All Questions
101
questions
0
votes
0
answers
68
views
Action principle dependent on spacetime-topology?
Consider the Lagrangian density
$$L(\phi, \nabla \phi, g) = g^{\mu \nu} \nabla_{\mu} \phi \nabla_{\nu} \phi$$
If one varies the action as usual, then one finds the equation
$$\delta S = \int_{\mathcal{...
- 743
2
votes
2
answers
96
views
Are $i^\pm$ and $i^0$ codimension 1 surfaces?
Standard textbooks like Carroll's say that spatial and temporal infinities in Minkowski space Penrose diagram are points. But on the footnote in pg. 3 of some draft notes on Celestial holography by ...
- 785
0
votes
1
answer
83
views
What objects are solutions to the Einstein Field Equations?
The usual way the solutions of the Einstien Field Equations are introduced is by saying they are (pseudo-) riemannian metrics that satiafy the diff equations for a given EM Tensor. My question is: ...
1
vote
2
answers
153
views
What is the manifold topology of a spinning Cosmic String?
Given the following metric which is that of a rotating Cosmic String:
$$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
1
vote
0
answers
48
views
JT gravity metric - solution to the dilaton equations of motion
I am reading Closed universes in two dimensional gravity by Usatyuk1, Wang and Zhao. The question is not too technical, it is about the solutions to the equations of motion that result from the ...
- 3,992
2
votes
0
answers
61
views
Examples of spacetimes that are asymptotically flat at future timelike infinity
There are interesting non-trivial examples of spacetimes which are asymptotically flat at null and spacelike infinities. For example, the Kerr family of black holes satisfies these conditions. However,...
- 22.2k
9
votes
1
answer
804
views
Mathematical anatomy of general relativity
I was always told that spacetime in general relativity was a Lorentzian manifold, that is, a Pseudo-Riemannian manifold $ (M, g) $ with metric signature $(+, -, -, -)$ or $(-, +, +, +)$ and that that ...
- 309
3
votes
2
answers
615
views
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 ...
- 743
6
votes
3
answers
2k
views
Is source of space-time curvature necessary?
Einstein field equations have vacuum solutions that (probably) assumes the source of curvature (either energy-momentum tensor or the cosmological constant term or both) is elsewhere. Like, in ...
- 1,161
2
votes
1
answer
85
views
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
votes
1
answer
205
views
A question about the topology of spacetime and the existence of CTCs
Let $(M, g)$ be a smooth Lorenzian time-oriented manifold.
Is it possible for the Lorenzian metric induced topology to be different from that of the manifold topology, without CTCs?
We know that the ...
- 1,268
5
votes
1
answer
209
views
Why should a Cauchy surface be closed?
A Cauchy surface is defined on any spacetime $M$ as a subset $S$ which is closed, achronal, and whose domain of dependence $D(S) = M$.
Why do we include the "closed" condition in the above ...
- 1,588
1
vote
1
answer
63
views
Non-Compactness in Penrose Singularity
I've been studying singularities in GR, and (obviously), came across PST.
Let us state it as the following:
Let $(M, g)$ be a connected globally hyperbolic
spacetime with a noncompact Cauchy ...
- 487
1
vote
1
answer
98
views
When a curve is future (past) inextendible?
Future (past) endpoint: We say that $p\in M$ is a future (past) endpoint of a curve $\lambda$ if for every neighborhood $O$ of $p$ there exists a $t_0$ such that
$\lambda(t)\in O$ for all $t>t_0$ (...
- 27
4
votes
2
answers
638
views
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