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.

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7
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1answer
193 views

Electromagnetic energy in Lovelock gravities

To fix ideas, let us recall that General Relativity describes gravitational phenomena on a 4-dimensional pseudo-Riemannian manifold $(X,g_{ab})$ with field equations that relate the energy-momentum ...
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0answers
46 views

Mathematical construction: ADM formulation in general relativity

I'm doing my undergraduate thesis and now I'm looking for references that presents ADM Formulation in general relativity mathematically. I studied the basics of general relativity theory by O'Neill ...
2
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1answer
60 views

Lower bound for domain of exponential map on Lorentzian manifolds

Let $M$ denote a manifold admitting a Lorentzian metric $g_{ab}$. Essentially, I would like to know the "minimum domain" on which the exponential map is defined at $p\in M$. To make this concrete, ...
5
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1answer
178 views

Completeness hypothesis in the positive mass theorem

I am trying to understand and further formalize Witten's proof of the positive mass theorem. Dan Lee, in his book "Geometric relativity" did a wonderful job with formalizing and carrying out the ...
5
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0answers
103 views

Gravity, connection, and curvature

Starting with Synge and Fock, many modern authors identify gravity with curvature. On the other hand, Einstein always emphasized that gravity should be equated with a connection, but not with ...
4
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1answer
288 views

Is the Wikipedia depiction of the ergosphere of a Kerr black hole a Cassini oval?

Disclaimer: this a cross post from MSE, where this question was asked on November 4th 2019 and has so far received no upvote, no comment and no answer whatsoever. Glancing at https://en.wikipedia.org/...
3
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1answer
282 views

Gauss-Bonnet-Chern Theorem [closed]

I am currently doing an undergraduate project about Gauss-Bonnet-Chern Theorem. Is there any particular books/papers regarding the application of the theorem in the theory of general relativity?
4
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1answer
136 views

Are quadrics the cones of maximal symmetry?

A paper by Ehlers, Pirani, and Schild axiomatizes the geometry of general relativity in what seems like a nice way. However, Jacobson criticizes one aspect of the system as not natural: One deep ...
3
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0answers
297 views

Maximal symmetry at the speed of light

Are there examples of 1 + 3 dimensional pseudo-Riemannian manifolds with 6 dimensional isometry group whose orbits are light-like (i.e., the metric restricted to each orbit is degenerate)? Here is a (...
16
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2answers
929 views

What's the “actual” shape of a black hole accretion disk?

[Warning: I have no expertise in general relativity, so this question might not be very rigorous] More and more often we come across science popularization articles like this one which show beautiful ...
2
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0answers
95 views

Spacetime symmetries

We know some nice space-time have a lot of symmetries. It is said that Minkowski spacetime has $$ISO(d-1,1)/SO(d-1,1),$$ de Sitter spacetime has $$SO(d,1)/SO(d-1,1)$$ and anti-de Sitter spacetime ...
2
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1answer
309 views

Definition of twisted geometries and existence of coordinate transformation for twisted $AdS_2 \times S^2$

In the paper Multiply Twisted Products by Yong Wang, general definitions for so called warped and twisted products are given: A (singly) warped product $B \times_b F$ of two pseudo-Riemannian ...
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1answer
192 views

Explanation for “Squashing” and “Stretching” (Lorentzian Analogue of Berger Spheres)

In the paper Anti-de Sitter space, squashed and stretched Bengtsson and Sandin introduce the Lorentzian analogue of the squashed 3-sphere. After looking up Berger spheres, it seems what is meant with "...
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0answers
86 views

Marginal surfaces in spacetimes

Is there some result on existence of marginally trapped surfaces in spacetime 4-manifolds? Am I right in saying that a marginal surface (like a trapped surface in general) is a compact spacelike 2-...
3
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2answers
392 views

Transformation Poincaré-coordinates to global coordinates in $\mathrm{AdS}_2$

For the two dimensional anti de-sitter space $\mathrm{AdS}_2$ one can consider the Poincaré-coordinates $\mathrm{d}s^2_P = -r^2 \mathrm{d}t^2 + \frac{1}{r^2} \mathrm{d}r^2$ which covers only half of ...
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96 views

Condition for Integrability of an Almost Complex Structure

The following question concerns a remark made in the paper: Lebrun, C., Complete Ricci-flat Kähler metrics on $\mathbb{C}^n$ need not be flat, Proceedings of Symposia in Pure Mathematics, Volume 52 ...
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1answer
59 views

Connection of the existence of Killing-Yano tensor and Killing tensor

Stephani states that in 4 dimensions a spacetime admits a non-reducible Killing-Yano tensor only if the Weyl tensor either is of Petrov type D or vanishes. Does this imply that the spacetime also ...
11
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1answer
493 views

Are there currently any plausible approaches to proving the Penrose сonjecture?

I have recently been reading some of the literature on the Penrose inequality, especially the papers by Bray and by Huisken and Ilmanen. One notices immediately that the existing proofs for the ...
2
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1answer
138 views

Gaussian null coordinates

I find it hard to find information on the so-called "Gaussian null coordinates", which Wikipedia says is used to describe "near horizon geometries". Can someone provide a reference where I can read ...
2
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0answers
79 views

Example Petrov Classification

I would like to calculate the Petrov type for a specific spacetime, unfortunately I am not able to find a step by step algorithm or example for the process, either using null-tetrad, nor calculating ...
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0answers
54 views

What exactly is a Cartan radius vector (and its role in Poincaré gauge theories)

I am studying approaches to gravity where the Poincaré group is "gauged". The original motivation of this is to understand what is meant on the statement that "Teleparallel gravity is a gauge theory ...
3
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0answers
159 views

Gauge structure of teleparallel gravity

I am interested in references that treat teleparallel gravity in a mathematically rigorous manner, especially in regards to it being a "gauge theory of the translation group". The standard reference ...
17
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1answer
1k views

Is Witten's Proof of the Positive Mass Theorem Rigorous?

I noticed that the only official reason given for awarding Edward Witten the Fields Medal was his 1981 proof of the positive mass theorem with spinors, so I was assuming that the proof was fully ...
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65 views

Stably causal spacetimes

Consider a stably causal Lorentzian spacetime $(\mathcal{M},g)$, so that a non-zero timelike vector field $t^a$ exists such that $$ \tilde{g}_{ab}=g_{ab}-t_a t_b $$ and $(\mathcal{M},\tilde{g})$ has ...
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1answer
105 views

Analytic approach to geodesic connectedness in Semi-Riemannian manifolds

Can you point out a reference (or references) that deal with analytical methods (rather than methods from differential geometry) for the study of geodesic connectedness on Semi-Riemannian manifolds?
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220 views

Mathematical Problems of General Relativity II

In the introduction of D. Christodoulou's book "Mathematical Problems of General Relativity I", he refers a few times to the second volume. My question is does it exists? Has it been (or will it be) ...
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125 views

The converse to the positive mass theorem

Let $(M^n,g)$ be an asymptotically flat manifold of decaying-order $\tau>\frac{n-2}{2}$, the positive mass theorem states that if the scalar curvature $S_g$ is non-negative, then the ADM mass $m_g$...
3
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2answers
231 views

Einstein warped product manifold Ricci flat

Let $(M,g)=(N,\ddot{g})\times f(B,\bar{g})$ be an Einstein warped-product manifold Ricci flat (i.e. $Ric=\lambda g$ with $\lambda=0$) where $f:N \rightarrow (0, \infty)$ (positive scalar function) and ...
6
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1answer
554 views

Physical (GR) Differential Geometry?

I am looking for problem lists or books which contain open problems in the area of mathematics motivated by physics. Ideally, I am looking for questions asking about which reduce to some calculation ...
6
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1answer
393 views

Manifolds with negative dimension – Definition, References

Does the concept of differential manifold with negative dimension make sense, in differential geometry? If yes, how is it defined? Do you have any reference to recommend? My problem was born in ...
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0answers
124 views

Timelike geodesic congruences covering the whole Schwarzschild spacetime

For some reason, I would need to know what are the timelike geodesic congruences which cover the entire exterior region of the Schwarschild spacetime. In fact the only thing I really need is the ...
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1answer
198 views

Schwartz distributions, Colombeau algebra and applications

I have studied "enough" the theory of distributions , I would like to deepen some topic with applications. With some research I arrived at this book: "Geometric Theory of Generalized Functions with ...
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5answers
3k views

Is there a mathematical book on general relativity that uses exclusively a coordinate free language even in practical computations?

I would also appreciate if it was as far from the physicists formalism as possible, no abstract indices ,etc. Also I don't consider using a basis or tetrads as coordinate free. The idea is to use ...
3
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0answers
342 views

Scalar curvature and warped-product manifolds - intuition

Let $(M, g) = (N_1, g_1) \times_f(N_2, g_2)$ be an Einstein warped-product manifold, with metric $g=g_1+f^2g_2$. What does it mean if the scalar curvature of its base-manifold $(N_1, g_1)$, equal to ...
3
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1answer
161 views

A problem about closed 2-forms on Minkowski space

The problem is: For any closed 2-form in the Minkowski space $\mathbb{R}^{3,1}$ satisfiying $dF=0$ and $\delta F \ne 0$ (with $\delta$ denoting the codifferential), does there exist a Lorentz ...
3
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0answers
151 views

Parallel frame for marginally trapped bi-harmonic surfaces in $\Bbb R^4_2$

I'm reading the paper Classification of marginally trapped Lorentzian flat surfaces in $\mathbb{E}^4_2$ and its applications to biharmonic surfaces by B. Y. Chen. Summarizing it quickly: he first ...
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1answer
379 views

Invariance of a vector under parallel transport along an infinitesimal orthogonal loop

I'm not very familiar with differential geometry and am coming from a general relativity background, so would appreciate help with a question from that context. If this question could be posed in a ...
6
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2answers
141 views

Why are they called “screen” distributions?

If $V$ is a vector space and $g$ is a symmetric degenerate bilinear form on $V$, every complementary subspace to the radical ${\rm rad}(V)$ is called a "screen subspace" of $V$: we have an orthogonal ...
8
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2answers
422 views

Some Mathematical Questions on Gravitational Waves and Numerical Relativity

Due to the recent spate of detections of gravitational waves by LIGO, my amateurish interest in the mathematics of general relativity has been revived. The wave-forms of the detected gravitational ...
3
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0answers
87 views

Poincare type inequality comparing a vector field and its Lie derviative

A function $f\in W^{1,2}_{loc}$ is in the space $W^{1,2}_{-\tau}$ if $\int_{\mathbb{R}^n} f^2|x|^{2\tau-n}<\infty$ and $\int_{\mathbb{R}^n} |\partial_kf|^2|x|^{2\tau+2-n}<\infty$ for all $k=1,2,\...
4
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1answer
601 views

Gravitational field in a spherical cavity inside a sphere of uniform density

It is well known that in Newtonian gravity if the center of a spherical cavity inside a sphere of uniform density is not concentric with the sphere then the gravitational field inside the cavity will ...
3
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0answers
68 views

Finding the particular and general solutions to Einstein Field Equations under generalized Vaidya Geometry

The problem I have is on finding the particular and general solutions to Einstein Field Equations under generalized Vaidya Geometry, which comes from the following paper : https://journals.aps.org/prd/...
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147 views

Reference request: recent progress in cosmic censorship, classification and evolution of singularities

After decades of inconclusive work, it seems that there may have been some dramatic progress within the last few years on the cosmic censorship conjecture (CCC). Joshi and Malafarina claim in a 2014 ...
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0answers
81 views

The dimension of the subspace of flat spin connections

I am interested in the the flat spin connections in a Riemann spacetime of dimension 4. They appear in the context of the frame formalism of metric gravity theories. I believe that they form a ...
4
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1answer
252 views

Killing fields for Yang-Mills

Physicists frequently talk about symmetries of a theory, and them being generated by Killing vectors. While this is clear to me in the context of gravity, where a Killing field $\xi$ is defined by $\...
46
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3answers
2k views

What results are immediately generalised to higher dimensions, in light of Schoen and Yau's recent preprint?

Many problems in geometric analysis and general relativity have been established in dimensions $3\leq n\leq 7$, as the regularity theory for minimal hypersurfaces holds up to dimension 7*. In a recent ...
2
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1answer
303 views

The characteristic initial value problem in general relativity in a double null foliation

In a Paper by Rendall, it is shown that the characteristic initial value problem for the Einstein equations is well-posed. In fact, if the data are specified in some coordinates, then one can extend ...
6
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1answer
514 views

A step in the proof on the uniqueness of mass

I am reading the survey paper "The Yamebe Problem" by Lee and Parker. In section 9, Theorem 9.6 in P.78, it was proved that the mass is well defined in the sense that $m(g)$ depends only on the metric ...
3
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1answer
359 views

Non-commutativity of the d'alambert operator acting on the covariant derivative of a scalar field in general relativity

Recently, I saw the following formula for the non-commutativity of the d'Alembert operator $\Box$ acting on the covariant derivative of a scalar field in general relativity, $\Box (\nabla_{\mu}\phi)-\...
4
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2answers
384 views

What exactly goes wrong with Schwarzschild coordinates at the event horizon?

It is well known that if one uses the Schwarzschild coordinates (t, r, $\theta$, $\phi$) to solve Einstein's equations, the components of the metric tensor blow up at the "event horizon", r = 2M (in ...