Twisted Lorentzian manifolds: a characterization with torse-forming time-like unit vectors

Abstract

Robertson–Walker and generalized Robertson–Walker spacetimes may be characterized by the existence of a time-like unit torse-forming vector field, with other constrains. We show that Twisted manifolds may still be characterized by the existence of such (unique) vector field, with no other constrain. Twisted manifolds generalize RW and GRW spacetimes by admitting a scale function that depends both on time and space. We obtain the Ricci tensor, corresponding to the stress–energy tensor of an imperfect fluid.

Appendix

$$\begin{aligned} (i,j,k,\ldots =0,1,\ldots ,n-1 ; \quad \mu ,\nu ,\rho ,\ldots = 1,2, \ldots , n-1) \end{aligned}$$

Christoffel symbols: $\Gamma _{ij}^k = \Gamma _{ji}^k = \tfrac{1}{2}g^{km} (\partial _i g_{jm} +\partial _j g_{im} -\partial _m g_{ij})$.

$$\begin{aligned} \Gamma _{i,0}^0= & {} 0,\quad \Gamma _{0,0}^k=0, \quad \Gamma ^\rho _{\mu ,0} = (\dot{f}/f ) \delta ^\rho _\mu , \quad \Gamma ^0_{\mu ,\nu } = f\dot{f} g^*_{\mu \nu }, \end{aligned}$$

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$$\begin{aligned} \Gamma ^\rho _{\mu ,\nu }= & {} \Gamma ^{*\rho }_{\mu ,\nu } + (f_\nu /f) \delta ^\rho _\mu + (f_\mu /f) \delta ^\rho _\nu - (f^\rho /f) g^*_{\mu \nu } \end{aligned}$$

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where $\dot{f} =\partial _t f$, $f_\mu = \partial _\mu f$ and $f^\mu = g^{*\mu \nu } f_\nu$.

Riemann tensor: $R_{jkl}{}^m = -\partial _j \Gamma ^m_{k,l} + \partial _k \Gamma ^m_{j,l} + \Gamma _{j,l}^p\Gamma ^m_{kp} - \Gamma _{k,l}^p \Gamma _{jp}^m$

$$\begin{aligned} R_{\mu 0\rho }{}^0= & {} (f{\ddot{f}}) g^*_{\mu \rho } \end{aligned}$$

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$$\begin{aligned} R_{\mu \nu \rho }{}^0= & {} g^*_{\mu \rho } (f \partial _\nu \dot{f} - \dot{f} f_\nu ) - g^*_{\nu \rho } (f \partial _\mu \dot{f} - \dot{f} f_\mu ) \end{aligned}$$

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$$\begin{aligned} R_{\mu \nu \rho }{}^\sigma= & {} \, R^*_{\mu \nu \rho }{}^\sigma + \left( {\dot{f}}^2 - \frac{f^\lambda f_\lambda }{f^2} \right) (g^*_{\mu \rho }\delta ^\sigma _\nu - g^*_{\nu \rho }\delta ^\sigma _\mu )\nonumber \\&+\frac{2}{f^2} ( f^\sigma f_\nu g^*_{\mu \rho } - f^\sigma f_\mu g^*_{\nu \rho } + f_\mu f_\rho \delta ^\sigma _\nu - f_\nu f_\rho \delta ^\sigma _\mu ) \nonumber \\&+\frac{1}{f} \left[ \nabla ^*_\mu (f^\sigma g^*_{\nu \rho } - f_\rho \delta ^\sigma _\nu ) - \nabla ^*_\nu ( f^\sigma g^*_{\mu \rho } - f_\rho \delta ^\sigma _\mu ) \right] \end{aligned}$$

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Ricci tensor: $R_{jl} = R_{jkl}{}^k$

$$\begin{aligned} R_{00}&= -(n-1) ({\ddot{f}} / f) \end{aligned}$$

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$$\begin{aligned} R_{\mu 0}&= -(n-2)\partial _\mu (\dot{f} / f) \end{aligned}$$

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$$\begin{aligned} R_{\mu \nu }&= R^*_{\mu \nu } + g^*_{\mu \nu } [(n-2){\dot{f}}^2 + f{\ddot{f}}] +2(n-3)\frac{f_\mu f_\nu }{f^2} -(n-4) \frac{f^\sigma f_\sigma }{f^2} g^*_{\mu \nu } \nonumber \\&\quad -(n-3) \frac{1}{f}\nabla ^*_\mu f_\nu - \frac{1}{f} g^*_{\mu \nu } \nabla ^*_\sigma f^\sigma \end{aligned}$$

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Curvature scalar: $R=R^k{}_k$

$$\begin{aligned} R=\frac{R^*}{f^2} +(n-1) \left[ (n-2)\frac{{\dot{f}}^2}{f^2} + 2\frac{{\ddot{f}}}{f}\right] - (n-2)(n-5)\frac{f^\sigma f_\sigma }{f^4} -2(n-2)\frac{\nabla ^*_\sigma f^\sigma }{f^3} \end{aligned}$$

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