All Questions
Tagged with spacetime differential-geometry
347
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Still having trouble understanding gravitational lensing [duplicate]
The normal diagram used to explain gravitational lensing shows a two-dimensional plane that is deflected by a heavy weight. This is a two dimensional description that requires an extra dimension to ...
- 319
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What is Dirac's reasoning when saying parallel displacement creates vector field with vanishing covariant derivative?
Section 12 of Dirac's book "General Theory of Relativity" is called "The condition for flat space", and he is proving that a space is flat if and only if the curvature tensor $R_{\...
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Examples of spacetimes that are asymptotically flat at future timelike infinity
There are interesting non-trivial examples of spacetimes which are asymptotically flat at null and spacelike infinities. For example, the Kerr family of black holes satisfies these conditions. However,...
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Mathematical anatomy of general relativity
I was always told that spacetime in general relativity was a Lorentzian manifold, that is, a Pseudo-Riemannian manifold $ (M, g) $ with metric signature $(+, -, -, -)$ or $(-, +, +, +)$ and that that ...
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Confused about spherically symmetric spacetimes
I'm following Schutz's General Relativity book and I am confused about his description and derivations of a spherically symmetric spacetime. I searched online and found that using Killing vectors is a ...
- 151
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Why do we call the Riemann curvature tensor the curvature of spacetime rather than the curvature tensor of its tangent bundle?
I was studying the mathematical description of gauge theories (in terms of bundle, connection, curvature,...) and something bothers me in the terminology when I compare it with general relativity.
In ...
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Geodesic beeing inextendible and incomplete
We say that a geodesic is (future ) inextendible if there exist no (future) endpoint for example. Wouldnt that imply the domain of the geodesic is $[p, \infty)$ with some beginning point $p$? And it ...
user390166
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Conjugate points on manifolds
My question is:
Why do conjugate points exist on globally hyperbolic manifolds, satisfying the strong energy condition?
We define M to be globally hyperbolic if it posseses a cauchy surface and a pair ...
user390166
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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 ...
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Why is spacetime pseudo-Riemannian manifold?
Forgive me for asking, This is a relatively naïve question, though, one i've had for a while now. I know that a pseudo-Riemannian manifold is a differentiable manifold with a metric tensor that is ...
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Are black holes 4-dimensional balls of spacetime? If so, will they have 3-sphere surfaces?
If black holes are 4-dimensional balls of spacetime, they will have a 3-sphere surface with a 3-dimensional volume. Would this allow infalling matter to remain within this surface?
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Null infinity reachable by timelike worldlines?
Usually, Penrose diagrams are marked with points and segments being named past/future timelike infinity $i^{-,+}$, past/future null infinity $\mathscr{I}^{-,+}$ and spacelike infinity $i^0$ -- see for ...
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Killing Tensors: What quantities do they preserve?
It is known that coordinate transformation generated by Killing vectors (KV) preserve the metric components, i.e. it generates an isometry transformation. Are there similar geometrical quantities that ...
- 1,752
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Can $\mathbb{R}^4$ be globally equipped with a non-trivial non-singular Ricci-flat metric?
I'm self-studying general relativity. I just learned the Schwarzschild metric, which is defined on $\mathbb{R}\times (E^3-O)$. So I got a natural question: does there exist a nontrivial solution (...
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What is a hypersurface?
What is the concept of hypersurface in general relativity? I know it could be characterized into three categories but how do we define hypersurface (in general) in physics? I didn't get what thing it ...
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