REVIEW
Spherical Solutions in Einstein-Aether Theory: Static Aether and Stars
General Relativity and Quantum Cosmology · Submitted 15 Mar 2006 · 36 references
Pith's verdict: The exact static aether solution is solid and genuinely new; the star matching worry is mostly a red herring, but the stability section overstates what is proven.
T0 · AI review
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Referee report
The paper
The paper claims
Gravity with a preferred frame yields a singular throat inside every static exterior
The reading
What carries the argument
The argument rides on a single reduced ODE. With $Y=rA'$, the metric function $B$ becomes $B=1+Y+\nu Y^2$ with $\nu=c_1/8$, and the field equations collapse to $dY/dr=-(Y/r)(1+Y+\nu Y^2)$, integrable by quadrature. The roots $Y_\pm$ of $B$ divide the solution branches: $Y_+$ is the naked-singularity endpoint for negative mass, $Y=0$ is spatial infinity, and $Y\to -\infty\sim Y_-$ is the internal area-infinity. The explicit expressions for $N=e^A$ and $r/r_{\min}$ in terms of $Y$, Eqs. (34) and (35), encode the throat, the extremal horizon, and the finite affine parameter; the star matching uses continuity of $A'$ at the zero-pressure surface.
Extended reading notes
Core claim
The paper establishes that in Einstein-aether theory the stationary spherically symmetric vacuum solutions form a three-parameter local family, of which asymptotic flatness leaves two parameters and alignment of the aether with the timelike Killing field leaves one, the total mass. For this static-aether family the metric functions are obtained analytically up to inversion of a transcendental equation. The positive-mass solution has a minimal-area 2-sphere that is not a horizon; inside it the sphere radius re-expands without bound, and the interior ends at a curvature singularity at an extremal Killing horizon with zero surface gravity. A radial null geodesic reaches the singular internal infinity in finite affine parameter for all admissible couplings, while the proper radial distance is finite only when $c_1<3/2$. The paper also proves that pure aether stars do not exist, and constructs regular perfect-fluid star solutions matched to the static-aether exterior, with constant-density stars having a maximum mass and an instability onset at that extremum.
Load-bearing premise
The fluid-star construction assumes the star's surface is not a characteristic surface of the field equations, so the interior can be matched uniquely to the static-aether vacuum exterior; the paper states this was not studied in detail.
Editorial extensions
If this is right
- The static-aether vacuum exterior is not Schwarzschild-like beyond leading order: at the same total mass, $r_{\min}$ is larger than $2M$, the throat is not a horizon, and the interior is singular.
- Because the interior is singular and the minimal sphere is not hidden by a horizon, any star matched to this exterior leaves the throat and singularity exposed to null observers within finite affine time.
- No asymptotically flat, regular pure-aether star (soliton) exists; regular static configurations require ordinary matter.
- Constant-density stars in aether theory have a lower maximum mass than in GR, and their mass-radius curves acquire extrema at which radial stability is lost; for $c_1=1$ instability begins at central pressure about $1.28$ times the density.
- The aether energy density is negative everywhere in the positive-mass static solution, yet the total mass is positive, with $R_{tt}=0$; the null energy condition is violated.
Reading between the lines
- If the static-aether interior were stable, gravitational collapse in this theory might produce a singular, horizonless compact object rather than a black hole—an observable distinction only if the aether couplings are large enough to matter.
- The two-parameter asymptotically flat family left after imposing flatness but not static aether is the natural arena for black holes, since a regular horizon forces the aether to tilt away from the Killing vector; exploring that branch could connect the star exteriors found here to horizon solutions.
- A direct test of the uniqueness assumption would be to linearize the equations about a matched star solution and look for a zero-speed spin-0 wave localized at the surface; if one exists, the surface is characteristic and the matching argument needs revision.
- The finite-affine-parameter singularity suggests that aether theory with these couplings may not admit globally hyperbolic, nonsingular static configurations, so questions of cosmic censorship and causal structure differ from GR in this sector.
Original abstract
The time independent spherically symmetric solutions of General Relativity (GR) coupled to a dynamical unit timelike vector are studied. We find there is a three-parameter family of solutions with this symmetry. Imposing asymptotic flatness restricts to two parameters, and requiring that the aether be aligned with the timelike Killing field further restricts to one parameter, the total mass. These "static aether" solutions are given analytically up to solution of a transcendental equation. The positive mass solutions have spatial geometry with a minimal area 2-sphere, inside of which the area diverges at a curvature singularity occurring at an extremal Killing horizon that lies at a finite affine parameter along a radial null geodesic. Regular perfect fluid star solutions are shown to exist with static aether exteriors, and the range of stability for constant density stars is identified.
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.
The abstract, sentence by sentence
The time independent spherically symmetric solutions of General Relativity (GR) coupled to a dynamical unit timelike vector are studied. We find there is a three-parameter family of solutions with this symmetry. Imposing asymptotic flatness restricts to two parameters, and requiring that the aether be aligned with the timelike Killing field further restricts to one parameter, the total mass. These "static aether" solutions are given analytically up to solution of a transcendental equation. The positive mass solutions have spatial geometry with a minimal area 2-sphere, inside of which the area diverges at a curvature singularity occurring at an extremal Killing horizon that lies at a finite affine parameter along a radial null geodesic. Regular perfect fluid star solutions are shown to exist with static aether exteriors, and the range of stability for constant density stars is identified.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | The time independent spherically symmetric solutions of General Relativity (GR) coupled to a dynamical unit timelike vector are studied. | contribution 0.80 | 0.73 | 0.05 | asserted 0.87 |
| s1 | We find there is a three-parameter family of solutions with this symmetry. | result 0.99 | 0.96 | 0.21 | asserted 1.00 |
| s2 | Imposing asymptotic flatness restricts to two parameters, and requiring that the aether be aligned with the timelike Killing field further restricts to one parameter, the total mass. | result 0.98 | 0.89 | 0.21 | asserted 1.00 |
| s3 | These "static aether" solutions are given analytically up to solution of a transcendental equation. | result 0.68 | 0.92 | 0.12 | asserted 1.00 |
| s4 | The positive mass solutions have spatial geometry with a minimal area 2-sphere, inside of which the area diverges at a curvature singularity occurring at an extremal Killing horizon that lies at a finite affine parameter along a radial null geodesic. | result 1.00 | 0.89 | 0.23 | asserted 1.00 |
| s5 | Regular perfect fluid star solutions are shown to exist with static aether exteriors, and the range of stability for constant density stars is identified. | result 0.89 | 0.94 | 0.25 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v4
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
Figures
Figures reproduced from arXiv:gr-qc/0603058 by the authors.
References
How the literature uses it
29 arXiv papers cite this in 71 sentences
Papers citing it · 54 sentences without a confident label
Uses its method or tool 1 paper
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However, for simplicity and in accordance with references [ref] , we consider a static æther field $u_μ$ of the form [equation]
Black hole solutions with a linear equation of state in Ho\v{r}ava gravity and Einstein--{\ae}ther theory · The equations of motion in our framework · 0.93
Compares with it 1 paper
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Upon closer inspection, we checked that these are identical to our algebraically special branch, specifically, [ref] and [ref] are equivalent to (25) and (26) of [ref] , as well as, (5.11) and (5.17) of [ref] , respectively.
Stealth black holes in Aether Scalar Tensor theory · Case $=1$ · 1.00
Cites it as support 3 papers
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However, as argued in Ref. [ref] , this boundary remains a physical singularity due to the divergence of $R_ab k^a k^b$, with $k^a$ being the tangent vector of an affinely parameterized radial null geodesic approaching the internal infinity.
Revisit Static Aether: Exact Vacuum Solution in Einstein-Aether Theory and Its Analytic Extension · Discussion and Summary · 0.97
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On the other hand, a spherically symmetric static vacuum solution is known explicitly [ref] which has positive energy despite having everywhere negative aether energy density, which suggests that there may be a general positive energy property.
A positive energy theorem for Einstein-aether and Ho\v{r}ava gravity · Introduction · 0.99
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The solution outside a static star is the unique vacuum solution for a given mass in which the æther is aligned with the Killing vector [ref] .
Einstein-aether gravity: a status report · Static vacuum æther · 0.96
Cites it as background 9 papers
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The rest of the paper is organized as follows: In Sec. [ref], we will give a brief review for the EJ solution [ref] and the OMW solution [ref] in Einstein-aether theory, and set up our convention of notations.
Shadow of the wormholelike static aether solution · Introduction · 0.97
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This static configuration was initially investigated by Eling and Jacobson in 2006 as a one-parameter family depending entirely on the coupling $c_14 = c_1 + c_4$ [ref] .
Revisit Static Aether: Exact Vacuum Solution in Einstein-Aether Theory and Its Analytic Extension · Introduction · 0.88
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Time independent spherically symmetric solutions in ae-theory were extensively studied recently in a pair of papers on fluid star [ref] and black hole [ref] configurations.
Neutron stars in Einstein-aether theory · Neutron Stars · 0.99
Discussion
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