cosmological non-linear hydrodynamics with post-newtonian...

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ournal of Cosmology and Astroparticle PhysicsAn IOP and SISSA journalJ

Cosmological non-linear hydrodynamicswith post-Newtonian corrections

Jai-chan Hwang1, Hyerim Noh2 and Dirk Puetzfeld3

1 Department of Astronomy and Atmospheric Sciences, Kyungpook NationalUniversity, Taegu, Korea2 Korea Astronomy and Space Science Institute, Daejon, Korea3 Department of Physics and Astronomy, Iowa State University, Ames,IA 50011, USAE-mail: [email protected], [email protected] and [email protected]

Received 19 September 2007Accepted 18 February 2008Published 12 March 2008

Online at stacks.iop.org/JCAP/2008/i=03/a=010doi:10.1088/1475-7516/2008/03/010

Abstract. The purpose of this paper is to present general relativisticcosmological hydrodynamic equations in Newtonian-like forms using the post-Newtonian (PN) method. The PN approximation, based on the assumptions ofweak gravitational fields and slow motions, provides a way to estimate generalrelativistic effects in the fully non-linear evolution stage of the large-scale cosmicstructures. We extend Chandrasekhar’s first-order PN (1PN) hydrodynamicsbased on the Minkowski background to the one based on the Robertson–Walkerbackground. We assume the presence of Friedmann’s cosmological spacetimeas a background. In the background we include the 3-space curvature, thecosmological constant and general pressure. In the Newtonian order and 1PNorder we include general pressure, stress, and flux. We show that the Newtonianhydrodynamic equations appear naturally in the 0PN order. The spatial gaugedegree of freedom is fixed in a unique manner and the basic equations are arrangedwithout taking the temporal gauge condition. In this way we can convenientlytry alternative temporal gauge conditions depending on the mathematicalconvenience. We investigate a number of temporal gauge conditions under whichall the remaining variables are equivalently gauge invariant. We show thatcompared with the action-at-a-distance nature of the Newtonian gravitationalpotential, 1PN corrections make the propagation speed of a perturbed potentialdependent on the temporal gauge condition; we show that to 1PN order thephysically relevant propagation speed of gravity is the same as the speed oflight. Our aim is to present the fully non-linear cosmological 1PN equationsin a form suitable for implementation in conventional Newtonian hydrodynamic

c©2008 IOP Publishing Ltd and SISSA 1475-7516/08/03010+47$30.00

mailto:[email protected]



http://stacks.iop.org/JCAP/2008/i=03/a=010

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Cosmological non-linear hydrodynamics with post-Newtonian corrections

simulations with minimal extensions. The 1PN terms can be considered asrelativistic corrections added to the well-known Newtonian equations. Theproper arrangement of the variables and equations in combination with suitablegauge conditions would allow for possible future 1PN cosmological simulations tobecome more tractable. Our equations and gauges are arranged for that purpose.We suggest ways of controlling the numerical accuracy. The typical 1PN orderterms are about 10−6–10−4 times smaller than the Newtonian terms. However,we cannot rule out the possible presence of cumulative (secular) effects due to thetime-delayed propagation of the relativistic gravitational field with finite speed,in contrast to the Newtonian case where changes in the gravitational field arefelt instantaneously. The quantitative estimation of such effects is left for futurenumerical simulations. If the reader is interested in the applications of 1PNequations, she/he may go directly to section 4 of the paper after reading theintroduction.

Keywords: cosmological perturbation theory, classical tests of cosmology,cosmological simulations

Contents

1. Introduction 3

2. Basic quantities 62.1. Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2. Energy–momentum tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3. Fluid-frame kinematic quantities . . . . . . . . . . . . . . . . . . . . . . . . 112.4. Normal-frame kinematic quantities . . . . . . . . . . . . . . . . . . . . . . 132.5. ADM quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3. Derivations 163.1. Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2. Einstein’s equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3. ADM approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4. Complete equations to 1PN order 224.1. Background order: Friedmann world model . . . . . . . . . . . . . . . . . . 224.2. 0PN order: Newtonian hydrodynamics . . . . . . . . . . . . . . . . . . . . 224.3. 1PN order: post-Newtonian cosmological hydrodynamics . . . . . . . . . . 234.4. Ideal fluid case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5. Geodesic equations 285.1. Geodesic for a massive particle . . . . . . . . . . . . . . . . . . . . . . . . . 285.2. Null geodesic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6. Gauge issue 316.1. Gauge transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.2. Chandrasekhar’s gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.3. Uniform-expansion gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.4. Transverse-shear gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Journal of Cosmology and Astroparticle Physics 03 (2008) 010 (stacks.iop.org/JCAP/2008/i=03/a=010) 2


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6.5. Harmonic gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

6.6. Transformations between different gauges . . . . . . . . . . . . . . . . . . . 39

7. Propagation of Weyl tensor 40

8. Discussion 41

Acknowledgments 45

References 45

1. Introduction

The non-linear evolution of large-scale cosmic structures is usually investigated withinthe framework of Newtonian theory either in analytic studies or numerical simulations.Without investigating general relativistic effects, however, it is not clear whether theNewtonian theory is sufficient for handling such large-scale cosmological structures.If we regard Einstein’s gravity theory to be the correct framework on such cosmicscales, the relativistic effects should exist always. The point is to which level we canpractically ignore relativistic correction terms considering currently available levels ofobservations and numerical experiments. Provided that relativistic or non-linear effectsare not large, we have two well-known ways to estimate relativistic effects: one is therelativistic perturbation approach [1]–[3], and the other is using the post-Newtonian (PN)approximation [4]–[9].

In the perturbation study, we have recently systematically investigated the weaklynon-linear regimes based on Einstein’s gravity. We showed that, except for thegravitational wave coupling, the relativistic perturbation equations for the density andvelocity perturbation equations of zero-pressure irrotational fluid in near flat backgroundcoincide exactly with the Newtonian ones up to the second order [10, 11]. In [12]we presented results of second-order perturbations relaxing all the assumptions madein our previous works in [11]. We derived the general relativistic correction termsarising due to (i) pressure, (ii) multi-components, (iii) background curvature, and (iv)rotation. In the case of multi-component, zero-pressure, irrotational fluids under theflat background, we effectively do not have relativistic correction terms in the densityand velocity perturbation equations; thus the relativistic equations of the density andvelocity perturbation equations again coincide with the Newtonian ones. In the otherthree cases we generally have pure general relativistic correction terms. In the case ofpressure, the relativistic corrections appear even at the level of background and linearperturbation equations. In the presence of background curvature, or rotation, purerelativistic correction terms directly appear in the Newtonian equations of motion ofdensity and velocity perturbations to the second order. In the small-scale limit (farinside the horizon), relativistic equations including the rotation coincide with the onesin Newton’s gravity. In the zero-pressure case, the pure relativistic correction termsappear in the third order. As long as the perturbation method is applicable, these puregeneral relativistic third-order terms in the density and velocity perturbation equationsturn out to be independent of the horizon scale and small (∼10−5 order) compared with



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the second-order Newtonian/relativistic terms [13]. This is again true even in the multi-component zero-pressure fluids situation [14]. All analyses in [10, 11, 13, 12, 14] includethe cosmological constant in the background world model.

Our studies have shown, from the viewpoint of Einstein’s gravity, that Newtoniangravity is practically reliable near the horizon scale where structures are supposedto be weakly non-linear. Therefore, our works on general relativistic weakly non-linear perturbation theory justify the use of Newtonian physics in current large-scalenumerical simulations which cover the Hubble volume. Considering the action-at-a-distance nature of Newton’s gravity our result is a rather surprising one because therelativistic perturbation theory is applicable in fully relativistic situations, and thus onall scales including the super-horizon scale. Thus, our perturbation study ensures thatto the weakly non-linear order, Newtonian theory is reliable even near and beyond thehorizon scale. Here, we note that in the perturbation approach we have shown therelativistic/Newtonian correspondence to the second order without using the Newtoniangravitational potential (or the relativistic metric potentials) which cannot be identified tothe second order. In the case where Newtonian simulations rely on using the gravitationalpotential directly, it is unclear whether using the Newtonian potential could lead todifference between relativistic and Newtonian approaches even to the second order. This isan interesting point which requires examination of the basic equations used in Newtoniansimulation in practice. However, from the analytic point of view, for an irrotational flowthe gravitational potential of Newton’s gravity can always be replaced by the density usingthe Poisson equation, and thus can be removed in the momentum conservation equation.Indeed, the well studied cosmological quasi-linear analysis in the Newtonian context doesnot involve the gravitational potential in the basic set of equations; see [15].

Now, how about situation in the non-linear regime on scales much smaller than thecurrent horizon? On such scales the structures could be in a fully non-linear stage.However, if the relativistic gravity effect is small we can apply another approximationscheme which is well developed for handling isolated bodies in Einstein’s theory: thePN approximation. The PN approximation has been important for testing Einstein’sgravity. It presents the relativistic equations in a form similar to that of the well-knownNewtonian equations. In the PN method, by assuming that the relativistic effects aresmall, we expand relativistic corrections in powers of v/c. In nearly virialized systemswe have GM/(Rc2) ∼ (v/c)2. Thus, the PN approximation is applicable in the slowmotion and weak gravitational field regime. The first-order PN (1PN) approximationcorresponds to adding relativistic correction terms of the order (v/c)2 to the Newtonianorder terms. In this approach we recover the well-known equations of Newtonianhydrodynamics in the 0PN order [5]. The PN approximation is suitable to use instudying systems in which Newtonian gravity plays a dominant role, while the relativisticeffects are small but non-negligible. Well-known applications include the precession ofMercury’s perihelion, and various solar system tests of Einstein’s gravity theory likelight deflection [16]. Recent successful applications include the generation of gravitationalwaves from compact binary objects, and the weakly relativistic evolution stages of isolatedsystems of celestial bodies [17]. Another important application is in relativistic celestialmechanics, which is required by recent technology driven precise measurements of thesolar system bodies [18, 19]. Notice that all these applications are based on the PNapproximation of isolated systems assuming the Minkowski background spacetime.



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The cosmological post-Newtonian approximation was studied in the literature in [20]–[23]. In this work on the cosmological PN approximation we assume the presence of aRobertson–Walker background. Thus, we have the Friedmann world model built in as thebackground spacetime. We will show that the Newtonian hydrodynamic equations comeout naturally in the 0PN order which is the Newtonian limit. In the context of cosmiclarge-scale structures PN effects could be essentially important in handling gravitationalwaves and gravitational lensing effects. It is well known that Newtonian order treatmentof the gravitational lensing shows only half of the result from Einstein’s theory which isdue to ignoring the 1PN correction in the metric. We also anticipate that PN correctionterms could affect the dynamic evolution of large-scale structures, especially consideringthe action-at-a-distance nature of the Newtonian theory. This can be compared with therelativistic situation in which the gravitational field propagates with finite speed; in thiswork we will show that it propagates with the speed of light; see section 7. In this workwe present the complete set of hydrodynamic equations for studying 1PN effects in thecosmological context. The PN approach can be compared with our previous studies ofthe weakly non-linear regime based on the perturbation approach which is applicable inthe fully relativistic regime and on all scales. In comparison, although the PN equationsare applicable in weak gravity regions inside the horizon, these are applicable in fullynon-linear situations. Thus, the two approaches are complementary in enhancing ourunderstanding of the relativistic evolutionary aspects of the large-scale structures in theuniverse.

In the PN approximation we attempt to study the equations of motion and the fieldequations in Einstein’s gravity in the Newtonian way as closely as possible. The Newtonianand post-Newtonian equations of motion follow from the energy–momentum conservation.The Newtonian order potential and 1PN order metric variables are determined byEinstein’s equations in terms of the Newtonian matter and potential variables. Thus, themetric (or relativistic) contributions are reinterpreted as small correction terms to the well-known Newtonian hydrodynamic equations. The Newtonian equations naturally follow inthe 0PN order; see equations (106)–(108). The form of 1PN equations is affected by ourchoice of the gauge conditions. In this work we take unique spatial gauge conditions whichfix the spatial gauge modes completely; under these spatial gauge conditions the remainingvariables are all equivalently spatially gauge invariant; see section 6. The temporal gaugecondition (slicing condition) will be deployed to handle the mathematical treatment of theequations conveniently. We show how to choose the temporal gauge condition which alsoremoves the temporal gauge mode completely. Under each such temporal gauge conditionthe remaining variables are equivalently (spatially and temporally) gauge invariant. Wearrange the equations and the potential temporal gauge conditions in such a way thatthe final form of equations is suitable for numerical implementation in conventionalcosmological hydrodynamic simulations.

We present basic quantities and the derivation of our 1PN equations in sections 2and 3. If the reader is more interested in the application of the 1PN equations, she/he maywish to note that the basic set of cosmological 1PN equations is summarized in section 4.Thus, unless the reader is interested in the detailed derivation of 1PN equations, she/hemay go directly to section 4 which is presented in a self-contained form. In section 5 wepresent the null geodesic equations and the geodesic equation of a massive body in thecontext of the 1PN metric. The gauge issue is expounded in section 6. In section 6 we



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find that the propagation speed of the gravitational potential depends on the temporalgauge choice. In section 7 we resolve the propagation speed issue of the relativistic gravityby showing that the naturally gauge-invariant electric part of the Weyl tensor propagateswith the speed of light; a striking similarity with electromagnetism is explained. Wediscuss our results and several issues in section 8.

2. Basic quantities

2.1. Curvature

As the cosmological metric valid to the 1PN order, we take

g00 ≡ −[1 − 1

c22U +

1

c4

(2U2 − 4Φ

)]+ O−6,

g0i ≡ − 1

c3aPi + O−5,

gij ≡ a2

(1 +

1

c22V

)γij + O−4,

(1)

where x0 ≡ ct, and a(t) is the cosmic scale factor of the background Friedmann worldmodel. Indices a, b, c, . . . indicate spacetime, and i, j, k, . . . indicate space. Tildes indicatespacetime covariant quantities, i.e. spacetime indices of quantities with tilde are raised andlowered with the spacetime metric gab. The spatial index of Pi is based on γij in raisingand lowering indices with γij , an inverse of γij. The metric γij is the comoving (time-independent) spatial part of the Robertson–Walker metric; for several representations; seeequation (2) in [24]. In a flat Robertson–Walker background γij could become δij if wetake Cartesian coordinates. (Compared with our previous notation used in perturbation

studies in [10], we set the comoving part of the 3-space background metric g(3)ij ≡ γij and

use some of the italic indices to indicate the space.)In the metric convention we are following Chandrasekhar and Nutku’s notation [5, 7]

extended to the cosmological situation; see also Fock [4]. The 2U/c2 term in g00

gives the Newtonian limit, and if we ignore all the Newtonian and post-Newtoniancorrection terms we have the Robertson–Walker spacetime; see equations (106)–(108) forthe Newtonian limit, and equations (101)–(104) for the Robertson–Walker limit. Thus,our PN formulation is built on the cosmological background spacetime. O−n indicates(v/c)−n and higher order terms that we ignore. The expansion in equation (1) is valid to1PN order [5]. The dimensions are as follows:

[gab] = [gab] = 1, [γij] = [γij ] = 1, [a] = 1, [c] = LT −1,

[U ] = [V ] = [c2], [Pi] = [P i] = [c3], [Φ] = [c4],(2)

where L and T indicate the length and the time dimensions, respectively.In our metric convention in equation (1) we have ignored the possible presence of

(1/c2)(2C,i|j + Ci|j + Cj|i)-like terms in gij with Ci|i ≡ 0 by choosing the spatial C-gauge

conditions (C ≡ 0 ≡ Ci) which remove the spatial gauge mode completely; this will beexplained in section 6; a vertical bar indicates the covariant derivative based on γij. Undersuch spatial gauge conditions, we can equivalently regard all our remaining PN variablesas spatially gauge-invariant ones; see section 6. We still have the freedom to take the



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temporal gauge condition which will be chosen later depending on the mathematicalsimplification or feasibility of physical interpretation of the problem under consideration;see section 6: in the perturbation approach we termed this a gauge-ready approach [25].We also have ignored the spatially trace-free–transverse part of the metric (Cij with

Cii ≡ 0 ≡ Cj

i|j) because gravitational waves are known to show up from the 2.5PN order [8].

For the most general form of the metric, see equation (158). While it is true that thegravitational waves arise from the weakly relativistic fluid only at the 2.5PN order, theprimordial gravitational waves generated in the early acceleration (inflation) phase can beindependently present at the linear perturbational order; here we ignore such gravitationalwaves with cosmological and other astrophysical origins.

The inverse metric becomes

g00 = −[1 +

1

c22U +

1

c4

(2U2 + 4Φ

)]+ O−6,

g0i = − 1

c3

1

aP i + O−5,

gij =1

a2

(1 − 1

c22V

)γij + O−4.

(3)

The determinant of the metric tensor g is

√−g = a3√γ

[1 +

1

c2(3V − U) + O−4

], (4)

where γ is the determinant of γij.The connection is

Γ000 = − 1

c3U +

1

c5

(−2Φ +

1

aP iU,i

)+ L−1O−7,

Γ00i = − 1

c2U,i −

1

c4(2Φ,i + aPi) + L−1O−6,

Γ0ij = a2

{1

c

a

aγij +

1

c3

[(V + 2

a

a(U + V )

)γij +

1

aP(i|j)

]}+ L−1O−5,

Γi00 =

1

a2

{− 1

c2U ,i +

1

c4

[2 (U + V ) U ,i − 2Φ,i −

(aP i

)·]}+ L−1O−6,

Γi0j =

1

c

a

aδij +

1

c3

[V δi

j −1

2a

(P i

|j − P|i

j

)]+ L−1O−5,

Γijk = Γ

(γ)ijk +

1

c2

(V,kδ

ij + V,jδ

ik − V ,iγjk

)+ L−1O−4,

(5)

where Γ(γ)i

jk is the connection based on γij; we introduce P(i|j) ≡ 12(Pi|j + Pj|i) and

P[i|j] ≡ 12(Pi|j − Pj|i). We have U,0 = (1/c)(∂U/∂t) ≡ (1/c)U .

The Riemann curvature is

R000i = − 1

c5

(aPi +

1

aU,i|jP

j

)+ L−2O−7,

R00ij = L−2O−6,



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R0i0j =

1

c2

(aaγij + U,i|j

)+

1

c4

{a2

[V + 2

a

a(U + V )

]·γij +

[−aaU + 2a2 (U + V )

]γij

− 2U,(iV,j) − U,iU,j + U ,kV,kγij + 2Φ,i|j +(aP(i|j)

)· }+ L−2O−6,

R0ijk =

1

c3a2

[2

(V +

a

aU

),[j

γk]i −1

aP[j|k]i +

2K

aγi[jPk]

]+ L−2O−5,

Ri00j =

1

c2

(a

aδij +

1

a2U ,i

|j

)+

1

c4

{[V +

a

a

(U + 2V

)]δij −

1

a2

[2 (U + V ) U ,i

|j

+ U ,iU,j + U ,iV,j + U,jV,i − U ,kV,kδ

ij − 2Φ,i

|j

]+

1

2a2

[a

(P i

|j + P|i

j

)]· }

+ L−2O−6,

Ri0jk =

1

c3

[2

(V +

a

aU

),[j

δik] −

1

aP i

[j|k]

]+ L−2O−5,

Rij0k =

1

c3

[V,jδ

ik − V ,iγjk +

a

a

(U,jδ

ik − U ,iγjk

)+

1

2a

(P i

|j − P|i

j

)|k

]+ L−2O−5,

Rijkl = −2Kγj[kδ

il] +

1

c22(−a2γj[kδ

il] + V,j|[kδ

il] − V ,i

|[kγl]j

)+ L−2O−4.

(6)

It is convenient to have

Ai|jk ≡ Ai

|kj − R(γ)i

ljkAl, Ai|jk = Ai|kj + R

(γ)lijkAl,

R(γ)i

jkl = K(δikγjl − δi

lγjk

), R

(γ)ij = 2Kγij, R(γ) = 6K,

(7)

where K indicates the comoving (time-independent) part of the background spatialcurvature with dimension [K] = L−2; using a, K can be normalized to become 0 or±1. The Ricci curvature and the scalar curvature become

R00 =

1

c2

(3a

a+

Δ

a2U

)+

1

c4

{3V + 3

a

a

(U + 2V

)+ 6

a

aU

− 1

a2

[U ,i (U − V ),i + 2V ΔU − 2ΔΦ −

(aP i

|i)·]}

+ L−2O−6,

R0i =

1

c3

[2

(V +

a

aU

),i

+1

2a

(P j

|ji − ΔPi − 2KPi

)]+ L−2O−5,

Ri0 = − 1

c3

1

a2

[2

(V +

a

aU

),i

+1

2a

(P j i

|j − ΔP i + 2KP i)]

+ L−2O−5,

Rij =

2K

a2δij +

1

c2

[(a

a+ 2

a2

a2− Δ + 4K

a2V

)δij +

1

a2(U − V )|i j

]+ L−2O−4,

R =6K

a2+

1

c2

[6

(a

a+

a2

a2

)+ 2

Δ

a2U − 4

Δ + 3K

a2V

]+ L−2O−4,

(8)

where Δ is the Laplacian based on γij.



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The Weyl curvature is introduced as [26]

Cabcd ≡ Ra

bcd − 12

(δac Rbd − δa

dRbc + gbdRac − gbcR

ad

)+ 1

6R (δa

c gbd − δad gbc) . (9)

To 1PN order we have

C000i = − 1

c3

K

aPi + L−2O−5,

C00ij = L−2O−6,

C0i0j =

1

c2

[1

2(U + V ),i|j −

1

6Δ (U + V ) γij

]+ L−2O−4,

C0ijk =

1

c3

a

2

[(P l

|l[k − ΔP[k − 2KP[k

)γj]i − 2P[j|k]i

]+ L−2O−5,

Ci00j =

1

c2

1

a2

[1

2(U + V )|i j −

1

6Δ (U + V ) δi

j

]+ L−2O−4,

Ci0jk =

1

c3

1

2a

[(P l

|l[k − ΔP[k + 2KP[k

)δij] − 2P i

[j|k]

]+ L−2O−5,

Cij0k =

1

c3

1

4a

[ (P l i

|l − ΔP i + 2KP i)γjk −

(P l

|lj − ΔPj + 2KPj

)δik

+ 2(P i

|j − P|i

j

)|k

]+ L−2O−5,

Cijkl =

1

c2

[2

3Δ (U + V ) δi

[kγl]j + (U + V ),j|[k δil] − (U + V ),i

|[k γl]j

]+ L−2O−4.

(10)

Here we encounter non-vanishing traces involving K terms, like

Cbb0i = − 1

c3

K

aPi + L−2O−5, Cc

0ci = − 1

c3

2K

aPi + L−2O−5, (11)

whereas

Cc0c0 = L−2O−4, Cb

bij = L−2O−4, Ccic0 = L−2O−5, Cc

icj = L−2O−4. (12)

Apparently, equation (11) looks inconsistent with the trace-free nature of the Weyl tensor:Cb

bcd = 0 = Ccbcd. This point will be resolved later as we show in section 3.2 that the

K terms should be regarded as O−2 higher order terms in the PN expansion; see belowequation (93). Until we reach such a conclusion about the character of the K term wewill keep the K terms in our equations.

2.2. Energy–momentum tensor

The normalized fluid 4-vector ua with uaua ≡ −1 gives, to 1PN order,

u0 = 1 +1

c2

(1

2v2 + U

)+

1

c4

[3

8v4 + v2

(3

2U + V

)+

1

2U2 + 2Φ − viPi

]+ O−6,

ui ≡ 1

c

1

aviu0,



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u0 = −{

1 +1

c2

(1

2v2 − U

)+

1

c4

[3

8v4 + v2

(1

2U + V

)+

1

2U2 − 2Φ

]}+ O−6,

ui = a

{1

cvi +

1

c3

[vi

(1

2v2 + U + 2V

)− Pi

]}+ O−5,

(13)

where the index of vi is based on γij and v2 ≡ vivi; vi corresponds to the peculiar velocityfield. The energy–momentum tensor is decomposed into fluid quantities as follows [26]:

Tab = �c2

(1 +

1

c2Π

)uaub + p (uaub + gab) + qaub + qbua + πab. (14)

The energy density μ (≡�c2 + �Π) is decomposed into the material energy density �c2 andthe internal energy density �Π; p, qa, and πab are the isotropic pressure, the flux, and theanisotropic stress, respectively. We have qau

a ≡ 0, πabub ≡ 0, πc

c ≡ 0, and πab ≡ πba; thus

q0 = −1

c

1

aqiv

i, π0i = −1

c

1

aπijv

j, π00 =1

c2

1

a2πijv

ivj, (15)

and

T = −�c2

(1 +

1

c2Π

)+ 3p. (16)

We introduce

qi ≡1

caQi, πij ≡ a2Πij , (17)

where indices of Qi and Πij are based on γij. We take qa and πab to have post-Newtonianorders as introduced in equation (17); see for example [27]. This will be justified by theenergy conservation equation in the Newtonian context which will be derived later; seeequation (109). In a strictly single-component situation we can always remove Qi byfollowing the fluid element. However, there exist situations where we have the additionalflux terms present even when we follow the fluid elements; the fundamental origin ofsuch flux terms can be traced to the presence of additional components in the energy–momentum tensor. Thus, in our case it is more general and convenient to keep the fluxterms separately. Up to the 1PN order the condition πc

c ≡ 0 gives

Πii =

1

c2Πijv

ivj. (18)

We also set

� ≡ �, Π ≡ Π, p ≡ p. (19)

The dimensions are as follows:

[ua] = [ua] = 1,

[Tab] = [T ab ] = [T ab] = [p] = [qa] = [qa] = [πab] = [μ] = [�c2] = ML−1T −2,

[vi] = [vi] = [c], [Π] = [c2], [Qi] = [Qi] = [cqa] = [�c3],

[Πij ] = [Πij ] = [Πij ] = [πab] = [�c2].

(20)



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We have

T 00 = −�c2

{1 +

1

c2

(v2 + Π

)+

1

c4

[v2

(v2 + 2U + 2V + Π +

p

�

)− P ivi

+1

�

(2Qiv

i + Πijvivj

) ]+ O−6

},

T 0i = a�c2

{1

cvi +

1

c3

[vi

(v2 + 2U + 2V + Π +

p

�

)− Pi +

1

�

(Qi + Πijv

j)]

+ O−5

},

T i0 = −1

a�c2

{1

cvi +

1

c3

[vi

(v2 + Π +

p

�

)+

1

�

(Qi + Πi

jvj)]

+ O−5

},

T ij = �c2

{1

c2

(vivj +

p

�δij +

1

�Πi

j

)+

1

c4

[vivj

(v2 + 2U + 2V + Π +

p

�

)− viPj

+1

�

(Qivj + Qjv

i)− 2

�V Πi

j

]+ O−6

},

T = −�c2

[1 +

1

c2

(Π − 3

p

�

)+ O−6

].

(21)

2.3. Fluid-frame kinematic quantities

The projection tensor hab based on the fluid-frame 4-vector ua is defined as

hab ≡ gab + uaub. (22)

The kinematic quantities are defined as [26]

θab ≡ hcah

db uc;d, θ ≡ ua

;a, σab ≡ θ(ab) − 13θhab,

ωab ≡ θ[ab], aa ≡ ˜ua ≡ ua;bub,

(23)

where we have uaθab ≡ 0, uaσab ≡ 0, uaωab ≡ 0, uaaa ≡ 0, and θ ≡ θaa. The kinematic

quantities θ, aa, σab, and ωab are the expansion scalar, the acceleration vector, the sheartensor, and the vorticity tensor, respectively.

To 1PN order, the projection tensor becomes

h00 = − 1

c2v2 − 1

c4

[v2

(v2 + 2U + 2V

)− viPi

]+ O−6,

h0i = a

{1

cvi +

1

c3

[vi

(v2 + 2U + 2V

)− Pi

]}+ O−5,

hi0 = −1

c

1

avi

{1 +

1

c2v2 +

1

c4

[v2

(v2 + 2U + 2V

)− vjPj

]}+ O−7,

hij = δi

j +1

c2vivj +

1

c4

[vivj

(v2 + 2U + 2V

)− viPj

]+ O−6.

(24)



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The kinematic quantities become

θij =1

ca

(vi|j + aγij

)+

1

c3a

{vj (avi)

· + vi|j

(1

2v2 + U + 2V

)+ vk

(vivk|j + vjvi|k

)

+ a

[V +

a

a(U + 2V ) +

1

avkV,k +

1

2

a

av2

]γij + 2v[i (U + V ),j] − P[i|j]

}

+ L−1O−5, (25)

θ =1

c

(3a

a+

1

avi

|i

)+

1

c3

[3V + 3

a

aU +

3

aV,iv

i +1

aUvi

|i +1

2a3

(a3v2

)·+

1

2a

(v2vi

)|i

]

+ L−1O−5, (26)

σij =1

ca

(v(i|j) −

1

3vk

|kγij

)+

1

c3a

{v(i

(avj)

)·+ v(i|j)

(1

2v2 + U + 2V

)

+ vk(vk|(ivj) + v(ivj)|k

)− vivj

(a + 1

3vk

|k)

− 13γij

[12a

(v2

)·+ (U + 2V ) vk

|k + 12

(v2vk

)|k

] }+ L−1O−5, (27)

ωij =1

cav[i|j] +

1

c3a

{v[j

(avi]

)·+ v[i|j]

(1

2v2 + U + 2V

)+ vk

(vk|[jvi] + v[jvi]|k

)

+ 2v[i (U + V ),j] − P[i|j]

}+ L−1O−5, (28)

ai = − 1

c2U,i

(1 +

1

c2v2

)− 1

c42Φ,i +

[1 +

1

c2

(1

2v2 + U

)](∂

∂t+

1

av · ∇

)1

cui

+1

c4

(−v2V,i + vjP

j|i

)+ L−1O−6. (29)

From ucac ≡ 0 and ubθab ≡ 0 we have

a0 = −1

c

1

aviai, θ0i = −1

c

1

avj θij , θ00 =

1

c2

1

a2vivj θij , (30)

and similarly for σ0a and ω0a.The electric and the magnetic parts of the Weyl (conformal) tensor are introduced

as [26]

Eab ≡ Cacbducud, Hab ≡ 1

2η ef

ac Cefbducud, (31)

where ηabcd ≡ (1/√−g)εabcd (or ηabcd ≡

√−gεabcd) is the totally antisymmetric tensor

density with εabcd, the totally antisymmetric Levi-Civita symbol with ε0123 ≡ 1 (orε0123 ≡ −1). Eab and Hab are symmetric, trace-free and uaEab = 0 = uaHab. To 1PNorder we have

Eij = −Ci

00j = − 1

c2

1

a2

[1

2(U + V )|i j −

1

6Δ (U + V ) δi

j

]+ L−2O−4, (32)



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H ij = −1

2u0u0

[ηi k

0l

(C l

k0j −1

c

1

avlC0

k0j

)+ ηi 0

lk

1

c

1

avlCk

00j

]

=1

c3

1

2a3ηikl

{[1

2

(P m

|ml − ΔPl + 2KPl

)+

1

3vlΔ (U + V )

]γkj

+ Pl|kj − vl (U + V )|kj

}+ L−2O−5, (33)

where we introduced ηijk ≡ (1/√

γ)εijk (or ηijk ≡ √γεijk) with εijk, the totally

antisymmetric Levi-Civita symbol with ε123 ≡ 1 (or ε123 ≡ 1). In equation (33)

symmetrization over i and j indices is implied. Thus η0ijk =√

γ/(−g)ηijk. Indices

of ηijk are based on γij. E0a follows from ubEab = 0 which gives E0a = −(ui/u0)Eia =

−(vi/ca)Eia, and thus E0i ∼ L−2O−3 and E00 ∼ L−2O−4; and similarly for H0a. Fornon-vanishing K, Hab is not trace-free to L−2O−3, but as we mentioned earlier the terminvolving K is already O−2 order higher in PN expansion, and thus consistent.

For later use in section 7, we present the quasi-Maxwellian equations satisfied by theelectric and magnetic parts of the Weyl tensor

hab h

cdE

bd;c − ηabcdubσ

ecHde + 3Ha

b ωb

=4πG

c4

(2

3habμ,b − ha

b πbc

;c − 3ωabq

b + σab q

b + πab a

b − 2

3θqa

), (34)

hab h

cdH

bd;c + ηabcdubσ

ecEde − 3Ea

b ωb

=4πG

c4

{2 (μ + p) ωa + ηabcdub [qc;d + πce (ωe

d + σed)]

}, (35)

hac h

bd

˜Ecd +

(Hf

d;eh(af − 2adH

(ae

)ηb)cdeuc + habσcdEcd + θEab − E(a

c

(3σb)c + ωb)c

)

=4πG

c4

[−(μ + p)σab − 2a(aqb) − h(a

c hb)d

(qc;d + ˜πcd

)−

(ω (a

c + σ(ac

)πb)c

− 13θπab + 1

3

(qc

;c + acqc + πcdσcd

)hab

], (36)

hac h

bd

˜Hcd −

(Ef

d;eh(af − 2adE

(ae

)ηb)cdeuc + habσcdHcd + θHab − H(a

c

(3σb)c + ωb)c

)

=4πG

c4

[(qeσ

(ad − πf

d;eh(af

)ηb)cdeuc + habωcq

c − 3ω(aqb)]. (37)

These equations follow from the Bianchi identity [26, 28, 10].

2.4. Normal-frame kinematic quantities

The normal-frame 4-vector na, which is normal to the space-like hypersurfaces, is definedas ni ≡ 0 with a normalization nana ≡ −1. To 1PN order we have

n0 = 1 +1

c2U +

1

c4

(1

2U2 + 2Φ

)+ O−6, ni =

1

c3

1

aP i + O−5,

n0 = −1 +1

c2U − 1

c4

(1

2U2 − 2Φ

)+ O−6, ni ≡ 0.

(38)



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On setting ui ≡ 0 in equation (13) we recover the normal vector; thus

vi =1

c2Pi + LT −1O−4, (39)

where vi is, say, the velocity of the normal-frame vector. The projection tensor based onna becomes

hij = gij, h0i = g0i, h00 = O−6, hij = δi

j ,

hi0 = − 1

c3

1

aP i + O−5, h0

i = 0 = h00.

(40)

Kinematic quantities based on na become

θ =1

c3a

a+

1

c3

(3V + 3

a

aU +

1

aP i

|i

)+ L−1O−5, (41)

σij =1

c3a

(P(i|j) −

1

3P k

|kγij

)+ L−1O−5, (42)

ωij = 0, (43)

ai = − 1

c2U,i −

1

c42Φ,i + L−1O−6. (44)

σ0c follows from σacnc ≡ 0, and thus σ0i ∼ L−1O−6 and σ00 ∼ L−1O−9; we have ωab = 0.

Similarly, a0 follows from acnc ≡ 0; thus a0 ∼ L−1O−5.

The electric and the magnetic parts of Weyl tensor based on na give

Eij = −Ci

00j = − 1

c2

1

a2

[1

2(U + V )|i j −

1

6Δ (U + V ) δi

j

]+ L−2O−4, (45)

H ij =

1

2n0n0η

0iklC0jkl =

1

c3

1

2a3ηikl

[1

2

(P m

|ml − ΔPl + 2KPl

)γjk + Pl|kj

]+ L−2O−5. (46)

Thus, Eij is the same in both frames to the 1PN order. Comparing with equation (33)

H ij differs; in equation (46) symmetrizations over i and j indices are implied. The non-

vanishing K term which causes Hab to be not trace-free to L−2O−3 can be ignored becausethe K term is of the O−2 order.

2.5. ADM quantities

In the Arnowitt–Deser–Misner (ADM) notation the metric and fluid quantities are [29]

g00 ≡ −N2 + N iNi, g0i ≡ Ni, gij ≡ hij , (47)

n0 ≡ −N, ni ≡ 0, n0 = N−1, ni = −N−1N i, (48)

E ≡ nanbTab, Ji ≡ −nbT

bi , Sij ≡ Tij , S ≡ hijSij , Sij ≡ Sij − 1

3hijS,

(49)

where Ni, Ji and Sij are based on hij as the metric, and hij is the inverse metric of hij .



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The extrinsic curvature is

Kij ≡1

2N(Ni:j + Nj:i − hij,0) , Ki

i ≡ hijKij , Kij ≡ Kij − 13hijK

kk , (50)

where indices of Kij are based on hij; a colon ‘:’ denotes the covariant derivative based

on hij with Γ(h)i

jk ≡ 12hil(hjl,k + hlk,j − hjk,l). The intrinsic curvatures are based on the

metric hij

R(h)i

jkl ≡ Γ(h)i

jl,k − Γ(h)i

jk,l + Γ(h)m

jlΓ(h)i

km − Γ(h)m

jkΓ(h)i

lm,

R(h)ij ≡ R

(h)kikj, R(h) ≡ hijR

(h)ij , R

(h)ij ≡ R

(h)ij − 1

3hijR

(h).(51)

To the 1PN order we have

N = 1 − 1

c2U +

1

c4

(1

2U2 − 2Φ

)+ O−6,

N i = − 1

c3

1

aP i

(1 − 1

c2U

)+ O−7, Ni = − 1

c3aPi

[1 +

1

c2(2V − U)

]+ O−7,

hij = a2

(1 +

1

c22V

)γij + O−4, hij =

1

a2

(1 − 1

c22V

)γij + O−4,

(52)

Γ(h)i

jk = Γ(γ)i

jk +1

c2

(V,kδ

ij + V,jδ

ik − V ,iγjk

)+ L−1O−4, (53)

R(h)ij = R

(γ)ij − 1

c2

(V,i|j + γijΔV

)+ L−2O−4, R(h) =

1

a2

[6K − 1

c24 (Δ + 3K)V

]

+ L−2O−4, (54)

Kij = −1

caaγij −

1

c3a2

{[V +

a

a(U + 2V )

]γij +

1

aP(i|j)

}+ L−1O−5,

Kii = −1

c3a

a− 1

c3

[3

(V +

a

aU

)+

1

aP i

|i

]+ L−1O−5, (55)

E = �c2

{1 +

1

c2

(v2 + Π

)+

1

c4

[v2

(v2 + 2U + 2V + Π +

p

�

)+ 2UΠ − 2Piv

i

+1

�

(2Qiv

i + Πijvivj

) ]+ O−6

},

Ji = a�c2

{1

cvi +

1

c3

[vi

(v2 + U + 2V + Π +

p

�

)− Pi +

1

�

(Qi + Πijv

j)]

+ O−5

},

Sij = a2�c2

{1

c2

(vivj +

p

�γij +

1

�Πij

)+

1

c4

[vivj

(v2 + 2U + 4V + Π +

p

�

)

+ 2p

�V γij − 2v(iPj) +

2

�Q(ivj)

]+ O−6

},

S = �c2

{1

c2

(v2 + 3

p

�

)+

1

c4

[v2

(v2 + 2U + 2V + Π +

p

�

)− 2Piv

i

+1

�

(2Qiv

i + Πijvivj

) ]+ O−6

}. (56)



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A complete set of ADM equations will be presented in section 3.3. As we will explainin that section, using the quantities in equations (52)–(56), the ADM equations provideanother (probably an easier) complementary way to derive the 1PN equations.

3. Derivations

3.1. Equations of motion

Using equation (21) the energy and momentum conservation equations give

0 = −1

cT b

0;b =1

a3

(a3σ

)·+

1

a

[σvi +

1

c2

(Qi + Πi

jvj)]

|i

+1

c2�

[V +

1

avi (V − U),i +

a

av2 − p

�

]+ �T −1O−4, (57)

0 =1

aT b

i;b =1

a4

{a4

[σvi +

1

c2

(Qi + Πijv

j)]}·

+1

a

{σviv

j + Πji +

1

c2

[Qjvi + Qiv

j − 2 (U + V ) Πji

]}|j

+1

a

(1 − 1

c22U

)p,i −

1

a

(σU,i +

1

c2�v2V,i

)

+1

c2�

{vi

(∂

∂t+

1

av · ∇

)(U + 3V ) +

2

a[(U + V )U,i − Φ,i]

− 1

a(aPi)

· − 2

avjP[i|j] +

1

�a(U + 3V ),j Πj

i

}+ �LT −2O−4, (58)

where

σ ≡ �

[1 +

1

c2

(v2 + 2V + Π +

p

�

)]. (59)

For Qi = 0 = Πij, V = U , a = 1 and γij = δij these equations reduce to equations (64)and (67) in Chandrasekhar [5].

Equation (57) can be written in another form as

1

a3

(a3�∗)· + 1

a

(�∗vi

)|i +

1

c2

[1

a

(Qi

|i + Πijv

j|i

)+ �

(∂

∂t+

1

av · ∇

)Π

+

(3a

a+

1

a∇ · v

)p

]+ �T −1O−4 = 0, (60)

where

�∗ ≡ �

√−g

a3√

γu0 ≡ �

[1 +

1

c2

(1

2v2 + 3V

)+ O−4

], (61)

with a3√γ the background part of√−g. This corresponds to equation (117) in

Chandrasekhar [5]; see also equation (44) in [6]. The mass conservation (continuity)



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equation 0 = (�uc);c = ˜� + θ� gives

0 = c (�uc);c = c1√−g

(�√

−guc)

,c

=1

a3

(a3�∗)· + 1

a

(�∗vi

)|i + �T −1O−4

=1

a3

(a3�

)·+

1

a

(�vi

)|i +

1

c2�

(∂

∂t+

1

av · ∇

) (1

2v2 + 3V

)+ �T −1O−4. (62)

Thus, if we assume the mass conservation, equation (60) gives(∂

∂t+

1

av · ∇

)Π +

(3a

a+

1

a∇ · v

)p

�= − 1

�a

(Qi

|i + Πijv

j|i

)+ T −1O−2. (63)

The specific entropy is introduced as T dS = d(Π/c2) + p/c2d(1/�); thus along the flowwe have

T˜S =

1

c2

[˜Π + p (1/�)·

]=

1

c3

[(∂

∂t+

1

av · ∇

)Π +

(3a

a+

1

a∇ · v

)p

�+ T −1O−2

]. (64)

This shows that for an adiabatic fluid flow the LHS of equation (63) vanishes; this alsomakes the RHS vanish, which is naturally required for an ideal fluid. According toChandrasekhar [6]: ‘the conservation of mass and the conservation of entropy are notindependent requirements in the framework of general relativity. And the reason for theirindependence in the Newtonian limit is that in this limit (‘c2 → ∞’) (our equation (62))reduces simply to the equation of continuity’.

In [30] Blanchet et al suggested using the following new combination of variablesinstead of vi:

v∗i ≡ 1

c

√−g

a4√γ

1

�∗ T 0i , (65)

which becomes

v∗i = vi +

1

c2

[vi

(1

2v2 + U + 2V + Π +

p

�

)− Pi +

1

�

(Qi + Πijv

j)]

+ LT −1O−4. (66)

Notice that v∗i is directly related to the ADM flux vector in equation (56)

Ji = a�cv∗i . (67)

Equation T bi;b = 0 can be written as follows:

1

c

(√−g√

γT 0

i

)·

+

(√−g√

γT j

i

)|j

=1

2

√−g√

γT abgab|i

=

√−g√

γ�

{U,i +

1

c2

[U,i

(v2 + Π

)+ V,i

(v2 + 3

p

�

)− vjPj|i + 2Φ,i

]

+ L2T −2O−4

}, (68)



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and we have

√−g√

γT j

i = a3vj�∗v∗i +

√−g√

γ

[pδj

i + Πji +

1

c2

(Qjvi − 2V Πj

i − Πikvjvk

)+ LT −1O−4

].

(69)

Thus, equation (68) can be arranged in the following form:

1

a(av∗

i )· +

1

av∗

i|jvj =

1

a

{U,i +

1

c2

[U,i

(1

2v2 − U + Π

)+ V,i

(v2 + 3

p

�

)− vjPj|i + 2Φ,i

]}

− 1

�∗a

{[1 +

1

c2(3V − U)

] [pδj

i + Πji +

1

c2

(Qjvi − 2V Πj

i − Πikvjvk

)]}|j

+1

c2

v∗i

�∗

[1

a

(Qj + Πj

kvk)|j + �

(∂

∂t+

1

av · ∇

)Π +

(3a

a+

1

a∇ · v

)p

]

+ LT −2O−4, (70)

where the third line vanishes in an adiabatic ideal fluid.To the O−0 order, equations (57) and (58) give the Newtonian mass and momentum

conservation equations, respectively

1

a3

(a3�

)·+

1

a∇i

(�vi

)= 0, (71)

1

a(avi)

· +1

avj∇jvi +

1

a�

(∇ip + ∇jΠ

ji

)− 1

a∇iU = 0. (72)

Thus, we naturally have the Newtonian equations to 0PN order; e.g. see [21, 31]. Thiscan be compared with the situation in the perturbation approach where the Newtoniancorrespondence can be achieved only after suitable choice of different gauges for differentvariables, thus being a non-trivial result even to the linear order in perturbationapproach [3, 32]. Equations (71) and (72), as well as all the 1PN equations in this work,are fully non-linear.

In the Friedmann background we set the PN variables U , Φ, Pi, V , vi, Qi, andΠij equal to zero, and set � = �b, Π = Πb, and p = pb, with μ ≡ �(c2 + Π) andσb ≡ �b[1 + (1/c2)(Πb + pb/�b)]. Equations (57) and (62) give

μb + 3a

a(μb + pb) = 0, (73)

�b + 3a

a�b = 0. (74)

On subtracting the background part, equation (57) becomes

1

a3

[a3 (σ − σb)

]·+

1

a

[σvi +

1

c2

(Qi + Πi

jvj)]

|i

+1

c2�

[V +

1

avi (V − U),i +

a

av2 − p − pb

�

]+ �T −1O−4 = 0. (75)



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3.2. Einstein’s equations

We take Einstein’s equations in the form

Rab =

8πG

c4

(T a

b − 1

2T δa

b

)+ Λδa

b . (76)

The dimensions are as follows:

[gab] = 1, [Rab] = [Rab ] = [Rab] = L−2,

[Tab] = [T ] = [�c2] = ML−1T −2, [Λ] = L−2, [G�] = T −2,(77)

where M indicates the dimension of mass. To 1PN order, equations (8) and (21) give

−Λ +1

c2

(3a

a+

Δ

a2U + 4πG�

)+

1

c4

{3V + 3

a

a

(U + 2V

)+ 6

a

aU

− 1

a2

[U ,i (U − V ),i + 2V ΔU − 2ΔΦ −

(aP i

|i)·]

+ 8πG�

(v2 +

1

2Π +

3

2

p

�

) }= 0, (78)

1

c3

[2

(V +

a

aU

),i

+1

2a

(P j


)− 8πG�avi

]= 0, (79)

(2K

a2− Λ

)δij +

1

c2

[(a

a+ 2

a2

a2− Δ + 4K

a2V − 4πG�

)δij +

1

a2(U − V )|i j

]

=1

c48πG�

[1

2

(Π − p

�

)δij + vivj +

1

�Πi

j

], (80)

where equations (78), (79), and (80) are R00, R0

i , and Rij parts of equation (76), respectively.

We kept the O−4 term on the RHS of equation (80) in order to get the correct Friedmannbackground equation. To be consistent, we also need to keep the LHS to O−4 which willexplicitly involve 2PN order variables. But we do not need such efforts for our backgroundsubtraction process because all the O−4 order terms involve PN order variables, which donot affect the Friedmann background.

Our PN approximation is based on the Friedmann background world model. In orderto get the Friedmann background we set U , Φ, Pi, V , vi, Qi, and Πij equal to zero, andset � = �b, Π = Πb, and p = pb. Then, equations (78) and (80) give

3

c2

{a

a+

4πG

3

[�b

(1 +

Πb

c2

)+

3pb

c2

]− Λc2

3

}= 0, (81)

1

c2

{a

a+ 2

a2

a2− 4πG

[�b

(1 +

Πb

c2

)− pb

c2

]+

2Kc2

a2− Λc2

}δij = 0. (82)

These can be arranged to give

a

a= −4πG

3

[�b

(1 +

Πb

c2

)+

3pb

c2

]+

Λc2

3, (83)

a2

a2=

8πG

3�b

(1 +

Πb

c2

)− Kc2

a2+

Λc2

3, (84)



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which are the well-known Friedmann equations. Notice that the spatial curvature termKc2/a2 (and the cosmological constant term Λc2/3) already includes a c2 factor, i.e., theKc2/a2 term is of the order a2/a2 or 8πG�b/3, and similarly for Λc2/3. Thus, a bare Kterm or bare Λ term (meaning, without the c2 factor) should be regarded as already O−2

in PN expansion; see below equation (93).On subtracting the background equations, equations (78)–(80) give

1

c2

[Δ

a2U + 4πG (� − �b)

]+

1

c4

{3V + 3

a

a

(U + 2V

)+ 6

a

aU

− 1

a2

[U ,i (U − V ),i + 2V ΔU − 2ΔΦ −

(aP i

|i)·]

+ 8πG[�v2 + 1

2(�Π − �bΠb) + 3

2(p − pb)

]}= 0, (85)

1

c3

[2

(V +

a

aU

),i

+1

2a

(P j


)− 8πG�avi

]= 0, (86)

1

c2

{−

[Δ + 4K

a2V + 4πG (� − �b)

]δij +

1

a2(U − V )|i j

}= 0. (87)

To O−2 order, equation (85) gives

Δ

a2U = −4πG (� − �b) , (88)

which is Poisson’s equation in Newton’s gravity. Notice that in the Newtonian limit of ourPN approximation the homogeneous part of the density distribution is subtracted. Thisrevised form of Poisson’s equation is consistent with Newton’s gravity, and in fact, is animproved form avoiding Jeans’ swindle and allowing a proper derivation of the Newtoniancosmology consistent with the relativistic cosmology [33].

The decomposition of equation (87) into trace and trace-free parts gives

Δ + 3K

a2V = −4πG (� − �b) , (89)

(U − V )|i j = KV δij . (90)

Thus, compared with equation (88), ignoring the K term (see below), we have

V = U, (91)

where we ignored any surface term S with ΔS ≡ 0 and S,i|j ≡ 0. As we mentioned belowequation (84), the K term can be regarded as O−2 higher in the PN expansion; thus, thebare K terms in our PN equations can be duly ignored to 1PN order. Notice that wehave V = U even in the presence of anisotropic stress; this differs from the situation inthe perturbation approach where U and V in the zero-shear gauge (setting P i

|i = 0; see

later) are different in the presence of the anisotropic stress [3]. For later use we take adivergence of equation (86) which gives

Δ

a2

(V +

a

aU

)= 4πG

1

a

(�vi

)|i +

K

a3P i

|i, (92)



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where the last term containing a K term can be ignored because it is O−2 higher orderin PN expansion.

Here, we would like to more properly address the issue related to the backgroundcurvature term in our PN approach. On taking a divergence of equation (86) and usingequations (88), (89), and (71) we have

1

c3

(−6KV − 2K

aP i

|i

)= 0, (93)

which is apparently inconsistent. This is because the K term is related to the c−2 higherorder terms in the background Friedmann equation in equation (84). On expanding theequation to 2PN order, the bare K terms (i.e., without the c2 factor) in the above 1PNequations can be removed by subtracting the background equation; this is true for all theK terms appearing in our 1PN equations. In this sense, in our PN approximation weshould regard the K term as already O−2 higher order, and thus the bare K terms canbe ignored to 1PN order. The remaining equations are valid to 1PN order consideringgeneral background spatial curvature. Recent observations favour the flat backgroundworld model with non-vanishing cosmological constant. We include both the backgroundspatial curvature and the cosmological constant in our PN approximation which appearexplicitly only in the background Friedmann equations.

3.3. ADM approach

The ADM equations are [29, 3, 10]

R(h) = KijKij −2

3

(Ki

i

)2+

16πG

c4E + 2Λ, (94)

Kji:j −

2

3Kj

j,i =8πG

c4Ji, (95)

Kii,0N

−1 − Kjj,iN

iN−1 + N :iiN

−1 − KijKij −1

3

(Ki

i

)2 − 4πG

c4(E + S) + Λ = 0, (96)

Kij,0N

−1 − Kij:kN

kN−1 + KjkNi:kN−1 − Ki

kNk:jN

−1

= Kkk Ki

j −(

N :ij −

1

3δijN

:kk

)N−1 + R

(h)ij −

8πG

c4Si

j, (97)

E,0N−1 − E,iN

iN−1 − Kii

(E + 1

3S)− SijKij + N−2

(N2J i

):i

= 0, (98)

Ji,0N−1 − Ji:jN

jN−1 − JjNj:iN

−1 − Kjj Ji + EN,iN

−1 + Sji:j + Sj

i N,jN−1 = 0. (99)

Using the ADM quantities presented in section 2.5 we can show that equations (94)–(99) give the same equations as we already have derived from Einstein’s equations andthe energy and momentum conservations. Equation (94) gives equations (84) and (89).Equation (95) gives equation (86). Equation (96) gives equation (78). Equation (97) givesthe trace-free part of equation (87). Equations (98) and (99) give equations (57) and (58),respectively. The ADM equations are often used in the PN approach [30, 22, 17, 23]. Wepresent the ADM approach because the ADM equations show the fully non-linear structureof Einstein’s gravity in a form suitable for numerical treatment. In fact, equation (70)can be derived from equation (99) using the identification made in equation (67).



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4. Complete equations to 1PN order

Here we summarize the complete set of equations valid to 1PN order in the cosmologicalsituation. We arrange this section so that the reader who is interested in the applicationscan easily reach the 1PN equations and the gauge issue by reading this section togetherwith the introduction and discussion sections only.

4.1. Background order: Friedmann world model

Ignoring all the 0PN (Newtonian) and 1PN order variables we have the backgroundFriedmann world model based on the Robertson–Walker metric. Equation (1) gives theRobertson–Walker metric

ds2 = −c2 dt2 + a2γij dxi dxj . (100)

Several representations of γij can be found in equation (2) of [24]. Equations (73), (74),(83), and (84) provide equations describing the background Friedmann world model

a

a= −4πG

3

[�b

(1 +

Πb

c2

)+

3pb

c2

]+

Λc2

3, (101)

a2

a2=

8πG

3�b

(1 +

Πb

c2

)− Kc2

a2+

Λc2

3, (102)

μb + 3a

a(μb + pb) = 0, (103)

�b + 3a

a�b = 0, (104)

where μb ≡ �bc2(1 + Πb/c

2). Apparently, we include the spatial curvature K and thecosmological constant Λ. In equations (101)–(103) only two are independent. Fromequations (103) and (104) we have �bΠb + 3(a/a)pb = 0. The background variables a,�b, and Πb can be derived by solving these equations; in order to close the system, thespatial curvature K and cosmological constant Λ should be specified, and the pressure pb

should be provided by an equation of state.

4.2. 0PN order: Newtonian hydrodynamics

To the 0PN (Newtonian) order, the metric in equation (1) becomes

ds2 = −(

1 − 1

c22U

)c2 dt2 + a2γij dxi dxj . (105)

To the 0PN order, equations (71), (72), and (88) give

1

a3

(a3�

)·+

1

a∇i

(�vi

)= 0, (106)

1

a(avi)

· +1

avj∇jvi +

1

a�

(∇ip + ∇jΠ

ji

)− 1

a∇iU = 0, (107)

Δ

a2U + 4πG (� − �b) = 0. (108)



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These are exactly the Newtonian hydrodynamic equations with gravity. In the Newtoniancase energy conservation and mass conservation provide an additional equation which is(equation (63))(

∂

∂t+

1

av · ∇

)Π +

(3a

a+

1

a∇ · v

)p

�+

1

�a

(Qi

|i + Πijv

j|i

)= 0. (109)

In the non-expanding background this equation gives the well-known energy conservationequation in the Newtonian theory; see for example equation (2.36) in [34]. It is remarkablethat the complete Newtonian hydrodynamics is already built in as the 0PN orderapproximation of Einstein’s gravity [5, 21, 31]. The above equations show that this is trueeven considering the pressure, anisotropic stress, flux, and internal energy. The Newtonianorder hydrodynamic equations are valid in the presence of general background curvatureK and the cosmological constant Λ. The K and Λ terms do not appear explicitly in thehydrodynamic equations, and appear only in the background equations; as we will show,this is true even to 1PN order.

4.3. 1PN order: post-Newtonian cosmological hydrodynamics

To the 1PN order the metric in equation (1) becomes

ds2 = −[1 − 1

c22U +

1

c4

(2U2 − 4Φ

)]c2 dt2 − 1

c22aPi dt dxi + a2

(1 +

1

c22V

)γij dxi dxj .

(110)

It is well known that in order to handle light propagation properly we need to include theV term which is a 1PN order term. On ignoring this 1PN order term, and thus consideringonly the Newtonian order U term as in equation (105), we have the well-known factor 2missing in the Newtonian theory of light deflection compared with the Einstein’s gravityprediction: see equation (149). To 1PN order we have V = U ; see equation (91).

The energy and the momentum conservation equations valid to 1PN order are derivedin equations (75) and (58). These are

1

a3

(a3σ

)·+

1

a

[σvi +

1

c2

(Qi + Πi

jvj)]

|i+

1

c2�

(U +

a

av2 − p

�

)= 0, (111)

1

a4

{a4

[σvi +

1

c2

(Qi + Πijv

j)]}·

+1

a

[σviv

j + Πji +

1

c2

(Qjvi + Qiv

j − 4UΠji

)]|j

+1

a

(1 − 1

c22U

)p,i −

1

a

(σ +

1

c2�v2

)U,i +

1

c2�

[4vi

(∂

∂t+

1

av · ∇

)U

+2

a

(U2 − Φ

),i− 1

a(aPi)

· − 2

avjP[i|j] +

4

�aU,jΠ

ji

]= 0, (112)

where

σ ≡ �

[1 +

1

c2

(v2 + 2U + Π +

p

�

)]. (113)



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These equations can be written directly using � and vi as

1

a3

(a3�

)·+

1

a

(�vi

)|i +

1

c2�

(∂

∂t+

1

av · ∇

) (1

2v2 + 3U

)

= − 1

c2

[1

a

(Qi

|i + Πijv

j|i

)+ �

(∂

∂t+

1

av · ∇

)Π +

(3a

a+

1

a∇ · v

)p

], (114)

1

a(avi)

· +1

avi|jv

j − 1

aU,i +

1

a

p,i

�+

1

c2

[−1

av2U,i +

2

a

(U2 − Φ

),i− 1

a(aPi)

· − 2

avjP[i|j]

− 1

a

(v2 + 4U + Π +

p

�

)p,i

�+ vi

(∂

∂t+

1

av · ∇

) (1

2v2 + 3U +

p

�

)

−(

3a

a+

1

avj

|j

)p

�vi

]

=1

a�Πj

i|j +1

c2

1

�

{1

a4

[a4

(Qi + Πj

ivj

)]·+

1

a

(Qjvi + QiV

j)|j −

4

aUΠj

i|j

},

(115)

where we used equation (62). To the Newtonian order, equations (114) and (115) for �and vi reduce to equations (106) and (107), respectively. From equations (60) and (70)we can derive alternative forms

1

a3

(a3�∗)· + 1

a

(�∗vi

)|i = − 1

c2

[1

a

(Qi

|i + Πijv

j|i

)+ �

(∂

∂t+

1

av · ∇

)Π

+

(3a

a+

1

a∇ · v

)p

], (116)

1

a(av∗

i )· +

1

av∗

i|jvj =

1

a

{U,i +

1

c2

[(3

2v2 − U + Π + 3

p

�

)U,i + 2Φ,i − vjPj|i

]}

− 1

�∗a

{(1 +

1

c22U

) [pδj

i + Πji +

1

c2

(Qjvi − 2V Πj

i − Πikvjvk

)]}|j

+1

c2

v∗i

�∗

[1

a

(Qj + Πj

kvk)|j + �

(∂

∂t+

1

av · ∇

)Π +

(3a

a+

1

a∇ · v

)p

],

(117)

where

�∗ ≡ �

[1 +

1

c2

(1

2v2 + 3U

)],

v∗i ≡ vi +

1

c2

[(1

2v2 + 3U + Π +

p

�

)vi − Pi +

1

�

(Qi + Πijv

j)]

.

(118)

The hydrodynamic and thermodynamic variables p, Qi and Πij should be provided byspecifying the equations of state and the thermodynamic state of the system underconsideration.

The metric variables U , Φ and Pi can be expressed in terms of the matter (fluid)variables �, vi, Π, p, Qi and Πij by using Einstein’s equations. Equations (85),



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and (86) give

Δ

a2U + 4πG (� − �b) +

1

c2

{1

a2

[2ΔΦ − 2UΔU +

(aP i

|i)·]

+ 3U + 9a

aU + 6

a

aU

+ 8πG[�v2 + 1

2(�Π − �bΠb) + 3

2(p − pb)

]}= 0, (119)

Δ

a2Pi = −16πG�vi +

1

a

(1

aP j

|j + 4U + 4a

aU

),i

. (120)

Notice that to 1PN order Einstein’s equations do not involve the anisotropic stress orflux term. The K terms disappear because these can be removed by subtracting thebackground order equations; see below equation (93). All the 1PN order equationsare valid for general K and Λ both of which appear explicitly only in the backgroundequations.

To 0PN order, equation (119) gives

Δ

a2U = −4πG (� − �b) , (121)

which determines the Newtonian gravitational potential U . After determining U , to1PN order equation (119) determines Φ. Our conservation equations in equations (111)and (112) contain U terms. In order to handle these terms it was suggested in [22] that thefollowing Poisson-type equation provides better numerical accuracy. From equation (92)we have

Δ

a2U = 4πG

[a

a(� − �b) +

1

a

(�vi

)|i

], (122)

which also follows on taking a time derivative of equation (108), and using equations (106)and (104). We also have a U term in equation (119) which can be handled by using thefollowing Poisson-type equation:

Δ

a2U = 4πG

{(a

a− 2

a2

a2

)(� − �b)

− 1

a

[4a

a�vi +

1

a

(�vj

)|j vi +

1

a�vjvi

|j +1

a

(p,i + Πij

|j

)− 1

a�U ,i

]|i

}. (123)

This equation follows by taking a time derivative of equation (122) and usingequations (106), (107), and (104). Notice that, whereas only the spatial gradient ofthe potential U appears in the Newtonian limit, we have bare U terms present in the 1PNorder which could be troublesome in the numerical treatment [35].

In order to handle these equations we have the freedom to take one temporal gaugecondition. This corresponds to imposing a condition on P i

|i or Φ—preferably on P i|i as

we will explain later; see below equation (180). For convenience, in table 1 we summarizeseveral temporal gauge conditions. In order to solve the 1PN equations one may take anyone temporal gauge condition suggested in table 1. As any one of these gauge conditionsfixes the temporal gauge transformation completely, any remaining variable corresponds toa unique temporally gauge-invariant combination using the variable and the combinationused as the temporal gauge condition; see section 6.1. Similarly, our equations are already



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Table 1. Gauge conditions to 1PN order.

Temporal gauge Definition Equation

General gauge 1aP i

|i + nU + m aaU = 0 (210)

Chandrasekhar’s gauge n = 3, m = arbitrary (196)Uniform-expansion gauge n = m = 3 (199)Transverse-shear gauge n = m = 0 (200)Harmonic gauge n = 4, m = arbitrary (204)

Table 2. Symbols used in 1PN equations.

Symbol Definition Equation

U 0PN order perturbed metric, Newtonian gravitational potential (105), (108)V 1PN order perturbed metric; V = U (110)Φ 1PN order perturbed metric (110), (119)Pi 1PN order perturbed metric (110), (120)� Material energy density (14), (106)–(108)�Π Internal energy density (14), (109)p Pressure (14), (107)vi Peculiar velocity (13), (106)–(107)Qi Momentum flux (17), (109)Πij Anisotropic stress (17), (107), (109)σ Density combination (113), (111)�∗ Density combination (118), (116)v∗i Velocity combination (118), (117)

based on certain spatial gauge conditions (there are three spatial gauge degrees of freedom)which fix the spatial gauge transformation properties completely. Thus all variables canbe equivalently regarded as spatially gauge-invariant ones; see equation (182). In thisway, after taking any one temporal gauge condition in table 1, all the variables in our setof 1PN equations can be regarded as (fully) gauge invariant. Gauge related issues will beaddressed in detail in section 6.

The presence of gauge degrees of freedom should not be regarded as problematic.On the contrary, our freedom to choose the gauge condition is rather an advantagebecause we can strategically use it to make the equations simpler depending on thesituation. Furthermore, since the value of any gauge-invariant variable should beindependent of the gauge conditions that we take, by solving a given problem in twodifferent gauges independently, we can check and control the numerical accuracy; seesection 6. As is often the case in the gauge theory, the best gauge choice is usuallynot known a priori. Thus, we suggest trying various gauge conditions even for agiven problem, and finding the best one by experience. In fact, it is often the casethat even for a given problem different aspects of the problem can be best describedusing different gauge conditions. Several suggested gauge choices are summarized intable 1.

For convenience we present the symbols used in our 1PN equations in table 2.



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4.4. Ideal fluid case

We consider an ideal fluid, i.e. Qi ≡ 0 ≡ Πij , and assume that the adiabaticity condition inequation (109) applies. Thus, the internal energy is determined by the energy conservationequation (

∂

∂t+

1

av · ∇

)Π +

(3a

a+

1

a∇ · v

)p

�= 0. (124)

To the background order we have equations (101)–(104). To the Newtonian order we haveequations (106)–(109). Equations (111) and (112) become

1

a3

(a3σ

)·+

1

a

(σvi

)|i +

1

c2�

(U +

a

av2 − p

�

)= 0, (125)

1

a4

(a4σvi

)·+

1

a

(σviv

j)|j −

1

a

(σ +

1

c2�v2

)U,i +

1

a

(1 − 1

c22U

)p,i

+1

c2�

[4vi

(∂

∂t+

1

av · ∇

)U +

2

a

(U2 − Φ

),i− 1

a(aPi)

· − 2

avjP[i|j]

]= 0,

(126)

where σ is given in equation (113). These equations can be written as

1

a3

(a3�

)·+

1

a

(�vi

)|i +

1

c2�

(∂

∂t+

1

av · ∇

) (1

2v2 + 3U

)= 0, (127)

1

a(avi)

· +1

avi|jv

j − 1

aU,i +

1

a

p,i

�+

1

c2

[−1

av2U,i +

2

a

(U2 − Φ

),i− 1

a(aPi)

· − 2

avjP[i|j]

− 1

a

(v2 + 4U + Π +

p

�

)p,i

�

+ vi

(∂

∂t+

1

av · ∇

) (1

2v2 + 3U + Π +

p

�

) ]= 0. (128)

Alternative forms follow from equations (116) and (117):

1

a3

(a3�∗)· + 1

a

(�∗vi

)|i = 0, (129)

1

a(av∗

i )· +

1

av∗

i|jvj = −1

a

(1 +

1

c22U

)p,i

�∗ +1

a

[1 +

1

c2

(3

2v2 − U + Π +

p

�

)]U,i

+1

c2

1

a

(2Φ,i − vjPj|i

), (130)

where

�∗ ≡ �

[1 +

1

c2

(1

2v2 + 3U

)], v∗

i ≡ vi +1

c2

[(1

2v2 + 3U + Π +

p

�

)vi − Pi

]. (131)

All these equivalent three sets of equations are written in Eulerian forms. Einstein’sequations in (119)–(122) will provide the metric perturbation variables U , Φ and Pi

in terms of the fluid variables (�, vi, Π, and p), after taking one temporal gauge



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condition summarized in table 1 or others. The pressure term should be provided byan equation of state. In the case of a zero-pressure fluid with vanishing internal energywe can set p = 0 = Π. We can take any set of equations of motion and temporalgauge condition depending on the physical interpretation or mathematical conveniencein numerical treatments. All the 1PN equations in this section are valid considering Kand Λ.

5. Geodesic equations

Here, we present the geodesic equation for a massive particle, and the null geodesicequation for the photon or a massless particle. Equations are presented without takingthe temporal gauge condition.

5.1. Geodesic for a massive particle

The geodesic equation is given as

ua;bub = 0. (132)

This equation follows from the energy–momentum conservation equation T ba;b = 0 without

pressure, stress and flux; thus p = 0 = Πab and qa = 0 in equation (14). Thus, wenaturally anticipate that the geodesic equation is a case of our momentum conservationequation in equation (58) in the limit of vanishing pressure, stress and the flux, i.e., for adust test particle. Still, in the following we derive the geodesic equation to 1PN order.

Using

ua ≡ dxa

ds, (133)

equation (132) can be written in a usually known form of the geodesic equation as

d2xa

ds2= −Γa

bc

dxb

ds

dxc

ds. (134)

Thus, ds = dx0/u0, and we have

d

ds= u0 d

dx0= u0

(∂

∂x0+

dxj

ds

∂

∂xj

), (135)

where x0 ≡ ct. Equation (134) leads to

d2xi

dt2= −Γi

ab

dxa

dt

dxb

dt+

1

cΓ0

ab

dxa

dt

dxb

dt

dxi

dt. (136)

From equation (13) we have

dxi

dt=

1

avi, (137)



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and using the connections in equation (5) we can derive

d2xi

dt2=

1

a2

(U ,i − 2avi − Γ

(γ)ijkv

jvk)− 1

c2

1

a2

[avi

(− a

av2 + U + 2V

)+ 2vivj (U + V ),j

+ 2 (U + V ) U ,i − v2V ,i − 2Φ,i −(aP i

)· − (P i

|j − P|i

j

)vj

]. (138)

To the Newtonian order in a flat Minkowski background, and thus with a = 1 and

Γ(γ)i

jk = 0, we have

d2xi

dt2= U ,i, (139)

thus recovering the Newton’s gravity. Using

d2xi

dt2=

1

a

(vi − a

avi +

1

avi

|jvj − 1

aΓ

(γ)ijkv

jvk

), (140)

equation (138) can be written as

vi +a

avi +

1

avi

|jvj =

1

aU ,i − 1

c2

[vi

(− a

av2 + U + 2V

)+

2

avivj (U + V ),j

+2

a(U + V )U ,i − 1

av2V ,i − 2

aΦ,i − 1

a

(aP i

)· − 1

a

(P i

|j − P|i

j

)vj

]. (141)

As it should do, this equation coincides with the momentum conservation equation inequation (115) with vanishing pressure, stress and flux. We may set V = U withoutlosing any generality.

5.2. Null geodesic equation

We introduce the null 4-vector

ka ≡ dxa

dλ, (142)

where λ is an affine parameter along the geodesic. The geodesic and null equations are,respectively,

ka;bk

b = 0, kaka = 0. (143)

We introduce

k0 ≡ ν, ki = ν1

c

dxi

dt, (144)

where ν is the frequency, and the latter relation follows from

ki =dxi

dλ=

dx0

dλ

dxi

dx0= ν

dxi

c dt. (145)

Using the metric in equation (1) the null equation gives

a2

c2γij

dxi

dt

dxj

dt= 1 − 1

c22 (U + V ) +

1

c42aPi

dxi

dt. (146)



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Notice that to the leading order dxi/dt ∼ O(c). The 0-component of the geodesic equationleads to

1

ν

dν

dt= − a

a+

1

c22U,i

dxi

dt+

1

c2

(U − V

)− 1

c4aPi|j

dxi

dt

dxj

dt+

1

c44Φ,i

dxi

dt. (147)

To the background order we have ν ∝ 1/a which is interpreted as the cosmological redshift.The i-component of the geodesic equation, together with the 0-component and the nullequations, leads to

d2xi

dt2= −Γ

(γ)ijk

dxj

dt

dxk

dt− a

a

dxi

dt+

1

a2(U + V ),i − 1

c22 (U + V ),j

dxj

dt

dxi

dt

− 1

c2

[(U + V

) dxi

dt− 1

a

(P i

|j − P|i

j

) dxj

dt+

1

c2aPj|k

dxj

dt

dxk

dt

dxi

dt

]. (148)

To the leading order in a flat Minkowski background, we have

d2xi

dt2= (U + V ),i = 2U ,i. (149)

Compared with the Newtonian equation of motion in equation (139) we have the well-known additional factor 2 in the case of light bending in Einstein’s gravity; notice thatthe additional factor is caused by the 1PN effect of the V term in equation (148). Weemphasize that to the proper 1PN order in the cosmological background the null geodesicis described by equations (146), (147) and (148). We may set V = U without losing any

generality. In a flat Minkowski background we set a = 1 and γij = δij; thus Γ(γ)i

jk = 0.

On introducing

dxi

dt≡ c

aei, (150)

we have

d2xi

dt2=

c

a

(ei − a

aei +

c

aei

,jej

),

dν

dt= ν +

c

aν,ie

i. (151)

Equations (146), (147) and (148) can be written as

eiei = 1 − 1

c24U +

1

c32Pie

i, (152)

ν

ν+

c

a

1

νν,ie

i = − a

a+

1

c

2

aU,ie

i − 1

c2

1

aPi|je

iej +1

c3

4

aΦ,ie

i, (153)

ei +c

aei

|jej =

1

c

2

a

(U ,i − 2U,je

jei)

+1

c2

[−2Uei +

1

a

(P i

|j − P|i

j

)ej +

1

aPj|ke

jekei

].

(154)



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6. Gauge issue

6.1. Gauge transformation

We consider the following transformation between two coordinates xa and xa:

xa ≡ xa + ξa(xe). (155)

For a tensor quantity we use the tensor transformation property, between xa and xa

coordinates,

tab(xe) =

∂xc

∂xa

∂xd

∂xbˆtcd(x

e). (156)

Comparing tensor quantities at the same spacetime point, xa, we can derive the gaugetransformation property of a tensor quantity (see equation (226) in [10]):

ˆtab(xe) = tab(x

e) − 2tc(aξc,b) − tab,cξ

c + 2tc(aξd,b)ξ

c,d + tcdξ

c,aξ

d,b

+ ξd(2ξc

,(atb)c,d + 2tc(aξc,b)d + 1

2tab,cdξ

c + tab,cξc,d

). (157)

We considered ξa to the second perturbational order which will turn out to be sufficientfor the 1PN order. In the following we will consider the gauge transformation propertiesof the metric (metric tensor) and the fluid (energy–momentum tensor) variables.

As the metric we consider the following more generalized form:

g00 ≡ −[1 − 1

c22U +

1

c4

(2U2 − 4Φ

)]+ O−6,

g0i ≡ − 1

c3aPi + O−5,

gij ≡ a2

[(1 +

1

c22V

)γij +

1

c2

(2C,i|j + 2C(i|j) + 2Cij

)]+ O−4,

(158)

where Ci is transverse (Ci|i ≡ 0), and Cij is transverse and trace-free (Cj

i|j ≡ 0 ≡ Cii);

indices of Ci and Cij are based on γij. In equation (158) we introduced 10 independentmetric components: U and Φ together (1-component; U and Φ correspond to theNewtonian and 1PN order metric variables, respectively), Pi (3-components), V (1-component), C (1-component), Ci (2-components), and Cij (2-components). It is knownthat gravitational waves show up in the 2.5PN order [8]. Thus, we ignore the transverseand trace-free part, i.e. set Cij ≡ 0 to 1PN order.

We wish to keep the metric in the form of equation (158) in any coordinate system.

Thus, we take the transformation variable ξa to be a PN order quantity. We considercoordinate transformations which satisfy

ξ0 ≡ 1

cξ(2)0 +

1

c3ξ(4)0 + · · · , ξi ≡ 1

c2

1

aξ(2)i + · · · , (159)

where the indices of the ξ(2)i s are based on γij .



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The gauge transformation of equation (157) applied to the metric gives

U = U + ξ(2)0, (160)

Φ = Φ +1

2ξ(4)0 − 1

2Uξ(2)0 − 1

2aU,iξ

(2)i − 1

2ξ(2)0ξ(2)0 − 1

4ξ(2)0ξ(2)0 − 1

2

(1

aξ(2)iξ

(2)0,i

)·,

(161)

ξ(2)0

,i = 0, (162)

Pi = Pi −1

aξ

(4)0,i + a

(1

aξ

(2)i

)·+

2

a

(U + ξ(2)0

)ξ

(2)0,i +

1

aξ

(2)0,iξ

(2)0 +1

a2

(ξ(2)jξ

(2)0,j

),i,

(163)

(Δ + 3K)ΔC = (Δ + 3K)

(ΔC − 1

aξ

(2)i|i

)+

Δ

4a2

(ξ(2)0,iξ

(2)0,i

)− 3

4a2

(ξ

(2)0,iξ

(2)0,j

)|ij,

(164)

(Δ + 2K) Ci = (Δ + 2K)

(Ci −

1

aξ

(2)i

)− 1

3aξ

(2)j|ji −

1

a2

(ξ

(2)0,iξ

(2)0,j

)|j

+1

a∇iΔ

−1

[4

3(Δ + 3K) ξ

(2)j|j +

1

a

(ξ

(2)0,kξ

(2)0,j

)|kj]

, (165)

(Δ + 3K) V = (Δ + 3K)

(V − a

aξ(2)0

)− Δ + 2K

4a2

(ξ(2)0,iξ

(2)0,i

)+

1

4a2

(ξ

(2)0,iξ

(2)0,j

)|ij.

(166)

Equations (160) and (161) follow from the transformation of g00; equations (162) and (163)follow from g0i; equations (164)–(166) follow from gij.

Equation (162) shows that ξ(2)0 is spatially constant,

ξ(2)0 = ξ(2)0(t). (167)

After this, equation (164) gives

ΔC = ΔC − 1

aξ

(2)i|i. (168)

Thus, by choosing C ≡ 0 (ΔC ≡ 0 is enough) as the gauge condition (this implies thatwe set C = 0 to be valid in all coordinates), we have

ξ(2)i

|i = 0. (169)

Imposing the conditions in equations (167) and (169), equation (165) gives

Ci = Ci −1

aξ

(2)i . (170)

Thus, by choosing Ci ≡ 0 to be valid in all coordinates, i.e. choosing Ci ≡ 0 as the gaugecondition, we have

ξ(2)i = 0. (171)



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Under the conditions in equations (167) and (171) the remaining metric perturbationvariables transform as

U = U + ξ(2)0, V = V − a

aξ(2)0, Pi = Pi −

1

aξ

(4)0,i,

Φ = Φ + 12ξ(4)0 − 1

2Uξ(2)0 − 1

2ξ(2)0ξ(2)0 − 1

4ξ(2)0ξ(2)0.

(172)

Notice that for non-vanishing ξ(2)0, U and V transform differently under the gaugetransformation even in the flat cosmological background. Since the spatially constant ξ(2)0

can be absorbed by a global redefinition of the time coordinate, without losing generalitywe can set ξ(2)0 equal to zero:

ξ(2)0 ≡ 0, (173)

thus allowing V = U in any coordinate.

If we set ξ(2)0 = 0 but do not take the spatial gauge which fixes ξ(2)i , from

equations (160) to (166), the metric variables transform as

U = U, (174)

Φ = Φ +1

2ξ(4)0 − 1

2aU,iξ

(2)i, (175)

Pi = Pi −1

aξ

(4)0,i + a

(1

aξ

(2)i

)·, (176)

ΔC = ΔC − 1

aξ

(2)i|i, (177)

(Δ + 2K) Ci = (Δ + 2K)

(Ci −

1

aξ

(2)i

)− 1

3aξ

(2)j|ji +

4

3a∇iΔ

−1 (Δ + 3K) ξ(2)j

|j, (178)

V = V. (179)

Taking the spatial gauge conditions C = 0 = Ci we have ξ(2)i = 0 = ξ(2)0, and the

remaining metric perturbation variables transform as

U = U, V = V, Φ = Φ + 12ξ(4)0, Pi = Pi −

1

aξ

(4)0,i. (180)

In the perturbation analysis we call C ≡ 0 ≡ Ci the spatial C-gauge [10]. Apparently,

under these gauge conditions the spatial gauge transformation function to 1PN order, ξ(2)i ,

is fixed completely, i.e. ξ(2)i = 0. On taking such gauge conditions, the only remaining

gauge transformation function to 1PN order is ξ(4)0. This temporal gauge transformationfunction affects only Φ and Pi.

If we take Φ ≡ 0 as the temporal gauge condition, we have ξ(4)0 = 0; thus ξ(4)0 doesnot vanish even after imposing the temporal gauge condition and has general dependenceon the spatial coordinate as ξ(4)0(x). Thus, in this gauge, even after imposing the temporalgauge condition the temporal gauge mode is not fixed completely; this situation is similarto the temporal synchronous gauge one which sets the perturbed part of g00 equal to



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zero [1, 10]. However, if we take P i|i ≡ 0 as the temporal gauge condition, we have

ξ(4)0 = 0; thus the temporal gauge condition is fixed completely. In fact, from the gaugetransformation properties of Φ and Pi we can make the following combinations:

Φ,i + 12(aPi)

· , ΔΦ + 12

(aP i

|i)·

, (181)

which are invariant under the temporal gauge transformation, i.e. temporally gaugeinvariant. Meanwhile, U and V are already temporally gauge invariant. Since our spatialC-gauge also has removed the spatial gauge transformation function completely, we canrelate each remaining variable to a unique (spatially and temporally) gauge-invariantcombination. Thus, in this sense, we can equivalently regard all our remaining variablesas gauge-invariant ones. For example, from equations (174)–(179) we can show that

Pi + a(Ci + C,i

)·= Pi + a (Ci + C,i)

· − 1

aξ

(4)0,i, (182)

where we ignored K terms considering their O−2 higher order nature. Thus, Pi+a(Ci+C,i)·

is spatially gauge invariant and becomes Pi under the spatial C-gauge. This impliesthat Pi under the spatial C-gauge is equivalent to a unique gauge-invariant combinationPi + a(Ci + C,i)

·. Similarly, in the case of the temporal gauge, from equation (181) wecan show that ΔΦ under the P i

|i = 0 gauge is the same as a unique gauge-invariant

combination ΔΦ + 12(aP i

|i)·. In this sense, under gauge conditions which fix the gauge

mode completely, the remaining variables can be regarded as equivalently gauge-invariantones with corresponding gauge-invariant combinations.

In equation (1) we began our 1PN analysis by choosing the spatial C-gauge

C ≡ 0 ≡ Ci. (183)

On examining equations (174)–(179) we notice that the spatial C-gauge is most economicin fixing the spatial gauge mode completely without due alternative. In this sense thespatial C-gauge can be regarded as a unique choice and we do not lose any mathematicalconvenience by taking this spatial gauge condition. In the literature these conditions areoften expressed in the following forms. The spatial component of the harmonic gaugecondition sets the 1PN part of

gabΓiab = − 1√

−g

(√−ggic

),c

=1

c2

Δ

a2

(C ,i + Ci

)+ L−1O−4, (184)

equal to zero. We can also set the 1PN part of

gjk

(gij,k −

1

3gjk,i

)=

1

c2Δ

(4

3C,i + Ci

)+ L−1O−4, (185)

equal to zero. In either case we arrive at equation (183); we note that in equations (184)and (185) we assumed a flat (K = 0) background. We have shown that under theconditions in equation (183), the spatial gauge transformation is fixed completely withoutlosing any generality or convenience.

Meanwhile, we have not chosen the temporal gauge condition which can be bestachieved by imposing a condition on P i

|i . We may call this a gauge-ready strategy becauseour equations are arranged so that we can readily impose many interesting temporal gaugeconditions [25, 10]. It is convenient to choose this remaining temporal gauge condition



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depending on the problem that we encounter; or to try many different ones in order tofind the most suitable one or ones.

Now, let us consider the gauge transformation property of the energy–momentumtensor. We set ξ(2)0 ≡ 0. The gauge transformation property of equation (157) applied tothe energy–momentum tensor in equation (21) gives

� = � +1

c2�(2), (186)

Π = Π − 1

a

�,i

�ξ(2)i − �(2)

�, (187)

vi = vi +1

c2v

(2)i , (188)

Qi = Qi − �

[v

(2)i − a

(1

aξ

(2)i

)·+

1

avjξ

(2)j|i +

1

avi|jξ

(2)j

], (189)

p = p − 1

c2

1

a

(p,iξ

(2)i +2

3pξ

(2)i|i +

2

3Πi

jξ(2)j

|i

), (190)

Πij = Πij −1

c2

1

a

[2p

(ξ

(2)(i|j) −

1

3γijξ

(2)k|k

)+ Πij|kξ

(2)k + 2Πk(iξ(2)k

|j) −2

3γijΠ

kl ξ

(2)l|k

]. (191)

Equations (186) and (187) follow from the transformation property of T00; equations (188)and (189) follow from T0i; equations (190) and (191) follow from Tij. The transformation

functions �(2) and v(2)i are not determined; see below.

If we take the spatial C-gauge, i.e. set ξ(2)i ≡ 0, we have

� = � +1

c2�(2), Π = Π − �(2)

�, vi = vi +

1

c2v

(2)i , Qi = Qi − �v

(2)i ,

p = p + O−4, Πij = Πij + O−4.

(192)

Thus, we have

�

(1 +

1

c2Π

)= �

(1 +

1

c2Π

), vi +

1

c2

1

�Qi = vi +

1

c2

1

�Qi. (193)

Although the gauge transformation properties of � and Π are not determined individually,the energy density μ ≡ �(1 + Π/c2) is gauge invariant under our C-gauge. For vanishingΠ we have �(2) = 0. Similarly, the gauge transformation properties of vi and Qi are notdetermined individually. For vanishing flux term in any coordinate, Qi = 0, we have

v(2)i = 0; thus vi = vi +O−4. Thus, the gauge transformation property of vi is determined

only for vanishing flux term. From equation (118) we have

�∗ = �∗ +1

c2�(2), v∗

i = v∗i +

1

c2

(1

aξ

(4)0,i −

�(2)

�vi

). (194)

We can check that up to the 1PN order Einstein’s equations and the energyand momentum conservation equations in section 4 are invariant under the gaugetransformation. In the following we introduce several temporal gauge conditions each of



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which fixes the temporal gauge mode completely. In relativistic gravity, gauge conditionsconsist of four non-tensorial relations imposed on the metric tensor or the energy–momentum tensor. Our purpose is to employ the temporal gauge (slicing) condition tomake the resulting equations simple for mathematical/numerical treatment. The spatialC-gauge takes care of three spatial gauge degrees of freedom. Depending on the situationwe can also take alternative spatial gauge conditions for that purpose. In the followingwe impose the spatial C-gauge in equation (183).

6.2. Chandrasekhar’s gauge

Chandrasekhar’s temporal gauge condition in his equation (24) of [5] corresponds tosetting the 1PN part of

gij

(g0i,j −

1

2gij,0

)= −1

c3a

a− 1

c3

(1

aP i

|i + 3U

)+ L−1O−5, (195)

equal to zero; in the literature this is often called the ‘standard PN gauge’ [30]. We take

1

aP i

|i + 3U + ma

aU = 0, (196)

as ‘Chandrasekhar’s gauge’. We used the freedom to add an arbitrary m(a/a)U term withm, a real number. In this case equations (119) and (120) give

Δ


1

a

[U − (m − 4)

a

aU

],i

, (197)

Δ

a2U + 4πG (� − �b) +

1

c2

{2Δ

a2Φ − (m − 3)

a

aU +

[(6 − m)

a

a− m

a2

a2

]U

+ 8πG[�v2 + 1

2(�Π − �bΠb) + U (� − �b) + 3

2(p − pb)

]}= 0. (198)

Therefore, U , Pi, and Φ are determined by equations (121), (197), and (198), respectively.The variable U can be determined from equation (122). This completes our 1PN schemebased on Chandrasekhar’s gauge. When we handle this complete set of 1PN equationsnumerically, we should monitor whether the chosen gauge condition is satisfied always;this could be used to check the numerical accuracy.

6.3. Uniform-expansion gauge

The expansion scalar of the normal-frame vector, θ ≡ nc;c, is given in equation (41). It

is the same as the trace of extrinsic curvature Kii with a minus sign; see equation (55).

Taking the 1PN part of Kii equal to zero, i.e.,

1

aP i

|i + 3U + 3a

aU ≡ 0, (199)

can be naturally called the ‘uniform-expansion gauge’. In the literature it is often calledthe ‘ADM gauge’ [30], or the ‘maximal slicing condition’ in numerical relativity [36]. Thiscondition corresponds to the m = 3 case of Chandrasekhar’s gauge in equation (196).



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6.4. Transverse-shear gauge

The shear of the normal-frame vector is given in equation (42). Thus, the gauge condition

P i|i ≡ 0, (200)

can be called the ‘transverse-shear gauge’. In this case equations (119) and (120) give

Δ


4

a

(U +

a

aU

),i

, (201)

Δ

a2U + 4πG (� − �b) +

1

c2

{2Δ

a2Φ + 3U + 9

a

aU + 6

a

aU

+ 8πG[�v2 + 1

2(�Π − �bΠb) + U (� − �b) + 3

2(p − pb)

]}= 0. (202)

Therefore, U , Pi, and Φ are determined by equations (121), (201), and (202), respectively.The U and U terms can be determined using the Poisson-type equations in equations (122)and (123).

6.5. Harmonic gauge

We have

gabΓ0ab = − 1√

−g

(√−gg0a

),a

=1

c3a

a+

1

c3

(1

aP i

|i + 4U + 6a

aU

)+ L−1O−5. (203)

The well-known ‘harmonic gauge’ condition sets the 1PN part of equation (203) equal tozero; thus we take

1

aP i

|i + 4U + ma

aU ≡ 0, (204)

where we used the freedom to add an arbitrary m(a/a)U term. Combined withequation (184) the full harmonic gauge condition can be expressed by setting the 1PN partsof gabΓc

ab equal to zero. The harmonic gauge condition is frequently used in the context ofgravitational waves. In the cosmological perturbation approach, however, the harmonicgauge (often called the de Donder gauge) is known to be a bad choice because the conditioninvolves time derivatives of the metric variables; this leads to an incomplete gauge fixingand consequently we would have to handle higher order differential equations which isan unnecessary complication; see the appendix in [37]. In this gauge equations (119)and (120) give

Δ

a2Pi = −16πG�vi − (m − 4)

a

a

1

aU,i, (205)

Δ

a2U + 4πG (� − �b) +

1

c2

{2Δ

a2Φ − U − (m − 1)

a

aU +

[(6 − m)

a

a− m

a2

a2

]U

+ 8πG

[�v2 +

1

2(�Π − �bΠb) + U (� − �b) +

3

2(p − pb)

]}= 0. (206)

Therefore, U , Pi, and Φ are determined by equations (121), (205), and (206), respectively.The U and U terms can be determined from equations (122) and (123). The presence



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of the U term in equation (206) makes the speed of light propagating nature of the 1PNorder metric fluctuations apparent in this gauge condition. This time-delayed propagationof the gravitational field in the 1PN approximation can be compared with the action-at-a-distance nature of Poisson’s equation in the Newtonian order in equation (206).We anticipate that the relativistic time-delayed propagation could lead to secular (timecumulative) effects. This could be important even in the case in which the 1PN correctionterms are small compared with the Newtonian terms; in the large-scale clustering regionswe would expect (v/c)2 ∼ GM/(Rc2) 10−6–10−4; see section 8. Using g00 ≡ −1+2U/c2

we have

U ≡ U +1

c2

(−U2 + 2Φ

)+ c2O−4, (207)

and thus

U;cc =

Δ

a2U − 1

c2

(U + 3

a

aU + 2U

Δ

a2U

)+ T −2O−4

=Δ

a2U − 1

c2

[Δ

a2

(U2 − 2Φ

)+ U + 3

a

aU + 2U

Δ

a2U

]+ T −2O−4, (208)

and equation (206) can be written as

U;cc + 4πG (� − �b) +

1

c2

{Δ

a2U2 − (m − 4)

a

aU +

[(6 − m)

a

a− m

a2

a2

]U

+ 8πG[�v2 + 1

2(�Π − �bΠb) + 3

2(p − pb)

]}= 0. (209)

Thus, in the harmonic gauge the propagation speed of the gravitational potential U isthe same as the speed of light c. (When we mention the propagation speed, we areconsidering the d’Alembertian part of the wave equation, ignoring the non-linear termsand background expansion. In this case we have U;c

c = a−2ΔU − c−2U, and for K = 0,Δ becomes the ordinary Laplacian in flat space.)

Apparently, the wave speed of the metric (potential) can take an arbitrary valuedepending on the temporal gauge condition that we choose. For example, if we take

1

aP i

|i + nU + ma

aU ≡ 0, (210)

as the gauge condition, with n and m real numbers, equations (119) and (120) give

Δ

a2Pi = −16πG�vi −

1

a

[(n − 4) U + (m − 4)

a

aU

],i

, (211)

Δ

a2U + 4πG (� − �b)

+1

c2

{2Δ

a2Φ − (n − 3) U − (2n + m − 9)

a

aU +

[(6 − m)

a

a− m

a2

a2

]U

+ 8πG[�v2 + 1

2(�Π − �bΠb) + U (� − �b) + 3

2(p − pb)

]}= 0. (212)



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In this case, for n ≥ 3, the speed of propagation corresponds to

c√n − 3

, (213)

which can take an arbitrary value depending on our choice of the value of n. It becomesc for n = 4 (e.g., the harmonic gauge), and infinity for n = 3 (e.g., Chandrasekhar’sgauge and the uniform-expansion gauge). In the case of the transverse-shear gauge wehave n = 0; thus equation (202) is no longer a wave equation. At this point one maywonder what is the actual propagation speed of gravity in Einstein’s relativistic gravitytheory. For a physical resolution of the propagation speed issue in Einstein’s gravity, seesection 7.

6.6. Transformations between different gauges

Now, let us show how we can relate the equations and solutions known in one gaugecondition to the ones in any other gauge condition. Our spatial C-gauge condition already

fixed the gauge transformation function ξ(2)i ≡ 0, and we have ξ(2)0 ≡ 0. Under the

remaining (temporal) gauge transformation

t = t +1

c4ξ(4)0, xi = xi, (214)

we have

Φ = Φ + 12ξ(4)0, Pi = Pi −

1

aξ

(4)0,i, (215)

and all the other variables are gauge invariant.As an example, let us consider the two gauge conditions used in sections 6.4 and 6.5.

Let xa denote coordinates in the transverse-shear gauge in section 6.4, and xa denotecoordinates in the harmonic gauge in section 6.5. Thus, we have P i

|i ≡ 0 in xa, and

4U + m(a/a)U + a−1P i|i ≡ 0 in xa. From equation (215) we have

−(

4U + ma

aU

)= 0 − Δ

a2ξ(4)0, (216)

thus

Δ

a2ξ(4)0 = 4U + m

a

aU. (217)

Thus, when we transform the result known in the transverse-shear gauge to the harmonicgauge, we use equation (215) with ξ(4)0 given in equation (217). All the othervariables are invariant under the gauge transformation. We can show that under thistransformation equations (201) and (202) become equations (205) and (206), respectively.In this way, if we know solutions in any gauge condition the rest of the solutionsin other gauge conditions can be simply derived without solving the equations again.Thus, as in other gauge theories (like the Maxwell or Yang–Mills theories) the gaugecondition should be strategically deployed to our advantage in handling the problemmathematically.



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7. Propagation of Weyl tensor

In section 6.5 we have shown that the propagation speed of the gravitational potential, Uor U, depends on the temporal gauge (hypersurface) condition; see equation (212). Onemay wonder about what the physical propagation speed of the gravity is. We naively knowthat the answer must be the speed of light in Einstein’s gravity. It is well known thatthe speed of the gravitational waves is the speed of the light. What we are referring to asthe propagation speed of gravity differs from that of the gravitational waves. Followinga suggestion by Lev Kofman [38] we present a wave equation for the electric part of theWeyl tensor which is a gauge-invariant combination of the potentials U and V . We willshow that the wave equation of Eij shows that Eij propagates with the speed of light.

We introduce

Eij ≡ Eij, Hij ≡ Hij, (218)

where the indices of Eij and Hij are based on γij as the metric. From equations (32)and (33) we have

Eij = − 1

c2

[1

2(U + V ),i|j −

Δ

6(U + V ) γij

]+ L−2O−4, (219)

Hij =1

c3

1

2aη kl

(i

{[1

2

(P m

|ml − ΔPl

)+

1

3vlΔ (U + V )

]γkj) + Pl|kj) − vl (U + V ),k|j)

}

+ L−2O−5, (220)

where symmetrization symbols apply only over i and j indices. Although, we have V = U ,we keep V in this section. From the gauge transformation properties in equations (160),(162), and (166) we can show that Eij is gauge invariant to 1PN order; this must be true toall orders, albeit that in the present work we have only shown this explicitly up to the 1PNorder. Although the Weyl tensor includes the gravitational waves, in the post-Newtonianapproximation the gravitational waves start contributing from 2.5PN order [8].

The covariant equations for the Weyl tensor are presented in equations (34)–(37).Equations (36) and (37) give

Eij +a

aEij +

1

aEij|kv

k +1

avk|lEklγij −

1

a

[(2v

|k(i + vk

|(i

)Ej)k − 2vk

|kEij

]+

c

aη kl

(i Hj)k|l

= −4πG

c2a�

(v(i|j) −

1

3vk

|kγij

), (221)

Hij +a

aHij +

1

aHij|kv

k +1

avk|lHklγij −

1

a

[(2v

|k(i + vk

|(i

)Hj)k − 2vk

|kHij

]

+ η kl(i

{− c

aEj)k|l +

1

c

[1

a(V + 2U),l Ej)k +

(Ej)l −

a

aEj)l

)vk

− a

aγj)lv

mEmk +1

avj)v

m|lEmk +

1

a

(vmEj)m

)|l vk

]}

= −4πG

c3aη kl

(i Πj)k|l. (222)



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Since our aim is to find the propagation speed of the Weyl tensor Eij , without losinggenerality, we may take the normal frame. In this frame we can ignore the velocity of thenormal vector because vi ∼ LT −1O−2; see equation (39). Thus, we have

Eij +a

aEij = − c

aη kl

(i Hj)k|l, (223)

Hij +a

aHij =

c

aη kl

(i

[Ej)k|l −

1

c2(V + 2U),l Ej)k −

4πG

c4a2Πj)k|l

]. (224)

By combining these two equations we can derive a wave equation for Eij :

Eij + 3a

aEij +

(a

a+

a2

a2

)Eij

=c2

a2

[ΔEij − Ek

(i|j)k −1

2E

|kk(i j) +

1

2

(Ekl

|kl − ΔEkk

)γij +

1

2Ek

k|ij

]

+1

a2

{[(V + 2U),(i E

kj)

]|k−

[(V + 2U),k Eij

]|k

+1

2

[(V + 2U),k Ek(i

]|j)

− 1

2

[(V + 2U),l Ek

l

]|k

γij

}+

4πG

c2

(−ΔΠij + Πk

(i|j)k), (225)

where we used

η0ijkη0lmn = −ηijkηlmn = −3!δi[lδ

jmδk

n]. (226)

Using equation (219) we can show that terms on the right-hand side of equation (225)vanish to the O0 order; the same is true for apparently O−1 order terms on the right-handsides of equations (222) and (224). Equation (225) apparently shows the d’Alembertianstructure of the Eij equation. Thus, Eij propagates with the speed of light c, and weresolved the issue of the speed of propagation of the gravitational field. The strikinganalogue with the situation in Maxwell’s theory of electromagnetism will be explained insection 8.

Using V = U , equation (219) becomes

Eij = − 1

c2

(∇i∇j −

Δ

3γij

)U, (227)

where U can be regarded as a perturbed gravitational potential in Newtonian theory;see equations (106)–(108). The magnetic part of the Weyl tensor Hij does not have ananalogue in Newtonian theory [26].

8. Discussion

Here, we compare the PN approximation with the relativistic perturbation approach.The relativistic perturbation analysis is based on the perturbation expansion of themetric and energy–momentum variables in a given background. All perturbation variablesare assumed to be small. In linear order perturbations we keep only the first-orderdeviations from the background [1]–[3], whereas in the weakly non-linear perturbationswe keep higher order deviations to the desired order [10]. Unlike in the PN approach the



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perturbation analysis is applicable in the strong (fully relativistic) gravity regime and onall cosmological scales as long as the perturbations are linear or weakly non-linear.

Meanwhile, in the PN approach, by assuming weak gravitational fields and slowmotions, we try to provide the general relativistic correction terms for the Newtonianequations of motion. Thus, in the PN approach, in fact, we abandon the geometric spirit ofgeneral relativity and recover the concept of absolute space and absolute time. Althoughthis could be regarded as a shortcoming of the PN approach, in this way it providesthe relativistic effects in the forms of correction terms for the well-known Newtonianequations, thus enabling us to use simpler conventional (numerical) treatment. We expandthe metric and the energy–momentum variables in powers of v/c in a given backgroundspacetime. In a nearly virialized system we have GM/(Rc2) ∼ (v/c)2 which is assumed tobe small. Thus, no strong gravity situation is allowed and the results are valid inside thehorizon

√GM/(Rc2) ∼ R/(c/H) < 1. Unlike in the perturbation approach, however, the

resulting equations in the 1PN approximation can be regarded as fully non-linear. Thus,we have

PN approximation Relativistic perturbation

Weakly relativistic Fully relativisticFully non-linear Weakly non-linear

and the two approaches are complementary for understanding the relativistic non-linearevolution stage of the cosmological structures. As in the perturbation case, even in ourcosmological PN approach we assume the presence of a Robertson–Walker cosmologicalbackground.

In our PN approach, we assumed the presence of the Robertson–Walker spacetimeas the cosmological background metric, and subtracted the equations of the Friedmannworld model; see section 4.1. Notice that our background world model includes thebackground curvature, the cosmological constant, and the pressure. To the 0PN orderwe exactly recovered the well-known Newtonian hydrodynamic equations including theenergy equation with the internal energy, flux, and stresses; see section 4.2. Comparedwith Newton’s gravity, our Poisson’s equation in equation (108) is an improved versionin which the background density is subtracted. Our 1PN order metric and equationsare summarized in section 4.3. At this point, one may wonder whether subtracting thebackground world model is necessary. Here we remind the reader that Chandrasekhar’soriginal work in [5] and most of the conventionally known PN approximations arebased on the Minkowski background. Subtracting the background world model isan important step in our cosmological PN approximation. Often in the literature‘averaging’ and ‘back-reaction’ prescriptions are made in the relativistic perturbationapproach. Although the linear perturbations are supposed to cancel out on average,the non-linear perturbations are supposed to give non-vanishing net contributionsafter the averages, and those are often reinterpreted as renormalized parameters ofthe background world model; we are, however, not aware whether the concepts ofaveraging and back-reaction prescriptions used in the literature are well established.At the moment, we do not know how to, or whether we could, handle properly theeffects of PN corrections to the background cosmological model. In this context, ourequations (57)–(70) and (78)–(80) are presented to 1PN order without subtracting



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equations of the background world model; in particular; see equations (57), (60), (62),(78), and (80).

As we have summarized in the introduction, our studies of the weakly non-linearregime of a zero-pressure cosmological medium showed that the Newtonian equations arequite successful even near the horizon scale where the fluctuations are supposed to be nearthe linear stage. We have shown that to the second order in perturbations, except forthe presence of gravitational waves, the relativistic equations of zero-pressure irrotationalfluids coincide exactly with the Newtonian ones [10]–[12]. The pure relativistic correctionterms appearing in the third order are smaller than the relativistic/Newtonian second-order terms by a factor of the ‘perturbed Newtonian gravitational potential divided byc2’, i.e., ∼δΦ/c2 ∼ GM/(Rc2), which is small in nearly all scales inside the horizon.The third-order correction terms are also small as compared with the second-order termsby a factor δΦ/c2 ∼ δT/T ∼ 10−5 near the horizon due to the low level of temperatureanisotropies of CMB, and thus negligible [13, 14]. Furthermore, the relativistic correctionsare independent of the presence of the horizon scale in the relativistic world model. Still,we would like to emphasize that, although the terms themselves are small, in [12]–[14]we did find a substantial number of pure general relativistic correction terms appearingfrom the pressure, rotation, gravitational waves, and the background curvature to thesecond order, and the pure relativistic corrections terms appearing in the third order inthe zero-pressure fluids.

Considering the action-at-a-distance nature of the Newtonian gravity theory it isimportant to check the domain of validity of the Newtonian theory in the non-linearevolution of cosmological structures. The PN approach provides a way to find therelativistic effects in such a regime. We anticipate that the propagating nature of thegravitational field with finite speed in relativistic gravity theory, compared with theinstantaneous propagation in Newton’s theory, could lead to accumulative (or secular)relativistic effects. In order to estimate relativistic effects it would be appropriate toconsider a single cold dark matter component as a zero-pressure fluid without internalenergy. In such a case our equations in section 4.4 with p = 0 = Π and the metricvariables U , Φ, and Pi presented in various gauge conditions in table 1 would provide acomplete set of equations expressed in various forms.

In order to properly estimate the relativistic effects in the evolution of large-scalecosmic structures we have to implement our equations in a hydrodynamic cosmologicalnumerical simulation. The 1PN correction terms are (v/c)2 or GM/(Rc2) ∼ δΦ/c2 orderssmaller than the Newtonian terms. In Newtonian numerical simulations the maximumlarge-scale velocity field of a cluster flow reaches nearly 3000 km s−1 and the typical valuefor the velocity is about an order of magnitude smaller than this [39]. Thus, we mayestimate the 1PN correction terms to be of the order (v/c)2 10−6–10−4, that is, quitesmall.

The current large-scale structures are still near the linear regime, and due to theenhanced amplitude of the initial mass power spectrum in the small scale the gravitationalevolution causes non-linear regions to begin in the small scale and to propagate to largerscales [40]. The current belief is that in small-scale structures, where the non-linearityis important, the dynamical timescale is much longer than the light travel time over thisscale; thus the time-delay effect from relativistic gravity is not important. The 1PN orderrelativistic effects could be important in the tidal interactions among clusters of galaxies,



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where the dynamical timescale could become substantial compared with the light crossingtime of the scales involved [35].

We can perform numerical simulations based on any two different gauge conditions.The results should coincide after making a gauge transformation between the two gaugesas in section 6.6. Also, the two simulations should give identical results for any givengauge-invariant variable. These might provide a way to check the numerical accuracyof the simulations. Analyses based on different gauge conditions would lead to differentresults; this is because a given variable evaluated in two different gauges is actually twodifferent variables, unless the original variable is gauge invariant. Even for the temporalgauge condition (after fixing the spatial C-gauge) there are infinitely many different gaugeconditions available. Since each of the gauge conditions displayed in sections 6.2–6.5,summarized in table 1, fixes the temporal gauge mode completely (and the spatial gaugemodes are already completely fixed by our spatial C-gauge), all remaining variables areequivalently gauge invariant. Thus, apparently, the gauge invariance does not guaranteeassociating the variables with physically measurable quantities. The identification ofphysically measurable quantities out of infinitely many gauge-invariant candidates is stillan open issue which remains to be addressed. The gauge invariance ensures that thevalue should not depend on the gauge that we take and this fact can be used to checkand control the numerical accuracy of the simulations.

In a related context, we have shown that the propagation speed of the gravitationalpotential depends on the temporal gauge condition that we take; see equation (213).A similar situation occurs in classical electrodynamics. The propagation speeds of theelectromagnetic scalar and vector potentials are the speed of light c in the Lorenzgauge, whereas that of scalar potential becomes infinite in the Coulomb gauge. Inelectromagnetism the issue is well resolved by showing that the electric and magnetic fieldspropagate with speed c independently of the gauge condition adopted [41]. In our 1PNapproach, we have resolved the issue quite similarly to in the electromagnetism case. Thatis, in section 7 we have shown that the electric part of the Weyl tensor propagates withspeed c independently of the gauge condition adopted which is natural because the Weyltensor is gauge invariant. One noticeable difference compared to the electromagnetismcase is that our 1PN approach addresses fully non-linear gravitational dynamics whereaselectromagnetism concerns linear processes.

The referee has pointed out that the cosmological weak lensing [42] is now consideredas one of the most promising methods for carrying out precision cosmology andvarious future surveys are being planned: SNAP (SuperNova Acceleration Probe),LSST (Large Synoptic Survey Telescope), etc. When we study the general relativisticeffects in the cosmological structure formation process, indeed, it is likely thatthe weak gravitational lensing effects would be more promising observationally thanthe interactions among gravitational bodies. The conventional gravitational lenseffect (including the weak lens) already considers part of the 1PN metric with aconsequent difference of a factor of two from the pure quasi-Newtonian calculation;see equation (149). In section 5 we presented the null geodesic equations in thecosmological 1PN spacetime and the geodesic equation of a massive particle; the full1PN order metric is in equation (110) and the null geodesic equation to 1PN order canbe found in equations (146)–(148), respectively. Applications will be made in a laterwork.



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The referee has also pointed out another practically important issue related to thepossible contribution of the massive neutrinos to the non-linear evolution of large-scalestructure in the PN regime. As the neutrinos have velocity dispersion close to the speedof light it should be handled using the Boltzmann equation. The presence of a substantialamount of neutrinos could contaminate our study of PN corrections (based on the colddark matter and the cosmological constant) using the cosmological observations. Thiswould be relevant if the neutrino distribution is also in the PN regime, i.e., weaklyrelativistic but (fully) non-linear; if it is in the linear regime, we can handle the case usingthe linear perturbation theory which is applicable in the fully relativistic case and welldeveloped in the literature [43]. In the case of a plausible cosmological massive neutrinocontributing to the structure formation, its gravitational contribution would be weakwhereas its velocity is relativistic, thus violating one of the post-Newtonian assumptions(the weak gravity and the slow motion). We would like to investigate the situation in afuture work.

Just like the solar system tests of Einstein’s gravity theory, the non-linear evolution ofthe large-scale cosmic structure could provide another regime where the gravitational fieldis weak and the motions are slow so that the post-Newtonian approximation would bepractically adequate for describing the ever-present relativistic effects using Newtonian-like equations. The issue is whether such effects are significant enough to be detectedin future observations and numerical experiments. This important issue is left for futurenumerical studies. We hope that our set of 1PN order equations and our strategy for usingthose equations will be useful for such studies in the cosmological context. In more realisticcosmological situations we have dust and cold dark matter which can be approximated bytwo zero-pressure ideal fluids. We can also include pressure and dissipation effects in thecase of dust. The multi-component situation of cosmological 1PN hydrodynamics will beconsidered in later extensions of this work. As we include the cosmological constant inour PN formulation, our formulation can handle recent acceleration of the universe usingthe cosmological constant as the dark energy. In the literature the acceleration is oftenmodelled using a scalar field with special potentials as the dark energy. Including thescalar field in our cosmological PN approximation will be investigated in a future work.Extending our formulation to higher order PN approximation is also an apparent nextstep which would be tedious but straightforward.

Acknowledgments

We wish to thank Professor Lev Kofman for insightful discussions on the topic. Hesuggested to us to study the conformal tensor in order to resolve the speed of propagationissue for gravity. We thank Professor Dongsu Ryu and Dr Juhan Kim for useful discussionsfrom the perspective of possible numerical implementations. DP wishes to thank ProfessorM Pohl for his support of this project. HN was supported by Grant No C0022 from theKorea Research Foundation. JH was supported by the Korea Research Foundation Grantfunded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2007-313-C00322).

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