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E force: the of although additional dynamics force, local Newtonian a the t of describes from description arising a provides characteristics ture Nonlocal anti-Newtonian. called ogy, force, (Newtonian) tidal t and counterparts. part fields, electrodynamical electric to gravitoelectric/-magnetic the into so-called the Weyl part, the split can one Additionally, 3113–3127 (2009) 38 No. 24, Vol. A Letters Physics Modern ∗ a propagates fie gravitomagnetic to similar concepts: Gravitation waves. main tional two predicts relativity general h ic esrdfie yteEnti qainadteWy curv Weyl the and curva equation Riemann Einstein the the split by can defined tensor one relativity, Ricci general the of theory the In Introduction 1. c rsn drs:Sho fMteaisadPyis Queens Physics, and of School Address: Present ol cetfi ulsigCompany Publishing Scientific World aut fPyis nvriyo riv,1 l .Cz Str Cuza I. Al. 13 Craiova, of University Physics, of Faculty ASNs:9.0-,9.0J,47.75.+f 98.80.Jk, 98.80.-k, Nos.: PACS Keywords nature nonlocal the for necessary are gravitation. fields a both models relativistic Therefore, with waves. inconsistent gravitational are field temp of i Newtonian The solutions Ricci fields. explicit Weyl and manifest the Bianchi tions for the additi propagations use and an we constraints and of paper, field this Newtonian In the anti-Newtonian. includes Weyl The NTESGIIAC FTEWY CURVATURE WEYL THE OF SIGNIFICANCE THE ON NARLTVSI OMLGCLMODEL COSMOLOGICAL RELATIVISTIC A IN eaiitccsooy elcraue oain forma covariant curvature; Weyl cosmology; Relativistic : 9 , 10 u h rvtmgei edhsn etna anal- Newtonian no has field gravitomagnetic the but SKI DANEHKAR ASHKBIZ [email protected] eevd1 ac 2008 March 15 Received eie uut2009 August 5 Revised 2 , 6 – 9 edsrb h rvteeti eda the as field gravitoelectric the describe We 1 9 , 11 h elcraueas nldsan includes also curvature Weyl The 5 eseta oeswt purely with models that see We ∗ dosrc onigsolutions. sounding obstruct nd en u osm similarity some to due being nvriy efs T N,UK. 1NN, BT7 Belfast University, ,208 riv,Romania Craiova, 200585 ., rleouin fpropaga- of evolutions oral n aitv ouin of solutions radiative and ette opoieaset a provide to dentities nlfil,teso-called the field, onal 2 lism. nfc,teter of theory the fact, In dnia speed, identical t d n gravita- and lds nti equation instein tr tensor. ature uetno into tensor ture eWy curva- Weyl he ascurrents mass emagnetic he 1 – 4 2 A. Danehkar that provides a sounding analysis and a radiative description of force. We notice the Weyl tensor encoding the tidal force, a new force by its magnetic part, and a treatment of gravitational waves. Determination of gravitational waves and gravitomagnetism (new force) is ex- perimental tests of . 12 Gravitational radiation of a binary system of compact objects has been proposed to be detected by a resonant bar 13 or a laser interferometer in space, 14,15 such as the LIGO 16 and VIRGO. 17 A non-rotating compact object produces the standard Schwarzschild field, whereas a rotating body also generates the gravitomagnetic field. It has been suggested as a mechanism for the jet formation in quasars and galactic nuclei. 18,19 The resulting action of the gravitomagnetic fields and of the viscous forces implies that the formation of the accretion disk into the equatorial plane of the central body while the jets are ejected along vector perpendicularly to the equatorial plane. 2,18 The gravitomagnetic field implies that a rotating body e.g. the Earth affects the motion of orbiting satellites. This effect has been recently measured using the LAGEOS I and LAGEOS II satellites. 20 However, we may need counting some possible errors in the LAGEOS data. 21 Using two recent orbiting geodesy satellites (CHAMP and GRACE), it has been reported confirmation of general relativity with a total error between 5% and 10%. 22–24 In this paper, we describe kinematic and dynamic equations of the Weyl curva- ture variables, i. e., the gravitoelectric field as the relativistic generalization of the tidal forces and the gravitomagnetic field in a cosmological model containing the relativistic fluid description of matter. We use the convention based on 8πG =1= c. We denote the round brackets enclosing indices for symmetrization, and the square brackets for antisymmetrization. The organization of this paper is as follows. In Sec. 2, we introduce the 3 + 1 covariant formalism, kinematic quantities, and dy- namic quantities in a hydrodynamic description of matter. In Sec. 3, we obtain con- straint and propagation equations for the Weyl fields from the Bianchi and Ricci identities. In Sec. 4, rotation and distortion are characterized as wave solutions. In Sec. 5, we study a Newtonian model as purely gravitoelectric in an irrotational static spacetime and a perfect-fluid model, and an anti-Newtonian model as purely grav- itomagnetic in a shearless static and perfect-fluid model. We see that both models are generally inconsistent with relativistic models, allowing no possibility for wave solutions. Section 6 provides a conclusion.

2. Covariant Formalism According to the pattern of classical hydrodynamics, we decompose the spacetime metric into the spatial metric and the instantaneous rest-space of a comoving ob- server. The formalism, known as the 3 + 1 covariant approach to general rela- tivity, 25–30 has been used for numerous applications. 10,31–35 In this approach, we rewrite equations governing relativistic fluid dynamics by using projected vectors and projected symmetric traceless instead of metrics. 10,34 On the Significance of the Weyl Curvature 3

We take a four-velocity vector ua field in a given (3 + 1)-dimensional spacetime a to be a unit vector field u ua = −1. We define a spatial metric (or projector tensor) hab = gab + uaub, where gab is the spacetime metric. It decomposes the spacetime metric into the spatial metric and the instantaneous rest-space of an observer moving with four-velocity ua. 2,33,36 We get some properties for the spatial metric

b c a habu =0, ha hcb = hab, ha =3. (1) We also define the spatial alternating tensor as d εabc = ηabcdu , (2) where ηabcd is the spacetime alternating tensor, η = −4! |g|δ0 δ1 δ2 δ3 , δ b = g gcb, |g| = det g . (3) abcd p [a b c d] a ac ab The covariant spacetime derivative ∇a is split into a covariant temporal deriva- tive b T˙a··· = u ∇bTa···, (4) and a covariant spatial derivative d c DbTa··· = hb ha ···∇dTc···. (5) The projected vectors and the projected symmetric traceless parts of rank-2 tensors are defined by

b c d 1 cd V ≡ ha Vb,S ≡ h h − h hab Scd. (6) hai habi  (a b) 3 The equations governing these quantities involve a vector product and its general- ization to rank-2 tensors:

b c b cd [V, W ]a ≡ εabcV W , [S,Q]a ≡ εabcS dQ , (7)

c d c d [V,S]ab ≡ εcd(aSb) V , [V,S]habi ≡ εcdha S bi V . (8) We define divergences and rotations as

a b div(V ) ≡ D Va, (divS)a ≡ D Sab, (9)

b c c d (curlV )a ≡ εabcD V , (curlS)ab ≡ εcd(aD Sb) , (10) c d (curlS)habi ≡ εcdha D S bi . ˙ d We know that Dchab =0=Ddεabc, hab = 2u(au˙ b), andε ˙abc = 3u[aεbc]du˙ , then a a a u h˙ ab = −u˙ b and u ε˙abc = −u˙ εabc. From these points one can also define the relativistically temporal rotations as c b c d [u, ˙ V ]a = −u ε˙abcV , [u,S ˙ ]ab = −u ε˙cd(aSb) , (11) c d [u,S ˙ ]habi = −u ε˙cdha S bi . 4 A. Danehkar

The covariant spatial distortions are

1 Dha V bi = D(aVb) − 3 (divV )hab, (12)

2 Dha S bci = D(aSbc) − 5 h(ab(divS)c). (13) We decompose the covariant derivatives of scalars, vectors, and rank-2 tensors into irreducible components

∇af = −fu˙ a + Daf, (14)

V V˙ u u u u V c 1 u V u σ V c u ω, V V , ∇b a = −  hai b + a b ˙ c − 3 Θ a b − a bc − a[ ]b + Da b (15)

S S˙ u u S udu 2 u S u S dσ ∇c ab = −  habi c +2 (a b)d ˙ c − 3 Θ (a b)c − 2 (a b) dc d e −2εcdeu(aSb) ω  + DaSbc, (16) where

1 c 1 c DaVb = 3 DcV hab − 2 εabccurlV + Dha V bi, (17)

3 d 2 d DaSbc = 5 D Sdha h bic − 3 εdc(acurlSb) + Dha S bci. (18) We also introduce the kinematic quantities encoding the relative motion of fluids:

∇bua = Dbua − u˙ aub, (19)

1 Dbua = 3 Θhab + σab + ωab, (20) b whereu ˙ a = u ∇bua is the relativistic acceleration vector, in the frames of instan- a taneously comoving observersu ˙ a =u ˙ hai,Θ=D ua the rate of expansion of flu- 1 c ids, σab = Dha u bi = D(aub) − 3 habDcu a traceless (σab = σ(ab), a σa = 0); the shear tensor describing the rate of distortion of fluids, and ωab = D[aub] a a skew-symmetric tensor (ωab = ω[ab], ωa = 0); the vorticity tensor describing the rotation of fluids. 27,33,37 38,39 The vorticity vector ωa is defined by

1 bc ωa = − 2 εabcω , (21) a b 2 1 ab where ωau = 0, ωabω = 0 and the magnitude ω = 2 ωabω ≥ 0 have been imposed. Accordingly, we obtain

1 b c ωa = − 2 εabcD u . (22) 1 ~ The sign convention is such that in the Newtonian theory ~ω = − 2 ∇× ~u. We denote the covariant shear and vorticity products of the symmetric traceless tensors as

b cd c d [σ, S]a = εabcσ dS , [ω,S]habi = εcdha S bi ω . (23) On the Significance of the Weyl Curvature 5

The energy density and pressure of fluids are encoded in the dynamic quanti- ties, which generally have the contributions from the energy flux and anisotropic pressure:

Tab = ρuaub + phab +2q(aub) + πab, (24)

a a b qau =0, π a =0, πab = π(ab), πabu =0, (25) a b a 1 ab where ρ = Tabu u is the relativistic energy density relative to u , p = 3 Tabh b c b a the pressure, qa = −Thaibu = −ha Tcbu the energy flux relative to u , and πab = c d c d 1 cd Thabi = Tcdh ha u bi = h (au b) − 3 habh Tcd the traceless anisotropic stress. a  Imposing q = πab = 0, we get the solution of a perfect fluid with Tab = ρuaub+phab. In addition p = 0 gives the pressure-free matter or dust solution. 27,33,37

3. Cosmological Field Equations In the theory of general relativity, we describe the local nature of gravitational field nearby matter as an algebraic relation between the and the matter fields, i. e., the Einstein field equations:

1 Rab = Tab − 2 Tgab, (26) where Rab is the Ricci curvature, Tab the energy–momentum of the matter fields, c and T = Tc the of the energy–momentum tensor. The successive contractions of Eq. (26) on using of Eq. (24) lead to a set of relations: a b 1 b c Rabu u = 2 (ρ +3p), ha Rbcu = −qa, (27) c d 1 ha hb Rcd = 2 (ρ − p)hab + πab,

a a R = Ra ,T = Ta = −ρ +3p, R = −T, (28) where R is the Ricci scalar. The Ricci curvature is derived from the once contracted c : Rab = R acb. The Riemann tensor is split into symmetric (massless) traceless Cabcd and trace- ful massive Mabcd parts:

Rabcd = Cabcd + Mabcd. (29) The symmetric traceless part of the Riemann curvature is called the Weyl conformal curvature with the following properties:

a Cabcd = C[ab][cd], C bca =0= Ca[bcd]. (30) The nonlocal (long-range) fields, the parts of the curvature not directly determined locally by matter, are given by the Weyl curvature; propagating the Newtonian (and anti-Newtonian) forces and gravitational waves. It can be shown that the Weyl 6 A. Danehkar

N tensor Cabcd is irreducibly split into the Newtonian Cabcd and the anti-Newtonian AN Cabcd parts: N AN Cabcd = Cabcd + Cabcd, (31)

ab [a [a b] CN cd =4{u u[c + h [c}E d], (32)

AN e e Cabcd =2εabeu[cHd] +2εcdeu[aHb] , (33)

c d 1 cd e where Eab = Cacbdu u is the gravitoelectric field and Hab = 2 εacdC beu the gravitomagnetic field. They are spacelike and traceless symmetric. The traceful massive part of the Riemann curvature consists of the matter fields and the characteristics of local interactions with matter

ab 2 [a b] 2 a b M cd = 3 (ρ +3p)u u[ch d] + 3 ρh [ch d] [a b] [a b] [a b] [a b] − 2u h [cqd] − 2u[ch d]q − 2u u[cπ d] +2h [cπ d], (34) Therefore, the Weyl curvature is linked to the matter fields through the Riemann curvature.

3.1. Dynamic Formulas To provide equations governing relativistic dynamics of matter, we use the Bianchi identities

∇[eRab]cd =0. (35) On substituting Eq. (29) into Eq. (35), we get the dynamic formula for the Weyl conformal curvature 3,40,41:

d 1 1 d ∇ Cabcd = −∇[a(Rb]c − 6 gb]cR)= −∇[a(Tb]c − 3 gb]cTd ) ≡ Jabc. (36) On decomposing Eq. (36) along and orthogonal to a 4-velocity vector, we obtain 1,2 1,2 constraint (C a) and propagation (P ab) equations of the Weyl fields in a form analogous to the Maxwell equations 10,42–44:

1 b 1 1 C a ≡(divE)a − 3ω Hab − [σ, H]a − 3 Daρ + 3 Θqa 1 b 3 1 − 2 σabq + 2 [ω, q]a + 2 (divπ)a =0, (37)

2 b C a ≡(divH)a +3ω Eab + [σ, E]a + ωa(ρ + p) 1 1 1 b + 2 curl(q)a + 2 [σ, π]a − 2 ω πab =0, (38)

1 ˙ P ab ≡curl(H)ab + 2[u,H ˙ ]habi − Ehabi − ΘEab + [ω, E]habi c 1 1 +3σcha E bi − 2 σab(ρ + p) − 2 Dha q bi − u˙ ha q bi 1 1 1 1 e − 2 π˙ habi − 6 Θπab + 2 [ω, π]habi − 2 σ ha π bie =0, (39) On the Significance of the Weyl Curvature 7

2 ˙ P ab ≡curl(E)ab + 2[u, ˙ E]habi + Hhabi +ΘHab − [ω,H]habi c 3 1 1 − 3σcha H bi − 2 ωha q bi − 2 [σ, q]habi − 2 curl(π)ab =0. (40) The twice contracted Bianchi identities present the conservation of the total energy momentum tensor, namely b b 1 ∇ Tab = ∇ (Rab − 2 gabR)=0. (41) It is split into a timelike and a spacelike momentum constraints: 3 a ab C ≡ ρ˙ + (ρ + p)Θ + div(q) + 2u ˙ aq + σabπ =0, (42)

4 4 b C a ≡(ρ + p)u ˙ a + Dap +q ˙hai + 3 Θqa + σabq b − [ω, q]a + (divπ)a +u ˙ πab =0. (43) They provide the conservation law of energy-momentum, i. e., how matter deter- mines the geometry, and describe the motion of matter.

3.2. Kinematic Formulas To provide the equations of motion, we use the Ricci identities for the vector field ua: d 2∇[a∇b]uc = Rabcdu . (44)

We substitute the vector field ua from the kinematic quantities, using the Einstein equation, and separating out the orthogonally projected part into trace, symmetric traceless, and skew symmetric parts. We obtain constraints and propagations for the kinematic quantities as follows 42: 3 ˙ 1 2 a ab ab 1 P ≡ Θ+ 3 Θ − div(u ˙) − u˙ u˙ a − (ωabω − σabσ )+ 2 (ρ +3p)=0, (45) 4 2 b 1 P a ≡ ω˙ hai + 3 Θωa − σa ωb + 2 curl(u ˙)a =0, (46)

5 c 2 P ab ≡Eab − Dha u˙ bi − u˙ ha u˙ bi +σ ˙ habi + σcha σ bi + 3 σabΘ+ ωha ω bi 1 − 2 πab =0. (47) Equation (45), called the Raychaudhuri propagation formula, is the basic equation of gravitational attraction. 45 In Eq. (46), the evolution of vorticity is conserved by the rotation of acceleration. Equation (47) shows that the gravitoelectric field is propagated in shear, vorticity, acceleration, and anisotropic stress. The Ricci identities also provide a set of constraints: 5 a C ≡ div(ω) − ωau˙ =0, (48)

6 2 C a ≡ 3 DaΘ − (divσ)a + curl(ω)a + 2[u, ˙ ω]a − qa =0, (49) 7 C ab ≡ Hab − curl(σ)ab + Dha ω bi + 2u ˙ ha ω bi =0. (50) Equation (48) presents the divergence of vorticity. Equation (49) links the divergence of shear to the rotation of vorticity. Equation (50) characterizes the gravitomagnetic field as the distortion of vorticity and the rotation of shear. 8 A. Danehkar

4. Gravitational Waves

1,2 We now obtain the temporal evolution of the dynamic propagations (P ab) in a a perfect-fluid model (q = πab = 0):

1 b 1 C a = (divE)a − 3ω Hab − [σ, H]a − 3 Daρ =0, (51)

2 b C a = (divH)a +3ω Eab + [σ, E]a + ωa(ρ + p)=0, (52)

1 ˙ P ab =curl(H)ab + 2[u,H ˙ ]habi − Ehabi − ΘEab + [ω, E]habi c 1 +3σcha E bi − 2 σab(ρ + p)=0, (53)

2 ˙ P ab =curl(E)ab + 2[u, ˙ E]habi + Hhabi +ΘHab − [ω,H]habi c − 3σcha H bi =0. (54) To the first order, the evolution of propagation is ˙ 1 2 ¨ 3 1 4 1 2 P ab =D Eab − Ehabi − 2 Dha C bi − 3 ΘP ab + curl(P )ab 4 2 7 ˙ ˙ c cd − 3 Θ Eab − 3 ΘEhabi − ΘEab − ΘEcha σ bi − σcdE σab cd cd ˙ c d c d + E σcaσbd − σ σc(aEb)d + εcd(aEb) ω + εcd(aEb) ω˙ 4 c c d c + 3 Θ[ω, E]habi + 4Θσcha E bi +ε ˙cd(aEb)ω + 3σ ˙ cha E bi ˙ c 1 c +3σcha E bi − 2curl([u, ˙ E])ab − 2 Dha ω H bic 3 8 − 2 Dha [σ, H] bi + 3 Θ[u, ˙ H]habi + curl([ω,H])ab c c e d c d + 3curl σ H − σe ε D H +2ε H˙ u˙ cha bi  cd(a b) cd(a b) c d c d 1 +2εcd(aHb) u¨ + 2ε ˙cd(aHb) u˙ − 2 σab(ρ ˙ +p ˙) 1 − 3 Θσab(ρ + p)=0. (55) We neglect products of kinematic quantities with respect to the undisturbed met- rics (unexpansive static spacetime). We can also prevent the perturbations that are merely associated with coordinate transformation, since they have no physical significance. In free space, we get ˙ 1 2 ¨ 3 1 4 1 2 P ab = D Eab − Ehabi − 2 Dha C bi − 3 ΘP ab + curl(P )ab =0. (56) 2 ¨ To be consistent with Eqs. (51)–(54), D Eab − Ehabi has to vanish. Similarly, the 2 2 ¨ evolution of P ab shows that D Hab − Hhabi = 0. The evolutions reflect that the divergenceless and nonvanishing rotation of the Weyl fields are necessary conditions for gravitational waves:

(divE)a =(divH)a =0, curl(E)ab 6=0 6= curl(H)ab. (57) Indeed, the rotation of the Weyl fields characterizes the wave solutions. The grav- itomagnetic field is explicitly important to describe the gravitational waves, and is comparable with the Maxwell fields. On the Significance of the Weyl Curvature 9

We use Eq. (18) to provide two more constraints:

8 3 d 2 d C abc ≡ DaEbc − Dha E bci − 5 D Edha h bic + 3 εdc(acurl(E)b) =0, (58)

9 3 d 2 d C abc ≡ DaHbc − Dha H bci − 5 D Hdha h bic + 3 εdc(acurl(H)b) =0. (59)

To first order, divergence of Eq. (58) is

a 8 2 a 3 a d D C abc =D Ebc − D Dha E bci − 5 D D Edha h bic 1 a d 1 a d + 3 D εdcacurl(E)b + 3 εdcbD curl(E)a =0. (60)

On substituting Eq. (54), it becomes

a 8 2 a 3 a d 3 a 1 D C abc =D Ebc − D Dha E bci − 5 D D Edha h bic − 5 D C ha h bic 2 a d 2 9 a d 3 a + 3 D ε c(aP b)d − 5 D ω Hdha h bic − 5 D [σ, H]ha h bic 1 a 2 a d ˙ 4 a d − 5 D Dha ρh bic − 3 D ε c(aHhb)di − 3 D ε c(a[u, ˙ E]hb)di 2 a d 2 a d − Θ 3 D ε c(aHb)d + 3 D ε c(a[ω,H]hb)di a d c + 2D ε c(aσchb) H di =0. (61)

We abandon products of kinematic quantities in the undisturbed metrics:

a 8 2 a 3 a d 3 a 1 D C abc =D Ebc − D Dha E bci − 5 D D Edha h bic − 5 D C ha h bic 2 a d 2 2 ˙ + 3 D ε c(aP b)d − 3 curl(H)ab =0. (62)

· To linearized order, we get curl(Sab) = curlS˙ab. Using the later point and the evolution of Eq. (53), we obtain:

a 8 2 a 3 a d 2 ¨ D C abc =D Ebc − D Dha E bci − 5 D D Edha h bic − 3 Ehabi 3 1 2 a d 2 2 ˙ 1 − 5 Dha C bi + 3 D ε c(aP b)d − 3 P ab =0. (63)

The result can be compared to the wave solution (56). Without the distortion parts, it is inconsistent with a generic description of wave. Distortion of the gravitoelectric

field (Dha E bci) must not vanish to provide the wave solution. We also obtain similar condition for the gravitomagnetic field. In free space, the divergence of the Weyl fields, determined by the matter, must be free. The temporal evolution decides that the rotation of the Weyl fields must be non-zero. Now, the distortion provides another condition to characterize the evolution of the Weyl fields:

Dha E bci 6=0 6= Dha H bci. (64)

The existence of rotation and distortion is necessary condition to maintain the wave solutions. 10 A. Danehkar

5. Newtonian and Anti-Newtonian Fields

We can associate a Newtonian model with purely gravitoelectric (Hab = 0). With- out the gravitomagnetism, the nonlocal nature of the Newtonian force cannot be retrieved from relativistic models. It also excludes gravitational waves. The New- tonian model is a limited model to show the characteristics of the gravitoelectric. We can also consider an anti-Newtonian model; a model with purely gravitomag- netic (Eab = 0). The anti-Newtonian model obstructs sounding solutions. In Ref. 43, it has been proven that the anti-Newtonian model shall include either shear or vorticity.

5.1. Newtonian Model

Let us consider the Newtonian model (Hab = 0) in an irrotational static spacetime a (ωa =u ˙ a = 0) and a perfect-fluid model (q = πab = 0). The constraints and propagations shall be

1 1 2 C a = (divE)a − 3 Daρ =0, C a = [σ, E]a =0, (65)

1 ˙ c 1 P ab = −Ehabi − ΘEab +3σcha E bi − 2 σab(ρ + p)=0, (66) 2 P ab = curl(E)ab =0,

6 2 7 C a = 3 DaΘ − (divσ)a =0, C ab = −curl(σ)ab =0. (67) To first order, divergence and evolution of Eq. (66b) are

b 2 1 b cd 1 b D P ab = 2 εabcD (DdE )+ 3 Θ[σ, E]a − σab[σ, E] 1 1 1 2 b 2 1 = 2 curl(C )a + 3 ΘC a − σa C b + 3 ωaρ,˙ (68)

˙ 2 1 c e d ˙ P ab = − 3 Θcurl(E)ab − σe εcd(aD Eb) + curl(E)ab 3 cd 1 4 2 3 c 6 d = − 2 ε (aσb)cC d − 3 ΘP ab + 2 ε d(aC cEb) 1 7 1 c − 2 (ρ + p)C ab − curl(P )ab + 3curl(σcha E bi ). (69) 1 The last parameter ( 3 ωaρ˙) in Eq. (68) vanishes because of irrotational condition. Equation (68) then conserves the constraints. Equation (69) must be consistent with Eqs. (65) and (67). Thus, the last parameters in Eq. (69) has to vanish:

c curl(σcha E bi )=0. (70) It is a necessary condition for the consistent evolution of propagation. This condition is satisfied with irrotational product of gravitoelectric and shear, but it is a complete contrast to Eq. (65b). Thus, the Newtonian model is generally inconsistent with generic relativistic models. Moreover, the temporal evolution of propagation shows no wave solutions. On the Significance of the Weyl Curvature 11

5.1.1. Newtonian Limit The Newtonian model obstructs wave solution, due to the instantaneous interaction. Following Refs. 28–30, we consider a model whose action propagates at infinite speed (c → ∞). This is compatible with lim Eab = Eab(t)|∞, where Eab(t)|∞ is an c→∞ arbitrary function of time. We define the Newtonian potential as

1 2 Eab ≡ Dha D biΦ = DaDbΦ − 3 habD Φ. (71) On substituting into Eq. (65a), we get

1 2 1 b 2 1 C a = DaD Φ − 3 D habD Φ − 3 Daρ =0. (72) In a spatial infinity, we obtain the Poisson equation of the Newtonian potential:

1 2 1 C ≡ D Φ − 2 ρ =0. (73) Equation (65a) generalizes the gravitoelectric as the Newtonian force in the gradient of the relativistic energy density. Moreover, Eq. (66a) gives 1 ˙ 1 2 ˙ 1 ˙ 2 P ab = −DaDbΦ+ 3 habD Φ − ΘDaDbΦ+ 3 (hab +Θhab)D Φ c c 2 1 +3σcha D biD Φ − σcha h bi D Φ − 2 σab(ρ + p)=0. (74) In the Newtonian theory, we could not find the temporal evolution of the Newtonian potential.

5.1.2. Acceleration Potential In an irrotational spacetime, Eq. (46) becomes

4 1 P a = 2 curl(u ˙)a =0. (75) It introduces a scalar potential:

u˙ a = DaΦ, (76) where Φ is the acceleration potential. This scalar potential corresponds to the New- tonian potential. In the irrotational Newtonian model, the linearized acceleration is characterized as the acceleration potential.

5.2. Anti-Newtonian Model

Let us consider the anti-Newtonian model (Eab = 0) in a shearless static spacetime a (ωa =u ˙ a = 0) and a perfect-fluid model (q = πab = 0). The constraints and propagations shall be

1 b 1 2 C a = −3ω Hab − 3 Daρ =0, C a = (divH)a + ωa(ρ + p)=0, (77)

1 2 ˙ P ab = curl(H)ab =0, P ab = Hhabi +ΘHab − [ω,H]habi =0, (78) 12 A. Danehkar

6 2 7 C a = 3 DaΘ + curl(ω)a =0, C ab = Hab + Dha ω bi + 2u ˙ ha ω bi =0. (79) To linearized order, divergence and evolution of Eq. (78a) are b 1 1 b cd D P ab = 2 εabcD (DdH ) 1 c b 2 1 6 1 = 2 εab D C c − 2 (ρ + p)C a + 3 (ρ + p)DaΘ, (80) ˙ 1 1 ˙ P ab = − 3 Θcurl(H)ab + curl(H)ab 4 1 2 = − 3 ΘP ab + curl(P )ab + curl([ω,H])habi. (81) Equation (80) is consistent only in the spacetime being free from either the gravita- tional mass and pressure or the gradient of expansion. According to Eqs. (77) and (79), the last term in Eq. (81) has to vanish:

curl([ω,H])habi =0. (82) It is a necessary condition for the consistent evolution of propagation. This condition is satisfied with irrotational vorticity products of gravitomagnetic, but it is not consistent with Eq. (79b): c 7 d 1 6 1 6 1 1 ε d(aC b)cω − 4 ωbC a − 4 ωaC b − 4 Db[ω,ω]a − 4 Da[ω,ω]b 1 1 c d c d + 6 ωbDaΘ+ 6 ωaDbΘ − ε dau˙ hb ω ciω − ε dbu˙ ha ω ciω =0. (83) Thus, the anti-Newtonian model is generally inconsistent with relativistic models. Furthermore, there is not a possibility of gravitational waves.

5.2.1. Vorticity Potential In an unexpansive spacetime, Eq. (79a) takes the following form: 6 C a = curl(ω)a =0. (84) It defines a vorticity scalar potential Ψ as

ωa = DaΨ. (85) In the unexpansive anti-Newtonian model, the linearized vorticity is characterized as the vorticity potential.

5.2.2. Anti-Newtonian Limit We may consider a gravitomagnetic model whose action propagates at infinite speed. Let us define the anti-Newtonian potential as 1 2 Hab ≡ Dha D biΨ = DaDbΨ − 3 habD Ψ. (86) We substitute Eqs. (85) and (86) into Eq. (77b): 2 2 1 b 2 C a = DaD Ψ − 3 D habD Ψ + DaΨ(ρ + p)=0. (87) In a spatial infinity, we derive the Helmholtz equation: 2 2 3 C ≡ D Ψ+ 2 (ρ + p)Ψ=0. (88)

Eq. (77b) associates the gravitomagnetic with the angular momentum ωa(ρ + p). REFERENCES 13

6. Conclusion The Weyl curvature tensor describes the nonlocal long-range interactions as en- abling gravitational act at a distance (tidal forces and gravitational waves). The gravitoelectric field is described as the relativistic generalization of the tidal (New- tonian) force. However, the gravitomagnetic (anti-Newtonian) force has no New- tonian analogue. We have no expression similar to E˙ ab in the Newtonian theory. This difference arises from the instantaneous action in the Newtonian theory, which excludes a sounding solution. In Sec. 4, the rotation and distortion of the Weyl fields characterize the . The gravitomagnetism is necessary to maintain the gravitational wave. In relativistic models, the Newtonian force is also inconsistent without the magnetic part of the Weyl curvature.

Acknowledgment I have been supported by a Marie Curie Action under FP6 from the EU contract MRTN-CT-2004-005104 during my stay at the University of Craiova. I am also indebted to a referee for valuable comments.

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