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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How does the Ricci tensor describe the changing separation of two airplanes flying from the equator? Conceptually understanding the Ricci tensor
I'm trying to understand the concept of the Ricci tensor and its physical implications using a concrete example involving two airplanes. Suppose two airplanes start at the equator, separated by a ...
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Does Mass Actually Displace Space-Time, or does Mass only Distort it?
1. Question
Given the plethora of space-time illustrations, there is a sense that space-time is actually being displaced by mass, (planets). But on its face, this doesn't really make sense because ...
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Probabilistic curvature of spacetime [duplicate]
I was wondering since matter tells space-time how to curve, and since matter is probabilistic in position (say hydrogen atom) is the curvature also probabilistic?
black holes slowly shrink by ...
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Spatial Curvature of Universe at recombination vs now
From my understanding, we use the CMB data to measure the spatial curvature of the universe today. Why is it the value for today if the CMB data reflects the universe at recombination (380K years ...
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Are there closed simply connected 2D manifolds that do not require a third dimension?
In the context of cosmology, space is commonly described as potentially having a global curvature that can be positive, zero, or negative. A common way that textbooks describe positive curvature is by ...
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Viable values for the $K$ parameter in the FLRW metric
The FLWR metric is sometimes given as $$c^2 d\tau^2 = c^2 dt^2 - \frac{a(t)^2}{(1-KX^2)} dX^2. $$
I am not interested in the tangential motion so I set $d \Omega = 0$ although it is of interest in ...
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Cone vs. small circle parallel transport
I'm having trouble reconcile the following two seemingly contradicting conclusions (in 2d space):
A cone is flat, because you can unfold it and it's a flat 2d surface.
A cone as shown in the picture ...
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Extrinsic Curvature Calculation on the Sphere
Given the following 2+1 dimensional metric:
$$ds^{2}=2k\left(dr^{2}+\left(1-\frac{2\sin\left(\chi\right)\sin\left(\chi-\psi\right)}{\Delta}\right)d\theta^{2}\right)-\frac{2\cos\left(\chi\right)\cos\...
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Derivation of Einstein-Cartan (EC) action for parametrized connection $A$ & introduction of torsion
I have some trouble with one missing step when I want to get the teleparallel action from general EC theory, which I am not fully understanding.
The starting form of action (3-Dimensional) is:
$$
S_{...
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Where is the mass in a Black Hole without a "central" curvature singularity?
Not all black holes have a curvature singularity at their center (an example). But in principle, I thought that the curvature singularity was a direct result of the fact that the mass is concentrated ...
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Extrinsic Curvature in a conformally-flat spacetime that is also asymptotically-flat spacetime
I would appreciate if someone can confirm or correct my understanding of extrinsic-curvature (as in the ADM 3+1 decomposition of spacetime) when dealing with a conformally-flat spacetime.
(I updated ...
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Does Matter Cause Curvature or Vice-Versa [closed]
From the way explanations about gravity-acceleration-curvature equivalence are usually phrased here or elsewhere, it would appear many or most think that matter causes space-time curvature.
I cannot ...
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Does (covariant) divergence-freeness of the stress-energy tensor ${T^{\mu\nu}}_{;\nu}=0$ follow from the Bianchi identity?
I'm working through Chap. $30$ of Dirac's "GTR" where he develops the "comprehensive action principle". He makes a very slick and mathematically elegant argument to show that the ...
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Homogeneous and Isotropic But not Maximally Symmetric Space
Is this statement correct: "In a homogeneous and Isotropic space the sectional curvature is constant, while in a maximally symmetric space the Riemann Curvature Tensor is covariantly constant in ...
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Constant curvature on a sphere?
$ds^2 = \frac{1}{1- r^2}dr^2 + r^2d\theta^2$ denotes a 2d spherical surface and it should have a constant curvature. The Riemann curvature tensor components are linear in their all 3 inputs. Since the ...
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