# 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 Calabi-Yau manifold.

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### ALL "forces" as manifestations of properties of space-time

I apologize if this seems like a quack question, but I need some insights by those who know much more than me in Physics. Anyway, the gravitational "force" (not really a force) is a manifestation of ...
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### Can the Ricci curvature tensor vary while we hold fixed the Ricci scalar and the Weyl tensor? (Nordstrom Gravity)

I am reading about Nordstrom's second theory of gravity and have become confused. On the wikipedia page and elsewhere it is said that the theory is completely captured by its equations for the Ricci ...
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### Are there any ways to conceptualize the relationship between gravity and space-time other than curvatures?

This might sound like a random question, but it came to me while I was trying to conceptualize the size of the universe and started thinking of entire galaxies resembling grands of sand floating ...
1 vote
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### Independent Components of the Riemann Curvature Tensor

I am struggling to understand a general method to calculate the independent components of the Riemann Curvature Tensor (RCT). Firstly, as far as I am aware the number of independent components of the ...
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### How can a triangle have a sum exceeding 180 degrees in a curved space?

I was reading a book to understand the limits of the euclidean space I understand that lines that are parallel in 2d can meet in 3d space like on a sphere but it is hard to imagine or fathom why the ...
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### No torsion with calculating the commutator of the covariant derivatives

For simplicity, I only calculated half of the commutator. I didn't leave everything in components because I'm uncomfortable considering (I previously messed up the indices. The following is the ...
1 vote
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### Is there a formula to work out how much the fabric of spacetime bends?

From my knowledge, a big mass (planet star etc) can bend the fabric of spacetime. Is there a formula that we can use to work out how much it bends?
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### How do black holes infinitely bend space-time when the bending is mass dependent and not density dependent?

According to Einstein, mass bends the fabric of space-time. And nothing in the universe has infinite mass to infinitely bend space-time. So how do remnants of supermassive stars, i.e black holes ...
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### Can the metric components be determined by the curvature value?

In JT gravity, we are called to solve the equation of motion with respect to the Dilaton field $\Phi$, which read $$R+2=0$$ Since the Ricci scalar $R$ is determined by the metric, then the equation ...
1 vote
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### Why does trajectory in the space curved by gravity, depend on the speed?

I am sorry about the probably naiive nature of this question (I am a software eng, not a physics student): I (think I) understand the popular curved "trampoline" model of 2-dimensional space,...
1 vote
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### What is spacetime like according to general relativity? [duplicate]

It is often said that matter curves space (or rather spacetime) in general relativity. But why should matter curve space one way and not the other way? So it seems like a metaphor, I guess. I read ...
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### Why does the Weyl tensor not show up in the Einstein field equations?

In the Einstein field equations, the only tensor that shows up is the Ricci tensor and the metric tensor, together with the Ricci scalar. The Weyl tensor though is a tensor that is a part of the ...
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### Flat space between colliding black holes

When 2 black holes approach each other, they both bend space in an opposite direction. There must always be a flat space between 2 colliding black holes. However, I heard that they actually merge, ...
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### How will two equal discs, rotating with equal but opposite angular velocities and put on top of each other affect the spacetime "surrounding" them?

In this article the Ehrenfest paradox is described. You can read in it that, according to Einstein's General Relativity (to which this paradox contributed), the spacetime around a rotating disc is non-...
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### Can a CTC contaning spacetime be purely electric?

Take a time-oriented Lorentzian manifold $(M, g)$ where $M$ is a topological 4-manifold and $g$ a Lorenzian metric. Assume such a spacetime contains a CTC. Since the manifold is time-oriented, one can ...
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### Extrinsic curvature of constant time hypersurfaces in Minkowski

Along the geodesic of a stationary observer in Minkowski spacetime we have the following tangent vector $$t^\mu = (1,0,0,0)$$ We have that hypersurfaces of constant time along this are just 3D ...
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### Does the Casimir effect only occur between flat plates?

What happens to the strength of the Casimir effect when the Casimir plates are curved instead of being completely flat. Does this have an effect on the negative vacuum pressure at different points ...
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### How we find the contorsion tensor?

I know that the formula for contorsion tensor is $$K^{\mu\nu}_a=\frac12({T_a}^{\mu\nu}+T^{\nu\mu}_a-T^{\mu\nu}_a)$$ I want to know how I can find ${T_a}^{\mu\nu}$. What kind of contraction do I follow ...
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### Understanding Riemann Curvature Tensor in Misner, Thorne and Wheeler's Gravitation

I'm trying to understand section 11.4 of Misner, Thorne and Wheeler's Gravitation textbook, which explains how the output of the Riemann Curvature Tensor $Riemann(...,A,u,v)$ is a vector describing ...
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### Is there any physical interpretation of the curvature in electromagnetism?

Electromagnetism can be modeled as a $U(1)$-principal bundle over Minkowski spacetime. The strength of the electromagnetic field is given by the 2-tensor $F_{\mu\nu}$. In differential geometry this is ...
1 vote
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### Definition of asymptotically flat spacetime

Following the definition in Wald's book on general relativity, in page 276 asymptotically flat spacetimes are defined using conformal isometry with conformal factor $Ω$. Then one of the requirements ...
1 vote
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### How does a curvature in time equate to Newtonian gravity? [duplicate]

I often read that a curvature in time (the rate at which clocks tick) near a massive object, is considered to be the source of Newtonian gravity. This got me wondering, does General Relativity use the ...
In cosmological perturbation theory, it's hard to find papers that would expose the general principle to perturb physical quantities (metric, fluid pressure and density, speed...) up to the $n$th ...