All Questions
Tagged with spacetime differential-geometry
80
questions with no upvoted or accepted answers
15
votes
0
answers
272
views
Is it known what the necessary and sufficient conditions are for the existence of a "3+1 split" (by means of a foliation) of a (Lorentzian) manifold?
When trying to do physics on a more general pseudo-Riemannian manifold we want to require that there is a foliation of this manifold into three-dimensional subspaces. By this I mean we would like to ...
- 301
8
votes
0
answers
460
views
Derivation of the Hypersurface Deformation Algebra
Let $({M},{g})$ be a smooth $4d$ spacetime manifold with lorentzian metric $g$ and local coordinates $\xi^{\alpha}$ and let further $({N},{q})$ be a smooth $3d$ manifold with metric $q$ and local ...
user272876
6
votes
1
answer
322
views
Pseudo-Riemannian 2D manifold (visualize time curvature)
My goal is to visualize somehow the curvature of time, as opposed to the curvature of space. I know that we generally talk about spacetime curvature altogether; however, the fact that spacetime has ...
- 719
5
votes
0
answers
128
views
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 ...
5
votes
0
answers
260
views
Help understanding the Cartan connection/formalism in General Relativity
I'm trying to understand the Cartan formalism in the context of General Relativity.
As I understand it given a pseudo-Riemannian spacetime manifold $M$ we can consider the group of spactime ...
- 2,847
5
votes
0
answers
782
views
Does any spacetime admit a global foliation in spacelike hypersurfaces?
In the comments of this question the following new questions came up: in general relativity, local coordinates can be found around any point, that single out a time coordinate and a three dimensional ...
- 9,304
4
votes
0
answers
92
views
Is the causal structure completely determined by the Weyl tensor alone?
By causal/conformal structure I mean the context of Malament's 1977 theorem. If I understand correctly this means that any two spacetimes which agree about all of the future-directed continuous ...
- 151
4
votes
1
answer
402
views
What is the geometry of light cones if space is curved/non-Euclidean?
In light cone diagrams, the plane corresponding to the present is always the Euclidean one, but what if space is curved? Now, I've also seen diagrams where spacetime is supposed to be regarded as ...
- 241
3
votes
0
answers
88
views
Intuition for the interior Killing vector fields in Schwarzschild?
The Schwarzschild metric represents a stationary (and static), spherically-symmetric, spacetime. These characteristics are manifested by the four Killing vector fields: one for time translation and ...
- 1,290
3
votes
0
answers
95
views
What does the Einstein-Hilbert action look like in terms of Riemannian metric of positive signature?
For a 4-manifold to admit a Lorentzian metric is equivalent to that manifold having vanishing Euler characteristic. Any spacetime that admits a Lorentzian metric $g^{\mathcal{L}}$ can have that metric ...
- 2,847
3
votes
0
answers
618
views
What is a Taub–NUT space?
I trying to read and understand the Taub–NUT space. Wikipedia introduced that it is associated with the Taub–NUT metric, an exact solution to Einstein's equations. However, I didn't find a direct ...
- 1,783
3
votes
0
answers
89
views
Are the spacelike foliations of a non-static spacetime topologically equivalent?
Assuming a stationary, globally hyperbolic spacetime, I can imagine that all spacelike foliations are topologically equivalent though not all will be identical since the spacetime is not static. Is ...
- 700
3
votes
0
answers
184
views
Doubt about energy conditions: the Time-like Convergence Condition
First of all, consider a congruence of smooth time-like geodesics parametrized by proper time $\tau$. So, a tangent vector to a time-like geodesic is indeed a four-velocity up to a factor constant; ...
- 3,085
3
votes
0
answers
145
views
Free-falling stationary observer in curved spacetime?
Let us consider the pseudo-riemannian manifold $(\mathcal{M},g)$ with $\mathcal{M}=\mathbb{R}\times\mathcal{N}$ with $\mathcal{N}$ being a maximally symmetric, 3-dimensional riemannian manifold and
$...
- 193
3
votes
1
answer
290
views
Abstract definition of conjugate points
Let $S$ be a Cauchy hypersurface of a globally hyperbolic spacetime $(\mathcal{M},\mathcal{O},\mathcal{A},g,T)$ with unit normal vector field $n$. Define the exponential map on a neighborhood $U\...
- 3,915