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 observer. The formalism, known as the 3 + 1 covariant approach to general relativity,[25,26,27,28,29,30] has been used for numerous applications.[10,31,32,33,34,35] In this approach, we rewrite equations governing relativistic fluid dynamics by using projected vectors and projected symmetric traceless tensors instead of metrics.[10,34]

We take a four-velocity vector $u^{a}$ field in a given (3 + 1)-dimensional spacetime to be a unit vector field $u^{a}u_{a}=-1$. We define a spatial metric (or projector tensor) $h_{ab}=g_{ab}+u_{a}u_{b}$, where $g_{ab}$ 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 $u^{a}$.[2,33,36] We get some properties for the spatial metric

$$\begin{array}[c]{ccc} {h_{ab}u^{b}=0,} & {~~~~~~~~h_{a}{}^{c}h_{cb}=h_{ab},} & {~~~~~~~~h_{a}{}^{a}=3.} \end{array}$$ (1)

We also define the spatial alternating tensor as

$$\varepsilon_{abc}=\eta_{abcd}u^{d},$$ (2)

where ${\eta_{abcd}}$ is the spacetime alternating tensor,

$$\begin{array}[c]{ccc} {\eta_{abcd}=-4!\sqrt{\vert g\vert}\del... ..._{ac}g^{cb},} & {~~~~~~~~\vert g\vert=\det g_{ab}.} \end{array}$$ (3)

The covariant spacetime derivative $\nabla_{a}$ is split into a covariant temporal derivative

$$\dot{T}_{a\cdots}=u^{b}\nabla_{b}T_{a\cdots},$$ (4)

and a covariant spatial derivative

$$\mathrm{D}_{b}T_{a\cdots}=h_{b}{}^{d}h_{a}{}^{c}\cdots\nabla_{d}T_{c\cdots}.$$ (5)

The projected vectors and the projected symmetric traceless parts of rank-2 tensors are defined by

$$\begin{array}[c]{cc} {V_{\left\langle a\right\rangle }\equiv ... ...tstyle{\frac {1}{3}}}h^{cd}h_{ab}}\right\} S_{cd}.} \end{array}$$ (6)

The equations governing these quantities involve a vector product and its generalization to rank-2 tensors:

$$\begin{array}[c]{cc} {[V,W]_{a}\equiv\varepsilon_{abc}V^{b}W^... ...S,Q]_{a}\equiv \varepsilon_{abc}S^{b}{}_{d}Q^{cd},} \end{array}$$ (7)

$$\begin{array}[c]{cc} {[V,S]_{ab}\equiv\varepsilon_{cd(a}S_{b)... ...e a\right. }S_{\left. b\right\rangle }{}^{c}V^{d}.} \end{array}$$ (8)

We define divergences and rotations as

$$\begin{array}[c]{cc} {\mathrm{div}(V)\equiv\mathrm{D}^{a}V_{a... ...~~~(\mathrm{div}S)_{a} \equiv\mathrm{D}^{b}S_{ab},} \end{array}$$ (9)

$$\begin{array}[c]{cc} ({\mathrm{curl}V)_{a}\equiv\varepsilon_{... ...\mathrm{D}^{c}S_{\left. b\right\rangle }{}^{d}.}&{} \end{array}$$ (10)

We know that $\mathrm{D}_{c}h_{ab}=0=\mathrm{D}_{d}\varepsilon_{abc}$, $\dot{h}_{ab}=2u_{(a}\dot{u}_{b)}$, and $\dot{\varepsilon}_{abc} =3u_{[a}\varepsilon_{bc]d}\dot{u}^{d}$, then $u^{a}\dot{h}_{ab}=-\dot{u}_{b}$ and $u^{a}\dot{\varepsilon}_{abc}=-\dot{u}^{a}\varepsilon_{abc}$. From these points one can also define the relativistically temporal rotations as

$$\begin{array}[c]{ccc} {[\dot{u},V]_{a}=-u^{c}\dot{\varepsilon... ...e a\right. }S_{\left. b\right\rangle }{}^{d}.} & {} \end{array}$$ (11)

The covariant spatial distortions are

$$\mathrm{D}_{\left\langle a\right. }V_{\left. b\right\rangle }=\mathrm{D} _{(a}V_{b)}-{\textstyle{\frac{1}{3}}}(\mathrm{div}V)h_{ab},$$ (12)

$$\mathrm{D}_{\left\langle a\right. }S_{\left. {bc}\right\rangle } =\mathrm{D}_{(a}S_{bc)}-{\textstyle{\frac{2}{5}}}h_{(ab}(\mathrm{div}S)_{c)}.$$ (13)

We decompose the covariant derivatives of scalars, vectors, and rank-2 tensors into irreducible components

$$\nabla_{a}f=-\dot{f}u_{a}+\mathrm{D}_{a}f,$$ (14)

$$\nabla_{b}V_{a}=-\left( {\dot{V}_{\left\langle a\right\rangle }u_... ...}V_{b} -u_{a}\sigma_{bc}V^{c}-u_{a}[\omega,V]_{b}}\right) +\mathrm{D}_{a}V_{b},$$ (15)

$$\nabla_{c}S_{ab}= -\Big({\dot{S}_{\left\langle {ab}\right\rangle }u_{c} +2u_{(a}S_{... ...{\textstyle{\frac{2}{3}}}\Theta u_{(a}S_{b)c} }-2u_{(a}S_{b)}{}^{d}\sigma_{dc}~$$

$${-2\varepsilon_{cde}u_{(a}S_{b)}{}^{d}\omega^{e}}\Big) +\mathrm{D} _{a}S_{bc},$$ (16)

where

$$\mathrm{D}_{a}V_{b}={\textstyle{\frac{1}{3}}}\mathrm{D}_{c}V^{c} ... ...thrm{curl}V^{c} +\mathrm{D}_{\left\langle a\right. }V_{\left. b\right\rangle },$$ (17)

$$\mathrm{D}_{a}S_{bc}={\textstyle{\frac{3}{5}}}\mathrm{D}^{d}S_{d\... ...}S_{b)}{}^{d}+\mathrm{D}_{\left\langle a\right. }S_{\left. {bc}\right\rangle }.$$ (18)

We also introduce the kinematic quantities encoding the relative motion of fluids:

$$\nabla_{b}u_{a}=\mathrm{D}_{b}u_{a}-\dot{u}_{a}u_{b},$$ (19)

$$\mathrm{D}_{b}u_{a}={\textstyle{\frac{1}{3}}}\Theta h_{ab}+\sigma_{ab} +\omega_{ab},$$ (20)

where $\dot{u}_{a}=u^{b}\nabla_{b}u_{a}$ is the relativistic acceleration vector, in the frames of instantaneously comoving observers $\dot{u}_{a} =\dot{u}_{\left\langle a\right\rangle }$, $\Theta=\mathrm{D}^{a}u_{a}$ the rate of expansion of fluids, $\sigma_{ab}=\mathrm{D}_{\left\langle a\right. }u_{\left. b\right\rangle }=\mathrm{D}_{(a}u_{b)}-{\textstyle{\frac{1}{3}} }h_{ab}\mathrm{D}_{c}u^{c}$ a traceless symmetric tensor ( $\sigma_{ab} =\sigma_{(ab)}$, $\sigma_{a}{}^{a}=0$); the shear tensor describing the rate of distortion of fluids, and $\omega_{ab}=\mathrm{D}_{[a}u_{b]}$ a skew-symmetric tensor ( $\omega_{ab}=\omega_{\lbrack ab]}$, $\omega_{a}{} ^{a}=0$); the vorticity tensor describing the rotation of fluids.[27,33,37]

The vorticity vector[38,39] $\omega_{a}$ is defined by

$$\omega_{a}=-{\textstyle{\frac{1}{2}}}\varepsilon_{abc}\omega^{bc},$$ (21)

where $\omega_{a}u^{a}=0$, $\omega_{ab}\omega^{b}=0$ and the magnitude $\omega^{2}={\textstyle{\frac{1}{2}}}\omega_{ab}\omega^{ab}\geq0$ have been imposed. Accordingly, we obtain

$$\omega_{a}=-{\textstyle{\frac{1}{2}}}\varepsilon_{abc}\mathrm{D}^{b}u^{c}.$$ (22)

The sign convention is such that in the Newtonian theory $\vec{\omega }=-{\textstyle{\frac{1}{2}}}\vec{\nabla}\times\vec{u}$.

We denote the covariant shear and vorticity products of the symmetric traceless tensors as

$$\begin{array}[c]{cc} {[\sigma,S]_{a} = \varepsilon_{abc} \sig... ... } S_{\left. b \right\rangle } {}^{c} \omega^{d} .} \end{array}$$ (23)

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

$$T_{ab}=\rho u_{a}u_{b}+ph_{ab}+2q_{(a}u_{b)}+\pi_{ab},$$ (24)

$$\begin{array}[c]{cccc} {q_{a}u^{a}=0,} & {~~~~~~~~\pi^{a}{}_{... ...\pi_{ab}=\pi_{(ab)},} & {~~~~~~~~\pi_{ab} u^{b}=0,} \end{array}$$ (25)

where $\rho=T_{ab}u^{a}u^{b}$ is the relativistic energy density relative to $u^{a}$, $p={\textstyle{\frac{1}{3}}}T_{ab}h^{ab}$ the pressure, $q_{a}=-T_{\left\langle a\right\rangle b}u^{b}=-h_{a}{}^{c}T_{cb}u^{b}$ the energy flux relative to $u^{a}$, and $\pi_{ab}=T_{\left\langle {ab} \right\rangle }=T_{cd}h^{c}{}_{\left\langle a\ri... ...{h^{c}{}_{(a}u^{d}{}_{b)}-{\textstyle{\frac{1}{3}} }h_{ab}h^{cd}}\right) T_{cd}$ the traceless anisotropic stress. Imposing $q^{a}=\pi_{ab}=0$, we get the solution of a perfect fluid with $T_{ab}=\rho u_{a}u_{b}+ph_{ab}$. In addition $p=0$ gives the pressure-free matter or dust solution.[27,33,37]