All Questions
22
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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 ...
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
2
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63
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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
3
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2
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615
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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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2
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Do the Einstein Field Equations force the metric to be Lorentzian?
In GR, we are working with Lorentzian metrics, which are examples of a pseudo-Riemannian metrics. That is, we are trying to find pseudo-Riemannian $g_{\mu\nu}$ that are solutions to the field equation ...
- 453
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1
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85
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Find an example of a closed, achronal set $S$ in Minkowski spacetime such that $J^+(S)$ is not closed
This is one of the exercises on Wald's General Relativity:
Chapter 8, Problem 8.b Find an example of a closed, achronal set $S$ in Minkowski spacetime such that $J^+(S)$ is not closed. (Hint: ...
- 22.2k
2
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2
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156
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Relationship between spacelike and timelike distances in General Relativity vs. Special Relativity
In Minkowski spacetime, the distance $d_S$ between two space-like separated events $x$ and $y$ can (up to constant) be given by a distance between the two time-like separated events $z$ and $w$ where $...
- 125
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93
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Spatial separation in analogy to time separation in Lorentzian geometry?
O'Neill (Semi-Riemannian Geometry With Applications to Relativity, 1983, p. 409) defines time separation between two events as follows:
"If $p, q \in M$, the time separation $\tau(p, q)$ from $p$...
- 125
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90
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Geometrically Impossible Spacetime
A result in math says that $S^n$ carries a Lorentzian metric iff $n$ is odd.
Using it we can observe that a 2-sphere spacetime is impossible, a 3-sphere spacetime is geometrically possible, but again ...
1
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1
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82
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What is a "timelike half-curve"?
I know what a timelike curve is. But what is a time-like half-curve, as in the definition of a Malament-Hogarth spacetime (below), which appears in this paper?
Definition: A spacetime $(M,g)$ is ...
- 121
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I need help with a proof in Hawking & Ellis [closed]
Here's a proof in Hawking and Ellis (1973) of proposition 6.4.6:
The definition of "strong causality" used in the book is that for every point $p$ and every neighborhood $U$ of $p$, there ...
- 863
3
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1
answer
530
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Confusion regarding Geodesics
Suppose we have a causal curve and we can cover the causal curve by convex normal neighborhoods. We also know that, in convex normal neighborhood there will exist a unique geodesic inside the ...
- 27
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2
answers
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?
- 159
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1
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157
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Characterizing compactness of the Alexandrov topology in a spacetime
This is perhaps more of a soft question and on the mathematical side of things, but I'm struggling to find references and to formulate a precise argument. There's of course the chance that what I'm ...
- 545
2
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1
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177
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Penrose diagrams for non-spherically symmetric spacetimes
As far as I have seen, Penrose diagrams are composed for spacetimes where there is spherical symmetry. The angular degrees of freedom are suppressed so as to understand the causal properties of ...
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