All Questions
Tagged with spacetime differential-geometry
347
questions
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
0
votes
0
answers
57
views
Commutation of exterior product in spacetime
I am following these notes (A Practical Introduction to
Differential Forms, by Alexia E. Schulz and William C. Schulz 2013) and on page 67 (pdf page number 71) there is an expression for the ...
- 147
2
votes
1
answer
201
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
1
vote
1
answer
464
views
Einstein tensor in 2d [duplicate]
Is the Einstein tensor in 2D or 1+1D always zero? If so, why?
I recently installed EinsteinPy and started playing wing different metrics - for the 2D cases the result turned out to be always zero.
- 1,161
7
votes
2
answers
2k
views
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
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
2
answers
156
views
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
1
vote
0
answers
93
views
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
-1
votes
1
answer
106
views
When doing general relativity in practice, how do we choose the appropriate manifold describing the scenario?
The theory only deals with the local curvatures, not the global topology. Hence any manifold with an allowed metric is allowed. These can be infinitely many, especially for negative curvature space-...
- 7,980
5
votes
2
answers
315
views
The limit of GR with infinite speed of light $c$
Just answer what you can. I don't mean the zero curvature flat space time version. I know that the Einstein Field equations use $c$ as a constant, but what would the universe be like if gravity was ...
- 83
0
votes
1
answer
71
views
Is curvature localised in General Relativity?
Is the curvature of spacetime in General Relativy localised?
- 476
8
votes
2
answers
826
views
How does the covariant derivative satisfy the Leibniz rule?
In Carroll's "Spacetime and Geometry", he states on page 95 (section 3.2) that the covariant derivative, $\nabla$, is a map from $\left(k, l\right)$ tensor fields to $\left(k, l+1\right)$ ...
- 103
0
votes
0
answers
90
views
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 ...
3
votes
2
answers
166
views
What is difference between an infinitesimal displacement $dx$ and a basis one-form given by the gradient of a coordinate function?
In general relativity, we introduce the line element as $$ds^2=g_{\mu \nu}dx^{\mu}dx^{\nu}\tag{1}$$ which is used to get the length of a path and $dx$ is an infinitesimal displacement But for a ...
- 374
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