I know that the Riemann tensor shows curvature in space. However, in the case of general relativity, where space AND time is curved, how would one use the Riemann tensor to show curvature in a temporal dimension?
2
-
$\begingroup$ Just use the Riemann tensor in four dimensions. What prevents you from doing so? $\endgroup$– Jeanbaptiste RouxCommented Aug 2, 2022 at 17:46
-
$\begingroup$ But would there be a way of using the Riemann tensor to show solely the curvature in time? $\endgroup$– RayCommented Aug 3, 2022 at 3:40
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
The idea of curvature solely in time doesn’t make sense. Curvature does not exist for a single dimension, it requires at least a two dimensional manifold.
However, what you can do is to look at the components of the Riemann curvature tensor $R_{\mu\nu\beta\zeta}$. Any of the components with a $t$ in any of the slots would qualify as giving information about curvature in the time dimension. So if you look at the standard Schwarzschild metric and coordinates there is a $R_{trtr}$, a $R_{t\phi t\phi}$, and a $R_{t\theta t\theta}$ component, and of course the components related to those by symmetry. All of those will tell you something about curvature in the time dimension.
