All Questions
Tagged with spacetime differential-geometry
347
questions
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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
2
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2
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305
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Derivation of the Schwarzschild metric: why are $g_{22}$ and $g_{33}$ the same as for flat spacetime?
I'm trying to understand the derivation of the Schwarzschild metric from Wikipedia, but I simply do not understand why, therein, $g_{22}$ and $g_{33}$ must be those of the flat spacetime.
Couldn't $g_{...
- 93
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128
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Is it possible to create a Nil geometry in real spacetime according to general relativity? (What metrics are possible in the real world?)
Background
I've heard that it is possible to construct a Penrose triangle in the 3D geometry Nil. And I wondered: Can we build a Penrose triangle in the real world if spacetime is appropriately ...
0
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361
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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 +...
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131
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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
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106
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Energy is the time component of 4-momentum in SR: Proof as per R. Wald's book
This is an excerpt fom R. Wald's book on General Relativity (page 61). I'm not able to understand how he deduces that $E$ must be the time component of $p^a$ with only the assertions made before this ...
- 107
6
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2
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433
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Confusion regarding bundle structure of Galilean spacetime in Penrose's The Road to Reality
I am reading Roger Penrose's The Road to Reality. In section 17.3, I encounter the following passage. To give a context, Penrose was explaining that even though an Aristotelian spacetime can be ...
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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
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638
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Characterising Minkowski spacetime as a flat manifold with some other property?
It is known that flat manifolds can be characterized as follows
If a pseudo-Riemannian manifold $M$ of signature $(s,t)$ has zero Riemann
curvature tensor everywhere on $M$, then the manifold is ...
- 1,588
2
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3
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221
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What is Dirac's reasoning when showing the curvature vanishing implies we can choose rectilinear coordinates?
In section 12 of Dirac's book "General Theory of Relativity" he is justifying the name of the curvature tensor, which he has just defined as the difference between taking the covariant ...
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133
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Simple distance calculation in General Relativity
So imagine a spacetime with the Schwarzschild metric:
$$ds^2=-c^2\left(1-\dfrac{2GM}{c^2r}\right)dt\otimes dt+\dfrac{1}{\left(1-\dfrac{2GM}{c^2r}\right)}dr\otimes dr+r^2\left(d\theta\otimes d\theta+\...
- 495
0
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144
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Does this theorem holds out for spacetime?
The theorem:
Let $F$ and $C$ be two finite geometric figures (those defined by two continuous functions in a given region $D$), where $F$ belongs to an $n$-dimensional Euclidean space and $C$ is the ...
- 495
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0
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39
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When is the Weyl tensor applied on null vectors a null vector?
Let $C^{\rho}_{~\alpha \beta \gamma}$ be the Weyl tensor of a spacetime $(M,g)$, that is a solution to Einstein's equation. Let $X^\alpha, Y^\alpha, Z^\alpha$ be null vector fields, i.e. $X_\alpha X^\...
- 384
4
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111
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On the Product Structure of Spacetimes after Compactification
I am currently looking into the compactification of spacetimes as it is often done in (super-)stringtheory.
So, say I start with a ten-dimensional Lorentz manifold $(N, g)$, where $N$ denotes the ...
- 357
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2
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158
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Does a straight line in flat space become a geodesic in curved space when the space becomes curved?
Flexible foam has shortest path from Point-A to Point-B. When the foam is not curved (space-time is not curved), the shortest path is Path-1 (straight line - before curving the foam). But if the foam ...
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