^2})]=0$ for the redshifted temperature. The paper solves this equation for radial heat flow on Schwarzschild, Schwarzschild--de Sitter, Kerr, and Kerr--de Sitter backgrounds and obtains explicit temperature and chemical-potential profiles. In the limit of vanishing heat flux the profiles reproduce the Tolman--Ehrenfest relation and Klein's law.

Load-bearing premise

The derivation assumes the fluid four-velocity is normal to $t=\text{constant}$ slices and that heat flows only radially; these restrictions are what make the $\tau_q$ terms in the heat-flux law vanish, and the paper explicitly notes that relaxing the normal-flow condition would require a separate analysis.

Editorial extensions

If this is right

On static and stationary backgrounds the conserved heat current is $\sqrt{N^2}\,q^a$, so the same redshift factor that fixes equilibrium temperature also controls heat transport away from equilibrium.
Vanishing heat flux reduces the heat equation to $T\sqrt{N^2}=\text{constant}$, recovering the Tolman--Ehrenfest relation, and the chemical-potential equation recovers Klein's $\mu/T=\text{constant}$.
Radial temperature profiles are obtained analytically for Schwarzschild, Schwarzschild--de Sitter, Kerr, and Kerr--de Sitter spacetimes, with the Kerr and de Sitter results reducing to the Schwarzschild and Schwarzschild--de Sitter expressions in the $a\to0$ and $\Lambda\to0$ limits.
The entropy production rate between two surfaces at different redshifted temperatures is proportional to $q_0(1/(T_2\sqrt{N^2_2})-1/(T_1\sqrt{N^2_1}))$, so second-law consistency is tied to the redshift factors.
The analytic profiles can be used as benchmarks for numerical codes in general-relativistic dissipative hydrodynamics.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same normal-flow reduction should apply to any static or stationary spacetime with a timelike Killing vector, so the profile method likely extends to neutron stars and accretion tori, a direction the paper hints at but does not work out.
Testing whether $\nabla_a(\sqrt{N^2}\,q^a)=0$ survives angular heat flux on an axisymmetric background would decide whether the conservation law is a general geometric identity or an artifact of radial slicing.
The paper's first-order-hydrodynamics profiles can be compared directly with the earlier static derivation in [24]; different predictions near the horizon would quantify how much causality and stability conditions alter heat transport.
Coupling the same heat equation to a charged fluid would add a source term from the electromagnetic field, which could produce temperature profiles relevant to magnetized accretion flows.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

First thing you should know: this paper does what it says. From BDN first-order hydrodynamics, it derives a conserved redshifted heat flux, ∇_a(√N² q^a)=0, and a heat equation ∇_a[κ∇^a(T√N²)]=0 for static and stationary backgrounds. The derivation is honest — Eq. (14) follows from the non-viscous reduction plus spacetime symmetries, and Eq. (19) follows from the simplified constitutive relation. The static part overlaps with the Eckart-based result in Miranda-Rinaldi-Faraoni [24], but the stationary treatment (Kerr, Kerr-de Sitter) and the BDN route are new. The analytic temperature profiles and the reduction to Tolman-Ehrenfest when q0→0 are genuinely useful benchmarks for numerical codes.

Where it gets soft: the stationary case is less general than it first appears. Eq. (19) relies on the heat flux being purely radial, because the proof of £_u q^a=0 in Appendix A.2 uses q^a=(0,q^r,0,0) and the absence of cross terms. The authors explicitly say in the Conclusions that results are for normal flow and radial heat flux, so it's an acknowledged scope restriction, not a hidden flaw. Still, the title-level claim 'heat flow' should be read as 'radial heat flow in normal flow'. Also there are notational slips: in Eqs. (41) and (49), they write 'g00 = ...' where they mean g^{00} or -1/N², and the relation between g00 and N² for stationary metrics is muddled. These are fixable. The constant-κ assumption is stated but not deeply justified; that's minor.

The math checks out as far as I can tell. The paper does not invoke an equation of state, which is a real strength. The citation pattern is fine; the self-citation to [22] is relevant and not padding.

Who benefits: people working on relativistic dissipative hydrodynamics in curved spacetime, especially those testing numerical codes against analytic solutions. It's not a landmark, but it is a solid, useful piece.

Recommendation: send it to peer review. A good referee can handle the typos and the scope caveats; there is a real result here that should be on record.

Referee Report

3 major / 5 minor

Summary. The paper derives a relativistic heat-flow equation for a non-viscous fluid using the BDN first-order hydrodynamics, restricted to static and stationary backgrounds, to a hypersurface-orthogonal 'normal flow' four-velocity, and to purely radial heat flux. The central results are the conservation law \nabla_a(\sqrt{N^2}q^a)=0 (Eq. 14) and the heat equation \nabla_a[\kappa\nabla^a(T\sqrt{N^2})]=0 (Eq. 19), together with analytic temperature profiles for Schwarzschild, Schwarzschild-de Sitter, Kerr, and Kerr-de Sitter spacetimes. The Tolman-Ehrenfest law is recovered as the q_0=0 limit, and chemical-potential profiles are discussed.

Significance. If the central derivation is correct, the paper provides explicit analytic benchmarks for heat transport in BDN first-order hydrodynamics on black-hole spacetimes, including stationary backgrounds, which goes beyond earlier Eckart-based treatments. A strength of the paper is that the main derivation is transparent and does not require an equation of state; the limitations of normal flow and radial heat flux are acknowledged in the conclusions. However, the manuscript contains a number of algebraic and notational errors that currently prevent the results from being taken at face value, especially in the chemical-potential section and in the labeling of inverse metric components for the stationary applications.

major comments (3)

[Sec. III.A, Eq. (29)] The second equality in Eq. (29) does not follow from Eqs. (26)-(28). For Schwarzschild, expanding the first equality of Eq. (29) to first order in q0 gives terms proportional to A/T0 and to T0, namely d(mu sqrt(-g00))/dr = -q0 A/(kappa T0 r^2 sqrt(f)) + q0 T0/(alpha r^2 f^{3/2}) + O(q0^2), whereas the displayed second equality is O(q0^2). Thus the simplified form is false unless the integration constants A and T0 are set to zero, which they are not. The chemical-potential results in Appendix C are based on Eq. (30), so they may survive, but the claimed simplification in Eq. (29) must be corrected or removed.
[Sec. III.B, Eqs. (41) and (49)] The quantity labeled g00 in Eq. (41) is actually the inverse time-time component g^00. Substituting the displayed expression into N^2 = -1/g^00 gives the correct Kerr lapse and reproduces the Schwarzschild limit N^2 = 1 - 2m/r only if the expression is read as g^00. As printed with g00, the definition of N^2 and all subsequent Kerr temperature formulas are inconsistent. The same care is needed for Eq. (49), whose a=0 limit appears to give the metric component g00 rather than g^00; the KdS formulas should be re-verified with the inverse metric component.
[Appendix A.2, Eq. (A11)] The argument that the last two terms of Eq. (A11) vanish is written with incorrect index placement: the text states qb xi^b_(t) = qr xi^r_(t) = 0 because xi^r_(t)=0, but q_b xi^b_(t) = q_t, not q_r xi^r_(t). The intended vanishing terms are q^b xi_(t)b = g_{0b} q^b = g_{0r} q^r and q^b xi_(phi)b = g_{phi r} q^r, which are zero only because the stationary metric (37) has no g_{0r} or g_{phi r} cross terms and q^a is purely radial. The conclusion \pounds_u q_c=0 is correct under these stated restrictions, but the proof should be rewritten and the no-r-cross-term condition made explicit.

minor comments (5)

[Sec. IV, Eqs. (62) and (63)] The proper-time factors mix g00 and g^00 notation: for a static metric d\tau = dt/u^0 = dt/\sqrt{-g^{00}} = dt\sqrt{-g_{00}}, but the text writes this in a way that appears to equate dt/\sqrt{-g_{00}} with dt\sqrt{-g_{00}}. Please use explicit superscripts to distinguish g^{00} from g_{00} throughout this section.
[Sec. III.A, Eq. (27)] The displayed formula is ambiguous in the typeset text: the correct factor is \kappa r^2\sqrt{-g_{00}}/\sqrt{g_{rr}} times d(T\sqrt{N^2})/dr, not \kappa r^2\sqrt{-g_{00}}\sqrt{g_{rr}}. Please check the typesetting so the denominator is unambiguous.
[Sec. III.B, text before Eq. (41)] After Eq. (39), the statement that N^2 = -1/g00 should read N^2 = -1/g^{00}; in static spacetimes the two coincide because g^{00}=1/g_{00}, but in stationary spacetimes they do not. This is directly related to the major comment on Eq. (41).
[Sec. I and Sec. V] The comparison with Ref. [24] is fair, but the abstract and title should state more explicitly that the stationary results are limited to normal flow and radial heat flux; the conclusions already acknowledge this, and the abstract should not imply general stationary heat flow.
[Appendix C, Eq. (C2)] The displayed expression for (mu sqrt(1-2m/r))_{sch} has a parenthesis structure that is hard to parse; please check the bracket matching and the sign of the q0/(m alpha) term against Eq. (C1).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the heat flow equation is derived from the BDN conservation equations, and the Tolman-Ehrenfest relation emerges as the q0=0 limit of the derived solutions rather than being assumed.

full rationale

The paper's derivation chain is self-contained relative to its stated inputs. It begins with the BDN constitutive relations (Eqs. 1-6), imposes a non-viscous fluid and normal flow u_a = -N grad_a t, and then uses Killing symmetries and the Frobenius hypersurface-orthogonality condition to show that L_u q^c = 0, giving Eq. (A10) and hence the simplified heat flux q^a = alpha grad^a(mu/T) in Eq. (16). Appendix B converts this to q^a = -kappa T grad^a ln(T sqrt(N^2)) in Eq. (18) using only thermodynamic identities and the already-derived Eq. (A10). Substitution into Eq. (12) yields the conservation law grad_a(sqrt(N^2) q^a) = 0 (Eq. 14) and the heat equation (Eq. 19). None of these steps assumes the target result: the Tolman-Ehrenfest relation is not an input but is recovered only in the q0 = 0 limit of the solved temperature profiles, e.g. Eq. (31) and Eq. (45), and Klein's law is similarly a q0 -> 0 consequence of Eq. (30). No parameter is fitted to a subset of data and then renamed a prediction; the integration constants T0 and q0 are genuine constants of integration. The only self-citation (Ref. 22) appears in the introduction as background on earlier work obtaining the TE and Klein relations from first-order hydrodynamics, and it is not used as a load-bearing premise in the derivation of Eqs. (14), (18), or (19). The paper explicitly acknowledges in the Conclusions that the results are restricted to normal flow and, in the stationary case, to radial heat flux; Appendix A.2 shows that the latter assumption is needed to make the last two terms of Eq. (A11) vanish. This is an honest scope restriction, not a circular step. The central derivation is therefore independent of any fitted output or self-citation chain, and no circularity is present.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

No new particles, fields, or entities are introduced. The free parameters are integration constants and the assumption that heat conductivity and alpha are constant. The main axioms are the BDN formalism, the non-viscous idealization, the normal-flow and radial-heat-flow restrictions, and the use of equilibrium thermodynamics for out-of-equilibrium fluid variables.

free parameters (4)

q0 (heat flux integration constant) = arbitrary real
Appears in Eq. (24) as the radial heat flux strength; fixed only by boundary conditions, and chosen positive in the entropy argument.
T0 (temperature integration constant) = positive constant
Integration constant in Eq. (31); for Schwarzschild the authors suggest matching to Hawking temperature at infinity.
kappa (heat conductivity) assumed constant = constant
Assumed constant to integrate Eq. (19), the same assumption as in [24]; not derived from a transport calculation.
alpha = sigma T (epsilon+p)/n assumed constant = constant
Assumed constant for the chemical potential integrations in Appendix C.

assumptions (7)

domain assumption BDN first-order constitutive relations (Eqs. 1-6) describe causal and stable relativistic hydrodynamics.
The entire derivation starts from these relations, which are cited from [31].
domain assumption The fluid is non-viscous, with dissipation only through heat flux.
Shear and bulk viscosity terms are set to zero in Section III.
ad hoc to paper The fluid four-velocity is hypersurface-orthogonal normal to t=constant slices.
The choice u_a=-N grad_a t restricts the analysis to 'normal flow'; acknowledged in the Conclusions.
domain assumption Fluid scalars respect the background spacetime symmetries, giving u^a grad_a X=0 and Lie derivatives of q along Killing vectors equal to zero.
Used in Section III and Appendix A to reduce Eqs. (9) and (10).
ad hoc to paper Heat flow is purely radial.
Allows analytic solutions and is stated in Section III and the Conclusions.
domain assumption Equilibrium thermodynamic relations (Euler relation and first law) apply to the out-of-equilibrium fluid.
Used in Appendix B to relate the pressure gradient to gradients of T and mu/T.
domain assumption The background metric is fixed and the fluid's backreaction is neglected.
The heat equation is solved on given black hole spacetimes without considering the fluid contribution to the metric.

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Pith. "Pith review of General relativistic heat flow from first order hydrodynamics." pith.science (2026). https://pith.science/paper/6LT62OYK

@misc{pith2026241202364,
  author       = {Pith},
  title        = {Pith review of: General relativistic heat flow from first order hydrodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6LT62OYK}},
  note         = {Machine review of arXiv:2412.02364}
}

read the original abstract

Following the recently proposed stable and causal first-order relativistic hydrodynamics by Bemfica, Disconzi, and Noronha, we find the heat flow equation in the presence of gravity for a non-viscous fluid, which suffers heat dissipation. The derivation is confined to static and stationary backgrounds. We find that in the presence of gravity, the heat flux times a redshift factor is conserved. Then for radial heat flow, the temperature profiles are obtained from the heat equation when the gravity is sourced by Schwarzschild, Schwarzschild-dS, Kerr and Kerr-dS black holes, respectively. Consequently, the chemical potential profile is also discussed.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Schwarzschild Geometry: – In this section, we take the Schwarzschild geometry as an example of the static spherically symmetric case

Application to static backgrounds To get a feel of the above results we apply them on a few speciﬁc static metrics. Schwarzschild Geometry: – In this section, we take the Schwarzschild geometry as an example of the static spherically symmetric case. On solving Eq. (26) for the Schwarzschild case with g00 = −1 + 2m r and grr = −1 g00 , we get T (r) = 1√ 1 ...

work page
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Application to stationary metrics Kerr Geometry: – Let us now turn to an example of a stationary Kerr metric. The metric elements in this case in Boyer-Lindquist coordinates are given by g00 = − ( 1 − 2mr ρ2 ) ; g0φ = − 2mar sin2 θ ρ2 ; grr = ρ2 ∆ ; gθθ = ρ2; gφφ = ( r2 + a2 + 2ma2r sin2 θ ρ2 ) sin2 θ,(39) with ρ2 ≡ r2 + a2 cos2 θ, ∆ ≡ r2 − 2mr + a2, and ...

work page 2020
[3]

(9) and Eq

Static background As discussed in Section III, the equations governing the conservation of the energy-momentum tensor, along with the conditions for a non-viscous ﬂuid, are given by Eq. (9) and Eq. (10). For a static background given by Eq. (11), we have ua∇aX = 0 and ∇aua = 0, where X represents any scalar ﬂuid variable. Consequently, under these conditi...

work page
[4]

(10) for the stationary background (37), simpliﬁes to Eq

Stationary background Again the non-viscous ﬂuid equation, given by Eq. (10) for the stationary background (37), simpliﬁes to Eq. (A1) due to ua∇aε = 0 and ∇aua = 0. Since in this case our choice of ua also satisﬁes the Frobenius theorem, we will have Eq. (A7) for the stationary background as well. The background has two Killing vectors, ξb (t) = (1, 0, 0...

work page
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(30) for the Schwarzschild case with g00 = −1/g rr = − ( 1 − 2m r ) and assuming α to be constant, we have µ T = A + q0 α ∫ 1 r2( 1 − 2m r ) 3/ 2 dr

Schwarzschild black hole Solving Eq. (30) for the Schwarzschild case with g00 = −1/g rr = − ( 1 − 2m r ) and assuming α to be constant, we have µ T = A + q0 α ∫ 1 r2( 1 − 2m r ) 3/ 2 dr . (C1) Further, on using Eq. (31), we have ( µ (r) √ 1 − 2m r ) (sch) = ( T0 − q0 2mκ ln ( 1 − 2m r ) ) × ( A − q0 m α 1√ 1 − 2m r ) , (C2) where A is the integration constant

work page
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(30) assuming α to be constant reduces to, µ T = A + q0 α ∫ 1 r2( 1 − 2m r − Λ r2 3 ) 3/ 2 dr , (C3) which doesn’t have a simpliﬁed or compact form

Schwarzschild-de Sitter black hole Again for the Schwarzschild-de Sitter case, Eq. (30) assuming α to be constant reduces to, µ T = A + q0 α ∫ 1 r2( 1 − 2m r − Λ r2 3 ) 3/ 2 dr , (C3) which doesn’t have a simpliﬁed or compact form. How- ever, it can be solved near the horizon, using r → r′ + rb, where rb is the black hole horizon. The above equation near ...

work page

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [24]

Rovelli and M

C. Rovelli and M. Smerlak, Class. Quant. Grav. 28, 075007 (2011), 1005.2985

arXiv 2011

[2] [22]

Gravitational instability caused by the weight of heat

Z. Roupas, Symmetry 11, 1435 (2019), 1809.04408

work page Pith review arXiv 2019

[3] [1]

Schwarzschild Geometry: – In this section, we take the Schwarzschild geometry as an example of the static spherically symmetric case

Application to static backgrounds To get a feel of the above results we apply them on a few speciﬁc static metrics. Schwarzschild Geometry: – In this section, we take the Schwarzschild geometry as an example of the static spherically symmetric case. On solving Eq. (26) for the Schwarzschild case with g00 = −1 + 2m r and grr = −1 g00 , we get T (r) = 1√ 1 ...

work page

[4] [2]

Application to stationary metrics Kerr Geometry: – Let us now turn to an example of a stationary Kerr metric. The metric elements in this case in Boyer-Lindquist coordinates are given by g00 = − ( 1 − 2mr ρ2 ) ; g0φ = − 2mar sin2 θ ρ2 ; grr = ρ2 ∆ ; gθθ = ρ2; gφφ = ( r2 + a2 + 2ma2r sin2 θ ρ2 ) sin2 θ,(39) with ρ2 ≡ r2 + a2 cos2 θ, ∆ ≡ r2 − 2mr + a2, and ...

work page 2020

[5] [3]

(9) and Eq

Static background As discussed in Section III, the equations governing the conservation of the energy-momentum tensor, along with the conditions for a non-viscous ﬂuid, are given by Eq. (9) and Eq. (10). For a static background given by Eq. (11), we have ua∇aX = 0 and ∇aua = 0, where X represents any scalar ﬂuid variable. Consequently, under these conditi...

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[6] [4]

(10) for the stationary background (37), simpliﬁes to Eq

Stationary background Again the non-viscous ﬂuid equation, given by Eq. (10) for the stationary background (37), simpliﬁes to Eq. (A1) due to ua∇aε = 0 and ∇aua = 0. Since in this case our choice of ua also satisﬁes the Frobenius theorem, we will have Eq. (A7) for the stationary background as well. The background has two Killing vectors, ξb (t) = (1, 0, 0...

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[7] [5]

(30) for the Schwarzschild case with g00 = −1/g rr = − ( 1 − 2m r ) and assuming α to be constant, we have µ T = A + q0 α ∫ 1 r2( 1 − 2m r ) 3/ 2 dr

Schwarzschild black hole Solving Eq. (30) for the Schwarzschild case with g00 = −1/g rr = − ( 1 − 2m r ) and assuming α to be constant, we have µ T = A + q0 α ∫ 1 r2( 1 − 2m r ) 3/ 2 dr . (C1) Further, on using Eq. (31), we have ( µ (r) √ 1 − 2m r ) (sch) = ( T0 − q0 2mκ ln ( 1 − 2m r ) ) × ( A − q0 m α 1√ 1 − 2m r ) , (C2) where A is the integration constant

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[8] [6]

(30) assuming α to be constant reduces to, µ T = A + q0 α ∫ 1 r2( 1 − 2m r − Λ r2 3 ) 3/ 2 dr , (C3) which doesn’t have a simpliﬁed or compact form

Schwarzschild-de Sitter black hole Again for the Schwarzschild-de Sitter case, Eq. (30) assuming α to be constant reduces to, µ T = A + q0 α ∫ 1 r2( 1 − 2m r − Λ r2 3 ) 3/ 2 dr , (C3) which doesn’t have a simpliﬁed or compact form. How- ever, it can be solved near the horizon, using r → r′ + rb, where rb is the black hole horizon. The above equation near ...

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