# Tagged Questions

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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I've been trying to grok General Relativity for a while now, and I've been having some trouble. Many physics textbooks gloss over the subject with an "it's too advanced for this medium", and many ...
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### A thought experiment on vision and curved spacetime

What follows is a long self-made example to deal with my conceptual issues of visualizing curved spacetime. Imagine an observer floating somewhere in space. He feels no strain on his body, ...
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### Space-time geometry and metric

I am confused in one question in general relativity, why we can always express a space-time geometry only by metric. It means a metric, which is just about distance in tangent space, can tell us all ...
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### Visualizing gravity in 3D

We've all seen the depiction of gravity bending space downwards, and so attracting objects into the dent it creates, cf. e.g. this and this Phys.SE posts. That's intuitive and makes a lot of sense, ...
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### What made Einstein think that gravity was caused by the curvature of spacetime?

What observation/thought experiment led him to think this?
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### Better explanation of the common general relativity illustration (stretched sheet of fabric)

I've seen many science popularisation documentaries and read few books (obviously not being scientist myself). I am able to process and understand basic ideas behind most of these. However for general ...
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### Why does the curvature of space decrease during inflation but increase after it?

According to the Friedmann equation, curvature of space will increase with time/expansion of space, but I've also read that during Inflation, the expansion caused the Universe to flatten out. What is ...
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### Exterior (covariant) derivatives and electromagnetism

I'm porting Maxwell's equations to curved spacetime and am having trouble reconciling the tensor and forms treatments. I think the problem boils down to a misunderstanding on my part concerning the ...
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### Equation of the saddle-like surface with constant negative curvature?

What is the equation for the saddle-like 2d surface (embeded in 3d Euclidean space with cartesian coordinates x, y and z) with constant negative curvature frequently used to illustrate open universe (...
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### How is the shape of the universe measured by scientists?

I would like to learn how scientists go about measuring the large-scale curvature of the universe to determine if the universe is closed 'i.e. spherical', flat, or open 'i.e. saddle shaped'. My ...
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### Curvature of electrostatic potential is zero

Could you please expound upon this claim? I found such claim on Zangwill's Classical Electrodynamics, which states that constraint coming from Laplacian equation implies electrostatic potential has ...
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### Did our Universe experience a curvature dominated phase?

So, my question is simple: did our Universe experience a curvature dominated phase? Or, rather, could our Universe have experienced a curvature dominated phase? This seems quite shruggish, at first ...
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### How/why can the cosmic background radiation measurements tell us anything about the curvature of the universe?

So I've read the Wikipedia articles on WMAP and CMB in an attempt to try to understand how scientists are able to deduce the curvature of the universe from the measurements of the CMB. The Wiki ...
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### Is the curvature of spacetime invariant? Could it be characterized as the ether?

I'm writing a paper for a Philosophy of Science course about GR/SR and I'm wondering if I can (1) characterize the curvature of spacetime as invariant and (2) argue that this is what Einstein referred ...
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### Curvature and edge state

If the boundary of quantum hall fluid has non-constant curvature, how will it affect the edge state which is usually described in chiral Luttinger fluid?
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### What is the variation of Gauss-Bonnet term a total derivative of?

What is the variation of Gauss-Bonnet term total derivative of? i.e. Variation of Gauss-Bonnet combination $= \nabla_{\mu} C^{\mu}$. What's $C^{\mu}$ in 4-dimensions?
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### Source term of the Einstein field equation

My copy of Feynman's "Six Not-So-Easy Pieces" has an interesting introduction by Roger Penrose. In that introduction (copyright 1997 according to the copyright page), Penrose complains that Feynman's "...
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### Killing tensor and Riemann tensor identity

I know that if we have a Killing vector then it's straightforward to show the identity: $$\nabla_a \nabla_b K_c = R_{cba}^k K_d$$ I'm now trying to show the following identity for a $(0,2)$ Killing ...
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### Metric of following spacetime and refractive index

Let's have metrics $$ds^{2} = f(\mathbf r)dt^{2} - h(\mathbf r )\delta_{ij}dx^{i}dx^{j}.$$ Hot to show that motion of light in spacetime with this metrics is equal to motion in continuous media with ...
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### Why geometrically four acceleration is a curvature vector of a world line? And what is proper acceleration?

Why geometrically four acceleration is a curvature vector of a world line? Geometrically, four-acceleration is a curvature vector of a world line. Therefore, the magnitude of the four-...
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### Curvature of spacetime and the equivalence principle

Assuming the Einstein equivalence principle, formulated as following: The outcome of any local non-gravitational experiment in a freely falling laboratory is independent of the velocity of the ...
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### How is $\Omega_0 = 1$ when the characteristic “teardrop” past light cone seems to admit curvature?

Introduction: The top graphic is just one I pulled from a page describing the process of detecting cosmic curvature. The second graphic is one I drew up to illustrate my misunderstanding. My ...
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### Euclidean black hole extrinsic curvature

I have read that the extrinsic curvature at the horizon of a euclidean black hole is zero? Does anybody know how this can be shown?
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### Space-Time Curvature Depends on Relative Speed

When the mass of a planet causes the curvature of space-time we see that an approaching free-falling object deviates its path towards the planet. We also see the amount of that deviation depends on it'...
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### Clarifying what metric counts as flat space

In (2D) Cartesian coordinates, the Euclidean metric... $$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$ ...is flat space. If the diagonal elements are exchanged for other real numbers ...
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### Gravitational self-interaction

Today, someone asked me why "the warped space-time warps itself" (he read it in Kip Thorne's: The Science of Interstellar). I guess this is related to the gravitational self-interaction. But I don't ...
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### What makes a coordinate curved?

Bear with me while I try to explain exactly what the question is. The question Can a curvature in time (and not space) cause acceleration? is imagining a coordinate system in which the curvature is ...
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### Is the scalar curvature of the Schwarzschild solution 0?

The Schwarzschild solution is meant to be a solution of the vacuum Einstein equations. That is $$R_{\mu\nu}=0.$$ So, the Ricci tensor must be null for $r>0$. Now, if the scalar curvature is ...
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### Physical meaning of harmonic function?

In complex numbers, we define a harmonic function as a twice continuously differentiable function such that the Laplace operator acting on it gives zero. Can anybody explain me the physical ...
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### Visualizing gravity

When you see graphics trying to help people visualize what gravity in Einstein's relativity "looks" like it will often be a mostly two-dimensional plane with a concave warp where a massive object is ...
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### Aside from experimental evidence, is there any reason to model space as Euclidean?

Obviously experiment is the end-all-be-all of any science, but I'm curious if there's any a priori reason to model space as Euclidean three-space (from a pre-relativity viewpoint, of course; I'm ...
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### How energy curves spacetime?

We know through General Relativity (GR) that matter curves spacetime (ST) like a "ball curves a trampoline" but then how energy curves spacetime? Is it just like matter curvature of ST?
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### Is the surface of a heavy sphere bigger than $4 \pi r^2$ due to general relativity?

I am unfortunately not familiar with the mathematics behind general relativity. However, on a heavy planet (say a sphere) gravity will bend space-time in a way that an object initially in rest, will ...