arxiv: 2604.09464 · v1 · submitted 2026-04-10 · 🌀 gr-qc

Recognition: 2 theorem links

· Lean Theorem

New Exact Vacuum Solutions in Extended Bumblebee Gravity

Jie Zhu , Hao Li

Authors on Pith no claims yet

Pith reviewed 2026-05-10 17:20 UTC · model grok-4.3

classification 🌀 gr-qc

keywords bumblebee gravityvacuum solutionsnon-minimal couplingblack holeswormholeszero entropyspherically symmetricvector-tensor gravity

0 comments

The pith

Varying the action before or after imposing the bumblebee vector's vacuum expectation value produces different vacuum solutions in extended bumblebee gravity.

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

The paper studies static spherically symmetric vacuum solutions in a generalized bumblebee gravity model that includes non-minimal couplings of the vector field to the curvature scalars. It finds that varying the action and enforcing the vector field's vacuum expectation value constraint do not commute as operations. This non-commutativity opens a larger set of exact solutions than earlier treatments assumed, including naked singularities, black holes, and wormholes, with up to ten solutions derived in total. Some of the new black hole solutions exhibit zero entropy under thermodynamic analysis. The authors conclude that any such non-minimally coupled vector-tensor theory is best formulated as a bumblebee model in which the vector field acquires a vacuum expectation value.

Core claim

In the extended bumblebee gravity model with non-minimal couplings B²R and B^μ B^ν R_μν, the variation of the action and the imposition of the vacuum expectation value constraint on the bumblebee field are non-commutative. Treating them as non-commutative yields a richer solution space containing ten exact static spherically symmetric vacuum solutions, among them black holes with vanishing entropy, wormholes, and naked singularities, whereas assuming commutativity recovers only the previously known restricted set.

What carries the argument

The non-commutativity between varying the full action and imposing the bumblebee vector field's vacuum expectation value constraint, which arises directly from the chosen non-minimal curvature couplings.

If this is right

New black hole solutions appear with zero entropy, violating the usual area law.
Wormhole and naked singularity solutions become possible that earlier commutative treatments missed.
The model admits a broader family of static spherically symmetric vacuum geometries.
Thermodynamic analysis of the new black holes reveals unusual properties such as vanishing entropy.
If the theory is fundamental, the universe is described by a bumblebee-type vector field that acquires a nonzero vacuum expectation value.

Where Pith is reading between the lines

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

The non-commutativity may signal that the theory possesses additional hidden degrees of freedom whose consistent quantization requires specifying an order of operations.
Similar order-of-operations subtleties could appear in other vector-tensor or non-minimally coupled gravity models when vacuum constraints are imposed.
The zero-entropy black holes offer a concrete target for numerical simulations of gravitational collapse in this framework.
These solutions could be tested against strong-field observations such as black-hole shadows or ringdown waveforms to see whether the non-commutative branch is realized.

Load-bearing premise

That the non-commutativity between varying the action and imposing the vector vacuum expectation value constraint yields physically meaningful solutions rather than artifacts of the chosen couplings or coordinates.

What would settle it

An astrophysical observation of a black hole whose entropy vanishes or whose metric exactly matches one of the ten derived solutions, or a clear absence of such configurations in all high-precision data on compact objects.

read the original abstract

We investigate the static spherically symmetric vacuum solutions in a generalized bumblebee gravity model characterized by non-minimal couplings $B^2 R$ and $B^\mu B^\nu R_{\mu\nu}$. We demonstrate that the variation of the action and the imposition of the vacuum expectation value constraint are non-commutative, leading to a richer solution space than previously explored. A diverse set of solutions, including naked singularities, black holes, and wormholes, is obtained, and as many as ten exact solutions are presented. The thermodynamic properties of the new black hole solutions are also analyzed, and a subset of these solutions is found to have zero entropy. We argue that if such a non-minimally coupled vector-tensor gravity provides a fundamental description of the universe, it is best described by a Bumblebee-type theory, where the vector field acquires a VEV.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The non-commutativity between varying the action and imposing the VEV constraint looks like it could be an artifact of skipping the Lagrange multiplier rather than a source of new physics.

read the letter

The non-commutativity between varying the action and imposing the VEV constraint looks like it could be an artifact of skipping the Lagrange multiplier rather than a source of new physics. The authors work with an extended bumblebee model that adds non-minimal couplings B²R and B^μ B^ν R_μν to the usual vector-tensor action. They report that the order in which one varies and then enforces B^μ B_μ = b₀² matters, and this ordering produces ten explicit static spherically symmetric vacuum solutions, among them black holes, naked singularities, and wormholes. They also compute the thermodynamics for the black-hole cases and find a subset with zero entropy. These concrete metrics are new relative to the earlier bumblebee literature they cite, and having explicit line elements makes it straightforward to check curvature scalars or geodesic behavior if one wants to use them for further calculations. The paper does a service by laying out the metrics in one place. The central worry is whether the solutions actually solve the theory as written. In a constrained vector theory the standard move is to add a multiplier λ(B² - b₀²) to the action and vary with respect to the metric, the vector, and λ at the same time; the resulting equations enforce the constraint on-shell. The manuscript does not appear to follow this route, so it remains unclear whether the ten metrics satisfy the full constrained equations or only a version in which the constraint is imposed after the fact. If the latter, the enlarged solution space is not a feature of the physical theory. The zero-entropy claim inherits the same uncertainty. This paper is for people who build and classify exact solutions in Lorentz-violating gravity. A reader who needs new black-hole metrics to test thermodynamic or stability properties will find usable examples here. It is worth sending to peer review. The explicit forms let a referee check the algebra directly, and the constraint-handling question is a sharp, answerable point that a careful referee can settle.

Referee Report

2 major / 2 minor

Summary. The paper investigates static spherically symmetric vacuum solutions in a generalized bumblebee gravity model with non-minimal couplings B²R and B^μ B^ν R_μν. It claims that the operations of varying the action and imposing the vacuum expectation value constraint B^μ B_μ = b₀² are non-commutative, yielding a richer solution space than in prior work. Up to ten exact solutions are presented, encompassing naked singularities, black holes, and wormholes. Thermodynamic analysis of the black hole solutions is performed, with a subset reported to have zero entropy. The authors conclude that non-minimally coupled vector-tensor gravity is best realized as a Bumblebee-type theory.

Significance. If the solutions are shown to satisfy the full set of constrained field equations, the work would expand the catalog of exact solutions in modified gravity and highlight potential thermodynamic anomalies such as zero-entropy black holes. The demonstration of non-commutativity, if physically justified rather than formal, could motivate further study of constrained vector fields in gravitational theories. However, the significance is tempered by the absence of explicit verification that the reported metrics solve the complete Euler-Lagrange equations derived from the constrained action.

major comments (2)

[Section 3 (field equations and solution ansatz)] The central procedure (variation of the action followed by post-hoc imposition of B^μ B_μ = b₀²) is presented without augmenting the action by a Lagrange multiplier λ(B^μ B_μ − b₀²) and without showing that the ten solutions satisfy the resulting on-shell equations for any finite λ. This directly affects the claim that non-commutativity generates physically new solutions rather than extraneous ones.
[Section 5 (thermodynamics)] The zero-entropy black hole solutions are stated to exist, yet the manuscript provides no explicit computation of the entropy (e.g., via Wald formula or Euclidean action) that would confirm the result is not an artifact of coordinate choice or horizon definition. This is load-bearing for the thermodynamic claims.

minor comments (2)

[Section 2] Notation for the non-minimal coupling coefficients is introduced without a clear table summarizing their values across the ten solutions.
[Section 4] Several metric functions are given in implicit form; explicit verification that the Ricci tensor components satisfy the vacuum equations would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback on our manuscript. We address each of the major comments in detail below, providing clarifications and indicating where revisions will be made to strengthen the paper.

read point-by-point responses

Referee: [Section 3 (field equations and solution ansatz)] The central procedure (variation of the action followed by post-hoc imposition of B^μ B_μ = b₀²) is presented without augmenting the action by a Lagrange multiplier λ(B^μ B_μ − b₀²) and without showing that the ten solutions satisfy the resulting on-shell equations for any finite λ. This directly affects the claim that non-commutativity generates physically new solutions rather than extraneous ones.

Authors: We appreciate this observation. While our approach of varying the action prior to imposing the constraint is deliberate to highlight the non-commutativity and follows precedents in the bumblebee gravity literature, we acknowledge the value of the Lagrange multiplier method for rigor. In the revised manuscript, we will introduce the multiplier term and explicitly verify that each of the ten solutions satisfies the constrained field equations for a suitable finite value of λ. This will substantiate that the solutions are not extraneous but arise naturally from the constrained theory. revision: yes
Referee: [Section 5 (thermodynamics)] The zero-entropy black hole solutions are stated to exist, yet the manuscript provides no explicit computation of the entropy (e.g., via Wald formula or Euclidean action) that would confirm the result is not an artifact of coordinate choice or horizon definition. This is load-bearing for the thermodynamic claims.

Authors: We agree that an explicit verification is necessary to support the thermodynamic claims. The zero entropy in our solutions stems from the vanishing horizon area in the specific metrics. In the revision, we will provide a detailed computation of the entropy using the Wald formula for the black hole solutions, demonstrating that it indeed vanishes for the identified cases and is independent of coordinate choices. revision: yes

Circularity Check

0 steps flagged

Derivation of exact solutions is self-contained with no circular reductions

full rationale

The paper starts from a specified action with non-minimal B²R and B^μB^νR_μν couplings, varies it to obtain field equations, and imposes the VEV constraint B^μB_μ = b₀² either before or after variation to generate explicit metrics. These steps are direct algebraic manipulations of the given Lagrangian; no fitted parameters are relabeled as predictions, no load-bearing uniqueness theorems are imported via self-citation, and no ansatz is smuggled in. The reported non-commutativity is exhibited by concrete solution sets rather than defined into existence. The ten metrics satisfy the paper's stated equations by construction of the derivation, which is the normal and non-circular outcome for an exact-solution paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The model assumes a specific non-minimal action with two curvature-vector couplings and the standard bumblebee VEV constraint; no new entities are introduced beyond the vector field already present in bumblebee gravity.

axioms (2)

domain assumption The action is varied before or after imposing the VEV constraint, and the two procedures are inequivalent.
This is the central technical claim of the paper and is not derived from more fundamental principles.
standard math Static spherically symmetric metric ansatz is sufficient to capture all vacuum solutions of interest.
Common assumption in GR but restricts the solution class.

pith-pipeline@v0.9.0 · 5436 in / 1404 out tokens · 37062 ms · 2026-05-10T17:20:10.704019+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We demonstrate that the variation of the action and the imposition of the vacuum expectation value constraint are non-commutative, leading to a richer solution space... as many as ten exact solutions are presented... a subset of these solutions is found to have zero entropy.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

52 extracted references · 28 canonical work pages · 2 internal anchors

[1]

By taking linear combinations of these four equations, we eliminate the term e2β, resulting in three independent equations
[2]

By taking linear combinations of the three equations from the last step, we eliminate the term α′(r)β′(r), resulting in two independent equations
[3]

After the three steps listed above, we obtain a simple relation 4b2κ (ℓ2 − 1) 2 κb2 + 2ℓ2 − 2 r3α′2 = 0

By taking linear combinations of the remaining two equations, we eliminate the term β′(r), resulting in one equation. After the three steps listed above, we obtain a simple relation 4b2κ (ℓ2 − 1) 2 κb2 + 2ℓ2 − 2 r3α′2 = 0. The solution in this case depends on which part of the relation is zero
[4]

By the redefinition of t, we can choose α(r) = 0

Case of α′(r) = 0 If ℓ2 ̸= 1 and κb2 + 2ℓ2 − 2 ̸= 0, we have α′(r) = 0, so α(r) is a constant, which implies that gtt = e2α(r) is also a constant. By the redefinition of t, we can choose α(r) = 0. Substitute α(r) = 0 into Eqs. (9), (10), (11), and (13), we have the following equations 2rβ ′ + e2β − 1 = 0, (ℓ2 + 1) rβ ′ + e2β − 1 = 0, 2rβ ′ + ℓ2 e2β − 1 = ...
[5]

Using the definition of ℓ1 and ℓ2, in this case, we have ℓ1 = 2ξ κ + 2λ , ℓ 2 = 2λ κ + 2λ , and b = r 2 κ + 2λ = r 2(1 − ℓ2) κ

Case of κb2 + 2ℓ2 − 2 = 0 Here, we consider the case κb2 + 2ℓ2 − 2 = 0. Using the definition of ℓ1 and ℓ2, in this case, we have ℓ1 = 2ξ κ + 2λ , ℓ 2 = 2λ κ + 2λ , and b = r 2 κ + 2λ = r 2(1 − ℓ2) κ . Substitute b = q 2(1−ℓ2) κ into Eqs. (9), (10), (11), and (13), and take linear combinations of these four equations to eliminate the term e2β, we obtain on...
[6]

In this case, we first replace Q with Q2, and the requirement that Ett = Err = Eθθ = 0 becomes κb2c2 2 + 2Q2(sℓ2 − 1) = 0, and we have c2 = Q b r 2(1 − sℓ2) κ

Case of ξ = 0 We first consider the case p = 2, or ξ = ℓ1 = 0. In this case, we first replace Q with Q2, and the requirement that Ett = Err = Eθθ = 0 becomes κb2c2 2 + 2Q2(sℓ2 − 1) = 0, and we have c2 = Q b r 2(1 − sℓ2) κ . In this case, the solution is ds2 = −f dt2 + dr2 f + r2dΩ2, f = 1 − Rs r + Q2 r2 , bt = P + r 2(1 − λsb2) κ Q r , b r = p b2 t − sb2f...
[7]

That is, the coupling between Bµ and gravity in the action is directly manifested as the coupling between Bµ and the Einstein tensor

Case of λ = −ξ/2 In this case, λ = −ξ/2, and the coupling in the action becomes 1 2κ − ξ 2 BµBµR + ξBµBνRµν = ξ 2κ GµνBµBν. That is, the coupling between Bµ and gravity in the action is directly manifested as the coupling between Bµ and the Einstein tensor. In this case, Eq. (28) becomes rG′(r) + G(r) − 1 = 0, and the solution is G(r) = 1 − Rs r . So in t...
[8]

Case of λ = ξ/2 In the case ξ ̸= 0, to let Eθθ be identically zero, we have s = c2
[9]

So in this case, we also only have solutions for timelike or lightlike VEVs

Again, since s = −1, 0, 1 and c1 is a real number, we have c1 = s = 1 or c1 = s = 0. So in this case, we also only have solutions for timelike or lightlike VEVs. We now consider the special case p = 1, which means that ℓ1 = 2ℓ2 and ξ = 2λ. In this case, the metric reduces to the Schwarzschild metric, and we can let Q = 0. The requirement that Ett = Err = ...
[10]

In this case, only solutions with timelike VEVs exist

Case of λ ̸= 0 and λ ̸= ±ξ/2 If Q ̸= 0, ℓ1 ̸= 0, ℓ1 + 2ℓ2 ̸= 0, and ℓ1 − 2ℓ2 ̸= 0, to ensure that Eθθ is identically zero, we have (ℓ1 + ℓ2)s = 1, s = c2 1, and the only solution is s = c1 = 1 and ℓ1 + ℓ2 = 1. In this case, only solutions with timelike VEVs exist. The relation ℓ1 + ℓ2 = 1 means that b is a fixed value as b = 1√λ + ξ . 9 Besides, we have o...
[11]

Substitute it into Eq

Solution IV In this solution, denote Rs = 2M, we have G = H = f = 1 − 2M r , δG = δH = δf = −2δM r , bt = b, b r = b f p 1 − f , δb t = 0, δb r = b(f − 2) 2f2√1 − f δf. Substitute it into Eq. (52), we have δQ − iξΘ = −1 − b2(λ + ξ) κr δf = (1 − ℓ1 − ℓ2) δM 4πr2 , (53) where ℓ1 = ξb2 and ℓ2 = λb2. The integration at asymptotic infinity gives δH∞ = δE = (1 ...
[12]

Solution V In this solution, the parameters satisfy the constraint ξ = κ/2. With Rs = 2 M, the field configuration can be expressed as G = H = f = 1 − 2M r , δG = δH = δf = −2δM r , bt = b + C r , b r = p b2 t − b2f f , δb t = δC r , δb r = btδbt f p b2 t − b2f + (−2b2 t + b2f)δf 2f2 p b2 t − b2f . Substitute it into Eq. (52), we have δQ − iξΘ = −1 − b2(λ...
[13]

This differs from Solutions IV and V only in the form of f

Solution VI and VII In these two solutions, we have G = H = f, δG = δH = δf, bt = b + C r , b r = p b2 t − b2f f , δbt = δC r , δb r = btδbt f p b2 t − b2f + (−2b2 t + b2f)δf 2f2 p b2 t − b2f . This differs from Solutions IV and V only in the form of f. Substitute it into Eq. (52), we have δQ − iξΘ = −1 − b2(λ + ξ) κr δf + (κ − 2ξ)(C + br) κr3 δC. (60) In...
[14]

Solution VIII Here we consider the special case ξ = 0, in which the solution can be viewed as the result of performing a gauge transformation on the vector field of the RN solution, followed by a redefinition of the parameter κ as κ′ = κ 1−λsb2 = κ 1−sℓ2 . With Rs = 2M, the field configuration is G = H = f = 1 − 2M r + Q2 r2 , δG = δH = δf = −2δM r + 2QδQ...
[15]

free theory

Solution IX Lastly, we consider the case when Solution IX is a black hole solution. From the relation br(r) = s bt(r)2 − b2G(r) G(r)H(r) , (69) we have δbr = b2G2δH + 2btGHδb t − b2 t (HδG + GδH) 2G3/2H3/2 p b2 t − b2G . (70) Varying the relation bt(r) = b r C + Z s G(r) H(r) dr ! , (71) we obtain δbt = b r δC + Z HδG − GδH 2H √ GH dr . (72) Substitute th...
[16]

V. A. Kostelecky and S. Samuel, Spontaneous Breaking of Lorentz Symmetry in String Theory, Phys. Rev. D 39, 683 (1989). 23

1989
[17]

Colladay and V

D. Colladay and V. A. Kostelecky, CPT violation and the standard model, Phys. Rev. D 55, 6760 (1997), arXiv:hep- ph/9703464

work page arXiv 1997
[18]

Lorentz-violating extensio n of the standard model

D. Colladay and V. A. Kostelecky, Lorentz violating extension of the standard model, Phys. Rev. D 58, 116002 (1998), arXiv:hep-ph/9809521

work page arXiv 1998
[19]

Colladay and V

D. Colladay and V. A. Kostelecky, Cross-sections and Lorentz violation, Phys. Lett. B 511, 209 (2001), arXiv:hep- ph/0104300

work page arXiv 2001
[20]

V. A. Kostelecky and R. Lehnert, Stability, causality, and Lorentz and CPT violation, Phys. Rev. D 63, 065008 (2001), arXiv:hep-th/0012060

work page arXiv 2001
[21]

V. A. Kostelecky, Gravity, Lorentz violation, and the standard model, Phys. Rev. D 69, 105009 (2004), arXiv:hep- th/0312310

work page arXiv 2004
[22]

Bluhm and V

R. Bluhm and V. A. Kosteleck´ y, Spontaneous lorentz violation, nambu-goldstone modes, and gravity, Phys. Rev. D 71, 065008 (2005)

2005
[23]

Bluhm, N

R. Bluhm, N. L. Gagne, R. Potting, and A. Vrublevskis, Constraints and stability in vector theories with spontaneous lorentz violation, Phys. Rev. D 77, 125007 (2008)

2008
[24]

An exact Schwarzschild-like solution in a bumblebee gravity model

R. Casana, A. Cavalcante, F. P. Poulis, and E. B. Santos, Exact Schwarzschild-like solution in a bumblebee gravity model, Phys. Rev. D 97, 104001 (2018), arXiv:1711.02273 [gr-qc]

work page Pith review arXiv 2018
[25]

Li and J

H. Li and J. Zhu, Static spherical vacuum solution to bumblebee gravity with time-like VEVs, Eur. Phys. J. C86, 2 (2026), arXiv:2506.17957 [gr-qc]

work page arXiv 2026
[26]

Full Classification of Static Spherical Vacuum Solutions to Bumblebee Gravity with General VEVs

J. Zhu and H. Li, Full Classification of Static Spherical Vacuum Solutions to Bumblebee Gravity with General VEVs, (2025), arXiv:2511.03231 [gr-qc]

work page internal anchor Pith review Pith/arXiv arXiv 2025
[27]

Liu, S.-P

J.-Z. Liu, S.-P. Wu, S.-W. Wei, and Y.-X. Liu, Exact Black Hole Solutions in Bumblebee Gravity with Lightlike or Spacelike VEVS, (2025), arXiv:2510.16731 [gr-qc]

work page arXiv 2025
[28]

¨Ovg¨ un, K

A. ¨Ovg¨ un, K. Jusufi, and˙I. Sakallı, Exact traversable wormhole solution in bumblebee gravity, Phys. Rev. D 99, 024042 (2019), arXiv:1804.09911 [gr-qc]

work page arXiv 2019
[29]

R. V. Maluf and J. C. S. Neves, Black holes with a cosmological constant in bumblebee gravity, Phys. Rev. D 103, 044002 (2021), arXiv:2011.12841 [gr-qc]

work page arXiv 2021
[30]

C. Ding, C. Liu, R. Casana, and A. Cavalcante, Exact Kerr-like solution and its shadow in a gravity model with spontaneous Lorentz symmetry breaking, Eur. Phys. J. C 80, 178 (2020), arXiv:1910.02674 [gr-qc]

work page arXiv 2020
[31]

A. F. Santos, A. Y. Petrov, W. D. R. Jesus, and J. R. Nascimento, G¨ odel solution in the bumblebee gravity, Mod. Phys. Lett. A 30, 1550011 (2015), arXiv:1407.5985 [hep-th]

work page arXiv 2015
[32]

S. K. Jha and A. Rahaman, Bumblebee gravity with a Kerr-Sen-like solution and its Shadow, Eur. Phys. J. C 81, 345 (2021), arXiv:2011.14916 [gr-qc]

work page arXiv 2021
[33]

A. A. A. Filho, J. R. Nascimento, A. Y. Petrov, and P. J. Porf´ ırio, Vacuum solution within a metric-affine bumblebee gravity, Phys. Rev. D 108, 085010 (2023), arXiv:2211.11821 [gr-qc]

work page arXiv 2023
[34]

R. Xu, D. Liang, and L. Shao, Static spherical vacuum solutions in the bumblebee gravity model, Phys. Rev. D 107, 024011 (2023), arXiv:2209.02209 [gr-qc]

work page arXiv 2023
[35]

C. Ding, Y. Shi, J. Chen, Y. Zhou, C. Liu, and Y. Xiao, Rotating BTZ-like black hole and central charges in Einstein- bumblebee gravity, Eur. Phys. J. C 83, 573 (2023), arXiv:2302.01580 [gr-qc]

work page arXiv 2023
[36]

Liu, W.-D

J.-Z. Liu, W.-D. Guo, S.-W. Wei, and Y.-X. Liu, Charged spherically symmetric and slowly rotating charged black hole solutions in bumblebee gravity, Eur. Phys. J. C 85, 145 (2025), arXiv:2407.08396 [gr-qc]

work page arXiv 2025
[37]

Q. G. Bailey, H. S. Murray, and D. T. Walter-Cardona, Bumblebee gravity: Spherically symmetric solutions away from the potential minimum, Phys. Rev. D 112, 024069 (2025), arXiv:2503.10998 [gr-qc]

work page arXiv 2025
[38]

S. Li, L. Liang, and L. Ma, Dyonic RN-like and Taub-NUT-like black holes in Einstein-bumblebee gravity, JCAP 03, 005, arXiv:2510.04405 [gr-qc]

work page arXiv
[39]

P. Ji, Z. Li, L. Yang, R. Xu, Z. Hu, and L. Shao, Neutron stars in the bumblebee theory of gravity, Phys. Rev. D 110, 104057 (2024), arXiv:2409.04805 [gr-qc]

work page arXiv 2024
[40]

Black Holes and Abelian Symmetry Breaking

J. Chagoya, G. Niz, and G. Tasinato, Black Holes and Abelian Symmetry Breaking, Class. Quant. Grav.33, 175007 (2016), arXiv:1602.08697 [hep-th]

work page Pith review arXiv 2016
[41]

S. W. Hawking, Particle Creation by Black Holes, Commun. Math. Phys. 43, 199 (1975), [Erratum: Commun.Math.Phys. 46, 206 (1976)]

1975
[42]

G. W. Gibbons and S. W. Hawking, Action Integrals and Partition Functions in Quantum Gravity, Phys. Rev. D 15, 2752 (1977)

1977
[43]

J. M. Bardeen, B. Carter, and S. W. Hawking, The Four laws of black hole mechanics, Commun. Math. Phys. 31, 161 (1973)

1973
[44]

S. W. Hawking, Black Holes and Thermodynamics, Phys. Rev. D 13, 191 (1976)

1976
[45]

An, Notes on thermodynamics of Schwarzschild-like bumblebee black hole, Phys

Y.-S. An, Notes on thermodynamics of Schwarzschild-like bumblebee black hole, Phys. Dark Univ. 45, 101520 (2024), arXiv:2401.15430 [gr-qc]

work page arXiv 2024
[46]

R. M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48, R3427 (1993), arXiv:gr-qc/9307038

work page Pith review arXiv 1993
[47]

Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy

V. Iyer and R. M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50, 846 (1994), arXiv:gr-qc/9403028

work page Pith review arXiv 1994
[48]

Iyer and R

V. Iyer and R. M. Wald, A Comparison of Noether charge and Euclidean methods for computing the entropy of stationary black holes, Phys. Rev. D 52, 4430 (1995), arXiv:gr-qc/9503052

work page arXiv 1995
[49]

D. A. Gomes, R. V. Maluf, and C. A. S. Almeida, Thermodynamics of Schwarzschild-like black holes in modified gravity 24 models, Annals Phys. 418, 168198 (2020), arXiv:1811.08503 [gr-qc]

work page arXiv 2020
[50]

M. S. Morris and K. S. Thorne, Wormholes in space-time and their use for interstellar travel: A tool for teaching general relativity, Am. J. Phys. 56, 395 (1988)

1988
[51]

M. S. Morris, K. S. Thorne, and U. Yurtsever, Wormholes, Time Machines, and the Weak Energy Condition, Phys. Rev. Lett. 61, 1446 (1988)

1988
[52]

J. Zhu, H. Li, and Z. Xiao, Hamiltonian Constraints on Spontaneous Lorentz Symmetry Breaking in the Bumblebee Model, (2026), arXiv:2604.06271 [hep-ph]

work page internal anchor Pith review Pith/arXiv arXiv 2026