pith. machine review for the scientific record. sign in

arxiv: 2602.00422 · v2 · submitted 2026-01-31 · 🌀 gr-qc

Recognition: 2 theorem links

· Lean Theorem

Jacobson's thermodynamic approach to classical gravity applied to non-Riemannian geometries: remarks on the simplicity of Nature

Jhan N. Martinez (1) , Jose F. Rodriguez-Ruiz (2) , Yeinzon Rodriguez (1 , 2) ((1) Universidad Industrial de Santander , (2) Universidad Antonio Narino)
Authors on Pith no claims yet

Pith reviewed 2026-05-16 09:30 UTC · model grok-4.3

classification 🌀 gr-qc
keywords Jacobson thermodynamic gravitynon-Riemannian geometriestorsion vectorEinstein-Hilbert actionLanczos-Lovelock theoriesnon-metricityenergy-momentum tensor
0
0 comments X

The pith

Jacobson's thermodynamic approach selects Einstein-Hilbert action plus quadratic torsion term for non-Riemannian geometries without non-metricity.

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

The paper extends Jacobson's result that gravitational equations follow from applying the first law of thermodynamics to local Rindler horizons. In Riemannian geometry this recovers the Einstein-Hilbert equations, but the same procedure fails to yield consistent equations in geometries that include torsion. Imposing the structural assumptions that define Lanczos-Lovelock theories then isolates one modification: the Einstein-Hilbert action supplemented by a quadratic term in the torsion vector. This modified theory satisfies both the thermodynamic requirement and the Lanczos-Lovelock hypotheses precisely when non-metricity is absent and the energy-momentum tensor is identified with its metric version; other choices produce mutual inconsistency between the two approaches.

Core claim

Jacobson's thermodynamic approach to classical gravity, when applied to non-Riemannian geometries, shows that the Einstein-Hilbert action does not belong to the pool of gravitational theories available for Nature's selection except in the Riemannian case. Considering the hypotheses from Lanczos-Lovelock theories of gravity, the two approaches together point to the theory derived from the Einstein-Hilbert action plus a term quadratic in the torsion vector as the one selected by Nature in the non-Riemannian case without non-metricity, when the energy-momentum tensor is identified as its metric version. The same strategy cannot be followed in the full non-Riemannian case or when using the canon

What carries the argument

Jacobson's thermodynamic relation applied to local Rindler horizons in non-Riemannian spacetimes, required to be consistent with Lanczos-Lovelock gravity hypotheses.

If this is right

  • The pure Einstein-Hilbert action is excluded as a candidate theory once torsion is present.
  • A quadratic term in the torsion vector must be added to restore consistency with both thermodynamic and Lanczos-Lovelock requirements.
  • The selection works only when non-metricity vanishes and the energy-momentum tensor is taken in its metric form.
  • The thermodynamic and Lanczos-Lovelock approaches become mutually inconsistent once non-metricity is allowed or the canonical energy-momentum tensor is used.
  • Nature would therefore select a unique modified theory rather than the standard Einstein-Hilbert action in the presence of torsion.

Where Pith is reading between the lines

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

  • If the selection rule is robust, torsion may be required to appear explicitly in any thermodynamically consistent gravitational action beyond Riemannian geometry.
  • The identified inconsistency when non-metricity is present suggests that a different thermodynamic construction would be needed to accommodate metricity violation.
  • The result could be tested by comparing predictions of the modified action against observations in cosmological models that allow torsion.
  • Similar thermodynamic derivations might be applied to other horizon types to see whether the same quadratic torsion term continues to be selected.

Load-bearing premise

The structural hypotheses used to formulate Lanczos-Lovelock theories can be applied directly to non-Riemannian geometries while preserving thermodynamic consistency only for specific energy-momentum identifications and in the absence of non-metricity.

What would settle it

An explicit derivation of the field equations from the thermodynamic first law on a horizon in a spacetime with torsion that yields equations different from those obtained from the Einstein-Hilbert action plus quadratic torsion term.

read the original abstract

Three decades ago, Ted Jacobson surprised us with a very appealing approach to classical gravity. According to him, the gravitational field equations are the consequence of the first law of thermodynamics applied to a Rindler observer. Jacobson's approach being formulated for Riemannian geometries, we have wondered what its consequences would be for non-Riemannian geometries. The results of our quest have been particularly appealing: we have found that the theory that derives from the Einstein-Hilbert action, arguably ``the simplest one'', does not belong to the pool of gravitational theories available for Nature's selection (except in the Riemannian case). In the search of a unique alternative, we have considered the hypotheses employed in the formulation of the Lanczos-Lovelock theories of gravity. Together, the two approaches point towards the theory that derives from the Einstein-Hilbert action plus a term quadratic in the torsion vector as the one that would be selected by Nature in the non-Riemannian case without non metricity (when the energy-momentum tensor is identified as its metric version). The same strategy cannot be followed in the full non-Riemannian case (and in the previous case when the energy-momentum tensor is identified as its canonical version) as the two approaches are mutually inconsistent.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript extends Jacobson's thermodynamic derivation of gravitational field equations from Rindler horizons to non-Riemannian geometries with torsion but no non-metricity. It concludes that the Einstein-Hilbert action is excluded from the set of theories selectable by Nature in this setting, but that supplementing it with a quadratic term in the torsion vector yields the unique theory consistent with both the thermodynamic first law and the Lanczos-Lovelock hypotheses, provided the energy-momentum tensor is identified with its metric version. The two approaches become mutually inconsistent when non-metricity is present or when the canonical energy-momentum tensor is used instead.

Significance. If the central selection claim is substantiated, the work would supply a thermodynamic criterion for choosing among torsion-inclusive gravitational actions, reinforcing the idea that Nature selects the simplest consistent extension beyond Riemannian geometry. The manuscript contains no machine-checked proofs or reproducible code, but it attempts a parameter-free derivation based on consistency requirements.

major comments (2)
  1. [Lanczos-Lovelock-type construction] The uniqueness of the Einstein-Hilbert plus quadratic-torsion term rests on extending the Lanczos-Lovelock axioms (diffeomorphism invariance, second-order equations of motion) unchanged to a connection with torsion. The manuscript does not demonstrate that the resulting Lovelock-like densities remain the only candidates once the torsion vector enters the curvature scalars and the variational principle is performed with respect to both metric and connection (see the load-bearing step identified in the skeptic note).
  2. [Energy-momentum tensor identification] Consistency between the thermodynamic and Lanczos-Lovelock approaches is asserted only for the metric version of the energy-momentum tensor. The paper must supply the explicit derivation showing mutual inconsistency for the canonical identification, including the relevant first-law application and variation, to support the abstract claim.
minor comments (2)
  1. [Abstract] The phrasing 'the results of our quest have been particularly appealing' in the abstract is informal; replace with a neutral statement of the findings.
  2. Clarify the precise definition of the quadratic torsion term (e.g., its coefficient and contraction) and ensure all equations are numbered for reference.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will revise the manuscript to provide the requested clarifications and explicit derivations.

read point-by-point responses

  1. Referee: The uniqueness of the Einstein-Hilbert plus quadratic-torsion term rests on extending the Lanczos-Lovelock axioms (diffeomorphism invariance, second-order equations of motion) unchanged to a connection with torsion. The manuscript does not demonstrate that the resulting Lovelock-like densities remain the only candidates once the torsion vector enters the curvature scalars and the variational principle is performed with respect to both metric and connection (see the load-bearing step identified in the skeptic note).

    Authors: We agree that the uniqueness argument requires a more explicit demonstration. In the revised version we will add a dedicated subsection that starts from the general curvature scalar built from the torsionful connection, imposes diffeomorphism invariance and second-order equations of motion, and shows by direct variation with respect to both metric and connection that the only admissible term quadratic in the torsion vector is the one we identified. This will make the load-bearing step fully transparent and substantiate the selection claim under the stated hypotheses. revision: yes

  2. Referee: Consistency between the thermodynamic and Lanczos-Lovelock approaches is asserted only for the metric version of the energy-momentum tensor. The paper must supply the explicit derivation showing mutual inconsistency for the canonical identification, including the relevant first-law application and variation, to support the abstract claim.

    Authors: We accept that the explicit calculation for the canonical energy-momentum tensor was only outlined. We will insert a new subsection that (i) applies the first law to the Rindler horizon using the canonical EMT, (ii) performs the metric and connection variations of the Einstein-Hilbert plus quadratic-torsion action, and (iii) demonstrates the resulting mismatch with the Lanczos-Lovelock conditions. The same explicit steps will be given both for vanishing non-metricity and for the general non-Riemannian case, thereby supporting the abstract statement. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper applies Jacobson's thermodynamic derivation to non-Riemannian geometries and then invokes the standard Lanczos-Lovelock hypotheses (diffeomorphism invariance, second-order equations) to identify a consistent extension containing a quadratic torsion term. This consistency is explicitly conditioned on the absence of non-metricity and a metric identification of the energy-momentum tensor; the resulting selection is presented as a conditional outcome rather than a tautology. No quoted step reduces by construction to the inputs, no parameter is fitted and relabeled as a prediction, and the LL axioms are treated as external rather than self-defined or smuggled via author self-citation. The central claim therefore retains independent content and remains open to external verification.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on extending Jacobson's thermodynamic law to non-Riemannian geometries and on adopting Lanczos-Lovelock hypotheses without new free parameters or invented entities.

axioms (2)
  • domain assumption The first law of thermodynamics applies to Rindler observers in non-Riemannian geometries.
    Core extension of Jacobson's original approach invoked throughout the abstract.
  • domain assumption Hypotheses employed in Lanczos-Lovelock theories remain valid for non-Riemannian geometries.
    Used to identify the unique alternative theory in the non-metricity-free case.

pith-pipeline@v0.9.0 · 5560 in / 1492 out tokens · 58401 ms · 2026-05-16T09:30:34.512720+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

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

80 extracted references · 80 canonical work pages · 37 internal anchors

  1. [1]

    A Model for Topological Fermions

    M. Faber, A Model for topological fermions, Few Body Syst.30, 149 (2001), arXiv:hep-th/9910221

    work page internal anchor Pith review Pith/arXiv arXiv 2001

  2. [2]

    Meinert and R

    J. Meinert and R. Hofmann, Electroweak Parameters from Mixed SU(2) Yang–Mills Thermodynamics, Symmetry16, 1587 (2024), arXiv:2401.03243 [hep-th]

    work page arXiv 2024

  3. [3]

    Saikumar, Exploring the Frontiers: Challenges and Theories Beyond the Standard Model, (2024), arXiv:2404.03666 [hep-ph]

    D. Saikumar, Exploring the Frontiers: Challenges and Theories Beyond the Standard Model, (2024), arXiv:2404.03666 [hep-ph]

    work page arXiv 2024

  4. [4]

    Meinert and R

    J. Meinert and R. Hofmann, Axial Anomaly in Galaxies and the Dark Universe, Universe7, 198 (2021), arXiv:2106.01457 [hep-ph]

    work page arXiv 2021

  5. [5]

    S. W. Hawking and G. F. R. Ellis,The Large Scale Structure of Space-Time, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 1973)

    work page 1973

  6. [6]

    C. P. Burgess,Introduction to Effective Field Theory (Cambridge University Press, 2020)

    work page 2020

  7. [7]

    C. P. Burgess, Quantum gravity in everyday life: General relativity as an effective field theory, Living Rev. Rel.7, 5 (2004), arXiv:gr-qc/0311082

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  8. [8]

    J. M. Lee,Introduction to Smooth Manifolds, Graduate Texts in Mathematics (Springer, 2013)

    work page 2013

  9. [9]

    Jacobson and R

    T. Jacobson and R. Parentani, Horizon entropy, Found. Phys.33, 323 (2003), arXiv:gr-qc/0302099

    work page arXiv 2003

  10. [10]

    Thermodynamics of Spacetime: The Einstein Equation of State

    T. Jacobson, Thermodynamics of space-time: The Einstein equation of state, Phys. Rev. Lett.75, 1260 (1995), arXiv:gr-qc/9504004

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  11. [11]

    J. D. Bekenstein, Black holes and entropy, Phys. Rev. D 7, 2333 (1973)

    work page 1973

  12. [12]

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

    work page 1973

  13. [13]

    S. W. Hawking, Black hole explosions?, Nature248, 30 (1974)

    work page 1974

  14. [14]

    Hawking, Particle creation by black holes, Commun

    S. Hawking, Particle creation by black holes, Commun. Math. Phys.43, 199 (1975)

    work page 1975

  15. [15]

    W. G. Unruh, Notes on black-hole evaporation, Phys. Rev. D14, 870 (1976)

    work page 1976

  16. [16]

    Non-equilibrium Thermodynamics of Spacetime

    C. Eling, R. Guedens, and T. Jacobson, Non-equilibrium thermodynamics of spacetime, Phys. Rev. Lett.96, 121301 (2006), arXiv:gr-qc/0602001

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  17. [17]

    J. M. Lee,Introduction to Riemannian manifolds, Graduate Texts in Mathematics (Springer, 2018)

    work page 2018

  18. [18]

    C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation(W. H. Freeman and Company, 1973)

    work page 1973

  19. [19]

    R. Dey, S. Liberati, and D. Pranzetti, Spacetime thermodynamics in the presence of torsion, Phys. Rev. D96, 124032 (2017), arXiv:1709.04031 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  20. [20]

    Spacetime Thermodynamics with Contorsion

    T. De Lorenzo, E. De Paoli, and S. Speziale, Spacetime Thermodynamics with Contorsion, Phys. Rev. D98, 064053 (2018), arXiv:1807.02041 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  21. [21]

    Cartan, Sur les vari´ et´ es ` a connexion affine et la th´ eorie de la relativit´ e g´ en´ eralis´ ee (premi` ere partie), Ann

    E. Cartan, Sur les vari´ et´ es ` a connexion affine et la th´ eorie de la relativit´ e g´ en´ eralis´ ee (premi` ere partie), Ann. Sci. de l’´Ecole Norm.3e s´ erie, 40, 325 (1923)

    work page 1923

  22. [22]

    Blagojevi´ c,Gravitation and Gauge Symmetries, Series in High Energy Physics, Cosmology and Gravitation (CRC Press, 2001)

    M. Blagojevi´ c,Gravitation and Gauge Symmetries, Series in High Energy Physics, Cosmology and Gravitation (CRC Press, 2001)

    work page 2001

  23. [23]

    Blagojevi´ c and F

    M. Blagojevi´ c and F. W. Hehl, eds.,Gauge Theories of Gravitation: A Reader with Commentaries(World Scientific, 2013)

    work page 2013

  24. [24]

    F. W. Hehl, P. Von Der Heyde, G. D. Kerlick, and J. M. Nester, General Relativity with Spin and Torsion: 15 Foundations and Prospects, Rev. Mod. Phys.48, 393 (1976)

    work page 1976

  25. [25]

    F. W. Hehl, Four lectures on Poincar´ e gauge field theory, Nato Science Series B58, 5 (1979)

    work page 1979

  26. [26]

    General relativity as the equation of state of spin foam

    L. Smolin, General relativity as the equation of state of spin foam, Class. Quant. Grav.31, 195007 (2014), arXiv:1205.5529 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  27. [27]

    Spacetime thermodynamics without hidden degrees of freedom

    G. Chirco, H. M. Haggard, A. Riello, and C. Rovelli, Spacetime thermodynamics without hidden degrees of freedom, Phys. Rev. D90, 044044 (2014), arXiv:1401.5262 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  28. [28]

    Cosmology from Group Field Theory Formalism for Quantum Gravity

    S. Gielen, D. Oriti, and L. Sindoni, Cosmology from Group Field Theory Formalism for Quantum Gravity, Phys. Rev. Lett.111, 031301 (2013), arXiv:1303.3576 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  29. [29]

    Generalized quantum gravity condensates for homogeneous geometries and cosmology

    D. Oriti, D. Pranzetti, J. P. Ryan, and L. Sindoni, Generalized quantum gravity condensates for homogeneous geometries and cosmology, Class. Quant. Grav.32, 235016 (2015), arXiv:1501.00936 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  30. [30]

    Horizon entropy from quantum gravity condensates

    D. Oriti, D. Pranzetti, and L. Sindoni, Horizon entropy from quantum gravity condensates, Phys. Rev. Lett.116, 211301 (2016), arXiv:1510.06991 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  31. [31]

    Emergent Friedmann dynamics with a quantum bounce from quantum gravity condensates

    D. Oriti, L. Sindoni, and E. Wilson-Ewing, Emergent Friedmann dynamics with a quantum bounce from quantum gravity condensates, Class. Quant. Grav.33, 224001 (2016), arXiv:1602.05881 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  32. [32]

    Time evolution as refining, coarse graining and entangling

    B. Dittrich and S. Steinhaus, Time evolution as refining, coarse graining and entangling, New J. Phys.16, 123041 (2014), arXiv:1311.7565 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  33. [33]

    Decorated tensor network renormalization for lattice gauge theories and spin foam models

    B. Dittrich, S. Mizera, and S. Steinhaus, Decorated tensor network renormalization for lattice gauge theories and spin foam models, New J. Phys.18, 053009 (2016), arXiv:1409.2407 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  34. [34]

    Coarse graining flow of spin foam intertwiners

    B. Dittrich, E. Schnetter, C. J. Seth, and S. Steinhaus, Coarse graining flow of spin foam intertwiners, Phys. Rev. D94, 124050 (2016), arXiv:1609.02429 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  35. [35]

    R. Dey, S. Liberati, and A. Mohd, Higher derivative gravity: field equation as the equation of state, Phys. Rev. D94, 044013 (2016), arXiv:1605.04789 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  36. [36]

    Horizon entropy and higher curvature equations of state

    R. Guedens, T. Jacobson, and S. Sarkar, Horizon entropy and higher curvature equations of state, Phys. Rev. D85, 064017 (2012), arXiv:1112.6215 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  37. [37]

    F. W. Hehl, J. D. McCrea, E. W. Mielke, and Y. Ne’eman, Metric affine gauge theory of gravity: Field equations, Noether identities, world spinors, and breaking of dilation invariance, Phys. Rept.258, 1 (1995), arXiv:gr-qc/9402012

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  38. [38]

    Jim´ enez-Cano, Metric-Affine Gauge theories of gravity: Foundations and new insights, Granada U., Theor

    A. Jim´ enez-Cano, Metric-Affine Gauge theories of gravity: Foundations and new insights, Granada U., Theor. Phys. Astrophys. (2022), arXiv:2201.12847 [gr- qc]

    work page arXiv 2022

  39. [39]

    Andrei, D

    I. Andrei, D. Iosifidis, L. J¨ arv, and M. Saal, Friedmann cosmology with hyperfluids, Phys. Rev. D111, 064063 (2025), arXiv:2411.19127 [gr-qc]

    work page arXiv 2025

  40. [40]

    Falk, Theory of elasticity of coherent inclusions by means of non-metric geometry, J

    F. Falk, Theory of elasticity of coherent inclusions by means of non-metric geometry, J. Elast.11, 359 (1981)

    work page 1981

  41. [41]

    The emergence of torsion in the continuum limit of distributed edge-dislocations

    R. Kupferman and C. Maor, The emergence of torsion in the continuum limit of distributed edge-dislocations, J. Geom. Mech.7, 361 (2015), arXiv:1410.2906 [math.DG]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  42. [42]

    Non-metricity in the continuum limit of randomly-distributed point defects

    R. Kupferman, C. Maor, and R. Rosenthal, Non- metricity in the continuum limit of randomly- distributed point defects, Isr. J. Math.223, 75 (2017), arXiv:1508.02003 [math.PR]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  43. [43]

    Lanczos, Elektromagnetismus als nat¨ urliche Eigenschaft der Riemannschen Geometrie, Z

    C. Lanczos, Elektromagnetismus als nat¨ urliche Eigenschaft der Riemannschen Geometrie, Z. Phys. 73, 147 (1932)

    work page 1932

  44. [44]

    Lanczos, A Remarkable property of the Riemann- Christoffel tensor in four dimensions, Annals Math.39, 842 (1938)

    C. Lanczos, A Remarkable property of the Riemann- Christoffel tensor in four dimensions, Annals Math.39, 842 (1938)

    work page 1938

  45. [45]

    Lovelock, The uniqueness of the Einstein field equations in a four-dimensional space, Arch

    D. Lovelock, The uniqueness of the Einstein field equations in a four-dimensional space, Arch. Ration. Mech. Anal.33, 54 (1969)

    work page 1969

  46. [46]

    Lovelock, Divergence-free tensorial concomitants, Aequat

    D. Lovelock, Divergence-free tensorial concomitants, Aequat. Math.4, 127 (1970)

    work page 1970

  47. [47]

    Lovelock, The Einstein tensor and its generalizations, J

    D. Lovelock, The Einstein tensor and its generalizations, J. Math. Phys.12, 498 (1971)

    work page 1971

  48. [48]

    Lovelock, The four-dimensionality of space and the Einstein tensor, J

    D. Lovelock, The four-dimensionality of space and the Einstein tensor, J. Math. Phys.13, 874 (1972)

    work page 1972

  49. [49]

    Limits on the number of spacetime dimensions from GW170817

    K. Pardo, M. Fishbach, D. E. Holz, and D. N. Spergel, Limits on the number of spacetime dimensions from GW170817, JCAP07(2018), 048, arXiv:1801.08160 [gr- qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  50. [50]

    Schouten,Ricci-Calculus: An Introduction to Tensor Analysis and Its Geometrical Applications, Grundlehren der mathematischen Wissenschaften (Springer, 1954)

    J. Schouten,Ricci-Calculus: An Introduction to Tensor Analysis and Its Geometrical Applications, Grundlehren der mathematischen Wissenschaften (Springer, 1954)

    work page 1954

  51. [51]

    The Raychaudhuri equation in spacetimes with torsion and non-metricity

    D. Iosifidis, C. G. Tsagas, and A. C. Petkou, Raychaudhuri equation in spacetimes with torsion and nonmetricity, Phys. Rev. D98, 104037 (2018), arXiv:1809.04992 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  52. [52]

    Dimensional Reduction in Quantum Gravity

    G. ’t Hooft, Dimensional reduction in quantum gravity, Conf. Proc. C930308, 284 (1993), arXiv:gr-qc/9310026

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  53. [53]

    The World as a Hologram

    L. Susskind, The World as a hologram, J. Math. Phys. 36, 6377 (1995), arXiv:hep-th/9409089

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  54. [54]

    Bombelli, R

    L. Bombelli, R. K. Koul, J. Lee, and R. D. Sorkin, Quantum source of entropy for black holes, Phys. Rev. D 34, 373 (1986)

    work page 1986

  55. [55]

    D. N. Blaschke, F. Gieres, M. Reboud, and M. Schweda, The energy–momentum tensor(s) in classical gauge theories, Nucl. Phys. B912, 192 (2016), arXiv:1605.01121 [hep-th]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  56. [56]

    M. R. Baker, N. Kiriushcheva, and S. Kuzmin, Noether and Hilbert (metric) energy-momentum tensors are not, in general, equivalent, Nucl. Phys. B962, 115240 (2021), arXiv:2011.10611 [math-ph]

    work page arXiv 2021

  57. [57]

    Mardones and J

    A. Mardones and J. Zanelli, Lovelock-Cartan theory of gravity, Class. Quant. Grav.8, 1545 (1991)

    work page 1991

  58. [58]

    Iosifidis and F

    D. Iosifidis and F. W. Hehl, Motion of test particles in spacetimes with torsion and nonmetricity, Phys. Lett. B 850, 138498 (2024), arXiv:2310.15595 [gr-qc]

    work page arXiv 2024

  59. [59]

    Iosifidis, The full quadratic metric-affine gravity (including parity odd terms): exact solutions for the affine-connection, Class

    D. Iosifidis, The full quadratic metric-affine gravity (including parity odd terms): exact solutions for the affine-connection, Class. Quant. Grav.39, 095002 (2022), arXiv:2112.09154 [gr-qc]

    work page arXiv 2022

  60. [60]

    Iosifidis, The Perfect Hyperfluid of Metric-Affine Gravity: The Foundation, JCAP04(2021), 072, arXiv:2101.07289 [gr-qc]

    D. Iosifidis, The Perfect Hyperfluid of Metric-Affine Gravity: The Foundation, JCAP04(2021), 072, arXiv:2101.07289 [gr-qc]

    work page arXiv 2021

  61. [61]

    Iosifidis, Cosmological Hyperfluids, Torsion and Non-metricity, Eur

    D. Iosifidis, Cosmological Hyperfluids, Torsion and Non-metricity, Eur. Phys. J. C80, 1042 (2020), arXiv:2003.07384 [gr-qc]

    work page arXiv 2020

  62. [62]

    Beltr´ an Jim´ enez and F

    J. Beltr´ an Jim´ enez and F. J. Maldonado Torralba, Revisiting the stability of quadratic Poincar´ e gauge gravity, Eur. Phys. J. C80, 611 (2020), arXiv:1910.07506 [gr-qc]

    work page arXiv 2020

  63. [63]

    Iosifidis and T

    D. Iosifidis and T. S. Koivisto, Hyperhydrodynamics: relativistic viscous fluids from hypermomentum, JCAP 05(2024), 001, arXiv:2312.06780 [gr-qc]

    work page arXiv 2024

  64. [64]

    A non-trivial connection for the metric-affine Gauss-Bonnet theory in $D = 4$

    B. Janssen, A. Jim´ enez-Cano, and J. A. Orejuela, A 16 non-trivial connection for the metric-affine Gauss-Bonnet theory inD= 4, Phys. Lett. B795, 42 (2019), arXiv:1903.00280 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  65. [65]

    Janssen and A

    B. Janssen and A. Jim´ enez-Cano, On the topological character of metric-affine Lovelock Lagrangians in critical dimensions, Phys. Lett. B798, 134996 (2019), arXiv:1907.12100 [gr-qc]

    work page arXiv 2019

  66. [66]

    Strong constraints on cosmological gravity from GW170817 and GRB 170817A

    T. Bakeret al., Strong constraints on cosmological gravity from GW170817 and GRB 170817A, Phys. Rev. Lett.119, 251301 (2017), arXiv:1710.06394 [astro- ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  67. [67]

    Dark Energy after GW170817 and GRB170817A

    P. Creminelli and F. Vernizzi, Dark Energy after GW170817 and GRB170817A, Phys. Rev. Lett.119, 251302 (2017), arXiv:1710.05877 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  68. [68]

    Implications of the Neutron Star Merger GW170817 for Cosmological Scalar-Tensor Theories

    J. Sakstein and B. Jain, Implications of the Neutron Star Merger GW170817 for Cosmological Scalar- Tensor Theories, Phys. Rev. Lett.119, 251303 (2017), arXiv:1710.05893 [astro-ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  69. [69]

    J. M. Ezquiaga and M. Zumalac´ arregui, Dark Energy After GW170817: Dead Ends and the Road Ahead, Phys. Rev. Lett.119, 251304 (2017), arXiv:1710.05901 [astro- ph.CO]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  70. [70]

    Polarization-based Tests of Gravity with the Stochastic Gravitational-Wave Background

    T. Callisteret al., Polarization-based Tests of Gravity with the Stochastic Gravitational-Wave Background, Phys. Rev. X7, 041058 (2017), arXiv:1704.08373 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  71. [71]

    B. P. Abbottet al.(LIGO Scientific, Virgo), Tests of General Relativity with GW170817, Phys. Rev. Lett. 123, 011102 (2019), arXiv:1811.00364 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  72. [72]

    Probing gravitational wave polarizations with Advanced LIGO, Advanced Virgo and KAGRA

    Y. Hagihara, N. Era, D. Iikawa, and H. Asada, Probing gravitational wave polarizations with Advanced LIGO, Advanced Virgo and KAGRA, Phys. Rev. D98, 064035 (2018), arXiv:1807.07234 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  73. [73]

    Abbottet al.(KAGRA, Virgo, LIGO Scientific), Upper limits on the isotropic gravitational-wave background from Advanced LIGO and Advanced Virgo’s third observing run, Phys

    R. Abbottet al.(KAGRA, Virgo, LIGO Scientific), Upper limits on the isotropic gravitational-wave background from Advanced LIGO and Advanced Virgo’s third observing run, Phys. Rev. D104, 022004 (2021), arXiv:2101.12130 [gr-qc]

    work page arXiv 2021

  74. [74]

    Binary Pulsar Constraints on the Parameterized post-Einsteinian Framework

    N. Yunes and S. A. Hughes, Binary Pulsar Constraints on the Parameterized post-Einsteinian Framework, Phys. Rev. D82, 082002 (2010), arXiv:1007.1995 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  75. [75]

    W. W. Zhuet al., Tests of Gravitational Symmetries with Pulsar Binary J1713+0747, Mon. Not. Roy. Astron. Soc. 482, 3249 (2019), arXiv:1802.09206 [astro-ph.HE]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  76. [76]

    H. B. Callen,Thermodynamics and an Introduction to Thermostatistics(John Wiley & Sons, 1985)

    work page 1985

  77. [77]

    R. M. Wald,General Relativity(The University of Chicago Press, 1984)

    work page 1984

  78. [78]

    Boundary Term in Metric f(R) Gravity: Field Equations in the Metric Formalism

    A. Guarnizo, L. Casta˜ neda, and J. M. Tejeiro, Boundary Term in Metricf(R) Gravity: Field Equations in the Metric Formalism, Gen. Rel. Grav.42, 2713 (2010), arXiv:1002.0617 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  79. [79]

    Einstein-Cartan Cosmologies

    S. Bravo Medina, M. Nowakowski, and D. Batic, Einstein-Cartan Cosmologies, Annals Phys.400, 64 (2019), arXiv:1812.04589 [gr-qc]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  80. [80]

    Ort´ ın,Gravity and Strings, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2004)

    T. Ort´ ın,Gravity and Strings, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2004)

    work page 2004