Questions tagged [general-relativity]
For questions about mathematical problems arising from general relativity, the branch of physics which provides and studies the currently accepted geometric description of gravity.
193
questions
5
votes
0
answers
155
views
A component of Bach tensor
First of all, this question may be more suited for the Math stack exchange site. If anyone finds this question irrelevant here, please feel free to transfer to the relevant site.
Recall that in terms ...
- 307
3
votes
0
answers
230
views
Categorical General Relativity
What are some good references for GR from a categorical point of view?
This is essentially just a big-list reference request.
I'm aware that the subject exists and can do some basic sleuthing to find ...
0
votes
1
answer
109
views
Non-inertial frames of reference in empty space
Imagine that somebody wants to generalize special relativity to non-inertial frames of reference. For example I am going around a point and the metrics of space is non-Euclidean from my point of view. ...
- 115
2
votes
1
answer
207
views
Mathematical explanation for connections on gauge bundles in curved spacetime for spinors
I asked this question https://physics.stackexchange.com/questions/820924/is-tetrad-postualte-independent-of-gauge-field
Here is what I know, $g_{\mu \nu} = e^{a}_{\mu} e^{b}_{\nu} \eta_{ab}$ and the ...
- 23
2
votes
1
answer
74
views
Quasilinear wave equations without (weak) null conditions and conjectures
I have found that most works on quasilinear wave equations require, at least, the (weak) null condition. There are only a few works without this condition, such as "Shock Formation in Small-Data ...
- 89
4
votes
0
answers
227
views
Possible obstructions to global Wick-rotation in distinguishing spacetimes
Take the time-orientable $3+1$ dimensional spacetime $(M,g)$ that is locally Wick-rotatable at any arbitrary point $p \in M$ to a Riemannian manifold $(N,h)$.
Local Wick-rotatability of $(M,g)$ ...
- 612
1
vote
0
answers
199
views
Are causally isomorphic spacetimes Wick-related?
Take the time-orientable spacetimes $(M_1,g_1)$ and $(M_2,g_2)$ that are locally(to be clarified below) Wick-related and both are globally Wick-rotatable(to be clarified below) to the same Riemannian ...
- 612
3
votes
2
answers
370
views
Does there exist an electromagnetic analogue of Einstein's field equations?
This will look like a physics question but it doesn't have anything to do with reality so its a vague math question if anything.
I recently learned about gravitoelectromagnetism which describes an ...
- 2,606
1
vote
0
answers
225
views
Is the topological dimension of spacetime fixed for causally isomorphic spacetimes?
Suppose time-oriented spacetimes $(M_1 , g_1)$ and $(M_2, g_2)$ are not homeomorphic under their manifold topologies $\mathcal{M}_1$ and $\mathcal{M}_2$ respectively.
The Lorentzian metrics $g_1$ and $...
- 612
3
votes
1
answer
68
views
Lorentzian norm of the covariant derivative of a vector field is zero
Let $Y$ be a vector field on a Riemannian manifold $(M, g)$. If $g(\nabla Y, \nabla Y)=0$, then $Y$ is covariantly constant, i.e. $\nabla Y=0$.
Now, let $V$ be a vector field on a Lorentzian manifold $...
- 149
1
vote
0
answers
166
views
Order isomorphism + manifold homeomorphism => path topology homeomorphism?
Suppose time-oriented spacetimes $(M_1 , g_1)$ and $(M_2, g_2)$ are homeomorphic under their manifold topologies $\mathcal{M}_1$ and $\mathcal{M}_2$ respectively.
Let's call this map $\phi: (M_1, \...
- 612
3
votes
1
answer
359
views
Topology and local isometry, spinning cosmic string
Suppose one is given the spacetime $(M,g)$ where $M$ is a fixed differentiable manifold and $g$ is a Lorentzian metric whose local expression is:
$$g= -(dt + a \, d \phi)^2 + d\rho^2 + \kappa^2 \rho^2 ...
- 612
1
vote
0
answers
107
views
What is the "intrinsic reason" for the failure of Schwarzschild coordinates in general relativity?
It is well known that the Schwarzschild metric fails at r = 2M (in units where c = G = 1) and this is the result of choosing "bad" coordinates. I find this surprising because the coordinates ...
- 411
3
votes
1
answer
138
views
Definitions of weak solutions for quasilinear wave equations
I am learning the shock problem for the balance system (perhaps not conserved, see, e.g., "Ingo Muller, Tommaso Ruggeri. Rational Extended Thermodynamics") and just have a question on the ...
- 89
0
votes
0
answers
46
views
On sub-maximally symmetric Riemannian spaces
Is there a 4-dimensional Riemannian manifold with 8-dimensional isometry group?
Context: Guido Fubini (Annali di Mat., ser. 3, 8 (1903) 54) shows that the dimension $n$ of the isometry group of a $d$-...
- 427
0
votes
0
answers
94
views
Prove the orthogonality of vector spherical harmonics
We define
$S_a^{lm} = \Big( - \frac{1}{\sin \theta} Y^{lm}_{,\varphi}, \sin \theta\ Y^{lm}_{,\theta} \Big)$
$Y_a^{lm} = \Big( Y^{lm}_{,\theta}, Y^{lm}_{,\varphi} \Big)$
to be the axial vector ...
- 95
0
votes
1
answer
59
views
Handling degenerate planes in pseudo-Riemannian geometry: impact on sectional curvature and comparison theorems
I've been studying Riemannian and pseudo-Riemannian manifolds and came across an intriguing point regarding the definition of sectional curvature in both geometries.
In pseudo-Riemannian geometry, for ...
- 45
3
votes
0
answers
121
views
On the linearized evolution equations in general relativity
The following puzzles me already for quite some time: In mathematical relativity, especially in the discussion of the Cauchy problem, one usually works in the so-called ADM-Formalism, in which one ...
- 1,299
3
votes
1
answer
609
views
What is the Lebesgue covering dimension of this topological space?
Take the 4 dimensional time-oriented spacetime $(M,g)$ such that it's not strongly causal.
Take the induced topology defined by the Lorentzian metric called Alexandrov topology.
This topology matches ...
- 612
14
votes
2
answers
1k
views
Is the Gödel universe Wick rotatable?
Take Wick rotatability being as the way defined in the following article by Helleland and Hervik:
Christer Helleland, Sigbjørn Hervik, Wick rotations and real GIT, Journal of Geometry and Physics 123 ...
- 612
1
vote
0
answers
99
views
How causal is a strongly causal purely electric spacetime?
Take a generic Lorentzian spacetime $(M, g)$ where $M$ is a time-oriented 4d manifold and $g$ is the Lorentzian metric that is strongly causal and purely electric.
According to this answer:
Is every ...
- 612
2
votes
0
answers
349
views
Is a Wick rotatable spacetime necessarily strongly causal?
There are a few viable ways to formulate Wick rotatability that preserve distinct features.
One is mentioned in the post:
Obtain Lorentzian manifolds from Riemannian ones by Wick rotation
There's also ...
- 612
3
votes
1
answer
240
views
Is every strongly causal spacetime purely electric?
Take a time 4-dimensional orinted Lorentzian manifold $(M,g)$.
A spacetime is called Strongly Causal at point $p$ if and only if for every neighbourhood $U$ of the point $p$ there exists a ...
- 612
3
votes
1
answer
354
views
reference for reading Schoen Yau positive mass theorem proof II
I am trying to read the paper by Schoen and Yau, Proof of the Positive Mass Theorem II. The notation is very different from what I am familiar with (basically Robert Wald's book on general relativity)....
- 61
0
votes
2
answers
281
views
Distinguishable under manifold topology but indistinguishable under the Alexandrov topology
Take the time-oriented Lorentzian spacetime $(M, g)$ that is not strongly causal.
In such case it is shown that the Alexandrov topology and the Manifolds topology deviate such that the manifold ...
- 612
5
votes
0
answers
104
views
Is it always possible to find a conjugate optical function?
Optical functions (functions with null gradients) and double null foliations (foliations of a spacetime by two related optical functions) play a large roll in modern mathematical relativity research. ...
- 419
2
votes
0
answers
361
views
What is the nerve of this category?
If $\mathcal{C}$ is a thin category, we call $U \subseteq \mathrm{Ob}(\mathcal{C})$ open if for every object $X \in U$ and any morphism $X \to Y$, we also have $Y \in U$. This declares an Alexandrov ...
- 612
1
vote
1
answer
279
views
Temporal evolution of a globally hyperbolic spacetime
Any globally hyperbolic spacetime can be assigned a global function of time as Hawking has demonstrated for stably causal spacetime. (Any globally hyperbolic spacetime is also stably causal).
For ...
- 612
3
votes
0
answers
786
views
Is this set a manifold?
Take a general spacetime that is not strongly causal.
Call this spacetime $(M, g) $ where $M$ is a connected time-oriented manifold and $g$ is the Lorentzian metric that satisfies the Einstein's Field ...
- 612
5
votes
2
answers
768
views
What is lost in General Relativity without Hahn-Banach axiom in the ZF+HB set theory?
In the same spirit of this question:
How much of mathematical General Relativity depends on the Axiom of Choice?
I want to go radically further ahead and ask for what remains of mathematical general ...
- 612
0
votes
1
answer
520
views
Non-diffeomorphic but homeomorphic (under Lorentzian topology) Lorentzian manifolds
$\newcommand{\lorentzian}{\mathrm{lorentzian}}\newcommand{\lorentzian}{\mathrm{lorentzian}}\newcommand{\diff}{\mathrm{diff}}\newcommand{\manifold}{\mathrm{manifold}}$Take a time-oriented Lorentzian ...
- 612
40
votes
3
answers
5k
views
How much of mathematical General Relativity depends on the Axiom of Choice?
One of the cornerstones of the mathematical formulation of General Relativity (GR) is the result (due to Choquet-Bruhat and others) that the initial value problem for the Einstein field equations is ...
- 655
4
votes
1
answer
360
views
Metric with a constant Chern–Pontryagin scalar
Do there exist 3+1 dimensional spacetimes (i.e. Lorentzian manifolds with signaure (1,3)), for which the Chern–Pontryagin scalar
\begin{equation}
K_2= \epsilon^{\mu\nu\rho\sigma}R^{\alpha}{}_{\beta\mu\...
- 628
2
votes
0
answers
93
views
Non-compactness on 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 ...
- 121
0
votes
1
answer
116
views
Curvature of an affine system
I find an interesting paper that mentioned the Definition of curvature of an affine optimal control system. It reminded me that many textbooks on Riemannian geometry only tell us about metrics, ...
- 111
4
votes
0
answers
99
views
Behavior of lapse function at infinity: stability of Minkowski
In the Stability of Minkowski Spacetime, Christodoulou and Klainerman prove a local existence proof for a particular class of quasilinear wave equation for a symmetric, traceless, covariant 2-tensor $...
- 419
2
votes
0
answers
135
views
Naked curvature singularity vs Cauchy horizon in stably causal space-time
There is a result
that says (theorem 2.11) that any stably causal space-time $M$ is either a product $\Sigma\times \mathbb{R}$ or the time-like gradient $\nabla f$ of a time function $f:M\rightarrow \...
13
votes
2
answers
423
views
Synthetic differential / conformal geometry of Lorentzian manifolds?
Let $M$ be a sufficiently nice Lorentzian manifold of dimension $\geq 3$. It's known [1] (see also [2]) that the differential and even conformal structure of $M$ is completely encoded in the causal ...
- 63.2k
1
vote
0
answers
57
views
Number of divergence free symmetric two tensor in dimension 4 [duplicate]
In a $4$ dimensional (semi)-Riemannian manifold $(M^{4}, g)$, both Einstein tensor $G= \operatorname{Ric}(g)- \frac{R(g)}{2}g$ and stress-energy tensor $T$ symmetric and divergence-free. Is there any ...
- 307
3
votes
1
answer
288
views
Definitions fundamental forms and their geometric Intuition
Let $(M^{n+1}, g)$ be a Lorentzian manifold (spacetime) that contains a Riemannian/spacelike hypersurface $(\Sigma ^{n},h).$ Then we can define the second fundamental form of the hypersurface in many ...
- 307
6
votes
1
answer
309
views
The Cauchy problem in general relativity, hyperbolic PDEs, and Sobolev spaces on manifolds
(I apologize in advance if this question is ill-posed or not suitable for Math Overflow, I am not yet a research mathematician, just a student.)
Let $(\Sigma,\bar{g})$ be an $n$-dimensional Riemannian ...
- 191
0
votes
1
answer
215
views
Numerical reconstruction of Einstein's field equations
A few analytic solutions are known to the Einstein field equations:
$$ R_{\mu\nu} - \frac{1}{2}g_{\mu\nu}R - kT_{\mu\nu} = 0$$
Taking a preexisting analytic solution such as Schwarzchild's solution:
$$...
- 109
1
vote
0
answers
90
views
Invariance signature in infinite dimension
Let $V$ be an infinite dimensional vector space and suppose we have a smooth family $\{g_t\}_{t\ge 0}$ of symmetric bi-linear forms such that:
$g_0$ is positive-definite
$g_t$ is non-degenerate for ...
- 395
0
votes
1
answer
278
views
Mathematical characterization of gravitational geons as reference request, and their properties as main question
I've edited (ten days ago) a question on Physics Stack Exchange, this Mathematical characterization of gravitational geons, post with identifier 726281 the users of the site were kind adding in the ...
2
votes
1
answer
338
views
Understanding the proof of lemma 1.1 from Fisher, Marsden, and Moncrief's paper
The following lemma is from Fisher, Marsden, and Moncrief's paper: the structure of the space of solutions of Einstein's equations:1
1.1. Lemma.
If Ein( $\left.{ }^{(4)} g\right)=0$, and ${ }^{(4)} h$ ...
- 307
5
votes
1
answer
844
views
On imaginary time
I've heard a few times that "the time was imaginary before the Big Bang".
I am guessing it means that at this stage, the space-time was a Riemannian $4$-manifold, but I am not sure this ...
- 14.2k
0
votes
1
answer
93
views
How to choose a set of non-orthonormal basis vectors for the absolute space of a stationary and axisymmetric space-time in General Relativity?
In General Relativity, the space-time is described by the metric tensor $g_{\mu\nu}$, where $\mu,\nu=0,1,2,3$ and the interval is written as $$ds^2=g_{\mu\nu}dx^\mu dx^\nu$$.
A 3+1 split allows to ...
- 51
3
votes
1
answer
142
views
Convergence of spheres in nonlinear stability of Minkowski space
My question is about Lemma 3.3.1 in Christodoulou and Klainerman's proof of nonlinear stability of Minkowski space. This lemma says the following: Consider a family of metrics $m_u$ on $S^2$ defined ...
- 419
7
votes
2
answers
277
views
In which dimensions is a strongly causal Lorentzian manifold determined conformally by its causal structure?
Let $M$ be a strongly causal Lorentzian manifold. If $M$ has dimension 4, a theorem of Hawking, King, and McCarthy (see Thm 5) says that $M$ is determined up to conformal isomorphism by its class of ...
- 63.2k
4
votes
1
answer
140
views
Example of a bicontinuous poset which is not jointly bicontinuous?
Recall that a poset $P$ is said to be continuous if, for every $p \in P$, the set $\{q \in P \mid q \ll p \}$ is directed with supremum $p$. Here $q \ll p$ is the "way below" relation (see ...
- 63.2k