Ricci tensor Definition and 59 Threads

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In Carrol's text, he shows that the covariant derivative of the Ricci scalar is zero along a Killing vector. He then goes on to say something about how this intuitively justifies our notion of geometry not changing along a Killing vector. This same informal reasoning would seem to imply that...

Looking for the Schwarzschild Solution for this equation: ds^2 = -A(r) / c^2 * dr^2 - r^2 / c^2 *(d\\theta^2 +(sin(\\theta))^2 *d\\phi^2) + B(r) * dt^2 where A(r) = 1 / (1-2*m/r) And B(r) = (1-2*m/r) From this can be calculated the co- and contra-varient metric tensors and...

The Ricci Tensor comes from the Riemann Curvature Tensor: R^{\beta}_{\nu\rho\sigma} = \Gamma^{\beta}_{\nu\sigma,\rho} - \Gamma^{\beta}_{\nu\rho,\sigma} + \Gamma^{\alpha}_{\nu\sigma}\Gamma^{\beta}_{\alpha\ rho} - \Gamma^{\alpha}_{\nu\rho}\Gamma^{\beta}_{\alpha\sigma} The Ricci Tensor just...

Hello; What does it mean physically if I have? R_{a b} = Ag_{a b} I think it means that my manifold is an n-sphere (i.e. if A is positive), or it is AdSn (i.e. if A is negative). Is this correct?

Starting with this definition of the Reimann tensor R^a_{mbn}=\Gamma ^{a}_{mn,b}-\Gamma ^{a}_{mb,n}+\Gamma ^{a}_{rb}\Gamma ^{r}_{mn}-\Gamma ^{a}_{rn}\Gamma ^{r}_{mb} Can I contract on indices a,b and r to get R_{mn} ? It bothers me that the expression on the right is not symmetric in...

Just wondering if Traces can be applied to tensors. If the Ricci tensor is Rii then is sums over diagonal elements. So technically, can you say the trace of the Riemann tensor is the Ricci tensor?

Hello, I wish to show that on 3-dimensional manifolds, the weyl tensor vanishes. In other words, I want to show that the curvature tensor, the ricci tensor and curvature scalar hold the relation Please, if anyone knows how I can prove this relation or refer to a place which proves the...

Producing the Ricci tensor On a pseudo-Riemannian manifold we can contract the Riemann curvature tensor to form the Ricci tensor. In this process of contraction we sum over two indices to make a (3-1)-tensor into a (2-0)-tensor. My question is, why must we contract two indices? Why can't we...