The general-relativity tag has no usage guidance.

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### Obtaining Killing fields from the tetrad

I'm reading the following article by Newman
http://scitation.aip.org/content/aip/journal/jmp/4/7/10.1063/1.1704018
about the generalization of the Schwarzschild metric. My question is the following: ...

**5**

votes

**0**answers

138 views

### Time-separation function on “globally hyperbolic” spacetimes with everywhere timelike boundary

It is well-known that, in globally hyperbolic spacetimes, the time separation function $\tau$ (aka Lorentzian distance function) enjoys the following property: fix a point $p$ and a point $q \in ...

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**5**answers

885 views

### Modern mathematical books on general relativity

I am looking for a mathematical precise introductory book on general relativity. Such a reference request has already been posted in the physics stackexchange here. However, I'm not sure whether some ...

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votes

**1**answer

98 views

### Conditions for existence of Penrose diagrams

A Penrose diagram (also known as a conformal diagram or Carter-Penrose diagram) is a technique for visualizing the causal (light-cone) structure of a 3+1-dimensional manifold. Usually the diagram is ...

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**1**answer

91 views

### Minkowski spacetime in Newman Penrose formalism

I have a rather basic question for which (surprisingly!) I cannot find a short and clear answer anywhere:
I'm currently looking at the Newman Penrose (NP) formalism (I use primarily Chandrasekhar's ...

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vote

**1**answer

78 views

### The momentum constraints in the ADM formulation of general relativity

Suppose that the space-time has a time function. Let $g_{ij}$ be
the Riemannian metrics of the time slices, and $K_{ij}$ be the second
fundamental forms. It is by Codazzi equation that
$$
...

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votes

**2**answers

172 views

### Conditions on a Lorentzian manifold to ensure existence of global proper-time foliation?

I am wondering what conditions a Lorentzian manifold $(M,g)$ must satisfy to ensure the existence of a global proper-time foliation (i.e. a decomposition of $M$ into spacelike Cauchy hypersurfaces and ...

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**0**answers

112 views

### Gauge freedom in the tetrad

I'm reading the following paper about Petrov type D space times called "Type D vacuum metrics":
http://scitation.aip.org/content/aip/journal/jmp/10/7/10.1063/1.1664958
by Kinnersley. I have a ...

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**0**answers

58 views

### Kerr metric affine parameter

I'm going through the chapter about Kerr space-time of Chandrasekhar's "Mathematical theory of black holes", and have a question about the following transformation:
the idea is, that one wants to ...

**1**

vote

**1**answer

148 views

### Null geodesic congruence

I came across a statement in Chandrasekhar's "Mathematical Theory of Black Holes" that I don't understand (rather say disagree):
Assume we have a Newman Penrose tetrad $\lbrace l, ...

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vote

**1**answer

98 views

### Petrov classification/Weyl scalars

There is one calculation in Chandrasekhar's "Mathematical Theory of Black Holes" that I cannot understand. Here is the setup:
We want to show that Petrov type D (i.e. two principal null directions) ...

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**2**answers

3k views

### *The* open problem in General Relativity?

Q. Is there a single, clear mathematical question that has emerged as
the open problem in General Relativity?
I ask this on the ~100th anniversary of Einstein's (4-page!) 1915 paper,
"Die ...

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votes

**1**answer

111 views

### Null tetrad transformation

I have been going through the Chandrasekhar's "The Mathematical Theory of Black holes", in particular the chapter on Newman Penrose formalism.
I have a question about what he calls a "class III ...

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votes

**1**answer

205 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 ...

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vote

**1**answer

152 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 ...

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**3**answers

322 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 ...

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**1**answer

385 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 ...

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**0**answers

120 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 ...

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**0**answers

92 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) := ...

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**0**answers

124 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 ...

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**1**answer

475 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 ...

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**1**answer

253 views

### Subset of causal spacetime+Imprisonment Condition+Compact Closure -> Stably Causal spacetime?

My question arose after studying the article "John K. Beem: Conformal Changes and Geodesic Completeness". (http://projecteuclid.org/euclid.cmp/1103899983) One of the results there is:
Let $(M,g)$ ...

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**0**answers

173 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 ...

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734 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 ...

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**0**answers

513 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, ...

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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 ...

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votes

**1**answer

223 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 ...

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votes

**1**answer

278 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**

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**1**answer

163 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. ...

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115 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 ...

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**1**answer

107 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... ...

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**1**answer

139 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 ...

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**1**answer

426 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
$$
g_{mn} = \left(
\begin{array}{cc}
g_{\mu\nu} & g_{\mu 5} \\
g_{5\nu} & g_{55} \\
\end{array}
\right)
$$
...

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**0**answers

172 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 ...

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715 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 ...

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### 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 $ ...

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**1**answer

210 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 ...

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84 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 ...

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votes

**1**answer

191 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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**0**answers

283 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 ...

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votes

**1**answer

294 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 ...

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**1**answer

547 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 ...

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**1**answer

718 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 ...

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**1**answer

627 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 ...

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**1**answer

281 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 ...

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1k views

### Is there a relation between 4-dimensional 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-Riemannian manifolds of signature $(n-1,1)$.
I've heard more than once people say that ...

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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 ...

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670 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. ...

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1k 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 ...