# Questions tagged [curvature]

Use this for questions pertaining to curvature of manifolds. Does not need to be specific to general relativity, but also for curvature of e.g. a [tag:calabi-yau] manifold.

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### Visualising/getting inuition for, the manifold resulting from these geodesics?

This is related to another post of mine. I know that the geodesics for flat Euclidean Space are straight lines. But I want to take these curved geodesics and tried to work backwards to determine the ...
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### Discontinuous changes in manifold's curvature?

First of all, there exists a question on PSE which does seem to pertain to my question below, but not exactly. This is one of those questions which requires perhaps an intuitive rather than ...
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### Derivative of the quadratic invariant term

I know that the derivative of the contraction of two Ricci tensors with respect to the Riemann tensor must be \begin{equation} \left(\frac{\partial \left(R^{\alpha\beta}R_{\alpha\beta}\right)}{\...
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### How curvature and field strength are exactly the same?

I am watching this lecture series by Fredric Schuller: "Curvature and torsion on principal bundles - Lec 24 - Frederic Schuller" @minute 34:00. In this part he discusses the Lie algebra valued one ...
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### Proving that the Berry curvature is always zero

If we have an energy eigenstate $|\psi(\boldsymbol \lambda )\rangle$ which is a function of some external parameter $\boldsymbol \lambda = (a,b)$, then the associated Berry connection is defined to be ...
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### How small could the universe be?

In the article Will the Universe expand forever?, NASA states: We now know (as of 2013) that the universe is flat with only a 0.4% margin of error. This suggests that the Universe is infinite in ...
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### About tests on the importance of general relativistic effects

I am dealing with the trajectories of charged particles in the vicinity of a Kerr black hole which is inmerse in an asymptotically uniform magnetic field. I pretend to make an estimation of the ...
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### Volume of an Universe with $k=+1$

I read in the Steven Weinberg’s book “Cosmology”: So far, we have considered only local properties of the spacetime. Now let us look at it in the large. For $k = +1$ space is finite, though like ...
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### How to get the Ricci tensor of an EM field?

We have the Einstein equations $$G_{\alpha\beta}=\frac{8\pi G}{c^4}T_{\alpha\beta}\\R_{\alpha\beta}-\frac12 g_{\alpha\beta}R=\frac{8\pi G}{c^4}T_{\alpha\beta}$$ I have been asked to show that for ...
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### How do we know that bending of light around stars is due to bending of space-time and not diffraction?

One question that popped up during the studies of special and general relativity (which I am forced to take unfortunately) is the following: How do we know that this is due to the bending of space-...
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### The invariance of BEC and curved space

I went to a lecture where the speaker explained that people could study space and cosmology and curved space by using BEC states. She tole me that the sound wave (mechanic wave) in BEC was equivalent ...
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### What (& why) exactly does a massive body curve/bend in General Relativity?

So, in General Relativity a massive body bends, curves the spacetime continuum... But WHAT exactly is it that "thing" that gets curved? What exactly is empty space? Does it have a real, physical ...
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### Would it be possible for a human to simply “step” through a traversable wormhole?

(For the purposes of a science-fiction story where the author is attempting to have as much basis in fact as possible) if one were to create or make use of a Morris-Thorne or similar traversable ...
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### Einstein field equations in empty space, question about non-zero curvature

After reading parts of Chapter 8 in Hobson, 'General Relativity: An introduction for Physicists,' I have a question regarding the observation on page 184 regarding the gravitational field equations in ...
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### What is the difference between Kretschmann and Ricci scalar curvature? [duplicate]

When you solve the vacuum field equations of gravity $R_{ab} = 0$ you come to the conclusion that the spacetime is flat in terms of scalar curvature when you have chosen a suitable metric tensor. ...
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### Significance of Kretschmann scalar to flat spaces?

If you are given a spacetime embedded with a particular metric tensor that satisfies the vacuum field equations of general relativity, how do you confirm that you aren’t simply dealing with a ...
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### Technically, what is a spacetime singularity? [duplicate]

In popular science books and articles, one often finds that the BigBang is a singularity of spacetime, and it is expected to be solved by a successful theory of Quantum gravity. Technically what is a ...
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### Why do objects “fall” along spacetime geodesic lines?

I'm working on a paper that also addresses the topic of general relativity (among other topics). The most common answer I get to the question above (why do objects fall) is that the objects are not ...
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### Ricci tensor derivation in Kaluza-Klein theory

I've been trying to follow the article Kaluza-Klein for Kids where the author derives the lagrangian density in the Kaluza-Klein theory. He takes scalar function $\Phi =1$, then he uses the "ansatz" ...