Holographic Dark Energy and the Universe Expansion Acceleration

J. G. Pereira; J. P. Beltran Almeida

REVIEW

Holographic Dark Energy and the Universe Expansion Acceleration

J. P. Beltran Almeida, J. G. Pereira

General Relativity and Quantum Cosmology · Submitted 24 Feb 2006 · 20 references

Pith's verdict: The advertised holographic bound in Eq. (25) is not derived by the energy conditions; the paper's real content is a consistent but parameter-input model that deserves a fixing revision rather than acceptance as-is.

T0 · AI review

How this review was made: An AI referee (deepseek-v4-flash) read the paper on August 28, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 3 major objections · 4 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv IDgr-qc/0602103 (version 1)

Pith IDALF7FKWB, Pith's permanent record of this paper and its review

CitationJ. G. Pereira, J. P. Beltran Almeida. "Holographic Dark Energy and the Universe Expansion Acceleration." arXiv:gr-qc/0602103 (2006). https://pith.science/paper/ALF7FKWB

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

Dear colleague,

The main advertised result doesn't hold up. The bound ω ≤ b/(1-b) in Eq. (25) is not derived from the energy conditions (23)-(24) the way the paper claims. The stress-test note is right: plug a unit timelike v^μ = u^μ into (23) and you get a condition involving 1+ω and the trace term, not the claimed upper bound; expanding (24) for a general timelike v^μ gives an absolute-value inequality such as |ω - b/(1-b)| ≤ 1/(1-b), not the one-sided bound. So the 'new physical bounds' in the abstract and conclusions are unsupported by the paper's own equations.

What survives is the model skeleton. Saturating the holographic bound with L = cH^{-1} closes the system (1)-(3), and the solution (6)-(8) with εm/εΛ = (1-b)/b is consistent. The acceleration condition (31) does follow from (12), and it allows an accelerated universe even for positive ω when b > 1/3. The geometric reading — no dark-energy equation of state — is a clean, standard presentation of a decaying-Λ model, and the explicit debt to Pavon and Zimdahl is acknowledged.

The soft spots are real. The coincidence is imposed: b is a free parameter tuned to make (8) match observations; the paper says so later but the abstract oversells it. No observational data are used. The entropy section is too sketchy to carry weight, and the authors themselves discount it. None of this kills the consistency demonstration, but it does mean the original contribution is much smaller than advertised — and the one genuinely new analytic result, the causal bound, is wrong.

This paper is for readers working on holographic dark energy or interacting Λ models who want a clean derivation of the geometric interpretation. It deserves peer review because the consistency model is coherent and the flaw is a fixable derivation error, not a poison in the overall idea. But a referee should require the energy-condition argument to be corrected or the (25) claim withdrawn.

Best

Referee report

Significance

If the derivations were correct, the paper would offer a simple, purely geometric dark-energy mechanism in which the holographic parameter b controls both the matter/dark-energy ratio and the condition for acceleration, with no dark-energy equation of state. The exact solutions and the scale-factor acceleration condition are internally consistent, and the authors are appropriately cautious in the Conclusions that the paper only demonstrates consistency, not a complete observational solution. However, the main advertised 'new physical bounds' rest on an incorrect derivation of Eq. (25), and the coincidence-ratio 'explanation' is a choice of the free parameter b rather than a prediction. No observational dataset is used. The acceleration condition (31) is a genuine algebraic consequence of the ansatz and may survive a correction of the energy-condition analysis, but the paper as written overstates its results.

Major objections (3)

[§3.1, Eqs. (23)-(25)] The advertised bound ω≤b/(1−b) does not follow from the weak and causal energy conditions. With the (−+++) signature and c≡b/(1−b), take a timelike v^μ=αu^μ+βe^μ, where u^μ is the fluid 4-velocity, e^μ is a unit spacelike vector, and β²=α²−1. Then Eq. (23) becomes ε_m[(1+ω)α²−(ω−c)]=ε_m[(1+ω)(α²−1)+1+c]≥0, which is automatically satisfied for all ω≥−1 and gives no upper bound. Eq. (24) becomes ε_m²[(1+ω)(ω−1−2c)α²−(ω−c)²]≤0; requiring this for all α≥1 and using the null condition ω≥−1 yields only ω≤1+2c=(1+b)/(1−b). For b=1/2 this is ω≤3 rather than ω≤1. Under the opposite signature convention the details differ, but the conclusion remains that (23)–(24) do not imply (25). The sentence 'both of them yield (25)' is therefore incorrect, and the claimed new holographically modified causal bound is unproven. This error propagates to §3.2, where the entropy bound (30) is dismissed as less restrictive than (25).
[§2, Eq. (8) and following paragraph] The 'explanation' of the coincidence problem is imposed rather than derived. Equation (8) fixes the ratio ε_m/ε_Λ=(1−b)/b, and b is a free parameter; the paper itself states that 'for an appropriate value of b ... relation (8) can yield the experimentally observed relation.' Choosing b≈0.7 to match Ω_m/Ω_Λ≈0.3/0.7 is a fit, not a prediction. The abstract's claim that the model 'may explain' the coincidence problem therefore overstates what the calculation shows. The concluding caveat that the paper 'only show[s] the consistency' is more accurate and should be the language used throughout.
[§3 and Conclusions] The paper contains no observational confrontation. No dataset, likelihood, or even an order-of-magnitude comparison is used to test the allowed range after Eq. (31); the only quantitative contact with observations is the statement that b can be chosen to match the density ratio. For a cosmology paper whose abstract promises that the model may explain both the coincidence problem and the acceleration, the absence of any fit to SNIa, CMB, or H(z) data leaves the central claim as an unfalsified consistency demonstration. The authors should either add such a comparison or explicitly restrict the claims to consistency, as the Conclusions already do.

Minor objections (4)

[§2] 'FRLW' should be 'FRW', and there are typographical errors such as 'with with an interacting' and 'highest energy energy density.'
[§3.2, Eq. (28)] The holographic entropy bound σ R_H^3 < (aR_H)^2 appears dimensionally inconsistent unless σ is defined in comoving units; please clarify the definition of σ and the volume factor.
[§2, Eqs. (11)-(12)] The constant H0 is introduced without definition; state that it is the present Hubble parameter and specify the sign conditions on β=3(1+ω)(1−b)/2, since the explicit solutions are written assuming β>0.
[§2, Eqs. (9)-(10)] The N-bound argument for b<1 is not logically necessary, because positivity of ε_m in Eq. (8) already requires b<1; as written, the inequality ε_Λ≤bκ^{-2}l^{-2} with b<1 does not follow from de Sitter being the maximum-entropy state, so this argument should be deleted or rewritten.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

Saturating a holographic bound on dark energy fixes the matter/dark-energy ratio and can drive cosmic acceleration with no dark-energy equation of state.

The reading

This letter argues that one holographic constraint can replace the missing equation of state in a universe with a time-dependent cosmological term. If the dark-energy density saturates the bound $\varepsilon_\Lambda = b\kappa^{-2}H^2$ at the Hubble radius, the Bianchi-consistent field equations force matter and dark energy into a constant ratio $\varepsilon_m/\varepsilon_\Lambda=(1-b)/b$, which directly addresses the coincidence problem. The same model changes the causal and energy bounds on the matter equation of state, permitting acceleration for $1/3

What carries the argument

The load-bearing device is the saturation of the holographic inequality $\varepsilon_\Lambda \le b\kappa^{-2}L^{-2}$ with the inverse Hubble radius, $L=cH^{-1}$. This turns the inequality into the algebraic relation $\varepsilon_\Lambda = b\kappa^{-2}H^2$, supplying the extra equation that closes the otherwise underdetermined system of Friedmann equations. From it follow the scaling solutions and the fixed ratio $\varepsilon_m=(1-b)/b\,\varepsilon_\Lambda$, and the same parameter $b$ enters the energy-condition bound $\omega \le b/(1-b)$ and the acceleration bound $\omega < b/(1-b)-1/[3(1-b)]$. The paper treats dark energy as geometric de Sitter-like curved-spacetime energy rather than a fluid, so no $\omega_\Lambda$ is introduced.

Extended reading notes

Core claim

The paper's central claim is that the holographic principle, applied to a decaying cosmological term, is enough to tie the main parameters of late-time cosmology together. Working with flat FRW equations and a total energy-momentum tensor $\Theta^{\mu\nu}=T^{\mu\nu}+\Lambda^{\mu\nu}$ that is covariantly conserved, the authors take $\varepsilon_\Lambda = b\kappa^{-2}H^2$ (with $\kappa^2=8\pi G/(3c^4)$) as an equality, $0<b<1$, and $L=cH^{-1}$ as the cutoff. The resulting exact solutions satisfy $\varepsilon_m=(1-b)/b\,\varepsilon_\Lambda$, so the two densities keep a fixed ratio while both scale as $a^{-3(1+\omega)(1-b)}$. Imposing the null, weak, and causal energy conditions on $\Theta^{\mu\nu}$ yields $\omega\ge -1$ and $\omega \le b/(1-b)$; requiring $\ddot{a}>0$ then gives $\omega < b/(1-b) - 1/[3(1-b)]$. Thus, for a band of allowed $b$ and $\omega$, the model produces both the observed order-of-magnitude density ratio and an accelerated expansion without any dark-energy equation of state, dark energy being a purely geometric entity.

Load-bearing premise

The whole chain depends on one premise: that the dark-energy density actually sits at the holographic bound with the Hubble radius as cutoff, so that if the bound is not tight, the fixed density ratio and the derived parameter bounds do not follow.

Editorial extensions

If this is right

If the model is correct, the coincidence problem is recast as one choice of the parameter $b$: the ratio $\varepsilon_m/\varepsilon_\Lambda=(1-b)/b$ is fixed for all epochs, so matter and dark energy do not drift apart as in the standard picture.
Acceleration does not require a negative-pressure dark-energy fluid: for $1/3<b<1$ the matter equation-of-state parameter $\omega$ can be positive while $\ddot{a}>0$.
The holographic parameter modifies causality: the usual bound $\omega<1$ is replaced by $\omega\le b/(1-b)$, so a decaying geometric dark energy changes which matter equations of state are physically allowed.
Because the combined matter-plus-geometry tensor is covariantly conserved, dark energy can convert into matter and back; the model can describe an initially empty, all-dark-energy cone-spacetime as a possible big-bang seed.

Reading between the lines

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

A sharp test follows immediately: equation (8) predicts $\Omega_m/\Omega_\Lambda=(1-b)/b$ at every redshift, whereas the standard constant-cosmological-term picture predicts this ratio growing as $(1+z)^3$; high-redshift measurements of the two densities can distinguish the models.
Because exact saturation is doing all the work, a natural extension would let $b$ vary with time or epoch; the density ratio would then drift, and the coincidence would no longer be automatic. Tracking such a drift in future surveys would test exact saturation.
If observations ever establish an evolving dark-energy equation of state $w_\Lambda(z)\neq -1$, the geometric interpretation advanced here would be hard to maintain, since the model's success depends on never having to introduce $\omega_\Lambda$ at all.

Original abstract

By incorporating the holographic principle in a time-depending Lambda-term cosmology, new physical bounds on the arbitrary parameters of the model can be obtained. Considering then the dark energy as a purely geometric entity, for which no equation of state has to be introduced, it is shown that the resulting range of allowed values for the parameters may explain both the coincidence problem and the universe accelerated expansion, without resorting to any kind of additional structures.

The compiled record

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

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,348 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.11 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.12 · compares with prior results 0.09 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

By incorporating the holographic principle in a time-depending Lambda-term cosmology, new physical bounds on the arbitrary parameters of the model can be obtained. Considering then the dark energy as a purely geometric entity, for which no equation of state has to be introduced, it is shown that the resulting range of allowed values for the parameters may explain both the coincidence problem and the universe accelerated expansion, without resorting to any kind of additional structures.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	By incorporating the holographic principle in a time-depending Lambda-term cosmology, new physical bounds on the arbitrary parameters of the model can be obtained.	contribution 0.61	0.81	0.18	asserted 0.84
s1	Considering then the dark energy as a purely geometric entity, for which no equation of state has to be introduced, it is shown that the resulting range of allowed values for the parameters may explain both the coincidence problem and the universe accelerated expansion, without resorting to any kind of additional structures.	result 0.63	0.89	0.77	hedged possibility 0.83

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.96.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

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

References

Works cited (20)

[1]

A.G. Riess et al , Astron. J. 116 (1998) 1009; S. Perlmutter et al , Astrophys. J. 517 (1999) 565

View on Pith 1998
[2]

A.D. Miller et al, Ap. J. Lett. 524, (1999) L1; P. de Bernardis et al, Nature 404 (2000) 955; S. Hanany et al , Ap. J. Letters 545 (2000) 5; N.W. Halverson et al , Ap. J. 568 (2002) 38; B.S. Mason et al , Ap. J. 591 (2003) 540; A. Benoit et al , Astron. Astrophys. 399 (2003) L25; D.N. Spergel et al , Astrophys. J. Suppl. 148 (2003) 175; L. Page et al , As...

View on Pith 1999
[3]

B. Ratra and P.J. E. Peebles, Phys. Rev. D 37 (1988) 3406; J . Frieman, C.T. Hill and R. Watkins, Phys. Rev. D 46 (1992) 1226; J. Frieman, C.T. Hill, A . Stebbins and I. Waga, Phys. Rev. Lett. 75 (1995) 2077; R.R. Caldwell, R. Dave and P. J. Steinhardt, Phys. Rev. Lett. 80 (1998) 1582; I. Slatev, L. Wang and P.J. Steinhardt, Phys. Rev. Lett. 82 (1999) 896...

View on Pith 1988
[4]

A.A. Starobinsky, Grav. Cosmol. 6 (2000) 157; A. Shulz an d M.J. White, Phys. Rev. D 64 (2001) 043514; R.R. Caldwell, Phys. Lett. B 545 (2002) 23; J. Hao and X. Li, Phys. Rev. D 67 (2003) 107303; S. Nojiri and S. D. Odintsov, Phys. Le tt. B562 (2003) 147; P. Singh, M. Sami and N. Dadhich, Phys. Rev. D 68 (2003) 023522; J . Hao and X. Li, Phys. Rev. D 68 (...

View on Pith 2000
[5]

T. Padmanabhan, Darker Side of the Universe, talk at the 2 9th International Cosmic Ray Conference, Pune, India, Aug, 3-10/2005, astro-ph/051049 2

View on Pith 2005
[6]

G.’t Hooft, Dimensional Reduction in Quantum Gravity, i n: A. Ali, J. Ellis, S. Randjbar- Daemi (Eds.), Salam Festschrift: A Collection of Talks (Wor d Scientiﬁc, Singapore, 1993) gr-qc/9310026; L. Susskind, J. Math. Phys. 36 (1995) 6377, h ep-th/9409089

View on Pith Pith review arXiv 1995
[7]

R. Fischler and L. Susskind, Holography and Cosmology, h ep-th/9806039

View on Pith arXiv
[8]

R. Bousso, JHEP 07 (1999) 004, hep-th/9906022

View on Pith Pith review arXiv 1999
[9]

A.G. Cohen, D.B. Kaplan and A.E. Nelson, Phys. Rev. Lett. 82 (1999) 4971

View on Pith 1999
[10]

M. Li, Phys. Lett. B603 (2004) 1

View on Pith 2004
[11]

D. Pav´ on and W. Zimdahl, Phys. Lett. B 628 (2005) 206; D. Pav´ on and W. Zimdahl, Holographic dark energy and present cosmic acceleration, P roceedings of the XXVIII Spanish Relativity Meeting, hep-th/0511053

View on Pith Pith review arXiv 2005
[12]

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

View on Pith 1973
[13]

O. Bertolami, N. Cim. B 93 (1986) 36; J.C. Carvalho, J.A. S. Lima and I. Waga, Phys. Rev. D 46 (1992) 2404; L P. Chimento and D. Pavon, Gen. Rel. Gra v. 30 (1998) 643; J.M. Overduin and F.I. Cooperstock, Phys. Rev. D 58 (1998) 043506 ; T. Harco and M.K. Mak, Gen. Rel. Grav. 31 (1999) 849; R.G. Vishwakarma, Class. Quan t. Grav. 18 (2001) 1159; R.G. Vishwak...

View on Pith Pith review arXiv 1986
[14]

S.M. Carroll, Why is the Universe Accelerating? in: W. L . Freedman (Ed.), Measuring and Modeling the Universe (Cambridge University Press, Cambri dge, 2003) astro-ph/0310342; P.J. Steinhardt, in: V.L. Fitch, D.R. Marlow (Eds.), Critic al Problems in Physics (Prince- ton University Press, Princeton, 1977)

View on Pith Pith review arXiv 2003
[15]

R. Aldrovandi, J. P. Beltr´ an Almeida and J. G. Pereira, Grav. Cosmol. 11 (2005) 277, gr-qc/0312017

View on Pith Pith review arXiv 2005
[16]

S. Thomas, Phys. Rev. Lett. 89 (2002) 081301

View on Pith 2002
[17]

R. Bousso, JHEP 11 (2000) 038

View on Pith 2000
[18]

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

View on Pith 1984
[19]

R. Aldrovandi, J. P. Beltr´ an Almeida and J. G. Pereira, J. Geom. Phys. 56 (2006) 1042, gr-qc/0403099

View on Pith Pith review arXiv 2006
[20]

W. Fischler, A. Loewy and S. Paban, JHEP 09 (2003) 024. 8

View on Pith 2003

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

A.G. Riess et al , Astron. J. 116 (1998) 1009; S. Perlmutter et al , Astrophys. J. 517 (1999) 565

View on Pith 1998

[2] [2]

A.D. Miller et al, Ap. J. Lett. 524, (1999) L1; P. de Bernardis et al, Nature 404 (2000) 955; S. Hanany et al , Ap. J. Letters 545 (2000) 5; N.W. Halverson et al , Ap. J. 568 (2002) 38; B.S. Mason et al , Ap. J. 591 (2003) 540; A. Benoit et al , Astron. Astrophys. 399 (2003) L25; D.N. Spergel et al , Astrophys. J. Suppl. 148 (2003) 175; L. Page et al , As...

View on Pith 1999

[3] [3]

B. Ratra and P.J. E. Peebles, Phys. Rev. D 37 (1988) 3406; J . Frieman, C.T. Hill and R. Watkins, Phys. Rev. D 46 (1992) 1226; J. Frieman, C.T. Hill, A . Stebbins and I. Waga, Phys. Rev. Lett. 75 (1995) 2077; R.R. Caldwell, R. Dave and P. J. Steinhardt, Phys. Rev. Lett. 80 (1998) 1582; I. Slatev, L. Wang and P.J. Steinhardt, Phys. Rev. Lett. 82 (1999) 896...

View on Pith 1988

[4] [4]

A.A. Starobinsky, Grav. Cosmol. 6 (2000) 157; A. Shulz an d M.J. White, Phys. Rev. D 64 (2001) 043514; R.R. Caldwell, Phys. Lett. B 545 (2002) 23; J. Hao and X. Li, Phys. Rev. D 67 (2003) 107303; S. Nojiri and S. D. Odintsov, Phys. Le tt. B562 (2003) 147; P. Singh, M. Sami and N. Dadhich, Phys. Rev. D 68 (2003) 023522; J . Hao and X. Li, Phys. Rev. D 68 (...

View on Pith 2000

[5] [5]

T. Padmanabhan, Darker Side of the Universe, talk at the 2 9th International Cosmic Ray Conference, Pune, India, Aug, 3-10/2005, astro-ph/051049 2

View on Pith 2005

[6] [6]

G.’t Hooft, Dimensional Reduction in Quantum Gravity, i n: A. Ali, J. Ellis, S. Randjbar- Daemi (Eds.), Salam Festschrift: A Collection of Talks (Wor d Scientiﬁc, Singapore, 1993) gr-qc/9310026; L. Susskind, J. Math. Phys. 36 (1995) 6377, h ep-th/9409089

View on Pith Pith review arXiv 1995

[7] [7]

R. Fischler and L. Susskind, Holography and Cosmology, h ep-th/9806039

View on Pith arXiv

[8] [8]

R. Bousso, JHEP 07 (1999) 004, hep-th/9906022

View on Pith Pith review arXiv 1999

[9] [9]

A.G. Cohen, D.B. Kaplan and A.E. Nelson, Phys. Rev. Lett. 82 (1999) 4971

View on Pith 1999

[10] [10]

M. Li, Phys. Lett. B603 (2004) 1

View on Pith 2004

[11] [11]

D. Pav´ on and W. Zimdahl, Phys. Lett. B 628 (2005) 206; D. Pav´ on and W. Zimdahl, Holographic dark energy and present cosmic acceleration, P roceedings of the XXVIII Spanish Relativity Meeting, hep-th/0511053

View on Pith Pith review arXiv 2005

[12] [12]

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

View on Pith 1973

[13] [13]

O. Bertolami, N. Cim. B 93 (1986) 36; J.C. Carvalho, J.A. S. Lima and I. Waga, Phys. Rev. D 46 (1992) 2404; L P. Chimento and D. Pavon, Gen. Rel. Gra v. 30 (1998) 643; J.M. Overduin and F.I. Cooperstock, Phys. Rev. D 58 (1998) 043506 ; T. Harco and M.K. Mak, Gen. Rel. Grav. 31 (1999) 849; R.G. Vishwakarma, Class. Quan t. Grav. 18 (2001) 1159; R.G. Vishwak...

View on Pith Pith review arXiv 1986

[14] [14]

S.M. Carroll, Why is the Universe Accelerating? in: W. L . Freedman (Ed.), Measuring and Modeling the Universe (Cambridge University Press, Cambri dge, 2003) astro-ph/0310342; P.J. Steinhardt, in: V.L. Fitch, D.R. Marlow (Eds.), Critic al Problems in Physics (Prince- ton University Press, Princeton, 1977)

View on Pith Pith review arXiv 2003

[15] [15]

R. Aldrovandi, J. P. Beltr´ an Almeida and J. G. Pereira, Grav. Cosmol. 11 (2005) 277, gr-qc/0312017

View on Pith Pith review arXiv 2005

[16] [16]

S. Thomas, Phys. Rev. Lett. 89 (2002) 081301

View on Pith 2002

[17] [17]

R. Bousso, JHEP 11 (2000) 038

View on Pith 2000

[18] [18]

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

View on Pith 1984

[19] [19]

R. Aldrovandi, J. P. Beltr´ an Almeida and J. G. Pereira, J. Geom. Phys. 56 (2006) 1042, gr-qc/0403099

View on Pith Pith review arXiv 2006

[20] [20]

W. Fischler, A. Loewy and S. Paban, JHEP 09 (2003) 024. 8

View on Pith 2003