All Questions
Tagged with spacetime differential-geometry
347
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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 ...
- 207k
2
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1
answer
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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 ...
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1
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What is the geometry of light cones if space is curved/non-Euclidean?
In light cone diagrams, the plane corresponding to the present is always the Euclidean one, but what if space is curved? Now, I've also seen diagrams where spacetime is supposed to be regarded as ...
CommunityBot
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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 ...
- 207k
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Spin connection raise and lower flat indices
The spin connection $\omega^a_{b\nu}$ is used to define the covariant derivative of a spinor in curved spacetime. I want to explicitly calculate the covariant derivative:
$$\nabla_\nu\Psi=(\partial_\...
CommunityBot
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1
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Finding the correct Christoffel-symbols in a 2+1D space-time
I'm trying to calculate the Christoffel Symbols in a 2+1D space-time with the following metric:
$$ds^2 = N^2(\vec r)c^2dt^2-\phi(\vec r)(dx^1)^2-\phi(\vec r)(dx^2)^2$$
To find the Christoffel ymbols ...
CommunityBot
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1
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Angular Deficit of a Conical Singularity
I'm currently studying the Bonnor solution starting with this paper on Black Diholes. The metric is given by :
$$ ds^2 = \left(1-\frac{2Mr}\Sigma\right)^2 \left[-dt^2 + \frac{\Sigma^4}{(\Delta + (M^2 +...
CommunityBot
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2
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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 ...
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2
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Does the expansion of the Universe into a higher dimensional space imply that 4D objects are real?
It is my understanding that objects in the Universe are not just getting farther apart but space itself is expanding and so in some real sense, higher-dimensional geometry is "real" -- if so, on a ...
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1
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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: ...
- 103k
98
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9
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What is a manifold? [closed]
For complete dummies when it comes to space-time, what is a manifold and how can space-time be modelled using these concepts?
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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
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2
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260
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Is a stationary spacetime automatically globally hyperbolic?
Is a stationary spacetime automatically globally hyperbolic? Can one construct a Cauchy-Surface by the existence of a global timelike Killing Vector field?
- 22.2k
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1
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Ricci scalar curvature in FLRW flat universe
I have a simple question about the relation between the Ricci scalar curvature and the $k$ constant in the Friedmann–Lemaître–Robertson–Walker solution. Assuming $k=0$, such that the space can be ...
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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 ...
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