All Questions
63
questions
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
2
votes
1
answer
124
views
Could the universe have a form of a $T^3$-torus?
Cosmological measurements suggest that we live in a flat universe. However, what might be less clear is its topology. So could the flat universe have the form of a $T^3$-torus, i.e. the torus whose ...
- 9,778
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
202
views
Is the celestial sphere we actually see the Riemann Sphere?
I've been watching a few lectures by R. Penrose where he seems to say that what we see around us is the Riemann sphere. He usually gives the example of an observer floating in deep space or if the ...
- 31
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
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
4
votes
2
answers
619
views
Can we just take the underlying set of the spacetime manifold as $\mathbb{R^4}$ for all practical purposes?
In mathematical GR and also in some informal GR presentations (eg: MTW), manifolds are always mentioned before talking about GR... but now I am starting to wonder.. if it even actually neccesary?
In ...
- 7,980
3
votes
1
answer
157
views
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
5
votes
1
answer
1k
views
What does Penrose mean when he talks about topology of spacetime?
Let us now set aside the question of the submicroscopic structure of space-time and concentrate, instead, on its large-scale properties. In this case, we may imagine that the smooth manifold picture ...
- 7,980