# 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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### Clarification regarding the transformation law of the Christoffel symbols

I'm learning about general relativity from Sean M. Carroll's textbook. I recently encountered the transformation law for the Christoffel symbols, and I'm confused, as it seems like I'm seeing two ...
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### Local cartesian coordinates on Riemannian manifold

I'm wondering is possible for every given metric $g=g_{ij}dx^i \otimes dx^j$ on $M$ and for every given $p\in M$ to find such chart $(U, \varphi)$ around $p\in U \subset M$ that the metric $g|_U$ in ...
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### Double Trace of the tensor product of the metric tensor with vector fields.

So I am currently preparing for an exam on General Relativity and while reading the notes I stumbled upon this: $$tr[tr[g \otimes X \otimes Y]]= g(X,Y)$$ Where $$g=g_{ij} dx^{i}\otimes dx^{j}$$ is ...
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### Conditions on positivity of Riemann squared products in Euclidean space?

I wonder if it is known whether there are simple conditions on $u,v$ such that: $$R_{abcd}R^{abcd}+ u R_{ab}R^{ab} + vR^2 = A_{abcd}A^{abcd}$$ and thus proving that the combination is always positive ...
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### A question about the definition of the normal variation of a tensor field

My question comes from the proof of Prop. 7.32 in Dan Lee's book Geometric relativity. The setting is we have a hypersurface $\Sigma$ with unit normal $\nu$ in an initial data set $(M, g, k)$. We now ...
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### General relativity problem from Wald's book

I'm self-studying General Relativity using Carroll's and Wald's book, and I'm confused about this problem from Wald (Chapter 3, Problem 2). Suppose that $M$ is a manifold with metric $g_{ab}$ and ...
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### Show that the conformally complete Schwarzschild spacetime is asymptotically flat at null infinity

I am trying to show that the conformal factor used to conformally complete the Schwarzschild spacetime renders it asymptotically flat at null infinity (according to the mathematical definition given ...
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1 vote
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### Derivation of a spin connection in general relativity

On my journey to understand the mathematical structure behind general relativity, I came across the concept of a spin connection. which (I understand) is a connection defined in a spinor bundle. The ...
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### Is time-orientability a condition on the metric, smooth or topological structure of a manifold?

I recently asked a question on Physics Stack Exchange about orientability and time-orientability of a manifold in the language of fiber bundles. This new question is related to, but independent, of ...
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### Counterexamples to show that "$p < q ⇒ t^{±}(p) < t^{±}(q)$", and "$t^{±}$ are continuous" are not true

About this proposition For a general spacetime $(M, g)$ the volume functions $t^{±}$ (a) $p < q ⇒ t^{±}(p) \le t^{±}(q)$, (b) $t^{±}$ are upper/lower semicontinuous. What counterexamples could I ...
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### How do I show that the region {r > 2m} of the Schwarzschild spacetime is globally hyperbolic?

How do I show that the region {r > 2m} of the Schwarzschild spacetime is globally hyperbolic? I am just starting with this so I don't really know how to lay out this arguments. I consulted Beem's ...
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