Raychaudhuri equation in spacetimes with torsion and nonmetricity

We introduce and develop the $1+3$ covariant approach to relativity and cosmology to spacetimes of arbitrary dimensions that have torsion and do not satisfy the metricity condition. Focusing on timelike observers, we identify and discuss the main differences between their kinematics and those of their counterparts in standard Riemannian spacetimes. At the center of our analysis lies the Raychaudhuri equation, which is the fundamental formula monitoring the convergence and divergence, namely the collapse and expansion, of timelike congruences. To the best of our knowledge, we provide the most general expression so far of the Raychaudhuri equation, with applications to an extensive range of nonstandard astrophysical and cosmological studies. Assuming that metricity holds, but allowing for nonzero torsion, we recover the results of analogous previous treatments. Focusing on nonmetricity alone, we identify a host of effects that depend on the nature of the timelike congruence and on the type of the adopted nonmetricity. We also demonstrate that in highly symmetric spaces one can recover the pure-torsion results from their pure nonmetricity analogues, and vice versa, via a simple ansatz between torsion and nonmetricity.

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