All Questions
281
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Hawking and Ellis Lemma 4.3.1 Proof
I have a few questions about Hawking and Ellis' proof of this lemma (pages 92-93):
Write the $(2, 0)$ stress-energy tensor in coordinates as
$\mathbf{T} = T^{ab} \partial_a \otimes \partial_b$ and ...
2
votes
1
answer
109
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Boundary conditions on transition maps on general relativity
On the initial courses of topology and differential geometry, we learn again and again about charts, and atlas, and transition maps. I feel that transition maps are a very powerful idea, because they ...
- 157
2
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60
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Under what circumstances can a 4D singularity occur in General Relativity?
I've tried to find on the literature about 4D (single point) singularities, but most of the theorems about singularities pertain to either space-like or time-like singularities, which always have some ...
- 157
2
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2
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96
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Are $i^\pm$ and $i^0$ codimension 1 surfaces?
Standard textbooks like Carroll's say that spatial and temporal infinities in Minkowski space Penrose diagram are points. But on the footnote in pg. 3 of some draft notes on Celestial holography by ...
- 785
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1
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83
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What objects are solutions to the Einstein Field Equations?
The usual way the solutions of the Einstien Field Equations are introduced is by saying they are (pseudo-) riemannian metrics that satiafy the diff equations for a given EM Tensor. My question is: ...
-1
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1
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53
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If an area in 2D cannot be curved and finite is the same regarding the space of our pressumed 3D universe?
Is the sentence in the title right that our universe is infinite? And if so does it mean that stars are not evenly distributed along our universe but they all move from a populated centre to a fairly ...
- 2,255
4
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92
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Is the causal structure completely determined by the Weyl tensor alone?
By causal/conformal structure I mean the context of Malament's 1977 theorem. If I understand correctly this means that any two spacetimes which agree about all of the future-directed continuous ...
- 151
1
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2
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153
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What is the manifold topology of a spinning Cosmic String?
Given the following metric which is that of a rotating Cosmic String:
$$g=-c^2 dt^2 + d\rho^2 + (\kappa^2 \rho^2 - a^2) d\phi^2 - 2ac d \phi dt + dz^2.$$
can one determine the manifold topology ...
- 1,268
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76
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How to motivate that in presence of gravity the spacetime metric must be modified to $ds^2=g_{ab}(x)dx^adx^b$?
In the presence of a gravitational field, the spacetime metric, $$ds^2=\eta_{ab}dx^a dx^b,$$ should be changed to, $$ds^2=g_{ab}(x)dx^adx^b.$$ What are the convincing physical arguments that motivate ...
- 11.7k
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57
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Does the Weyl tensor amount to tidal effects of gravity?
The Ricci tensor, for the spacetime surrounding the Earth, is zero, so the spacetime around the Earth is Ricci-flat.
The Riemann tensor though is not zero since spacetime certainly is curved. This ...
- 450
2
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1
answer
100
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Reducing Tensor-rank by fixing an argument
Assume for example that you are given a (2,0) tensor $T^{\mu\nu}$ and you want to create
a vector, i.e., a (1,0) tensor out of it. Is it possible to just fix an index of
$T^{\mu\nu}$ while keeping the ...
- 56
1
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0
answers
32
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Example of lightlike curve that's not a geodesic in Lorentz spacetime [duplicate]
Let $(M,g)$ be a 4 dimensional Lorentz spacetime. A smooth curve $\alpha:\ I\to M$ is called lightlike if $\alpha'(s)\in TM_{\alpha(s)}$ is lightlike for all $s\in I$, which means
$$g_{\alpha(s)}\big(\...
- 143
3
votes
0
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88
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Intuition for the interior Killing vector fields in Schwarzschild?
The Schwarzschild metric represents a stationary (and static), spherically-symmetric, spacetime. These characteristics are manifested by the four Killing vector fields: one for time translation and ...
- 1,290
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32
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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
1
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2
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85
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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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