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Black Holes in Einstein-Aether Theory

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Christopher Eling, Ted Jacobson

General Relativity and Quantum Cosmology · Submitted 20 Apr 2006 · 19 references

Pith's verdict: A real step forward in Einstein-Aether black holes, but the one-parameter family is rigorously shown only for special couplings; the abstract overstates the generic case.

T0 · AI review

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arXiv IDgr-qc/0604088 (version 4)
Pith IDLYEVPGD3, Pith's permanent record of this paper and its review
CitationChristopher Eling, Ted Jacobson. "Black Holes in Einstein-Aether Theory." arXiv:gr-qc/0604088 (2006). https://pith.science/paper/LYEVPGD3
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The paper

The paper claims

Einstein-aether theory admits a one-parameter family of regular spherical black holes that closely mimic Schwarzschild outside the horizon but oscillate inside.

The reading

This paper asks whether Einstein-aether theory, a modified gravity with a preferred timelike vector field, admits black holes that trap every causal influence. It argues that a genuine aether black hole must trap not just metric modes but also the spin-0, spin-1, and spin-2 aether waves, each with its own effective horizon. The central result is a parameter count: local static spherical solutions form a three-parameter family, regularity at the metric horizon costs nothing, regularity at the spin-0 horizon removes one parameter, and asymptotic flatness removes another, leaving a one-parameter family. Numerically, these black holes are nearly identical to Schwarzschild outside the horizon, possess a spacelike singularity near r equals zero, and display oscillatory metric and aether quantities in the interior.

What carries the argument

The load-bearing object is the effective metric g^(i)_ab = g_ab + ($s_i^{2}$ - 1)u_a u_b for each wave mode, whose null surfaces define the spin-0, spin-1, and spin-2 horizons. In spherical symmetry the aether has one radial degree of freedom, and a spin-0 horizon occurs where the combination f = N $a^{2}$ equals (1-s0)/(1+s0) or its inverse. The argument runs on power-series solutions of the field equations about the metric horizon and about the spin-0 horizon; a field redefinition makes the spin-0 and metric horizons coincide, reducing the coupling space and allowing the series to be solved in special cases. This machinery converts 'regularly extends as a static spherical solution' into algebraic conditions on free parameters at the horizon.

Extended reading notes

Core claim

In Einstein-aether theory, black holes exist as a one-parameter family of asymptotically flat, static, spherically symmetric solutions with regular metric, spin-0, spin-1, and spin-2 horizons. The paper establishes this by expanding field equations about candidate horizons: a metric horizon is generically regular, but a spin-0 horizon is generically singular unless one initial-data parameter is tuned; asymptotic flatness then fixes one further parameter. For the special coupling class c3 equals zero, the solutions are very close to Schwarzschild outside, while inside the horizon the metric function N and aether quantities oscillate, and a spacelike curvature singularity appears at or very near r equals zero. The aether flows inward but lags free-fall until near the horizon, and the black-hole first-law coefficient alpha depends on the couplings.

Load-bearing premise

The whole parameter count is verified only for the reduced theory and, at the spin-0 horizon, only for the special cases c3 equals zero and c3 equals plus or minus c1; the claim that this represents the generic four-coupling theory is assumed, not proven.

Editorial extensions

If this is right

  • Any nonsingular spherical collapse that settles into a static black hole must emit a burst of spin-0 aether radiation while the exterior adjusts, since the aether cannot remain aligned with the Killing vector and must be infalling.
  • For fixed mass, the horizon radius is larger than in Schwarzschild: in units with horizon radius 1, the mass parameter r0 decreases with the coupling c1, while the surface gravity increases by up to 25 percent over the studied range.
  • The ratio of mass parameter to horizon radius is constant for each coupling set, so the first law of black hole mechanics keeps the general-relativistic form but with a coupling-dependent dimensionless coefficient alpha.
  • The aether congruence differs measurably from free-fall: it barely falls outside the horizon, catches up inside, and then oscillates in relative 3-velocity near the singularity.
  • The interior is not Schwarzschild-like in detail: the Kretschmann scalar slope alternates between roughly -6 and -4.5, and the quantities f and Ba oscillate out of phase, as in Einstein-Yang-Mills black holes.

Reading between the lines

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

  • If the parameter count is generic, the spin-0 horizon acts as a selection rule analogous to requiring regularity at the center in general relativity; one could test whether generic collapse with generic couplings produces the same counting or whether the special-case reliance hides a different family.
  • Because the exterior is nearly Schwarzschild for a wide coupling range, observational tests of aether theory are more likely to come from the horizon-radius versus mass relation or from interior physics than from the weak-field exterior.
  • The interior oscillations may indicate that the static solutions approach a limiting cycle near the singularity; analytic control near r equals zero could reveal whether the number of oscillations is finite or infinite.
  • Since field redefinitions can make any chosen mode horizon coincide with the metric Killing horizon, the identification of the black-hole entropy surface is not fixed by the theory itself, suggesting that aether black-hole thermodynamics needs additional physical input.
Original abstract

We study black hole solutions in general relativity coupled to a unit timelike vector field dubbed the "aether". To be causally isolated a black hole interior must trap matter fields as well as all aether and metric modes. The theory possesses spin-0, spin-1, and spin-2 modes whose speeds depend on four coupling coefficients. We find that the full three-parameter family of local spherically symmetric static solutions is always regular at a metric horizon, but only a two-parameter subset is regular at a spin-0 horizon. Asymptotic flatness imposes another condition, leaving a one-parameter family of regular black holes. These solutions are compared to the Schwarzschild solution using numerical integration for a special class of coupling coefficients. They are very close to Schwarzschild outside the horizon for a wide range of couplings, and have a spacelike singularity inside, but differ inside quantitatively. Some quantities constructed from the metric and aether oscillate in the interior as the singularity is approached. The aether is at rest at spatial infinity and flows into the black hole, but differs significantly from the the 4-velocity of freely-falling geodesics.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,869 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.54 · human participants 0.01 · controlled comparison 0.71 · randomized assignment 0.01 · states quantitative result 0.57 · compares with prior results 0.95 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; .

We study black hole solutions in general relativity coupled to a unit timelike vector field dubbed the "aether". To be causally isolated a black hole interior must trap matter fields as well as all aether and metric modes. The theory possesses spin-0, spin-1, and spin-2 modes whose speeds depend on four coupling coefficients. We find that the full three-parameter family of local spherically symmetric static solutions is always regular at a metric horizon, but only a two-parameter subset is regular at a spin-0 horizon. Asymptotic flatness imposes another condition, leaving a one-parameter family of regular black holes. These solutions are compared to the Schwarzschild solution using numerical integration for a special class of coupling coefficients. They are very close to Schwarzschild outside the horizon for a wide range of couplings, and have a spacelike singularity inside, but differ inside quantitatively. Some quantities constructed from the metric and aether oscillate in the interior as the singularity is approached. The aether is at rest at spatial infinity and flows into the black hole, but differs significantly from the the 4-velocity of freely-falling geodesics.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We study black hole solutions in general relativity coupled to a unit timelike vector field dubbed the "aether". contribution 0.85 0.83 0.04 asserted 0.92
s1 To be causally isolated a black hole interior must trap matter fields as well as all aether and metric modes. background 0.92 0.16 0.08 asserted 0.99
s2 The theory possesses spin-0, spin-1, and spin-2 modes whose speeds depend on four coupling coefficients. background 0.99 0.16 0.04 asserted 1.00
s3 We find that the full three-parameter family of local spherically symmetric static solutions is always regular at a metric horizon, but only a two-parameter subset is regular at a spin-0 horizon. result 1.00 0.96 0.37 asserted 1.00
s4 Asymptotic flatness imposes another condition, leaving a one-parameter family of regular black holes. result 0.99 0.86 0.15 asserted 1.00
s5 These solutions are compared to the Schwarzschild solution using numerical integration for a special class of coupling coefficients. method 0.97 0.92 0.06 asserted 0.99
s6 They are very close to Schwarzschild outside the horizon for a wide range of couplings, and have a spacelike singularity inside, but differ inside quantitatively. result 1.00 0.92 0.22 asserted 1.00
s7 Some quantities constructed from the metric and aether oscillate in the interior as the singularity is approached. result 1.00 0.90 0.07 asserted 0.96
s8 The aether is at rest at spatial infinity and flows into the black hole, but differs significantly from the the 4-velocity of freely-falling geodesics. result 1.00 0.86 0.09 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.94 · evidence from simulation or computation 0.70.

What was examined, and what was not

sources read
  • arXiv listing abstract v4
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Solution for reduced theory (23) with c ′ 1 = 0.051 and c ′ 2 = 0.116, determined by data N1 = −1 and a2 = −0.1 at spatial infinity. There is both an outer and inner metric horizon where N vanishes, but f does not decrease enough to reach a spin-0 horizon. The functions N and f go to zero slightly inside r = 1 which would be the horizon radius of the corresponding Schwarzschild solution. As a2 increases the minimum of f(r) decreases until the solution acquires a spin-0 horizon, where is it generically singular. there is a curvature singularity. In contrast, note that in
Figure 2. Plots of f, N, B, a, and S (the Schwarzschild version of N) vs. z = r − rh for c1 = 0.3, in units with r0 = 2. The horizon radius is rh ≈ 2.07 for the ae-theory black hole, so S = 0 at z ≈ −0.07. The solutions agree closely outside the horizon. Deviations are noticeable near the horizon and become significant in the interior, where N blows up more rapidly. Near the singularity f begins to oscillate rapidly. 5.2. Asymptotically flat black holes To produce asymptotically flat solutions we numerically integrate outward, starting with the horizon data and matching onto (30), (31), and…
Figure 3. Oscillations of f = N a2 near the singularity inside the black hole solution shown in figure 2, plotted vs. z = r − rh in units with M = 1. While N decreases monotonically and B increases monotonically, a goes to zero, so the oscillations of Na2 and Ba arise because of variations in the magnitude of their derivatives. Since the oscillations inside z = −2 are not clearly visible in
Figure 4. Plot of ln RabcdRabcd vs. ln r for c1 = 0.3 ae-theory black hole (wiggly curve) and Schwarzschild black hole (stright line) of the same mass, in units with M = 1. r is plotted on a logarithmic scale. The slope for the GR case is −6, while for the ae-theory case it alternates between roughly −6 and −4.5. 5.2.5. Aether congruence The aether field defines a congruence of radial timelike curves at rest at infinity and flowing into the black hole. It is interesting to compare this with the static frame and with the congruence of freely falling radial geodesics with 4-velocity v a that…
Figure 5. Radial proper velocity dr/dτ of free-fall (lower, solid curve) and aether (upper, dashed curve) vs. z = r − rh, in units with M = 1, for c1 = 0.3. In contrast to the free-falling geodesics, the aether does not begin to fall significantly inward until close to the horizon. is still about fifteen times smaller than that of free-fall. Inside the horizon the aether proper velocity is equal to the free-fall one around z = −1.3, but the 4-velocities do not agree there. The aether is still going inward faster, but its proper time is “running slower” so it can have the same proper velocity.…
Figure 6. Inward 3-velocity of the aether relative to free-fall inside the horizon for c1 = 0.3. The velocity is initially negative and the aether lags behind the free-fall. Near the singularity the velocity oscillates between between faster and slower than free-fall. always the Killing horizon, but recall that by a field redefinition we can make any one of these horizons be the Killing horizon. We shall not try to shed any light on these puzzling issues here. Rather, we just briefly examine the variational relation between mass, surface gravity and area of the spin-0 horizon, for possible…

Figures reproduced from arXiv:gr-qc/0604088 by the authors.

References
How later papers used it

58 later papers cite it

Evidence: substantial. 54 of the citing papers share no author with it, and 6 of those carry a confident label from Pith's citation classifier.

Pith read 12 of the 106 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

Compares with it 1 of 2 papers with this label
How the literature uses it

53 arXiv papers cite this in 100 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 72 sentences without a confident label

Uses its method or tool 3 papers
Compares with it 2 papers
  • Ref. [ref] has obtained spherically symmetric BHs in the Einstein-aether and khronometric theories by numerically solving the coupled system of equations for the metric and aether (khronon) field[For previous studies of BHs in the Einstein-aether model see [ref] .].

    Horava gravity vs. thermodynamics: the black hole case · Introduction and summary · 0.88

  • Since we assume an asymptotic flatness, the asymptotic values of the variables are given by [equation] In order to investigate the asymptotic behavior of the solution, we perform an asymptotic expansion around Minkowski spacetime, that is, the functions $T(r)$, $B(r)$ and $a(r)$ are expanded as a series of $1/r$ as [equation] Substituting these series into the basic equations, and solving them order by order, we find the expansion coefficients as [equation] [equation] The asymptotic behavior of the function $b(r)$ is given by [equation] with [equation] The important point is every order of these functions, at least up to the eighth order, is described only by two arbitrary coefficients, $T_1$ and $a_2$ as in the case of the æ-theory [ref] .

    Black Holes and Thunderbolt Singularities with Lifshitz Scaling Terms · Asymptotic behaviour · 0.86

Cites it as support 1 paper
  • It has been demonstrated for several Lorentz violating theories including the Einstein-aether [ref] , ghost condensate [ref] and massive gravity [ref] that spherically symmetric solutions with given mass are unique.

    Horava gravity vs. thermodynamics: the black hole case · Introduction and summary · 0.97

Cites it as background 15 papers

Discussion

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