The general-relativity tag has no wiki summary.
5
votes
1answer
109 views
Foliations of Lorentzian manifolds by Spacelike Hypersurfaces
Suppose that $M$ is a Lorentzian manifold (not necessarily satisfying Einstein's equations). What conditions do we need in order to guarantee that $M$ admits a foliation by codimension-$1$ spacelike ...
1
vote
1answer
124 views
Generalized spin connection and dreibein in higher spin gravity
I am studying higher spin gravity and I would like to know the mathematical and physical meaning of generalized spin connection and generalized dreibein that appear in this theory.
It is well known ...
2
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0answers
124 views
Conjugate or focusing points on null geodesics imply chronality
Theorem
Let $\beta\colon [0,1] \to M$ be a null geodesic. If $\beta(t_0)$ is conjugate to $\beta(0)$ along $\beta$ for some $t_0\in (0,1)$, then there is a timelike curve from $\beta(0)$ to ...
5
votes
1answer
277 views
Killing vector fields on sphere
Let $u$ be a smooth function on $\mathbb S^2$, and assume that for every killing vector field $V$ on $\mathbb S^2$.
$$\int_{\mathbb S^2} V(u) x_j dS=0\text{,}\forall j=1,2,3$$
Is $u$ necessarily ...
2
votes
0answers
94 views
Differential equation related to the Schwarzschild metric
How can one find solutions of the following second-order differential equation
$$\frac{d^2W}{dr^2}-\frac{1}{r}\frac{dW}{dr}=\frac{C}{W^2}\frac{dW}{dr}$$
with the boundary condition $W(r)\to r^2$ at ...
3
votes
0answers
75 views
Generalized Hawking Mass
This is a fairly general question. Let $(M^3,g)$ be a Riemannian 3-manifold. Let $\Sigma^2$ be a dimension-2 submanifold of $M$. The Hawking mass of $\Sigma^2$ is defined as
$m(\Sigma^2) := ...
7
votes
0answers
117 views
Topological restrictions from mean curvature bounds
Alexandrov's Theorem says that a compact constant mean curvature hypersurface embedded in $\mathbb{R}^{n+1}$ must be a round sphere.
What happens when the mean curvature is small, or bounded? (For ...
5
votes
0answers
256 views
Geometric meaning of the black hole horizon
It is widely accepted that the singularity of the Schwarzschild metric at the event horizon is purely an artifact of the coordinates and no physical singularity exists at the horizon. However, as ...
2
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0answers
153 views
The Cauchy Problem in General Relativity: Existence of a Hausdorff Development
This is related to a problem that I posed about a year ago. I was given several references by a number of experts who were kind enough to entertain my rather arcane question. Those references were ...
10
votes
2answers
566 views
Obtain Lorentzian manifolds from Riemannian ones by Wick rotation
In some cases, Wick rotation of a metric, formally consisting in substituting a coordinate with i times the coordinate itself, allows one to construct a Riemannian manifold starting from a Lorentzian ...
3
votes
0answers
386 views
Existence of diagonalizing coordinates for the metric tensor
Solving for metrics that are Einstein, i.e that satisfy $R_{\mu \nu} = \Lambda g_{\mu \nu}$ is highly non-trivial as soon as $g_{\alpha \beta}$ is allowed to have off-diagonal components. However, ...
5
votes
5answers
1k views
Can anyone give an example of Ricci flat Riemannian or Lorentzian Manifold that is not flat?
Does there exist a Ricci flat Riemannian or Lorentzian manifold which is geodesic complete but not flat? And is there any theorm about Ricci-flat but not flat?
I am especially interset in the case ...
2
votes
1answer
213 views
What is the meaning of Yang-Mills action evaluated on Levi-Civita connection?
On a Riemannian manifold $M$ with riemann curvature tensor $R_{\mu\nu\rho\sigma}$ written as (endomorphism valued) curvature two-tensor of the Levi-Civita connection $R=R_{\mu\nu}dx^\mu\wedge ...
2
votes
1answer
262 views
Euler characteristic of Cauchy surface in Lorentz manifold
Are there any known topological restrictions on what kinds of manifolds can form the Cauchy hypersurface of a Lorentzian manifold? I'm particularly interested about restrictions on Euler ...
1
vote
1answer
149 views
manifolds whose charts are maps to Minkowski space
I'm doing a project involving tilings of Minkowski space. For instance in 2d I have rectangular tiles determined by a spacelike line segment: the rectangle is the region caused by the line segment. ...
0
votes
0answers
105 views
Hitchin–Thorpe inequality for Lorentzian manifold
I've recently read the following:
For which $b$ it is possible that $S^n$ can have a Lorentz metric? Why?
An answer shows that a compact, oriented, simply connected manifold carries a Lorentz metric ...
2
votes
1answer
97 views
Dimension of the space of null geodesics
So that is my question. If I have a manifold with Lorentz metric, how do I know the dimension of the space of null geodesics. For example, in the general relativity the space of null geodesics is 5... ...
3
votes
1answer
121 views
Global conformal equivalence of two regions of Minkowski spacetime
I am wondering whether the region $H:=\{(t,x):x^2−t^2<1\}$ of $(1+1)$-dimensional Minkowski spacetime, equipped with the restriction $g_H$ of the standard Minkowski metric $g=−\mathrm{d} t\otimes ...
5
votes
1answer
416 views
Why does closed string theory have only one dilaton field instead of $22$? [closed]
Looking at $5D$ Kaluza-Klein theory, the Kaluza-Klein metric is given by
$$
g_{mn} = \left(
\begin{array}{cc}
g_{\mu\nu} & g_{\mu 5} \\
g_{5\nu} & g_{55} \\
\end{array}
\right)
$$
...
2
votes
0answers
143 views
Solving ODE with negative expansion power series
I'm moving this here, as suggested from physics.stackexchange. The original is here.
So, I need to solve a system of ODE, using negative power expansion. I will give all the necessary equations and ...
11
votes
4answers
619 views
Einstein field equations in perspectives from PDE and functional analysis
The Einstein field equations have been subject of research in theoretical physics, and differential geometry, apparently with methods from classical analysis and geometry. In particular, solutions in ...
11
votes
1answer
665 views
Does the gluing procedure in Robert Wald’s book *General Relativity* yield a Hausdorff spacetime?
Before I state my problem, let me provide some definitions pertaining to the Cauchy Problem in General Relativity.
Definition 1: A triplet $ (\Sigma,h,k) $ is called an initial data set if $ ...
-3
votes
1answer
189 views
Can Hartogs' extension theorem be used to prove there's no naked singularity?
Maybe for the first time my question doesn't deal with number theory. Tonight a friend of mine told me about Hartogs' extension theorem and said his work in analytic microlocal analysis was somehow ...
0
votes
0answers
71 views
Null vector fields given Bondi metric
I'm trying to understand how to compute the null future-directed vector fields if I have a given (Bondi) metric
$g=-e^{2\nu}du^{2}-2e^{\nu+\lambda}dudr+r^{2}d\Omega$
with $d\Omega$-standard metric ...
2
votes
1answer
160 views
naked singularity and null coordinates
I'm trying to understand the notion of a naked singularity on a more mathematical level (intuitively, it's a singularity "one can see and poke with a stick", but I'm having troubles on how to actually ...
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votes
0answers
232 views
Functional Analysis and Differential Manifold incompatibility
From the mathematical point of view, where is the idea of the incompatibility between functional analysis, at the basis of quantum mechanics, and differential geometry, at the basis of general ...
3
votes
1answer
253 views
Are all null curves of a Lorentzian metric extrema?
Suppose $g$ is a Lorentzian metric on $\mathbb{R}^4$, then consider the variational problem of finding extrema of
$$F(\gamma) = \int_a^b \| \dot{\gamma}(s) \|_g ds$$
The so-called "null curves" are ...
8
votes
1answer
491 views
The Speed of Gravitational Waves in General Relativity
Is it possible to mathematically prove that the speed of gravitational waves in general relativity equals the speed of light, without linearizing the Einstein Field Equations? The approach via the ...
4
votes
1answer
650 views
Geometric derivation of the Einstein’s field equation from the Hilbert action.
It is well-known that the equation for stationary solutions of the Einstein-Hilbert functional is given by the Einstein field equation (for a statement, see previous question). The standard derivation ...
7
votes
1answer
558 views
Coordinate-free derivation of the Einstein's field equation from the Hilbert action.
It is well-known that the equation for stationary solutions of the Einstein-Hilbert functional (without matter and cosmological constant, which is irrelevant here):
$$S = \int_M R \mu_g,$$
is given by ...
4
votes
1answer
262 views
Work on an Einstein-Hilbert type action but with the *absolute value* of scalar curvature?
This is only my second question on mathoverflow, so my apologies if this would be more appropriate at a physics site. My question concerns a modification to the Einstein-Hilbert action. The standard ...
9
votes
5answers
969 views
Is there a relation between 4-dimentional general relativity and exotic smooth structures on $\mathbb{R}^4$?
Let's say General Relativity is the study of the Einstein equation on smooth Lorentzian manifolds, i.e. pseudo-Riemanniann manifolds of signature $(n-1,1)$.
I've heard more than once people say that ...
7
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2answers
821 views
22
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1answer
3k views
Math and Wormholes
Hopefully, MathOverflow is the correct place for this. I had a student approach me and ask me what kinds of mathematics goes into the study of wormholes. She specifically asked whether there is any ...
10
votes
1answer
624 views
Regge calculus: Questions of consistency resolved?
Hello,
Regge calculus is an approximation scheme for General Relativity, which has been introduced in early-sixties and has been adopted both in numerical relativity and numerical quantum relativity. ...
11
votes
4answers
993 views
Singular semi-Riemannian Geometry: usefulness and state of the art
My question has two parts, one concerning the state of the art of the subject, and the other the usefulness.
1. State of the art.
Can someone provide references reflecting the state of the art in ...
2
votes
2answers
2k views
Killing vectors and Ricci Tensor
Hi all,
We all know that the lie derivative of the metric tensor along a Killing Vector vanishes, by definition. I am trying to show that the Lie derivative of the Ricci tensor along a Killing vector ...
4
votes
1answer
718 views
Christodoulou's paper on naked singularities in inhomogeneous dust collapse
I have been studying of late about formation of naked singularities in certain collapse scenarios in Einstein's theory. It seems to me that the canonical paper to read about how such a formation is ...
