All Questions
8
questions
1
vote
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_{\...
- 43
2
votes
2
answers
261
views
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
3
votes
2
answers
171
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Does covariant derivative include magnitude change of a vector as well as direction change of the same vector?
Does covariant derivative include magnitude change of a vector as well as direction change of the vector? In some explanations I followed I have not noticed mentioning of magnitude change along with ...
- 1,815
3
votes
2
answers
409
views
Globally constant vector field in a curved spacetime
Is it possible to define a globally constant vector field in a curved spacetime, that is a vector field for which the covariant derivative vanishes along every world line? The vector field $V^{\mu}=0$ ...
- 487
1
vote
1
answer
87
views
Determining the orbit of a vector field in curved space
Suppose one is working in Kruskal space-time with metric in the usual Kruskal $(U,V)$ coordinates:
$$ds^2 = -\dfrac{32M^3e^{r/2M}}{r}dUdV+r^2d\Omega_2^2$$
Where $r$ now defined implicitly in terms ...
- 113
9
votes
3
answers
2k
views
Intuition behind differential operators as the basis vectors of a manifold (space-time)
I understand that in order to provide a basis for every point in space-time, the differential operators, $\partial_\mu$ (or partial derivative operator with respect to each one of the curvilinear ...
- 902
1
vote
0
answers
104
views
Is it obvious, if an excision from Minkowski spacetime breaks isometry...?
Strongly related to the question Prove isometry preserving excision is Killing-like?
There the question was (loosely): if I drill a smooth 4D hole through Minkowski spacetime, is a timelike isometry ...
- 772
4
votes
1
answer
391
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Prove isometry preserving excision is Killing-like?
(If you think thia is e.g. not well expressed you already understand the request for help.)
Theorem: Given a manifold $M$ equipped with a metric $g$ and possessing at least one non-trivial isometry $\...
- 772