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5
votes
5answers
817 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
154 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
236 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
128 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
82 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
86 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
68 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
0answers
247 views

Why does closed string theory have only one dilaton field instead of $22$?

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
103 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 ...
10
votes
4answers
424 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 ...
10
votes
1answer
558 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 $ ...
-2
votes
1answer
160 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
60 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
137 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 ...
0
votes
0answers
202 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
193 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
437 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
555 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
457 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
229 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 ...
7
votes
5answers
897 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
votes
2answers
737 views

A survey on positive mass theorem?

Could you suggest a good survey paper on positive mass theorem?
10
votes
1answer
565 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. ...
10
votes
4answers
912 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
662 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 ...