Pith. sign in

REVIEW 4 major objections 3 minor 47 references

G\"{o}del-type universes in unimodular gravity

T0 review · 4 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Gödel-type universes survive unimodular gravity, but only in the linear class of metrics.

desk verdict A genuinely new question, but the central derivation is algebraically inconsistent—the claimed UG Gödel-type solutions don't satisfy the paper's own equations. read the letter →

arxiv 2506.11717 v1 pith:5NAKKFOT submitted 2025-06-13 gr-qc

R. Raimarda , A. F. Santos , R. Bufalo This is my paper · ORCID
classification gr-qc MSC 83C0583C15 PACS 04.20.-q04.20.Jb
keywords GödeluniverseunimodulargravityclosedtimelikecurvescausalityGödel-typemetricscosmologicalconstantrotatingdustperfectfluid
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 asks whether the rotating, causality-violating Gödel universes can exist in unimodular gravity, where the cosmological constant is an integration constant rather than a coupling constant. It shows that the original Gödel metric, when rewritten in an unimodular gauge, fails to satisfy the unimodular field equations for rotating dust. By generalizing to Gödel-type metrics, the paper finds consistent unimodular solutions for rotating dust, a perfect fluid, and a perfect fluid plus a scalar field. In every case the field equations force the radial function $D(r)$ to the constant value $1$, i.e. the linear class $m=0$, and this restriction makes the causal structure coincide with that of general relativity: the two non-scalar sources remain non-causal, while the combined source is causal.

What carries the argument

The key object is the unimodular Gödel-type line element (55), obtained by imposing $\sqrt{-g}=1$ on the homogeneous Gödel-type metric with functions $H(r)$ and $D(r)$ obeying $H'/D=2\omega$ and $D''/D=m^2$. The vierbein fields in (56)-(57) turn the unimodular field equations into a set of algebraic relations in which every component is multiplied by $D(r)$. Requiring the energy density, pressure, and cosmological constant to be constant and demanding equivalence of the causal structure with general relativity forces $D(r)=1$, which via $D''/D=m^2$ selects the linear class $m=0$ and yields $H(r)\sim\omega r$ and the critical radius $r_c=1/\omega$ for non-causal regions.

What would settle it

Search for any solution of the unimodular field equations with $D(r)$ not constant, $m\neq 0$, and constant energy density, pressure, and cosmological constant; the existence of even one such hyperbolic- or trigonometric-class solution would falsify the paper's conclusion that only the linear class $D(r)=1$ is consistent.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central result is that Gödel-type universes are admitted as exact solutions of unimodular gravity for three matter sources—rotating dust, a perfect fluid, and a perfect fluid combined with a scalar field—provided the unimodular metric is written in the form (55) and the function $D(r)$ is restricted to the linear class $D(r)=1$ ($m=0$). Under that restriction the solutions become equivalent to the general-relativistic Gödel-type solutions: they share the same cosmological-constant relations and the same causal structure, with non-causal regions for the rotating-dust and perfect-fluid cases and a causal universe for the perfect-fluid-plus-scalar-field case. The paper also establishes that the original Gödel metric, in its unimodularized form, is not a solution of unimodular gravity, since the field equations cannot be solved simultaneously for the vorticity parameter.

Load-bearing premise

The load-bearing premise is that the unimodular gauge metric (55) is the correct physical representation of the Gödel-type spacetime and that the resulting field equations are algebraically consistent for the sources considered.

Editorial extensions

If this is right

  • In unimodular gravity the original Gödel universe is absent; only the generalized Gödel-type metrics admit solutions, and only in the linear class $m=0$.
  • For rotating dust and perfect fluid sources, the unimodular Gödel-type solutions possess a non-causal region $r>r_c$ with $r_c=1/\omega$, exactly as in general relativity.
  • For a perfect fluid plus a scalar field source, the unimodular field equations give $m^2=4\omega^2$, producing an infinite critical radius and a fully causal universe.
  • The cosmological constant, which in unimodular gravity is an integration constant, satisfies relations that reduce to the general-relativity ones when $D(r)=1$.
  • The paper's summary table shows that both general relativity and unimodular gravity are non-causal for rotating dust and perfect fluid, and causal for the combined source.

Reading between the lines

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

  • If the $D(r)=1$ restriction is taken as a selection rule, unimodular gravity would exclude the hyperbolic and trigonometric classes of Gödel-type metrics entirely, a stronger constraint than general relativity imposes.
  • The inconsistency that arises for general $D(r)$ suggests a possible resolution in which the source fields are allowed to vary with $r$; the paper's constant-density assumption might be relaxed rather than the metric class.
  • The same unimodular-gauge construction could be applied to other rotating or stationary spacetimes to test whether the linear-class restriction is a general feature of unimodular gravity rather than an artifact of the Gödel-type ansatz.
  • A direct check of whether the field equations can be satisfied by any nonconstant $D(r)$ with constant matter sources would settle whether the paper's restriction is exhaustive.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 3 minor

Summary. The paper analyses Gödel-type universes in unimodular gravity (UG). It first shows that the original Gödel metric, cast in determinant-one form, does not satisfy the traceless UG field equations; it then considers the generalized metric (55) and claims to find UG solutions for rotating dust, perfect fluid, and perfect fluid plus scalar field, with the same causal structure as in GR once the function D(r) is restricted to the linear class D=1 (m=0). The central claim is that UG admits Gödel-type solutions with critical radius r_c=1/ω in that restricted class.

Significance. If the derivations were correct, the paper would contribute a useful comparison between GR and UG for rotating cosmologies, since UG promotes Λ to an integration constant and the causal structure of Gödel-type metrics depends on Λ. The GR review in Section IV is standard, and the identification of the obstruction for the original Gödel metric in Section IIIB is a clear result. However, the new UG solutions in Section IVA are internally inconsistent: the printed field equations contradict the claimed solutions, and the proposed D=1 restriction does not cure the contradictions. The central claims are therefore unsupported, and the paper in its present form cannot be accepted.

major comments (4)
  1. [IV A 1, Eqs. (58)-(60)] The claimed solution (62)-(63) does not satisfy the printed field equations. With m²=2ω², Eq. (58) gives 8ω²D=24πGρ, while Eq. (59) gives 4ω²D=8πGρ; these require 4ω²D=12πGρ and 4ω²D=8πGρ respectively, which are incompatible for any nonzero ρ. Using instead the relation m²D=4πGρ from Eq. (63) leaves Eq. (58) requiring ω²D=2.8πGρ while Eq. (59) requires ω²D=2πGρ, so no parameter choice resolves the contradiction.
  2. [IV A 1 and Section V] The proposed restriction D(r)=1, m=0 does not remove the inconsistency. For D=1 and m=0, Eqs. (58)-(60) reduce to 10ω²=24πGρ, 6ω²=8πGρ, and −2ω²=8πGρ; the last equation is impossible for positive ρ and nonzero ω. Thus the paper's central conclusion that the linear class m=0 gives consistent physical solutions is not supported by the manuscript's own equations.
  3. [IV A 2, Eqs. (64)-(66)] For the perfect fluid, the same structural inconsistency appears. Since the right-hand sides of Eqs. (64)-(66) are all proportional to p+ρ, Eq. (64) versus Eq. (65) forces m²=4ω², while Eq. (65) versus Eq. (66) forces m²=2ω²; no value of m² can satisfy all three for p+ρ≠0. The reported solution m²=2ω²=4πG(ρ+p)D^{-1} in Eq. (68) therefore does not solve the stated equations.
  4. [IV A 3, Eqs. (70)-(73)] Substituting the claimed m²=4ω² into Eqs. (70)-(72) gives 6ω²D=8πG[3(p+ρ)+ε²D], 2ω²D=8πG[(p+ρ)−ε²D], and 10ω²D=8πG[(p+ρ)+3ε²D]. Solving the first two yields ε²D=0 and 8πG(p+ρ)=2ω²D, which makes the third equation reduce to 2ω²D=10ω²D and hence ω=0. The stated solution (74)-(75) is therefore inconsistent. Moreover, the final restriction to D=1 and m=0 contradicts Eq. (74) itself, since m=0 would force ω=0 and would eliminate the claimed critical radius r_c=1/ω.
minor comments (3)
  1. [Section IV, Eqs. (26)-(29)] For the linear class m=0 with D=1, Eq. (26) yields H'=2ω, hence H(r)=2ωr up to a constant; the CTC boundary W(r)=D²−H²=1−4ω²r² then gives r_c=1/(2ω), not r_c=1/ω as stated in Eq. (29) and repeated in Sections IVA1 and V. The assertion H(r)∼ωr is only consistent with Eq. (26) if D=1/2, in which case W=0 still gives r_c=1/(2ω).
  2. [Throughout] There are numerous grammatical and typographical errors, including 'is must be emphasized', 'the more appropriated', 'hyperbbolic', and inconsistent hyphenation of 'Gödel-type'; these should be corrected in any revision.
  3. [Table I] Table I summarizes the claimed causal structure for GR and UG, but because the underlying UG solutions are inconsistent, the table should be revisited after the field equations are corrected.

Circularity Check

1 steps flagged · score 6.0 of 10

The claimed D(r)=1 restriction of the physical solution space is imposed as an input requirement and then reported as a derived result, making the central conclusion self-definitional; the rest of the calculation is self-contained but algebraically inconsistent.

  1. self definitional [Section IV A 1 (bullet on D(r)=1) and Section V (Conclusion)]
    "the requirements that the physical solutions were equivalent to GR and that the field equations were mathematically consistent imposed strong constraints upon the function D(r). Hence, we have shown that in order to satisfy all these conditions the choice D(r) = 1 is the more appropriated, showing that the set of physical solutions is restricted only to a special class of functions D(r), i.e. to the linear class m = 0 of solutions."

    The conclusion that the physical UG solutions are restricted to D(r)=1 (m=0) is presented as a demonstrated result ('we have shown'), but the paper's own preceding text states that this restriction is imposed by requiring UG to share GR's causal structure and to be consistent, and that these requirements 'can be satisfied for the simplest choice D(r)=1'. Thus the 'physical solution space' is effectively defined as the class satisfying the imposed equivalence-plus-consistency condition, and the claimed restriction to m=0 is just that input choice restated as an output.

full rationale

Most of the paper's formal apparatus is self-contained or rests on standard external references: the unimodular action and traceless field equations are standard, the Gödelype metric and Killing conditions are taken from Rebouças-Tiomno, and the GR comparisons are internal to the manuscript. There is no load-bearing self-citation: the authors' own prior works are cited only as related literature, not to justify the central derivation. However, the central advertised result -- that the unimodular theory admits Gödelype solutions only in the linear class D(r)=1 (m=0) -- is not derived from the unimodular field equations. It is first imposed as a requirement ('if one requires the equivalence between UG and GR solutions... all these conditions can be satisfied for the simplest choice D(r)=1') and then reported in the conclusion as a restriction that has been 'shown'. This is a self-definitional step: the output restriction is the input choice restated. Separately, the manuscript's field equations for rotating dust, perfect fluid, and scalar field sources are algebraically inconsistent under the claimed solutions, e.g. substituting m^2=2omega^2 into (58) and (59) gives incompatible relations 8omega^2D=24pi G rho and 4omega^2D=8pi G rho. That inconsistency is a soundness defect rather than circularity, but it reinforces that the D=1 restriction is not a solution of the equations as written. Overall, one genuinely circular step affecting the central claim warrants a score of 6.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the standard Gödel-type classification, the chosen unimodular gauge, and the algebraic consistency of the field equations. The only hand-input model parameter is the scalar-field amplitude epsilon. No new particles, forces, or dimensions are introduced.

free parameters (1)
  • epsilon (scalar field amplitude) = unspecified
    Introduced by hand as phi=epsilon*z+epsilon in the scalar-field source (47)-(49); it is an input parameter of the source, not fitted to data. The paper's equations would fix its value, but the equations are inconsistent.
assumptions (4)
  • domain assumption The homogeneous Gödel-type metric conditions H'/D=2ω and D''/D=m² from ref. [6] characterize all homogeneous Gödel-type spacetimes.
    Invoked in Eqs. (26)-(27) and used to parametrize H(r) and D(r) in the hyperbolic class (34)-(35). If this classification were incomplete, the solution space considered would be too narrow.
  • domain assumption The unimodular coordinate representation (55) with vierbein (56)-(57), obtained by imposing √-g=1, is physically representative of any Gödel-type spacetime in UG.
    The paper states there are other unimodular forms but asserts without proof that they lead to the same conclusions. The central result depends on this gauge choice.
  • domain assumption The matter sources (dust, perfect fluid, scalar field) have vanishing off-diagonal stress in the local Lorentz frame and are compatible with stationarity and homogeneity.
    The field equations (58)-(73) assume T_(0)(2)=0 and constant energy density/pressure for a homogeneous spacetime. The paper does not revisit this assumption when D(r) is nonconstant.
  • standard math The traceless unimodular field equations (5) and the integration-constant relation (7) are the correct classical equations of unimodular gravity.
    These are derived in Section II from the action (1). They are standard in the UG literature and are not the source of the inconsistency.

how reviews work

0 comments
Cite this review

Pith. "Pith review of G\"{o}del-type universes in unimodular gravity." pith.science (2026). https://pith.science/paper/5NAKKFOT

@misc{pith2026250611717,
  author       = {Pith},
  title        = {Pith review of: G\"odel-type universes in unimodular gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5NAKKFOT}},
  note         = {Machine review of arXiv:2506.11717}
}
read the original abstract

In this paper, the G\"{o}del-type universes are examined within the framework of unimodular gravity. Since the existence of G\"{o}del solutions is intrinsically related to the presence of a cosmological constant in general relativity, one can naturally wonder how the acausal structure of the G\"{o}del solutions behaves in different cosmological scenarios. One of the simplest, but important, frameworks within this context is the unimodular gravity, in which the cosmological constant emerges as an integration constant rather than a coupling constant. Hence, the validity of G\"{o}del-type solutions is scrutinized within the unimodular approach, examining whether this theory can address the known issue of causality violation in G\"{o}del universes. In detail, it is demonstrated that for certain gravitational sources, both causal and non-causal regions are permissible.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

47 extracted references · 42 canonical work pages

  1. [1]

    Hence, we recall that the case for the rotating-dust has an energy-momentum tensor TAB given by (36)

    Rotating-dust Since we will study the unimodular theory (55) in the vierbein basis, we have defined previously the sources. Hence, we recall that the case for the rotating-dust has an energy-momentum tensor TAB given by (36). Let us now examine the unimodular field equations. The components of the UG field equations 14 (5) for a G¨ odel-type metric (55) s...

    work page

  2. [2]

    Perfect fluid We now consider a perfect fluid as the gravitational source of the unimodular gravity in the G¨ odel-like metric, in which its energy-momentum tensor is described by (41) with trace given by T=ρ−3p. Thus, the field equations for the unimodular framework (5) and the condition (7), for the G¨ odel-type universe (55), yield the relations 10ω2 −...

    work page

  3. [3]

    Perfect fluid plus scalar field We now approach the unimodular theory sourced by a combination between perfect fluid and a scalar field given by (47) and (48). This is part of our endeavour to determine how different types of matter source can induce a causal G¨ odel solution, as well as affects the equivalence between the unimodular approach and GR. In t...

    work page 2023

  4. [4]

    The Cosmological Constant Problem,

    S. Weinberg, “The Cosmological Constant Problem,” Rev. Mod. Phys.61(1989), 1-23

    work page 1989

  5. [5]

    Cosmological constant: The Weight of the vacuum,

    T. Padmanabhan, “Cosmological constant: The Weight of the vacuum,” Phys. Rept.380(2003), 235-320

    work page 2003

  6. [6]

    TASI Lectures on the Cosmological Constant,

    R. Bousso, “TASI Lectures on the Cosmological Constant,” Gen. Rel. Grav.40(2008), 607-637

    work page 2008

  7. [7]

    Lectures on the Cosmological Constant Problem,

    A. Padilla, “Lectures on the Cosmological Constant Problem,” arXiv:1502.05296 [hep-th]

    arXiv

  8. [8]

    G¨ odel, “An Example of a New Type of Cosmological Solutions of Einstein’s Field Equations of Gravitation’,’ Rev

    K. G¨ odel, “An Example of a New Type of Cosmological Solutions of Einstein’s Field Equations of Gravitation’,’ Rev. Mod. Phys.21, 447 (1949)

    work page 1949

Show all 47 references
  1. [9]

    M. J. Rebou¸ cas and J. Tiomno, “Homogeneity of Riemannian space-times of G¨ odel type’,’ Phys. Rev. D28, 1251 (1983)

    1983

  2. [10]

    The Existence of godel, Einstein and de Sitter universes,

    T. Clifton and J. D. Barrow, “The Existence of godel, Einstein and de Sitter universes,” Phys. Rev. D 72(2005), 123003

    2005

  3. [11]

    G¨ odel and G¨ odel-type universes in Brans–Dicke theory,

    J. A. Agudelo, J. R. Nascimento, A. Y. Petrov, P. J. Porf ´ ırio and A. F. Santos, “G¨ odel and G¨ odel-type universes in Brans–Dicke theory,” Phys. Lett. B762(2016), 96-101 19

    2016

  4. [12]

    Chern-Simons modified gravity and closed timelike curves,

    P. J. Porfirio, J. B. Fonseca-Neto, J. R. Nascimento, A. Y. Petrov, J. Ricardo and A. F. Santos, “Chern-Simons modified gravity and closed timelike curves,” Phys. Rev. D94(2016) no.4, 044044

    2016

  5. [13]

    Causality aspects of the dynamical Chern-Simons modified gravity,

    P. J. Porfirio, J. B. Fonseca-Neto, J. R. Nascimento and A. Y. Petrov, “Causality aspects of the dynamical Chern-Simons modified gravity,” Phys. Rev. D94(2016) no.10, 104057

    2016

  6. [14]

    G¨ odel-type solutions within the f(R,Q) gravity,

    F. S. Gama, J. R. Nascimento, A. Y. Petrov, P. J. Porfirio and A. F. Santos, “G¨ odel-type solutions within the f(R,Q) gravity,” Phys. Rev. D96(2017) no.6, 064020

    2017

  7. [15]

    Causal G¨ odel-type metrics in non-local gravity theories,

    J. R. Nascimento, A. Y. Petrov and P. J. Porf ´ ırio, “Causal G¨ odel-type metrics in non-local gravity theories,” Eur. Phys. J. C81(2021) no.9, 815

    2021

  8. [16]

    First-order perturbations of G¨ odel-type metrics in non-dynamical Chern–Simons modified gravity,

    B. Altschul, J. R. Nascimento, A. Y. Petrov and P. J. Porf ´ ırio, “First-order perturbations of G¨ odel-type metrics in non-dynamical Chern–Simons modified gravity,” Class. Quant. Grav.39(2022) no.2, 025002

    2022

  9. [17]

    Closed Timelike Curves, Singularities and Causality: A Survey from G¨ odel to Chrono- logical Protection,

    J. P. Luminet, “Closed Timelike Curves, Singularities and Causality: A Survey from G¨ odel to Chrono- logical Protection,” Universe7(2021) no.1, 12

    2021

  10. [18]

    G¨ odel-type solutions within the f(R, Q, P) gravity,

    J. R. Nascimento, A. Y. Petrov, P. J. Porfirio and R. N. da Silva, “G¨ odel-type solutions within the f(R, Q, P) gravity,” arXiv:2302.09935 [gr-qc]

    arXiv

  11. [19]

    The Foundation of the General Theory of Relativity

    A. Einstein, “The Foundation of the General Theory of Relativity”, Annalen Phys.49, 769 (1916). Translated and included in The Principle of Relativity, by H.A. Lorentz et al. (Dover Press, New York, 1923)

    1916

  12. [20]

    Do gravitational fields play an essential part in the structure of the elementary particles of matter?

    A. Einstein, “Do gravitational fields play an essential part in the structure of the elementary particles of matter?”, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.)1919, 433 (1919). Translated and included in The Principle of Relativity, by H.A. Lorentz et al. (Dover P...

    1919

  13. [21]

    How unimodular gravity theories differ from general relativity at quantum level

    R. Bufalo, M. Oksanen and A. Tureanu, “How unimodular gravity theories differ from general relativity at quantum level”, Eur. Phys. J. C75, 477 (2015)

    2015

  14. [22]

    A note on classical and quantum unimodular gravity,

    A. Padilla and I. D. Saltas, “A note on classical and quantum unimodular gravity,” Eur. Phys. J. C 75, 561 (2015)

    2015

  15. [23]

    Unimodular Gravity vs General Relativity: A status report

    R. Carballo-Rubio, L. J. Garay and G. Garc ´ ıa-Moreno, “Unimodular Gravity vs General Relativity: A status report”, Class. Quantum Grav.39, 243001 (2022)

    2022

  16. [24]

    Unimodular Approaches to the Cosmological Constant Problem,

    P. Jirouˇ sek, “Unimodular Approaches to the Cosmological Constant Problem,” Universe9(2023) 131

    2023

  17. [25]

    A Primer on Unimodular Gravity,

    E. Alvarez and E. Velasco-Aja, “A Primer on Unimodular Gravity,” arXiv:2301.07641 [gr-qc]

    arXiv

  18. [26]

    A clarification on prevailing misconceptions in unimodular gravity,

    G. R. Bengochea, G. Leon, A. Perez and D. Sudarsky, “A clarification on prevailing misconceptions in unimodular gravity,” JCAP11(2023), 011 arXiv:2308.07360 [gr-qc]

    2023 arXiv

  19. [27]

    UV structure of quantum unimodular gravity,

    I. D. Saltas, “UV structure of quantum unimodular gravity,” Phys. Rev. D90(2014) no.12, 124052

    2014

  20. [28]

    Sequestering the Standard Model Vacuum Energy,

    N. Kaloper and A. Padilla, “Sequestering the Standard Model Vacuum Energy,” Phys. Rev. Lett.112 (2014) no.9, 091304

    2014

  21. [29]

    Manifestly Local Theory of Vacuum Energy Sequestering,

    N. Kaloper, A. Padilla, D. Stefanyszyn and G. Zahariade, “Manifestly Local Theory of Vacuum Energy Sequestering,” Phys. Rev. Lett.116(2016) no.5, 051302

    2016

  22. [30]

    Unimodular Gravity Redux,

    E. ´Alvarez, S. Gonz´ alez-Mart ´ ın, M. Herrero-Valea and C. P. Mart ´ ın, “Unimodular Gravity Redux,” Phys. Rev. D92(2015) no.6, 061502 20

    2015

  23. [31]

    Quantum Corrections to Uni- modular Gravity,

    E. ´Alvarez, S. Gonz´ alez-Mart ´ ın, M. Herrero-Valea and C. P. Mart ´ ın, “Quantum Corrections to Uni- modular Gravity,” JHEP08(2015), 078

    2015

  24. [32]

    Unimodular F(R) Gravity

    S. Nojiri, S. D. Odintsov and V. K. Oikonomou, “Unimodular F(R) Gravity”, JCAP05, 046 (2016)

    2016

  25. [33]

    Unimodular quantum gravity and the cosmological constant,

    R. Percacci, “Unimodular quantum gravity and the cosmological constant,” Found. Phys.48(2018) no.10, 1364-1379

    2018

  26. [34]

    Can quantum fluctuations differentiate between standard and unimodular gravity?,

    G. P. de Brito, O. Melichev, R. Percacci and A. D. Pereira, “Can quantum fluctuations differentiate between standard and unimodular gravity?,” JHEP12(2021), 090

    2021

  27. [35]

    Embedding Unimodular Gravity in String Theory

    L. J. Garay and G. Garc ´ ıa-Moreno, “Embedding Unimodular Gravity in String Theory”, JHEP03, 027 (2023)

    2023

  28. [36]

    Brans-Dicke unimodular gravity

    A. M. R. Almeida, J. C. Fabris, M. H. Daouda, R. Kerner, H. Velten and W. S. Hip´ olito-Ricaldi, “Brans-Dicke unimodular gravity”, Universe8, 429 (2022)

    2022

  29. [37]

    Extension of unimodular gravity and the cosmological constant,

    F. R. Klinkhamer, “Extension of unimodular gravity and the cosmological constant,” Phys. Rev. D 106(2022) no.12, 124015

    2022

  30. [38]

    The Cosmological Constant and General Covariance,

    M. Henneaux and C. Teitelboim, “The Cosmological Constant and General Covariance,” Phys. Lett. B222(1989), 195-199

    1989

  31. [39]

    The Quantization of unimodular gravity and the cosmological constant problems,

    L. Smolin, “The Quantization of unimodular gravity and the cosmological constant problems,” Phys. Rev. D80(2009), 084003 arXiv:0904.4841 [hep-th]

    2009 arXiv

  32. [40]

    Cosmological Perturbations in Unimodular Grav- ity

    C. Gao, R. H. Brandenberger, Y. Cai and P. Chen, “Cosmological Perturbations in Unimodular Grav- ity”, JCAP09, 021 (2014)

    2014

  33. [41]

    Using cosmological perturbation theory to distinguish be- tween General Relativity and Unimodular Gravity

    M. H. Alvarenga, J. C. Fabris and H. Velten, “Using cosmological perturbation theory to distinguish be- tween General Relativity and Unimodular Gravity”, Symmetry15(2023) no.7, 1392 arXiv:2301.12464 [gr-qc]

    2023 arXiv

  34. [42]

    Nonconservative unimodular gravity: a viable cosmological scenario?,

    J. C. Fabris, M. H. Alvarenga, M. Hamani-Daouda and H. Velten, “Nonconservative unimodular gravity: a viable cosmological scenario?,” Eur. Phys. J. C82(2022) no.6, 522 arXiv:2212.06644 [gr-qc]

    2022 arXiv

  35. [43]

    New Homogeneous Solutions of Einstein’s Field Equations with Incoherent Matter Ob- tained by a Spinor Technique

    I. Ozsvath, “New Homogeneous Solutions of Einstein’s Field Equations with Incoherent Matter Ob- tained by a Spinor Technique”, J. Math. Phys.6, 590 (1965)

    1965

  36. [44]

    Stationary distributions of dust and electromagnetic fields in general relativity

    A. Banerjee and S. Banerji, “Stationary distributions of dust and electromagnetic fields in general relativity”, J. Phys. A1, 188 (1968)

    1968

  37. [45]

    Cylindrically symmetric charged dust distributions in rigid rotation in general relativity

    M. M. Som and A. K. Raychaudhuri, “Cylindrically symmetric charged dust distributions in rigid rotation in general relativity”, Proc. R. Soc. London A304, 81 (1968)

    1968

  38. [46]

    Homogeneous space-times of the G¨ odel type

    A. K. Raychaudhuri and S. N. G. Thakurta, “Homogeneous space-times of the G¨ odel type”, Phys. Rev. D22, 802 (1980)

    1980

  39. [47]

    A note on G¨ odel’s metric

    F. Bampi and C. Zordan, “A note on G¨ odel’s metric”, Gen. Relativ. Gravit.9, 393 (1978)

    1978

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.