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# 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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### Proof that the scalar curvature of a two-dimensional space can be expressed by only one component of the Riemann tensor

So I'm working on a question that asks for a proof that in two-dimensional space, the scalar curvature is given by: $$R = \frac{2R^1{}_{212}}{{g}_{22}}$$ Now, I've been playing around with the ...
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### Is curved spacetime a real thing or just math

I was curious if the curving of spacetime by mass/energy was actually a real thing or is it just a mathematical construct, a way of visualizing the force of gravity and explaining it and that there is ...
112 views

### Is there a limitation on the values ​that Einstein tensor $G_{\mu\nu}$ can take?

Is there a limitation on the values ​​that Einstein tensor $G_{\mu\nu}$ can take? For example: Is it always bigger than zero? What is the highest amount that can be taken by it? What is the ...
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### How spacetime distortion can ever be noticed from inside spacetime itself?

this is a naive question from a non-physicist. It is my understanding that gravitational waves are a deformation of spacetime. However, those are noticeable through, for example, laser interferometry. ...
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### Is curvature the exterior covariant derivative of the connection?

Let $P\to M$ be a $G$-principal bundle, $G$ a topological group, $\omega$ the connection and $V$ a vector space. We define $d_\omega: \Omega^k_G(P, V)\to\Omega^{k+1}_G(P, V)$ the exterior covariant ...
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### Conventions in the FRW metric

So the FRW metric is $$ds^2=-c^2dt^2+a(t)^2\left(\frac{dr^2}{1-kr^2}+r^2(d\theta^2+\sin^2\theta d\phi^2)\right).$$ I have seen books mention that $k=1,0,-1$ - depending one of the values listed ...
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### Can we envision the curvature of a 2d spacetime with the help of a second space dimension near the big bang?

Consider a 2-dimensional spacetime. For a 3-, let alone a 4-dimensional spacetime it's impossible to envision the curvature of spacetime near (at the beginning or just after) the big bang. Is it ...
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### Riemann tensor Contracted with full antisymmetric tensor

I'm not able to show that $\epsilon^{abcd} R_{bcae} = 0$ Note: Properly, I have to show that $\epsilon^{Iabc} R_{abIL} = 0$, where $I,L$ are tetrad index and $a,b,c$ are spacetime index, but it ...
60 views

### Is space in a rotating frame flat?

Apparently in papers like "Space geometry of rotating platforms: an operational approach" https://arxiv.org/abs/gr-qc/0207104 (page 21) and https://www.amherst.edu/.../view/10267/original/reden05.pdf (...
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### Bending of space and time

we say massive object bends spacetime, but space is same in all direction, then how do u decide which direction it will bend spacetime, for showing the movement of planet around sun is ok, wat about ...
227 views

### Make $\pi = 3$ again [closed]

The value of $\pi$, or the circumference divided by the diameter of a circle, is known with absurd precision, but I want it to be 3. The circumference around a black hole outside the Schwarzschild ...
64 views

### Curvature scalars and singularities

Without resorting to the singularity theorems, can we say that there is a singularity at a particular $r=\textrm{constant}$ if the value of the Ricci and Kretschmann scalars get infinitely large at ...
35 views

### Is intrinsic curvature of an embedded surface a covariant quantity from the embedding space point of view?

Suppose I have a $(d+1)$-dimensional manifold with metric $g_{\mu\nu}$. In it I have an embedded codimension-$1$ surface, $\Gamma$, with induced metric $\gamma_{ab}$. Is Ricci scalar defined in terms ...
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### Meaning of $M$ in Schwarzschild metric

In the Schwarzschild metric $$ds^2 = - \left(1-\frac{2M}{r} \right) dt^2 + \frac{dr^2}{1-\frac{2M}{r} }+ r^2 d\Omega^2.$$ Is it safe to call $M$ the mass of the source of curvature? Or should I ...
173 views

### Is there is a concensus among physicists if spacetime actually curves and if so what is it?

Going off from what others have told me on here, and based on the Wikipedia page for Quantum Gravity, General Relativity can be described mathematically in a way different than the geometrical curved ...
98 views

### Evidence for zero curvature of universe (besides CMB)

There is a beautiful argument, based on the spherical harmonics of the cosmic microwave background, which calculates the curvature of the universe. Are there any other methods of computing the ...
47 views

### Riemann curvature tensor components having 3 or 4 distinct components

When, if ever, will we see Riemann curvature tensor (RCT) components having 3 or 4 distinct indices?, like for $R_{txxy}$ or $R_{txyz}$ for ex. How this came about was I that I was reading that ...
86 views

### Gauge dependence of the Einstein tensor and the Riemann/Ricci curvature tensors in non-linear general relativity

The Einstein field equations are given by (with assuming $\Lambda = 0$), $$R_{ab} - \frac{1}{2} R g_{ab} = \kappa T_{ab}.$$ The principle of general covariance states that the form of these ...
160 views

### Does a vacuum solution to the Einstein equation imply flat spacetime?

I have read that a solution to the vacuum Einstein equation has a vanishing Einstein tensor, and therefore a vanishing stress-energy tensor. This means that there is no matter to generate spacetime ...
29 views

### Is the warping of spacetime proportional to the mass/energy/momentum of an object in GR [duplicate]

Just wondering if the warping of spacetime proportional to the mass/energy/momentum of an object in general relativity?
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### Ricci Scalar as Curvature

So I understand that the Ricci scalar represents the curvature of the space. Since any manifold can be considered locally flat, is Ricci scalar always zero locally for any manifold? On one hand it ...
131 views

### Is spacetime-curvature relative?

Velocity is relative, which means kinetic energy is. Since, according to general relativity, energy bends spacetime around it, wouldn't this mean observers moving in different inertial frames measure ...
I am trying to figure out the calculation which leads to the geodesic deviation on this site. So far I understood all steps until (14.7) and managed to show that (14.6) = (14.7), namely  \ddot\xi^\...