Semi-tetrad decomposition of spacetime with conformal symmetry

Abstract

We study the kinematical and dynamical properties of spacetimes that admit a conformal Killing vector. A 1 + 1 + 2 decomposition of the spacetime is performed using the fluid 4-velocity and a preferred spatial direction. This provides new insights into the behaviour of the acceleration, expansion, shear and vorticity scalars. The energy momentum tensor for an anisotropic fluid with no sheet components is considered and decomposed using the semi-tetrad covariant approach. This makes it possible to generate a set of constraint equations in the new geometrical variables. All the geometrical and thermodynamical quantities are written in terms of the 1 + 1 + 2 decomposition variables. We also find the constraints that must be satisfied by the thermodynamical variables when conformal symmetry exists in a perfect fluid. Noteworthily we show that the conformal factor satisfies a damped wave equation.

Appendices

Appendix A: Additional 1 + 3 equations and definitions

We write down the definitions of important components including the kinematical, Weyl and matter quantities in the 1 + 3 formalism.

$$\begin{aligned} \varepsilon _{abc}= & {} \sqrt{|\text {det } g|}\delta ^0{}_{[a} \delta ^1{}_{b}\delta ^2{}_{c}\delta ^3{}_{d]} u^{d}, \end{aligned}$$

(A1)

$$\begin{aligned} \varepsilon _{abc}\varepsilon ^{def}= & {} 3! h^d{}_{\left[ a \right. } h^e{}_{b} h^f{}_{\left. c \right] }, \nonumber \\ \varepsilon _{abc}\varepsilon ^{dec}= & {} 2 h^d{}_{\left[ a \right. } h^e{}_{\left. b\right] }, \nonumber \\ A_{b}= & {} \dot{u_{b}}, \end{aligned}$$

(A2)

$$\begin{aligned} \Theta= & {} D_{a} u^{a}, \end{aligned}$$

(A3)

$$\begin{aligned} \sigma _{ab}= & {} \left( h^{c}{}_{(a} h^{d}{}_{b)} - \frac{1}{3}h_{ab} h^{cd}\right) D_{c} u_{d}, \end{aligned}$$

(A4)

$$\begin{aligned} \omega ^{a}= & {} \varepsilon ^{abc} D_{b} u_{c}, \end{aligned}$$

(A5)

$$\begin{aligned} E_{ab}= & {} C_{abcd} u^{c} u^{d} = E_{<ab>}, \end{aligned}$$

(A6)

$$\begin{aligned} H_{ab}= & {} \frac{1}{2} \varepsilon _{ade} C^{de}{}_{bc} u^{c} = H_{<ab>}, \end{aligned}$$

(A7)

$$\begin{aligned} \mu= & {} T_{ab} u^{a} u^{b}, \end{aligned}$$

(A8)

$$\begin{aligned} p= & {} \frac{1}{3} h_{ab} T^{ab}, \end{aligned}$$

(A9)

$$\begin{aligned} q_{a}= & {} q_{<a>} = - h^{c}{}_{a} T_{cd} u^{d}. \end{aligned}$$

(A10)

Angle brackets denote orthogonal projections of covariant time derivatives along ${u^{a}}$ as well as represent the projected, symmetric and trace-free part of tensors as follows

$$\begin{aligned} v_{<a>}= & {} h^{b}{}_{a}{\dot{V}}_{b}, \end{aligned}$$

(A11)

$$\begin{aligned} Z_{<ab>}= & {} \left( h^{c}{}_{(a} h^{d}{}_{b)} - \frac{1}{3}h_{ab} h^{cd}\right) Z_{cd}. \end{aligned}$$

(A12)

Appendix B: Additional 1 + 1 + 2 equations and definitions

We write down the definitions of important components in the 1 + 1 + 2 formalism.

$$\begin{aligned} \varepsilon _{ab}\equiv & {} \varepsilon _{abc} e^{c} = \sqrt{|\text {det } g|}\delta ^0{}_{[a} \delta ^1{}_{b}\delta ^2{}_{c}\delta ^3{}_{d]} e^{c} u^{d}, \end{aligned}$$

(B1)

$$\begin{aligned} E_{ab}= & {} \mathcal {E} \left( e_{a} e_{b} - \frac{1}{2} N_{ab}\right) + 2\mathcal {E}_{(a} e_{b)} + \mathcal {E}_{ab}, \end{aligned}$$

(B2)

$$\begin{aligned} H_{ab}= & {} \mathcal {H} \left( e_{a} e_{b} - \frac{1}{2} N_{ab}\right) + 2\mathcal {H}_{(a} e_{b)} + \mathcal {H}_{ab}, \end{aligned}$$

(B3)

$$\begin{aligned} \sigma ^{2}= & {} \frac{1}{2}\sigma _{ab}\sigma ^{ab} = \frac{3}{4}\Sigma ^{2} + \Sigma _{a}\Sigma ^{a} + \frac{1}{2}\Sigma _{ab}\Sigma ^{ab}, \end{aligned}$$

(B4)

$$\begin{aligned} a_{a}\equiv & {} e^{c} D_{c} e_{a} = {\hat{e}}_{a}, \end{aligned}$$

(B5)

$$\begin{aligned} \phi\equiv & {} \delta _{a} e^{a}, \end{aligned}$$

(B6)

$$\begin{aligned} \varsigma\equiv & {} \frac{1}{2}\varepsilon ^{ab}\delta _{a} e_{b}, \end{aligned}$$

(B7)

$$\begin{aligned} \zeta _{ab}\equiv & {} \delta _{\{a} e_{b\}}. \end{aligned}$$

(B8)

The irreducible set of geometric variables

$$\begin{aligned}&\left\{ \Theta , \mathcal {A}, \Omega , \Sigma , \mathcal {E}, \mathcal {H}, \phi , \varsigma , \mathcal {A}_{a}, \Omega _{a}, \Sigma _{a}, \varphi _{a}, \right. \nonumber \\&\quad \left. a_{a}, \mathcal {E}_{a}, \mathcal {H}_{a}, \Sigma _{ab}, \zeta _{ab}, \mathcal {E}_{ab}, \mathcal {H}_{ab}\right\} , \end{aligned}$$

(B9)

together with the irreducible set of thermodynamic variables

$$\begin{aligned} \{\mu , p, Q, \Pi , Q_{a}, \Pi _{a}, \Pi _{ab}\}, \end{aligned}$$

(B10)

make up the key variables in the 1 + 1 + 2 formalism of first order gravity for a given equation of state.

Any spacetime 3-vector ${\Phi ^{a}}$ can be irreducibly split into ${\chi }$, a scalar component along ${e^{a}}$, and a 2-vector ${\chi ^{a}}$, which is a sheet component orthogonal to ${e^{a}}$, as follows

$$\begin{aligned}&\Phi ^{a} = \chi e^{a} + \chi ^{a} \quad \text {where} \quad \chi \equiv \Phi _{a} e^{a} \quad \nonumber \\&\quad \text {and} \quad \chi ^{a} \equiv N^{ab} \Phi _{b} \equiv \Phi ^{{\bar{a}}}, \end{aligned}$$

(B11)

where the bar on a particular index denotes projection with ${N_{ab}}$ on that index such that the vector or tensor lies on the sheet. Similarly we can split a projected, symmetric, trace-free tensor ${\Phi _{ab}}$ into scalar, 2-vector and 2-tensor parts as follows

$$\begin{aligned} \Phi _{ab} = \Phi _{<ab>} = \chi \left( e_{a} e_{b} - \frac{1}{2} N_{ab}\right) + 2\chi _{(a} e_{b)} + \chi _{ab}, \end{aligned}$$

(B12)

where the components

$$\begin{aligned} \chi\equiv & {} e^{a} e^{b} \Phi _{ab} = -N^{ab}\Phi _{ab}, \nonumber \\ \chi _{a}\equiv & {} N_{a}{}^{b} e^{c}\Phi _{bc}, \nonumber \\ \chi _{ab}\equiv & {} \chi _{\{ab\}} = \left( N_{(a}{}^{c} N_{b)}{}^{d} - \frac{1}{2} N_{ab} N^{cd}\right) \Phi _{cd}, \end{aligned}$$

(B13)

are defined. The curly brackets denote the part of the tensor that is projected, symmetric and trace-free, with respect to ${e^{a}}$.

The Ricci identities for ${e^{a}}$ are given by

$$\begin{aligned} R_{abc} \equiv 2\nabla _{[a}\nabla _{b]} e_{c} - R_{abcd} e^{d} = 0, \end{aligned}$$

(B14)

where ${R_{abcd}}$ is the Riemann curvature tensor. Splitting this rank three tensor using ${u^{a}}$ and ${e^{a}}$, we can obtain the propagation and evolution equations for the covariant parts of the derivative of ${e^{a}}$. We note that not all information needed to determine the complete set of 1 + 1 + 2 equations is contained in ${R_{abc}}$. Hence, a 1 + 1 + 2 decomposition of the standard 1 + 3 equations is performed and we can write down the energy and momentum conservation equations given by

$$\begin{aligned} {\dot{\mu }} + {\hat{Q}}= & {} -\Theta \left( \mu + p\right) - \left( \phi + 2\mathcal {A}\right) Q \nonumber \\&-\, \frac{3}{2}\Sigma \Pi + \left( a_{a} - 2\mathcal {A}_{a}\right) Q^{a} \nonumber \\&-\, \delta _{a} Q^{a} - 2\Sigma _{a}\Pi ^{a} - \Sigma _{ab}\Pi ^{ab}, \end{aligned}$$

(B15)

$$\begin{aligned} {\dot{Q}} + {\hat{p}} + {\hat{\Pi }}= & {} -\delta _{a}\Pi ^{a} - \left( \frac{3}{2}\phi + \mathcal {A}\right) \Pi \nonumber \\&-\, \left( \frac{4}{3}\Theta + \Sigma \right) Q - \left( \mu + p\right) \mathcal {A} \nonumber \\&+\, \left( \varphi _{a} - \Sigma _{a} + \varepsilon _{ab}\Omega ^{b}\right) Q^{a} \nonumber \\&+\, \left( 2 a_{a} - \mathcal {A}_{a}\right) \Pi ^{a}+ \zeta _{ab}\Pi ^{ab}, \end{aligned}$$

(B16)

$$\begin{aligned} {\dot{Q}}_{{\bar{a}}} + {\hat{\Pi }}_{{\bar{a}}}= & {} -\delta _{a}p + \frac{1}{2}\delta _{a}\Pi - \delta ^{b}\Pi _{ab} \nonumber \\&-\, \frac{3}{2}\Pi a_{a} - Q\left( \varphi _{a}+ \Sigma _{a} + \varepsilon _{ab}\Omega ^{b}\right) \nonumber \\&-\, \left( \frac{4}{3}\Theta - \frac{1}{2}\Sigma \right) Q_{a} + \Omega \varepsilon _{ab} Q^{b} \nonumber \\&-\, \left( \frac{3}{2}\phi + \mathcal {A}\right) \Pi _{a} + \varsigma \varepsilon _{ab}\Pi ^{b} \nonumber \\&-\, \left( \mu + p - \frac{1}{2}\Pi \right) \mathcal {A}_{a} - \Sigma _{ab} Q^{b} \nonumber \\&-\,\zeta _{ab}\Pi ^{b} + \Pi _{ab}\left( a^{b} - \mathcal {A}^{b}\right) . \end{aligned}$$

(B17)

We write down the 1 + 1 + 2 double derivative expression for a scalar ${\kappa }$ as follows

$$\begin{aligned} \nabla ^{a} \nabla ^{b}\kappa= & {} -{\dot{\kappa }}\left\{ \frac{1}{3}\Theta \left( N^{ab} + e^{a} e^{b}\right) \right. \nonumber \\&\left. +\, \Sigma \left( e^{a} e^{b} - \frac{1}{2} N^{ab}\right) + 2\Sigma ^{(a} e^{b)} \right. \nonumber \\&+\, \Sigma ^{ab} \left. + e^{a}\varepsilon ^{bc}\Omega _{c} - e^{b}\varepsilon ^{ac}\Omega _{c} + \varepsilon ^{ab}\Omega \right\} \nonumber \\&+\, u^{b}\left\{ \frac{1}{3}\Theta \left( {\hat{\kappa }} e^{a} + \delta ^{a}\kappa \right) \right. \nonumber \\&+\, \left[ \Sigma \left( e^{a} e^{c} - \frac{1}{2} N^{ac}\right) + 2\Sigma ^{(a} e^{c)} +\, \Sigma ^{ac}\right] \left( {\hat{\kappa }} e_{c} + \delta _{c}\kappa \right) \nonumber \\&+ \left[ e^{a} \varepsilon ^{cd}\Omega _{d} - e^{c}\varepsilon ^{ad}\Omega _{d} + \varepsilon ^{ac}\Omega \right] \left( {\hat{\kappa }} e_{c} + \delta _{c}\kappa \right) \nonumber \\&\left. +\, u^{a}\ddot{\kappa } - \left( \hat{{\dot{\kappa }}} e^{a} + \delta ^{a}{\dot{\kappa }}\right) \right\} - u^{a}\left\{ \left( N^{cb} + e^{c} e^{b}\right) \left( {\hat{\kappa }} e_{c} + \delta _{c}\kappa \right) \right. \nonumber \\&+\, u^{b}\left( \mathcal {A} e^{c} + \mathcal {A}^{c}\right) \left( {\hat{\kappa }} e_{c} + \delta _{c}\kappa \right) \left. -{\dot{\kappa }}\left( \mathcal {A} e^{b} + \mathcal {A}^{b}\right) \right\} \nonumber \\&+\frac{1}{3}\left\{ \hat{{\hat{\kappa }}} + \phi {\hat{\kappa }} -\, \delta ^{c}\kappa a_{c} + \delta ^{c}\delta _{c}\kappa \right\} \left( N^{ab} + e^{a} e^{b}\right) \nonumber \\&+\, \frac{1}{3}\left\{ 2\hat{{\hat{\kappa }}} - \phi {\hat{\kappa }} - 2\delta ^{c}\kappa a_{c} - \delta ^{c}\delta _{c}\kappa \right\} \left( e^{a} e^{b} - \frac{1}{2} N^{ab}\right) \nonumber \\&+\, \left\{ 2\delta ^{(a}{\hat{\kappa }} - \left( \Sigma ^{(a} + \Omega _{c}\varepsilon ^{c(a}\right) {\dot{\kappa }} \right. \nonumber \\&\left. -\,\phi \delta ^{(a}\kappa + 2\delta _{c}\kappa \left( \varsigma \varepsilon ^{c(a} - \zeta ^{c(a}\right) \right\} e^{b)} \nonumber \\&+\, {\hat{\kappa }}\zeta ^{ab} + \delta ^{\{a}\delta ^{b\}} + \frac{1}{2}\left( e^{a}\varepsilon ^{bc} - e^{b}\varepsilon ^{ac}+ e^{c}\varepsilon ^{ab}\right) \nonumber \\&\left\{ \left( 2\varsigma {\hat{\kappa }} + \varepsilon _{mn}\delta ^{m}\delta ^{n}\kappa \right) e_{c} \right. \nonumber \\&\left. +\, \varsigma \delta _{c}\kappa + \varepsilon _{cm}\left( \Sigma ^{m}{\dot{\kappa }} - \varepsilon ^{mc}\Omega _{c}{\dot{\kappa }} \right. \left. + \varepsilon ^{mc}\varsigma \delta _{c}\kappa \right) \right\} . \end{aligned}$$

(B18)

We mention that in general the dot, hat and delta derivatives do not commute. The commutation relations for any scalar ${\kappa }$ are

$$\begin{aligned} \hat{{\dot{\kappa }}} - \dot{{\hat{\kappa }}}= & {} -\mathcal {A}{\dot{\kappa }} + \left( \frac{1}{3}\Theta + \Sigma \right) {\hat{\kappa }} \nonumber \\&+\, \left( \Sigma _{a} + \varepsilon _{ab}\Omega ^{b} - \varphi _{a} \right) \delta ^{a}\kappa , \end{aligned}$$

(B19)

$$\begin{aligned} \delta _{a}{\dot{\kappa }} - \left( \delta _{a}\kappa \right) ^{\cdot }_{\perp }= & {} -\mathcal {A}_{a}{\dot{\kappa }} + \left( \varphi _{a} + \Sigma _{a} -\varepsilon _{ab}\Omega ^{b} \right) {\hat{\kappa }}\nonumber \\&+\, \left( \frac{1}{3}\Theta - \frac{1}{2}\Sigma \right) \delta _{a}\kappa + \left( \Sigma _{ab} + \Omega \varepsilon _{ab}\right) \delta ^{b}\kappa , \end{aligned}$$

(B20)

$$\begin{aligned} \delta _{a}{\hat{\kappa }} - \left( \widehat{\delta _{a}\kappa }\right) _{\perp }= & {} -2\varepsilon _{ab}\Omega ^{b}{\dot{\kappa }} + a_{a}{\hat{\kappa }} \nonumber \\&+\, \frac{1}{2}\phi \delta _{a}\kappa + \left( \zeta _{ab} + \varsigma \varepsilon _{ab}\right) \delta ^{b}\kappa , \end{aligned}$$

(B21)

$$\begin{aligned} \delta _{[a}\delta _{b]}\kappa= & {} \varepsilon _{ab}\left( \Omega {\dot{\kappa }} - \varphi {\hat{\kappa }}\right) , \end{aligned}$$

(B22)

where ${\perp }$ denotes projection onto the sheet.

Appendix C: Extended calculations

In the 1 + 1 + 2 formalism, the Herrera et al. [6] expression for the Lie derivative along a CKV ${\xi ^{a}}$ of Einstein’s field Eq. (74) equates to

$$\begin{aligned} \mathcal {L}_{\varvec{\xi }} G_{ab}= & {} 2\left[ -\Theta {\dot{\Psi }} - \ddot{\Psi } + {\hat{\Psi }}\left( \mathcal {A} + \phi \right) + \hat{{\hat{\Psi }}} - \delta ^{c}\delta _{c}\Psi \right] \left[ N_{ab} - u_{a} u_{b} + e_{a} e_{b}\right] \nonumber \\&-\, 2 \left[ -{\dot{\Psi }}\left\{ \frac{1}{3}\Theta \left( N_{ab} + e_{a} e_{b}\right) \right. \right. +\, \Sigma \left( e_{a} e_{b} - \frac{1}{2} N_{ab}\right) + 2\Sigma _{(a} e_{b)}\nonumber \\&+\, \Sigma _{ab} + e_{a}\varepsilon _{bc}\Omega ^{c} \left. - e_{b}\varepsilon _{ac}\Omega ^{c} + \varepsilon _{ab}\Omega \right\} +\, u_{b}\left\{ \frac{1}{3}\Theta \left( {\hat{\Psi }} e_{a} + \delta _{a}\Psi \right) \right. \nonumber \\&+\, \left[ \Sigma \left( e_{a} e_{c} - \frac{1}{2} N_{ac}\right) + 2\Sigma _{(a} e_{c)} + \Sigma _{ac}\right] \left( {\hat{\Psi }} e^{c} + \delta ^{c}\Psi \right) \nonumber \\&+ \left[ e_{a} \varepsilon _{cd}\Omega ^{d}- \,e_{c}\varepsilon _{ad}\Omega ^{d} + \varepsilon _{ac}\Omega \right] \left( {\hat{\Psi }} e^{c} + \delta ^{c}\Psi \right) \nonumber \\&\left. + \,u_{a}\ddot{\Psi }- \left( \hat{{\dot{\Psi }}} e_{a} + \delta _{a}{\dot{\Psi }}\right) \right\} - u_{a}\left\{ \left( N_{cb} + e_{c} e_{b}\right) \left( {\hat{\Psi }} e^{c} + \delta ^{c}\Psi \right) \right. \nonumber \\&+\, u_{b}\left( \mathcal {A} e_{c} + \mathcal {A}_{c}\right) \left( {\hat{\Psi }} e^{c} + \delta ^{c}\Psi \right) - \left. {\dot{\Psi }}\left( \mathcal {A} e_{b} + \mathcal {A}_{b}\right) \right\} \nonumber \\&+ \frac{1}{3}\left\{ \hat{{\hat{\Psi }}} + \phi {\hat{\Psi }} -\, \delta ^{c}\Psi a_{c} + \delta ^{c}\delta _{c}\Psi \right\} \left( N_{ab} + e_{a} e_{b}\right) \nonumber \\&+\, \frac{1}{3}\left\{ 2\hat{{\hat{\Psi }}} - \phi {\hat{\Psi }} - 2\delta ^{c}\Psi a_{c} - \,\delta ^{c}\delta _{c}\Psi \right\} \left( e_{a} e_{b} - \frac{1}{2} N_{ab}\right) \nonumber \\&+\, \left\{ 2\delta _{(a}{\hat{\Psi }} - \left( \Sigma _{(a} + \Omega ^{c}\varepsilon _{c(a}\right) {\dot{\Psi }} \right. \left. -\phi \delta _{(a}\Psi + 2\delta ^{c}\Psi \left( \varsigma \varepsilon _{c(a} - \zeta _{c(a}\right) \right\} e_{b)}\nonumber \\&+\, {\hat{\Psi }}\zeta _{ab} + \delta _{\{a}\delta _{b\}} + \frac{1}{2}\left( e_{a}\varepsilon _{bc} - e_{b}\varepsilon _{ac} + e_{c}\varepsilon _{ab}\right) \left\{ \left( 2\varsigma {\hat{\Psi }} + \varepsilon _{mn}\delta ^{m}\delta ^{n}\Psi \right) e^{c} \right. \nonumber \\&+\, \varsigma \delta ^{c}\Psi + \varepsilon ^{cm}\left( \Sigma _{m}{\dot{\Psi }} - \varepsilon _{mc}\Omega ^{c}{\dot{\Psi }} \left. \left. + \varepsilon _{mc}\varsigma \delta ^{c}\Psi \right) \right\} \right] . \end{aligned}$$

(C1)