# Questions tagged [general-relativity]

Questions related to the mathematical aspects of Einstein's theory of relativity. For the physics and its interpretations, please ask at the physics.SE. You may also consider the tags (differential-geometry) and (pde).

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### If $\mathcal{M}$ is a manifold, what is meant by "the standard pairing on $\mathcal{M}^2$"?

I've been taking a look at the paper arXiv:2202.07580 [math-ph] , which discusses the so-called Wetterich equation in Lorentzian manifolds. I believe the Physics details behind the Wetterich equation ...
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### Can the equivalence of the covariant and contravariant "connection coefficients" be proved simply by using the flatness of the tangent plane?

Do the derivations below prove the equivalence of the "connection coefficients" (5.78) and (5.78) for Riemannian and pseudo-Riemannian manifolds? Are they correct, and justified by my stated ...
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### Showing metric is coordinate independent implies Killing vector field.

I am studying general relativity out of Sean Carroll's textbook, and I had trouble proving the following assertion (which is more about differential geometry than physics per se). Suppose we have a ...
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### The killing vectors of flat space in polar coordinates

The metric of flat three-dimensional space is written in usual Cartesian coordinates ds^2 = dx^2 + dy^2 + dz^2 There are 3 killing vectors (1,0,0) , (0,1,0) and (...
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### Checking Metric Tensor relations

I need to check some simple tensor relations for continuing my calculations. Please write me if you think anyone is incorrect. P.S.: I know this is very simple question, but i need to be assured about ...
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### Killing Vector Fields of Schwarzschild

It is known that the vector fields $\partial_t$, $\partial_\phi$, $\sin \phi \partial_\theta+\cot \theta \cos\phi \partial_\phi$, and $\cos\phi \partial_\theta-\cot \theta\sin \phi\partial_\phi$ are ...
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### Inconsistency in the definition of the connection coefficients

I am new to general relativity and I am currently facing an apparent inconsistency in the definition of the connection coefficients. Some references I've been consulting (e.g. the lecture notes by S. ...
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### About the Christoffel Symbols

I have a quick but important question about the concept of the Christoffel Symbols, and it is if you can apply another tensor rather than the metric tensor to find the Christoffel Symbols. I am trying ...
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