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 Ricci curvature and the matter fields, i.e., the Einstein field equations:

$$R_{ab}=T_{ab}-{\textstyle{\frac{1}{2}}}Tg_{ab},$$ (26)

where $R_{ab}$ is the Ricci curvature, $T_{ab}$ the energy-momentum of the matter fields, and $T=T_{c}{}^{c}$ the trace of the energy-momentum tensor.

The successive contractions of Eq. (26) on usage of Eq. (24) lead to a set of relations:

$$\begin{array}[c]{ccc} {R_{ab}u^{a}u^{b}={\textstyle{\frac{1}{... ...{\textstyle{\frac{1}{2}} }(\rho-p)h_{ab}+\pi_{ab},} \end{array}$$ (27)

$$\begin{array}[c]{ccc} {R=R_{a}{}^{a},} & {T=T_{a}{}^{a}=-\rho+3p,} & {R=-T,} \end{array}$$ (28)

where $R$ is the Ricci scalar. The Ricci curvature is derived from the once contracted Riemann curvature tensor: $R_{ab}=R^{c}{}_{acb}$ .

The Riemann tensor is split into symmetric (massless) traceless $C_{abcd}$ and traceful massive $M_{abcd}$ parts:

$$R_{abcd}=C_{abcd}+M_{abcd}.$$ (29)

The symmetric traceless part of the Riemann curvature is called the Weyl conformal curvature with the following properties:

$$\begin{array}[c]{cc} {C_{abcd}=C_{[ab][cd]},} & {C^{a}{}_{bca}=0=C_{a[bcd]}.} \end{array}$$ (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 tensor $C_{abcd}$ is irreducibly split into the Newtonian $C_{abcd}^{\mathrm{N}}$ and the anti-Newtonian $C_{abcd}^{\mathrm{AN}}$ parts:

$$C_{abcd}=C_{abcd}^{\mathrm{N}}+C_{abcd}^{\mathrm{AN}},$$ (31)

$$C_{\mathrm{N}}^{ab}{}_{cd}=4\{u^{[a}u_{[c}+h^{[a}{}_{[c}\}E^{b]}{}_{d]},$$ (32)

$$C_{abcd}^{\mathrm{AN}}=2\varepsilon_{abe}u_{[c}H_{d]}{}^{e}+2\varepsilon _{cde}u_{[a}H_{b]}{}^{e},$$ (33)

where $E_{ab}=C_{acbd}u^{c}u^{d}$ is the gravitoelectric field and $H_{ab}={\textstyle{\frac{1}{2}}}\varepsilon_{acd}C^{cd}{}_{be}u^{e}$ 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

$$M^{ab}{}_{cd} ={\textstyle{\frac{2}{3}}}(\rho+3p)u^{[a}u_{[c}h^{b]}{} _{d]}+{\textstyle{\frac{2}{3}}}\rho h^{a}{}_{[c}h^{b}{}_{d]}$$

$$-2u^{[a}h^{b]}{}_{[c}q_{d]}-2u_{[c}h^{[a}{}_{d]}q^{b]}-2u^{[a}u_{[c}\pi {}^{b]}{}_{d]}+2h^{[a}{}_{[c}\pi^{b]}{}_{d]},$$ (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

$$\nabla_{\lbrack e}R_{ab]cd}=0.$$ (35)

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

$$\nabla^{d}C_{abcd}=-\nabla_{\lbrack a}(R_{b]c}-{\textstyle{\frac{... ...\lbrack a}(T_{b]c}-{\textstyle{\frac{1}{3}}}g_{b]c}T_{d} {}^{d})\equiv J_{abc}.$$ (36)

On decomposing Eq. (36) along and orthogonal to a 4-velocity vector, we obtain constraint ( $C^{1,2}{}_{a}$ ) and propagation ( $P^{1,2} {}_{ab}$ ) equations of the Weyl fields in a form analogous to the Maxwell equations[10,42,43,44]:

$$C^{1}{}_{a}\equiv{(\mathrm{div}E)_{a}}-3\omega^{b}H_{ab}-[\sigma ... ...extstyle{\frac{1}{3}}}\mathrm{D}_{a}\rho+{\textstyle{\frac{1}{3}} }\Theta q_{a}$$

$$-{\textstyle{\frac{1}{2}}}\sigma_{ab}q^{b}+{\textstyle{\frac{3}{2}}} [\omega,q]_{a}+{\textstyle{\frac{1}{2}}(\mathrm{div}\pi)_{a}} =0,$$ (37)

$$C^{2}{}_{a}\equiv{(\mathrm{div}H)_{a}}+3\omega^{b}E_{ab}+[\sigma,E]_{a} +\omega_{a}(\rho+p)$$

$$+{\textstyle{\frac{1}{2}}}\mathrm{curl}(q)_{a}+{\textstyle{\frac{1}{2}} }[\sigma,\pi]_{a}-{\textstyle{\frac{1}{2}}}\omega^{b}\pi_{ab} =0,$$ (38)

$$P^{1}{}_{ab}\equiv\mathrm{curl}(H)_{ab}+2[\dot{u},H]_{\left\langl... ... {ab}\right\rangle }-\Theta E_{ab}+[\omega,E]_{\left\langle {ab}\right\rangle }$$

$$+3\sigma_{c\left\langle a\right. }E_{\left. b\right\rangle }{} ^{... ...t. b\right\rangle }-\dot {u}_{\left\langle a\right. }q_{\left. b\right\rangle }$$

$$-{\textstyle{\frac{1}{2}}}\dot{\pi}_{\left\langle {ab}\right\rang... ...frac{1}{2}}}\sigma^{e} {}_{\left\langle a\right. }\pi_{\left. b\right\rangle e} =0,$$ (39)

$$P^{2}{}_{ab}\equiv\mathrm{curl}(E)_{ab}+2[\dot{u},E]_{\left\langl... ... {ab}\right\rangle }+\Theta H_{ab}-[\omega,H]_{\left\langle {ab}\right\rangle }$$

$$-3\sigma_{c\left\langle a\right. }H_{\left. b\right\rangle }{} ^{... ...eft\langle {ab}\right\rangle }-{\textstyle{\frac{1}{2}}}\mathrm{curl}(\pi)_{ab} =0.$$ (40)

The twice contracted Bianchi identities present the conservation of the total energy momentum tensor, namely

$$\nabla^{b}T_{ab}=\nabla^{b}(R_{ab}-{\textstyle{\frac{1}{2}}}g_{ab}R)=0.$$ (41)

It is split into a timelike and a spacelike momentum constraints:

$$C^{3}\equiv\dot{\rho}+(\rho+p)\Theta+{\mathrm{div}(q)}+2\dot{u}_{a} q^{a}+\sigma_{ab}\pi^{ab}=0,$$ (42)

$$C^{4}{}_{a}\equiv(\rho+p)\dot{u}_{a}+\mathrm{D}_{a}p+\dot{q}_{\le... ...\sigma_{ab} q^{b}-[\omega,q]_{a}+{(\mathrm{div}\pi)_{a}}+\dot{u}^{b}\pi_{ab}=0.$$ (43)

They provide the conservation law of energy-momentum, i. e., how matter determines 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 $u_{a}$ :

$$2\nabla_{\lbrack a}\nabla_{b]}u_{c}=R_{abcd}u^{d}.$$ (44)

We substitute the vector field $u_{a}$ 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]:

$$P^{3}\equiv\dot{\Theta}+{\textstyle{\frac{1}{3}}}\Theta^{2}-{\mat... ..._{ab}\omega^{ab}-\sigma_{ab} \sigma^{ab})+{\textstyle{\frac{1}{2}}}(\rho+3p)=0,$$ (45)

$$P^{4}{}_{a}\equiv\dot{\omega}_{\left\langle a\right\rangle }+{\te... ...a_{a}{}^{b}\omega_{b}+{\textstyle{\frac{1}{2}} }\mathrm{curl}{(}\dot{u})_{a}=0,$$ (46)

$$P^{5}{}_{ab}\equiv E_{ab}-\mathrm{D}_{\left\langle a\right. }\dot... ...a\right. }\omega_{\left. b\right\rangle }-{\textstyle{\frac{1}{2}}}\pi_{ab} =0.$$ (47)

Eq. (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. Eq. (47) shows that the gravitoelectric field is propagated in shear, vorticity, acceleration, and anisotropic stress.

The Ricci identities also provide a set of constraints:

$$C^{5}\equiv{\mathrm{div}(\omega)}-\omega_{a}\dot{u}^{a}=0,$$ (48)

$$C^{6}{}_{a}\equiv{\textstyle{\frac{2}{3}}}\mathrm{D}_{a}\Theta-{(... ...hrm{div} \sigma)_{a}}+\mathrm{curl}(\omega)_{a}+2[\dot{u},\omega]_{a}-q_{a} =0,$$ (49)

$$C^{7}{}_{ab}\equiv H_{ab}-\mathrm{curl}(\sigma)_{ab}+\mathrm{D}_{... ...ht\rangle }+2\dot{u}_{\left\langle a\right. }\omega_{\left. b\right\rangle }=0.$$ (50)

Eq. (48) presents the divergence of vorticity. Eq. (49) links the divergence of shear to the rotation of vorticity. Eq. (50) characterizes the gravitomagnetic field as the distortion of vorticity and the rotation of shear.