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REVIEW 3 major objections 4 minor 41 references

Specifying a linear equation of state rather than a density profile yields exact black hole metrics in Hořava–æther gravity, including an extremal degenerate horizon and a remnant whose horizon encloses a central singularity.

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2026-08-03 14:23 UTC pith:WZUUR6FT

load-bearing objection New exact BH metrics from linear EoS in Horava/Einstein-Aether, but the derivation skips the aether field equations and the entropy claim rests on unproven Wald/Komar assumptions. the 3 major comments →

arxiv 2512.20445 v2 pith:WZUUR6FT submitted 2025-12-23 gr-qc

Black hole solutions with a linear equation of state in Hov{r}ava gravity and Einstein--{ae}ther theory

Milko Estrada This is my paper
classification gr-qc PACS 04.70.-s04.50.Kd
keywords Hořava gravityEinstein-æther theoryblack hole solutionslinear equation of stateextremal black holestiff fluidarea law entropyblack hole remnant
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper turns the usual black-hole construction around: instead of prescribing an energy-density profile (which in general relativity implies the equation of state ρ = −p_r), it prescribes a linear equation of state and solves the modified field equations of Hořava–æther gravity for the metric function f(r). It shows that three simple linear equations of state close the system exactly. The resulting metrics include a Reissner–Nordström-like solution sourced by exotic anisotropic matter, an extremal black hole whose horizon is odd-fold degenerate with zero temperature and area-law entropy, and a stiff-fluid solution with a repulsive short-scale potential whose evaporation halts at a remnant enclosing a central singularity. The point is that the Hořava/æther terms change the effective equation of state, so linear equations of state become a natural generating device for new exact solutions.

Core claim

For a static, spherically symmetric spacetime with a static aether, the Hořava–æther field equations reduce to three ordinary differential equations plus a conservation condition. The paper's central claim is that choosing a linear equation of state—p_r + p_θ = 0, ρ = −p_r − N p_θ, or ρ = p_r—determines f(r) exactly. Case I gives f(r) = 1 − C1/r + C2/r², formally like Reissner–Nordström but with an exotic anisotropic source. Case II gives f(r) = (1 − 2M/r)^{n_odd}, where n_odd is an odd integer; the horizon is n_odd-fold degenerate, the temperature vanishes, and the entropy still satisfies the area law. Case III gives f(r) = 1 − 2M/r + C2/r^{4ξ/η}; the repulsive term makes the temperature re

What carries the argument

The engine is the ansatz of a static, spherically symmetric metric with g_tt = −1/g_rr and a static unit timelike aether field. Under this ansatz, the Hořava/Einstein–æther equations collapse to three metric equations plus conservation, with the single combination η/ξ controlling the aether contributions. Substituting a chosen linear equation of state into these equations turns the system into an ordinary differential equation for f(r) that integrates in closed form. The theory's parameters then become exponents or couplings in the metric: N scales with η in the extremal case, and n = 4ξ/η sets the repulsive power in the stiff-fluid case.

Load-bearing premise

The whole construction rests on the reduction of the full Hořava/Einstein–æther equations to the three metric equations (10)–(12) with a static aether; if the unsolved aether field equations impose further constraints, the displayed metrics are not solutions of the complete theory.

What would settle it

Take any of the three metrics and vary the action with respect to the aether field u^μ; if the resulting equations are not satisfied for the static ansatz with nonzero η/ξ, the claimed solutions fail. Concretely, substituting f(r) = (1 − 2M/r)^3 into the full aether equations and checking whether all components vanish would decide Case II.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The extremal solution in Case II is an exactly degenerate horizon with T = 0 yet entropy A/4, showing the area law can survive in a matter-sourced modified-gravity setting.
  • In Case III the aether/Hořava terms act as a short-scale repulsion that cuts off the divergent Schwarzschild temperature, producing a maximum temperature and a final remnant state.
  • Case III also exhibits a heat-capacity phase transition between a large-scale unstable branch and a small-scale stable branch, with the crossover moving outward as n = 4ξ/η decreases.
  • For n = 3 or n = 4, the repulsive correction reproduces known GUP and semiclassical quantum corrections to the Newtonian potential, giving these corrections a concrete gravitational realization.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The full aether field equations are never written down; until they are checked, the metrics are solutions of the truncated metric system rather than proven solutions of the complete theory. Verifying them is the immediate next step.
  • The method—prescribe a linear equation of state, solve for f(r)—is not tied to the three cases; it should generate further exact families for other linear combinations, including cases with non-integer exponents.
  • If the Case III remnant is real, it differs from regular-black-hole remnants: it hides a singularity rather than a de Sitter core, which would sharpen observational searches for evaporation endpoints.
  • The Case II combination of vanishing temperature with nonzero area-law entropy is a useful toy model for discussions of extremal black hole thermodynamics and the information puzzle.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper proposes a method to construct static, spherically symmetric black-hole solutions in the low-energy covariant Hořava gravity / Einstein-æther framework by fixing a linear equation of state rather than prescribing a density profile. Three cases are analyzed: p_r + p_θ = 0 yields f(r) = 1 - C1/r + C2/r^2 (an analogue charged black hole); ρ = -p_r - N p_θ yields, after tuning N, f(r) = (1 - 2M/r)^{n_odd} with n_odd odd (a degenerate extremal black hole); and ρ = p_r yields f(r) = 1 - 2M/r + C2/r^n with n = 4ξ/η (an ultra-relativistic stiff fluid with a short-scale repulsive term). For each case the authors discuss horizon structure, temperature, entropy, heat capacity, and the possibility of a remnant.

Significance. If the solution families are genuine solutions of the full Hořava/Einstein-æther field equations, they would be useful exact examples that connect linear equations of state to exotic black-hole thermodynamics in a Lorentz-violating gravity theory. The explicit metric functions (15), (22), and (30) and the temperature/heat-capacity analysis in Case III are concrete and could be followed up by further work. However, the central claims rest on two unverified pillars: the completeness of the reduction in §III and the correctness of the Wald/Noether entropy computation. Neither is demonstrated in the present manuscript.

major comments (3)
  1. [§III, Eqs. (10)–(13)] The full aether/khronon field equations are never written or solved. The paper asserts, citing Refs. [18,19], that for a static aether the complete set of field equations reduces to Eqs. (10)–(12) plus the conservation equation (13). This reduction is load-bearing: if the full aether equations impose additional constraints on f(r) or on the couplings, then the metric families (15), (22), and (30) are not solutions of the covariant Hořava/Einstein-æther theory. The authors should either state and prove the reduction theorem (or point to a derivation with enough detail for the reader to verify it) and show that the ansatz (9) is compatible with it. This is a reproducibility gap in the central construction.
  2. [§IV.B, Eqs. (17), (20)–(21)] Solving Eq. (20) for N gives N = [4(1−ξ) − 2 n_odd + 4 n_odd ξ]/n_odd, which is independent of η. Equation (21) instead contains η explicitly in the denominator, so the two equations are algebraically inconsistent. Moreover, the claimed behavior 'N∼η' and 'N→0 as η→0' does not follow from the exponent in Eq. (17), since that exponent is independent of η after cancellation; e.g., n_odd=1 forces N=2 for any η. The tuning of N through Eq. (21) should be corrected and the consequences for the Case II parameter space reexamined.
  3. [§IV.B, Eqs. (25)–(28)] The entropy computation is not a Wald computation for the theory under consideration. The Noether charge Q(∂_t) in Eq. (27) is taken from the GR Komar expression, but the action (1)–(3) contains additional aether/khronon terms whose Noether charges can contribute to the entropy. The paper neither writes the full Noether charge nor proves that the aether contributions vanish on the horizon for the static-aether ansatz. Therefore the area-law claim S=area/4 for the extremal solution (and the analogous claim in Case III) is not established. A correct derivation should start from the Wald charge of the full HG-EA action or cite a known result for this theory.
minor comments (4)
  1. [§IV.B, Eq. (19)] The temperature formula for the 1/\bar n_odd branch is written inconsistently with the metric (17): the power of (1−2M/r_h) and the prefactor are not what one obtains from f'(r_h)/(4π) for that metric. Since this case is excluded from the paper's final claims, the text should either derive the formula correctly or explicitly label it as a heuristic expression.
  2. [§IV.B, after Eq. (16)] The statement that 'in the vacuum case, Eq. (16) is also satisfied for all values of N' is trivial for ρ=p_r=p_θ=0. It would be clearer to state that the Schwarzschild limit is obtained after also setting η=0 and that N is then undetermined.
  3. [§IV.C, Eq. (30)] The paper writes n=4ξ/η without stating the condition η≠0. Since the GR limit is η=0, the reader should be told explicitly that Case III is defined only in the Lorentz-violating sector with η≠0, and that n must satisfy the constraints listed after Eq. (32).
  4. [General presentation] There are several typos and stylistic inconsistencies: 'bad defined' should be 'badly defined'; the spellings 'Hoˇrava', 'æther', and 'Einstein-Æther' are inconsistent; in Fig. 1 the heat capacity is labeled 'C' without defining it in the caption; and the notation around Eq. (18) ('exponent = 1/\bar n_odd') is confusing even though the intended meaning is recoverable.

Circularity Check

1 steps flagged

Case II's extremal horizon is engineered by the choice of N; Cases I and III are genuine ODE constructions.

specific steps
  1. self definitional [Section IV.B, Eqs. (20)-(22)]
    "The exponent takes an integer and odd value: exponent = n_odd ... N= 2η·(2−n_odd)/(n_odd ·η+ 4(1−n_odd)ξ) ... The solution is given by f(r) = (1− 2M/r)^{n_odd} ... Thus we note that, in this case, the value of the event horizon is n_odd-fold degenerate."

    Equation (21) fixes the EoS parameter N by requiring the exponent in Eq. (17) to be the odd integer n_odd. The metric is then written as (1−2M/r)^{n_odd}, so the n_odd-fold degenerate root and the consequent T=0 extremality are immediate algebraic consequences of this parameter choice. The advertised 'non-trivial extremal BH' is therefore a selected parameter case of the linear EoS, not an independent prediction from the EoS alone.

full rationale

Most of the paper is a straightforward solution-generating procedure: the EoS is specified and the metric is obtained by solving the ODEs (10)-(12) with (13). I verified that Cases I and III do satisfy the corresponding linear EoS by direct substitution; those derivations are self-contained and not circular. The reduced equations (10)-(13) are imported from Refs. [18,19] as an external assumption; this is a correctness risk (the full aether field equations are never checked) but it is not a circularity, since those references are not self-citations and the assumption is stated rather than disguised. The only step that reduces to its own input is Case II: the EoS parameter N is solved from the requirement that the exponent equal an odd integer, so the degenerate horizon and vanishing temperature are put in by construction. Because this affects one of the three headline results while the others remain independent, the overall circularity is moderate.

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 0 invented entities

The construction depends on several domain assumptions: the metric and aether ansätze, completeness of the reduced field equations, validity of GR-style horizon thermodynamics in a Lorentz-violating theory, and positivity/range conditions on theory parameters. The only free parameter deliberately tuned to force a headline result is N (and the discrete n_odd) in Case II; C2 in Case III scales the repulsive term. No new fundamental entities are introduced.

free parameters (3)
  • N = N = 2η(2−n_odd)/(n_odd η + 4(1−n_odd)ξ)
    The parameter in the EoS ρ=-p_r-N p_θ is chosen through Eq. (21) so that the metric exponent is an odd integer; this choice is what makes the horizon degenerate and the temperature vanish.
  • n_odd = 3, 5, 7, ...
    A discrete integer selected in Case II to engineer the claimed extremal degenerate horizon; it is not derived from the theory parameters alone.
  • C2 = arbitrary > 0
    Integration constant in Case III; chosen positive so that the repulsive term produces an extremal radius and remnant. Its value sets the scale of the short-distance correction.
axioms (6)
  • domain assumption The metric has the form ds² = f(r) dt² − dr²/f(r) − r² dΩ².
    Assumed in Eq. (6); the entire reconstruction depends on this static, spherically symmetric ansatz with g_tt = −1/g_rr.
  • domain assumption The aether is static and has the form u^α = (1/√f(r), 0, 0, 0).
    Assumed in Eq. (9) following [18,19]. This is a restricted aether configuration; a non-static or more general aether would change the equations.
  • domain assumption Equations (10)-(12) plus the conservation equation (13) are the complete equations of motion for the matter+aether system.
    The paper states that only three of the four equations are independent but never writes or solves the aether field equations; completeness relies on [18].
  • ad hoc to paper The Hawking temperature is the standard Killing-horizon surface gravity f'(r_h)/(4π) and the Wald entropy can be computed from the GR Komar formula.
    Used in Eqs. (24), (27)-(28) and (33). In a Lorentz-violating theory the physical horizon and Noether charge may include aether/universal-horizon contributions, which are not considered.
  • domain assumption The Hořava/Einstein-æther parameters take values such that n = 4ξ/η > 1 (Case III) or such that N in Eq. (21) is real (Case II).
    Positivity/range constraints on η, ξ are not derived from the theory or observations; they are imposed so that the constructed solutions have real horizons and signs as claimed.
  • ad hoc to paper The chosen linear equations of state (14), (16), and (29) are admissible matter models.
    The EoS are selected to make the problem solvable; no derivation from fundamental matter fields is given, and energy conditions are not checked.

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We provide a procedure to obtain black hole (BH) solutions in Ho\v{r}ava gravity and Einstein--{\ae}ther theory (HG--EA) for the spherically symmetric (SS) case with a static {\ae}ther. This procedure consists of first specifying the form of the equation of state (EoS), rather than prescribing an energy density profile. The usual EoS for the static and SS case, $\rho = -p_r$, is no longer satisfied due to the presence of the HG--EA terms. We study three linear EoS associated with: an analogue charged BH, a non-trivial extremal BH, and an ultra-relativistic stiff fluid, respectively. The HG--EA terms lead to exotic behaviors, both in the physical properties of the solutions and in their thermodynamics. In Case I, the matter sources can be interpreted as an exotic anisotropic matter distribution, giving rise to an effective electric-potential term in the geometry. In Case II, we obtain a non-trivial extremal BH solution for which the event horizon is $n_{\text{odd}}$-fold degenerate. In Case III, we find a solution with a non-trivial repulsive potential, where the influence of the HG--EA terms at short scales leads to the formation of a BH remnant whose horizon encloses a central singularity (instead of a de Sitter core as occurs in regular BHs

Figures

Figures reproduced from arXiv: 2512.20445 by Milko Estrada.

Figure 1
Figure 1. Figure 1: FIG. 1. First panel: Temperature for [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗

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Reference graph

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