Propagation of Gravitational Waves in Anisotropic Universe

Sarbari Guha; Sucheta Datta

REVIEW 2 major objections 3 minor 59 references

Propagation of Gravitational Waves in Anisotropic Universe

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read In an anisotropic toy universe, gravitational wave speed depends on direction and falls below c.

desk verdict Sign error in background equations plus an invalid plane-wave ansatz sink the paper's central dispersion relations; the topic is fine but the execution is not sound. read the letter →

arxiv 1908.06743 v1 pith:MSWMCCHA submitted 2019-08-14 gr-qc hep-th

Sucheta Datta , Sarbari Guha This is my paper · ORCID

classification gr-qchep-th PACS 04.30.-w98.80.-k

keywords gravitationalwavesanisotropicuniversetoymodellinearperturbationsdispersionrelationtransversetracelessgaugetidalaccelerationcosmological

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

This paper studies how gravitational waves propagate in a simple anisotropic universe whose spatial metric is isotropic in x and y but stretched by a factor b(z) along z. The authors' central claim is that the wave number of a gravitational wave picks up an explicit dependence on b(z) when the wave travels along the anisotropy axis, while waves travelling perpendicular to that axis have wave numbers that depend only on the scale factor a(t). The paper also argues that the transverse-traceless gauge condition is not an extra assumption: the linearized Einstein equations themselves force the perturbation to be traceless, in both propagation directions, through equations (83) and (85). If correct, the model shows that cosmic anisotropy would leave a directional imprint on gravitational-wave dispersion, making wave speed direction-dependent and lower than c, which is qualitatively different from the FLRW case. The analysis is deliberately a toy model: pressure must vanish (p≃0) unless the anisotropy factor equals one, so the results are an illustration of mechanism rather than a realistic cosmology.

What carries the argument

The load-bearing device is the toy background metric (1), together with the plane-wave ansatz $h_{11}(t,z) = \mathrm{Re}[\varepsilon_{11} \exp\{i(\omega t - k_{11} z)\}]$ (and its $\times$-polarization counterpart) inserted into the linearized Einstein equations. The background metric supplies $a(t)$, which sets the overall cosmological expansion, and $b(z)$, which breaks symmetry along one spatial direction; the plane-wave ansatz converts the partial differential equations (34) and (35) into algebraic dispersion relations for $k_{11}$ and $k_{12}$. The same procedure applied to $x$-propagation turns the perturbed equations (58), (63), and (64) into the wave-number formulas (69), (74), and (76). The tracelessness argument works through the extra Ricci perturbations, e.g. $\delta R_{03}$ and $\delta R_{01}$, which vanish only when the trace $h$ vanishes; equations (83) and (85) are the Einstein equations that enforce that vanishing.

What would settle it

For a specific anisotropy $b(z) = 1 + \varepsilon \sin(z/\lambda)$, substitute the plane-wave form $h_{11} = \varepsilon_{11} \exp\{i(\omega t - kz)\}$ into Eq. (34) and check whether a single complex $k$ satisfies the equation at every $z$; if the equation demands $k(z)$, Eq. (38) is not a valid dispersion relation.

Watch

Extended reading notes

Core claim

The central discovery is that the propagation of linear gravitational-wave perturbations in the metric $ds^2 = dt^2 - a^2(t)(dx^2 + dy^2) - a^2(t)b^2(z)\,dz^2$ is direction-sensitive in a precise way. Along the anisotropy axis $z$, the two polarizations have distinct dispersion relations, $k_{11} = -ib'/b \pm (1/b)\sqrt{-b'^2 + a^2b^4(\omega^2 + 8\pi(\rho-3p))}$ and $k_{12} = -ib'/b \pm (1/b)\sqrt{-b'^2 + a^2b^4\omega^2}$, so both wave numbers depend on $b(z)$. Along a perpendicular direction $x$, the wave numbers $k_{22}$ and $k_{33}$ are equal in the pressureless case and independent of $b(z)$; they depend only on $a(t)$, through $\omega$, $\dot{a}$, $\ddot{a}$, and $\rho$. The same equations also yield the tracelessness of $h_{\mu\nu}$ as a consequence of particular components of Einstein's equations rather than as a gauge assumption, with the trace condition modified to $h_{33} + b^2 h_{22} = 0$ for $x$-propagation. The paper concludes that the speed of gravitational waves in this background is lower than $c$, that the lowering depends on both $a(t)$ and $b(z)$ along $z$ but only on $a(t)$ along $x$, and that in vacuum ($a \to 1$, $b \to 1$) the Minkowski results are recovered.

Load-bearing premise

The plane-wave ansatz inserts a wave with constant frequency $\omega$ and wave number $k$ into equations whose coefficients depend on $t$ and $z$; whenever $a(t)$ or $b(z)$ varies on scales comparable to the wavelength, the exponential is not a solution and the derived dispersion relations do not hold.

Editorial extensions

If this is right

Along the anisotropy axis the two polarizations travel with slightly different wave numbers, so an anisotropic background acts as a birefringent medium for gravitational waves.
Perpendicular to the anisotropy axis the wave number is independent of $b(z)$, so the directional dependence of dispersion could in principle be used to locate the anisotropy axis from gravitational-wave observations.
Because the wave speed is below $c$ in both directions, any observation of gravitational waves arriving with speed $c$ would constrain the allowed size of $a(t)$ and $b(z)$ in such a model.
The tracelessness of $h_{\mu\nu}$ follows from the field equations rather than being imposed, which means the synchronous transverse traceless gauge is consistent in this background.
In vacuum, $a(t)\to 1$ and $b(z)\to 1$ reproduce the standard Minkowski result, so the model's new effects are entirely due to the anisotropic expansion.

Reading between the lines

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

If $b(z)$ varies on scales comparable to the gravitational-wave wavelength, the constant-$k$ plane-wave ansatz is no longer an exact solution; a numerical or slowly-varying-wave treatment would be needed to see whether the $b$-dependence in equations (38)-(39) survives as a genuine dispersion effect or is an artifact of the ansatz.
The same calculation could be repeated for non-pressureless matter only if the constraint $p(b^2-1)=0$ is avoided, for instance by adding an anisotropic stress term; that would show whether the direction-dependent slowing is robust beyond dust.
A natural observational analogue is a gravitational wave crossing a region with mild anisotropic expansion: the model predicts a differential arrival time or phase between polarizations correlated with the direction to the source, though the effect would be tiny for realistic anisotropies.
The tidal-acceleration formulas imply that a detector's response depends on the anisotropy factor $b(z)$ when the wave travels perpendicular to the anisotropy axis; simulating the response for a prescribed $b(z)$ would give a concrete, testable waveform prediction.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear Colleague,

The short version: this paper has a sign error in the background Einstein equations and its main dispersion relations are not actual solutions of the wave equations, so the headline claims about direction-dependent GW speed are unsupported. I'd desk-reject it.

That said, the authors are working on a legitimate topic. The toy metric—an LRS-like Bianchi I with a z-dependent anisotropy function b(z)—is a reasonable place to test how anisotropy affects GW propagation. They cite the relevant older literature (Hu, Miedema & van Leeuwen, Cho & Speliotopoulos, Adams et al.) and set up linear perturbation theory in a transparent way. The Ricci tensor computation for the background (Eqs. 2-3) is correct.

The problems start with Eq. (6). With their metric, G00 = R00 - R/2 = 3(ȧ/a)^2, so the dust density should be 8πρ = 3(ȧ/a)^2. The paper has the opposite sign, which forces ρ negative and contradicts their own perturbed equation (29). That's a clear internal inconsistency.

The bigger issue is the plane-wave ansatz. In Sec. IV, they substitute h ∝ e^{i(ωt - kz)} with constant k and ω into equations whose coefficients depend on t through a(t) and on z through b(z). The resulting dispersion relations, Eqs. (38)-(39), make k a function of z (and t), so the exponential is no longer a solution: derivatives of exp[-i k(z) z] produce k'(z) and k''(z) terms, and the t-derivatives of a time-dependent amplitude produce ȧ and ä terms. No WKB or slowly-varying justification is given. The same defect appears in the x-direction calculation, where the a^4(t) amplitude in Eq. (68) contributes extra terms to ̈h_{23}, so Eq. (69) is an algebraic relation that does not solve the PDE. Since the direction-dependent b(z)-dependence of the wave vector is the paper's central result, this is load-bearing.

There's also a logical error in the tracelessness validation. From ∂0∂3(h/a^2)=0 one cannot conclude h=0; h = a^2(t) B(z) is a nonzero solution. The same issue appears in Eq. (85).

I don't see a hidden gem here. The paper is honest in presentation, and the errors are visible, but they are fundamental. The authors should fix the sign error and redo the propagation analysis with a proper mode decomposition or eikonal ansatz. As it stands, it doesn't deserve a referee's time.

Best.

Referee Report

2 major / 3 minor

Summary. The paper studies linearized gravitational waves in a simple anisotropic toy universe with metric ds^2 = dt^2 - a^2(t)(dx^2+dy^2) - a^2(t)b^2(z) dz^2. It imposes synchronous, transverse, traceless gauge conditions, computes first-order Ricci perturbations, writes the linearized Einstein equations for a perfect fluid, and proposes plane-wave solutions for GWs propagating along the anisotropy direction z and along the perpendicular direction x. The central claims are that the wave vector for propagation along z depends explicitly on the anisotropy function b(z) (Eqs. 38-39), that the wave vector for propagation along x is independent of b(z) (Eqs. 74-76), that tracelessness of the perturbations follows from the field equations (Eqs. 83 and 85), and that the GW speed is lower than c with the reduction controlled by a(t) and b(z) along z but only by a(t) along x. The paper ends with a computation of the tidal acceleration induced by the waves.

Significance. If the results were sound, the paper would provide a concrete analytical example of how a background anisotropy can modify the dispersion relation and speed of gravitational waves, complementing earlier studies of Bianchi type I universes. The paper is self-contained, does not fit parameters to data, and explicitly presents many intermediate Christoffel and Ricci perturbation expressions, which is a useful feature. However, the central propagation claims rest on a plane-wave ansatz that is not a valid solution of the variable-coefficient wave equations, and the background field equations contain a sign error. The significance is therefore currently overshadowed by technical issues that affect the main conclusions.

major comments (2)

[§II.A, Eq. (6)] The background G00 equation has the wrong sign. Combining R00 = -3ä/a from Eq. (2) and R = -6ä/a - 6(ȧ/a)^2 from Eq. (3) with G00 = R00 - R/2 gives G00 = 3(ȧ/a)^2, so Einstein's equations imply 3(ȧ/a)^2 = 8πρ, not (ȧ/a)^2 = -8πρ/3. As written, Eq. (6) yields negative energy density for a standard expanding universe and is inconsistent with the perturbed 00 equation (29), which is the standard ï¿½/a = -4π/3(ρ+3p). This internal inconsistency affects the use of ρ in the dispersion relations such as Eq. (38).
[§VI, Eqs. (83) and (85)] The claimed derivation of tracelessness is incorrect. From (1/2)∂0∂3(h/a^2) = 0 it follows that h/a^2 = f(z) + g(t), so h = a^2[f(z)+g(t)] is a general nonzero solution; the conclusion that a(t) being time-dependent forces h = 0 does not follow without additional boundary or initial conditions. Similarly, ∂0∂1(h/(a^2 b^2)) = 0 does not imply h = 0, since the most general solution is h = a^2 b^2[F(t,z) + G(x,z)] (or an equivalent separable form). Thus the statement that tracelessness follows from the perturbed Einstein equations in the manner claimed is not established.

minor comments (3)

[Throughout] There are numerous typographical errors, including 'Univers e' in the title, 'eqaution' in §V.B, and mislabelled references (e.g., 'arXiv:1608.01982 [gr-gc]' should be '[gr-qc]').
[§II.A, Eqs. (7)-(8)] The spatial components of the background Einstein equations appear to be missing factors of a^2 and b^2 relative to the stated index convention: with Tij = p gij and g11 = -a^2, Eq. (7) should contain an a^2 on the right-hand side, and Eq. (8) an a^2 b^2 factor. Please clarify the index convention used.
[§V, Eqs. (65)-(72)] The notation is confusing: the same symbol k23 is used both as a constant wavenumber in the exponential and as a solution that depends on a(t) through Eq. (69). It would be helpful to state explicitly that Eq. (69) is a local algebraic relation at fixed t, not a global dispersion relation, if that is intended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained and does not reduce any prediction to a fitted input or self-citation; the paper's main defects are mathematical validity issues, not circular reasoning.

full rationale

The paper is a self-contained analytic exercise on the toy background metric (1). It fits no parameters to data; a(t), b(z), rho, and p are free background inputs, and the dispersion relations (38)-(39), (69), (74), and (76) are obtained by substituting the stated exponential trial solutions into the linearized Einstein equations. The claimed b(z) dependence along the anisotropy direction is an algebraic consequence of the equation's coefficients, not a quantity inserted as an input and then relabelled as a prediction. There are no load-bearing self-citations: the cited works are used as methodological background or comparison, not as a uniqueness theorem or as authority for the ansatz, and the authors cite no prior work of their own as the basis for the central result. The tracelessness section is not circular: the paper defines a nonzero trace h and attempts to show that the perturbed Einstein equations force h=0; the inference is mathematically incorrect (Eq. (83) gives h/a^2 = F(t)+G(z), not h=0, and similarly for Eq. (85)), but this is a flawed deduction, not an assumption of the conclusion. Likewise, the plane-wave ansatz with constant omega and k is inconsistent with z- and t-dependent coefficients unless a slow-variation or WKB approximation is stated, but that is an internal validity defect rather than a circular reduction. The central results are model-dependent because they depend on the hand-chosen background and ansatz, but model-dependence is not circularity. Therefore the circularity score is 0.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

No new particles, fields, or forces are introduced. The only inputs are the arbitrary functions a(t) and b(z), the perfect fluid assumption, and the plane-wave ansatz. The axioms list the standard GR framework plus the specific modeling choices; the ad hoc items are the unjustified plane-wave treatment and the flawed trace argument.

free parameters (4)

Scale factor a(t)
Chosen by hand as the isotropic expansion function. Its time dependence enters all dispersion relations and tidal responses, but the paper never solves for it from the field equations.
Anisotropy function b(z)
Introduced ad hoc as the z-direction distortion. It controls the claimed anisotropy effects in the dispersion relations and tidal acceleration, but its functional form is never specified or derived.
GW frequency ω
A mode parameter inserted in the plane-wave ansatz; it is part of the model input rather than a derived quantity, but the dispersion relations are expressed in terms of it.
Integration constants m and n
Arbitrary constants appearing in the x-direction perturbation solutions (67) and (71). They set wave amplitudes but are not fitted to data.

assumptions (7)

standard math General relativity with Einstein field equations Gμν = 8πTμν in geometrized units.
The entire calculation assumes classical GR with c = G = 1.
domain assumption The matter content is a perfect fluid with Tμν = (ρ+p)uμuν - p gμν.
Introduced in Sec. II.A, Eq. (4). The fluid four-velocity, density, and pressure define the background and perturbation source.
domain assumption The model requires p ≃ 0 (dust) to keep b² ≠ 1.
From Eq. (9), p(b² - 1) = 0, so to preserve anisotropy the pressure is set to zero. This is a strong modeling constraint and, with the sign error in Eq. (6), forces negative energy density.
domain assumption Gravitational waves are non-material perturbations, so δTμν = 0.
Stated in Sec. III.B following Miedema and van Leeuwen. It lets the paper ignore first-order perturbations of density, pressure, and velocity.
domain assumption Linear perturbation theory with first-order truncation and synchronous, transverse, traceless gauge is valid.
Assumed throughout Sec. III and used to compute δRμν.
ad hoc to paper A plane wave with constant k and ω is a solution to equations with nonconstant coefficients a(t) and b(z).
The trial solutions (36)-(37) and the analogous x-direction ansätze insert constant-frequency plane waves into wave equations with coordinate-dependent coefficients. No WKB or slowly-varying justification is provided, making the derived dispersion relations invalid as stated.
ad hoc to paper From ∂0∂3(h/a²) = 0 and a(t)-dependent a, it follows that h = 0.
In Sec. VI, Eq. (83), the paper infers tracelessness. A nonzero time-independent h(z) satisfies this equation, so the inference is false without additional assumptions.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Propagation of Gravitational Waves in Anisotropic Universe." pith.science (2026). https://pith.science/paper/MSWMCCHA

@misc{pith2026190806743,
  author       = {Pith},
  title        = {Pith review of: Propagation of Gravitational Waves in Anisotropic Universe},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MSWMCCHA}},
  note         = {Machine review of arXiv:1908.06743}
}

read the original abstract

In this paper, we have considered a toy model of an anisotropic universe and studied the propagation of gravitational waves in such a universe. The consideration of this toy model simplifies the analysis and helps us to illustrate the effects of anisotropy. Incorporating linear perturbations on this anisotropic background, we have considered the synchronous, transverse, traceless gauge conditions and evaluated the perturbations of the Ricci tensor. The energy-momentum tensor is that of a perfect fluid, for which the Einstein's field equations are determined in presence of perturbations. We arrive at the set of linearised Einstein's equations explicitly and find solutions for gravitational waves propagating along the direction of anisotropy. We also study the propagation along a direction perpendicular to the direction of anisotropy. Subsequently we have validated the assumption of the tracelessness of the linear perturbations. Finally we determine the amount of tidal acceleration caused by the propagation of gravitational waves in this background spacetime.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

59 extracted references · 54 canonical work pages

[1]

Einstein, Sitzungsber

A. Einstein, Sitzungsber. Preuss. Akad. Wiss. Berlin (M ath. Phys.) 1916, 688-696 (1916)

work page 1916
[2]

Einstein, Sitzungsber

A. Einstein, Sitzungsber. Preuss. Akad. Wiss. Berlin (M ath. Phys.) 1918, 154-167 (1918)

work page 1918
[3]

R. A. Hulse and J. H. Taylor, Astrophys. J. 195, L51-L53 (1975)

work page 1975
[4]

J. H. Taylor and J. M. Weisberg, Astrophys. J. 253, 908-920 (1982)

work page 1982
[5]

B. F. Schutz and F. Ricci, ‘Elements of gravitational wav es’, Chapter 2 of ‘Gravitational waves’, edited by I. Ciufol ini, V. Gorini, U. Moschella and V. Fre, published by IOP Publishing Ltd., pp.15-23, (2001)

work page 2001
[6]

(90) Similarly, for χ2, we obtain the solution χ2(t) = (log a)χ2 0 + 1 2 ( h22χ2 0 +h21χ1 0 ) +χ1 0 +χ2

work page
[7]

ˆχ(t) = 1 2 hD(t) ˆχ0

(91) The eigenvalues of χi can be determined from those of the diagonalised perturbati on matrix hD, i.e. ˆχ(t) = 1 2 hD(t) ˆχ0. (92) The perturbation matrix is of the form h(t) = ( ε11 ε12 ε12 −ε11 ) e{i(wt−kz)}, (93) having the eigenvalues λ = ± √ (ǫ11)2 + (ε12)2, so that its diagonalised form is hD(t) = ( √ (ε11)2 +ε12)2 0 0 − √ (ε11)2 + (ε12)2 ) e{i(w...

work page
[8]

(95) • For ‘ ×’ polarization, ε11 = 0, and therefore ˆχ1(t) = 1 2ε12 exp{i(wt −kz)} ˆχ1 0 , ˆχ2(t) = − 1 2ε12 exp{i(wt −kz)} ˆχ2

work page

Show all 59 references

[9]

Propagation along x-direction Here, i and j will be 2 or 3

(96) B. Propagation along x-direction Here, i and j will be 2 or 3. The non-zero components of the perturbed Riema nn tensor are obtained as R2 020 = −∂2 0 (loga − 1 2a2h22), R 2 030 = −∂2 0(− 1 2a2h23), R3 020 = −∂2 0 (− 1 2a2b2h23) and R3 030 = −∂2 0(loga − 1 2a2b2h33). Proc...
[10]

(98) 12 Similarly, we can ﬁnd χ3(t) = (log a)χ3 0 + 1 2 ( h33χ3 0 +h32χ2 0 ) +χ2 0 +χ3
[11]

we have h22 = 1 a4h22, h 23 = 1 a4b2h23, h 33 = 1 a4b4h33. (43) The divergenceless condition ( 13) gives rise to two non-trivial relations (on expressing the covariant derivatives in terms of partial derivatives and Christoﬀel connections of the background metric) : ∇3h32 = 0 ...
[12]

Linearised Einstein’s equations and their solutions The explicit form of the Einstein’s equations (

(28) B. Linearised Einstein’s equations and their solutions The explicit form of the Einstein’s equations (
[13]

(102) • For ‘+’ polarization, ε23 = 0, and we have ˆχ2(t) = 1 2a4ε2 2e{i(wt−kx)} ˆχ2 0 , ˆχ3(t) = − 1 2a4b2ε2 2e{i(wt−kx)} ˆχ3

(99) As before, we can write ˆχ(t) = 1 2 hD(t) ˆχ0, (100) where the perturbation matrix h(t) = ( ε22 ε23 ε23 −b2ε22 ) e{i(wt−kx)}, (101) has the eigenvalues λ = 1 2 [ −(b2 − 1)ε2 2 ± √ (b2 − 1)2(ε22)2 + 4[(ε23)2 +b2(ε22)2] ] = 1 2 [ −(b2 − 1)ε22 ± √ (b2 + 1)2(ε22)2 + 4(ε23)2 ]...
[14]

(103) • For ‘ ×’ polarization, ε22 = 0, which means that ˆχ2(t) = 1 2a4ε23e{i(wt−kx)} ˆχ2 0 , ˆχ3(t) = − 1 2a4ε23e{i(wt−kx)} ˆχ3
[15]

Summation is carried over k

(104) Hence the non-zero components of tidal acceleration are giv en by χi(t) = (log a)χi 0 + Σ k [ 1 2hikχk 0 +χk 0 ] , (105) where i and k take up values 1 , 2 in case of propagation along z-direction, and 2 , 3 in case of propagation along x-direction. Summation is carried ...
[16]

L. D. Landau and E. M. Lifshitz, ‘The Classical Theory of Fields’, Third English edition, Pergamon Press (1971)

1971
[17]

Prasanna, ‘Gravitation’, CRC Press (2017)

A.R. Prasanna, ‘Gravitation’, CRC Press (2017)

2017
[18]

Dunya and M

O. Dunya and M. Arik, Gen. Relativ. Gravit. 51, 94 (2019), arXiv:1707.01169 [gr-qc]

2019 arXiv
[19]

C. Chen, J. M. Nester and W. Ni, Chinese J. of Phys., Vol. 1, Issue 55, (2017) 142-169, arXiv:1610.08803 [gr-qc] (2016 )

2017 arXiv
[20]

Abbott and R

B.P. Abbott and R. Abbott et al., Phys. Rev. Lett. 116, 061102 (2016)

2016
[21]

Abbott and R

B.P. Abbott and R. Abbott et al., Phys. Rev. Lett. 116, 241103 (2016)

2016
[22]

(33) Subtracting equation (

are as follows: R00 equation : ¨a a = − 4π 3 (ρ + 3p), (29) R11 equation : ( a¨a + 2 ˙a2) + ( − 1 2 ¨h11 + 1 2a2b2h′′ 11 − b′ a2b3h′ 11 ) = 8π [ p + a2 2 (ρ − 3p) − 1 2 (ρ − 3p)h11 ] , (30) R22 equation : ( a¨a + 2 ˙a2) + ( 1 2 ¨h11 − 1 2a2b2h′′ 11 + b′ a2b3h′ 11 ) = 8π [ p + ...
[23]

Abbott and R

B.P. Abbott and R. Abbott et al., Phys. Rev. Lett. 118, 221101 (2017)

2017
[24]

Abbott and R

B.P. Abbott and R. Abbott et al., Phys. Rev. Lett. 119, 141101 (2017)

2017
[25]

Abbott and R

B.P. Abbott and R. Abbott et al., ApJL 851, L35 (2017)

2017
[26]

Abbott and R

B.P. Abbott and R. Abbott et al., Phys. Rev. Lett. 119, 161101 (2017)

2017
[27]

Weinberg, ‘Gravitation and cosmology: principles a nd applications of the general theory of relativity’, Repri nt, Wiley India Pvt.Ltd., 2016 from Wiley, New York, 1972

S. Weinberg, ‘Gravitation and cosmology: principles a nd applications of the general theory of relativity’, Repri nt, Wiley India Pvt.Ltd., 2016 from Wiley, New York, 1972

2016
[28]

Schutz, ‘A First Course in General Relativity’, Seco nd edition, Cambridge University Press (2009)

B. Schutz, ‘A First Course in General Relativity’, Seco nd edition, Cambridge University Press (2009)

2009
[29]

M. P. Hobson, G. P. Efstathiou and A. N. Lasenby, ‘Genera l Relativity – An Introduction for Physicists’, Cambridge University Press (2006)

2006
[30]

Martel and E

K. Martel and E. Poisson, Phys. Rev. D 71, 104003 (2005)

2005
[31]

(34) 6 Equation ( 33) can be rewritten as ¨h12 − 1 a2b2h′′ 12 + 2b′ a2b3h′ 12 = 0

from equation ( 30), we have ¨h11 − 1 a2b2h′′ 11 + 2b′ a2b3h′ 11 = 8π(ρ − 3p)h11. (34) 6 Equation ( 33) can be rewritten as ¨h12 − 1 a2b2h′′ 12 + 2b′ a2b3h′ 12 = 0. (35) In order to solve equations (
[32]

B. L. Hu, Phys. Rev. D 18, 968 (1978)

1978
[33]

P. G. Miedema and W. A. van Leeuwen, Phys. Rev. D 47, 3151 (1993)

1993
[34]

and ( 35), let us assume the trial solutions: h11(t,z ) = Re [ε11 exp {i(ωt −k11z)}], (36) h12(t,z ) = Re [ε12 exp {i(ωt −k12z)}], (37) so that a plane gravitational wave of frequency ω is characterised by wave-vectors k11 and k12 in its ‘+’ and ‘ ×’- polarization modes respec...
[35]

H. T. Cho and A. D. Speliotopoulos, Phys. Rev. D 52, 5445-5458, (1995), arXiv:gr-qc/9504046

1995 arXiv
[36]

L. H. Ford and L. Parker, Phys. Rev. D 16, 1601 (1977) 14

1977
[37]

J. M. Bardeen, Phys. Rev. D 22, 1882-1905 (1980)

1980
[38]

J. M. Stewart, Class. Quant. Grav. 7, 1169-1180 (1990)

1990
[39]

V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger, Ph ys. Rept. 215, 203-333 (1992)

1992
[40]

P. G. Miedema and W. A. van Leeuwen, Phys. Rev. D 54, 7227 (1996)

1996
[41]

Regge and J

T. Regge and J. A. Wheeler, Phys. Rev. 108, 1063 (1957)

1957
[42]

U. H. Gerlach and U. K. Sengupta, Phys. Rev. D 19, 2268 (1979); Phys. Rev. D 22, 1300 (1980)

1979
[43]

Gundlach and J

C. Gundlach and J. M. Martin-Garcia, Phys. Rev. D 61, 084024 (2000)

2000
[44]

Clarkson, T

C. Clarkson, T. Clifton and S. February, JCAP 06 (2009) 0 25, arXiv:0903.5040 [astro-ph.CO]

2009 arXiv
[45]

February, J

S. February, J. Larena, C. Clarkson and D. Pollney, Clas s. Quantum Grav. 31 (2014) 175008, arXiv:1311.5241 [astro-ph.CO] (2013)

2014 arXiv
[46]

Meyer, M

S. Meyer, M. Redlich and M. Bartelmann, JCAP 2015(03) 05 3-053, arXiv:1412.3012 [astro-ph.CO] (2014)

2014 arXiv
[47]

S. Meyer, Thesis: ‘The evolution of linear perturbatio ns in Lemaître-Tolman-Bondi void models and the eﬀect on lig ht propagation’, Ruperto-Carola-University of Heidelberg, Germany, 2015

2015
[48]

P. J. Adams, R. W. Hellings, R. L. Zimmerman, H. Farhoosh , D. I. Levine and S. Zeldich, Astrophys. J. 253, 1-18 (1982)

1982
[49]

P. J. Adams, R. W. Hellings and R. L. Zimmerman, Astrophy s. J. 288, 14-21 (1985)

1985
[50]

Sacchetti and D

F. Sacchetti and D. Travese, General Relativity and Gra vitation, Vol. 10, No. 11 (1979), pp. 947-952

1979
[51]

Bettoni, J

D. Bettoni, J. M. Ezquiaga, K. Hinterbichler and M. Zuma lacárregui, Phys. Rev. D 95, 084029 (2017), arXiv:1608.01982 [gr-gc] (2016)

2017 arXiv
[52]

C. M. Will, Phys. Rev. D 57, 2061 (1998), arXiv:gr-qc/9709011

1998 arXiv
[53]

Mirshekari, N

S. Mirshekari, N. Yunes, and C. M. Will, Phys. Rev. D 85, 024041 (2012), arXiv:1110.2720 [gr-qc]

2012 arXiv
[54]

Lazkoz, Phys

R. Lazkoz, Phys. Rev. D 60, 104008 (1999)

1999
[55]

Ehlers, A

J. Ehlers, A. R. Prasanna, and R. A. Breuer, Classical Qu antum Gravity 4, 253 (1987)

1987
[56]

Ehlers and A

J. Ehlers and A. R. Prasanna, Classical Quantum Gravity 13, 2231 (1996)

1996
[57]

A. R. Prasanna, Phys. Lett. A 257, 120 (1999)

1999
[58]

Flauger and S

R. Flauger and S. Weinberg, Phys. Rev. D 97, 123506
[59]

Cusin, C

G. Cusin, C. Pitrou and J.-P. Uzan, Phys. Rev. D 97, 123527 (2018)

2018

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Einstein, Sitzungsber

A. Einstein, Sitzungsber. Preuss. Akad. Wiss. Berlin (M ath. Phys.) 1916, 688-696 (1916)

work page 1916

[2] [2]

Einstein, Sitzungsber

A. Einstein, Sitzungsber. Preuss. Akad. Wiss. Berlin (M ath. Phys.) 1918, 154-167 (1918)

work page 1918

[3] [3]

R. A. Hulse and J. H. Taylor, Astrophys. J. 195, L51-L53 (1975)

work page 1975

[4] [4]

J. H. Taylor and J. M. Weisberg, Astrophys. J. 253, 908-920 (1982)

work page 1982

[5] [5]

B. F. Schutz and F. Ricci, ‘Elements of gravitational wav es’, Chapter 2 of ‘Gravitational waves’, edited by I. Ciufol ini, V. Gorini, U. Moschella and V. Fre, published by IOP Publishing Ltd., pp.15-23, (2001)

work page 2001

[6] [6]

(90) Similarly, for χ2, we obtain the solution χ2(t) = (log a)χ2 0 + 1 2 ( h22χ2 0 +h21χ1 0 ) +χ1 0 +χ2

work page

[7] [7]

ˆχ(t) = 1 2 hD(t) ˆχ0

(91) The eigenvalues of χi can be determined from those of the diagonalised perturbati on matrix hD, i.e. ˆχ(t) = 1 2 hD(t) ˆχ0. (92) The perturbation matrix is of the form h(t) = ( ε11 ε12 ε12 −ε11 ) e{i(wt−kz)}, (93) having the eigenvalues λ = ± √ (ǫ11)2 + (ε12)2, so that its diagonalised form is hD(t) = ( √ (ε11)2 +ε12)2 0 0 − √ (ε11)2 + (ε12)2 ) e{i(w...

work page

[8] [8]

(95) • For ‘ ×’ polarization, ε11 = 0, and therefore ˆχ1(t) = 1 2ε12 exp{i(wt −kz)} ˆχ1 0 , ˆχ2(t) = − 1 2ε12 exp{i(wt −kz)} ˆχ2

work page