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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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Critical growth and geodesic connectedness in Lorentz manifold

What is the deep ("heuristic") reason why the quadratic growth of $\beta$ is critical for the study of geodesic connectedness in standard static Lorentz spacetime $\mathcal M = \mathcal M_0 \times \...
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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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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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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 $m_g$...
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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 ...
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452 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 ...
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333 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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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
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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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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 ...
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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 ...
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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 ...
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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
271 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 ...
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2answers
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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 ...
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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 ...
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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,\...
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279 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 ...
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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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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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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 ...
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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 $\...
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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 ...
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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 ...
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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 ...
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1answer
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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)-\...
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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 ...
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Conformal Transformations that are Ricci Positive Invariant

Is there any known class of conformal transformations $\phi : M \to M$ of a riemannian/semi-riemanian manifold $(M,g)$ that have the property: $g$ is ricci-positive iff $\phi^* g$ ricci positive? ...
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142 views

Maximal symmetry and isometries not connected to the identity

A pseudo-Riemannian manifold $M$ of dimension $n$ is said to be maximally symmetric if the space of its Killing vector fields has $n(n+1)/2$ dimensions. If $M$ is maximally symmetric, then we have ...
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1answer
200 views

Induced connection on null hypersurfaces

I will use a local coordinate formalism here, since this is related to research in general relativity, and my supervisor only tolerates local coordinate formalisms. Plus the research papers I base my ...
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Singularity theorems for semiclassical gravity

The semi-classical Einstein equations (without a cosmological constant) are $G^{\mu \nu} = 8\pi \langle T^{\mu \nu} \rangle$. I am told that there are serious objections as to why these equations ...
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Clarification on tetrad indices

In a set of notes, I came across the following few lines involving the covariant derivative, and just wanted to make sure I understood the notation correctly: Let $\lbrace F_{1},F_{2},F_{3},F_{4}\...
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Penrose transform and general wave equations

In the late 1960's Penrose developed twistor theory, which (amongst other things) led to an exceptional description for solutions to the wave equation on Minkowski space via the so-called Penrose ...
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An question about Cauchy Problem in General Relativity [closed]

Yesterday, in Brazilian School on Differential Geometry, a friend asked me the question: Given an (non-trivial) initial data set $(M,g,k)$ for the Cauchy problem in General Relativity. Is there ...
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Geometrical point of view of the harmonic constraints ($\Delta g_{ij}=0$) in General Relativity

What does it mean, from the geometrical point of view, use (in General Relativity) of the constraints on the metric tensor's coefficients such that $\Delta g_{ij}=0$? (where $\Delta$ is the Beltrami-...
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General Relativity and Differential Geometry intuitions of Second Bianchi Identity

In General Relativity, one uses the Riemann Tensor in its coordinate form $R_{abcd}$, and proves the Second Bianchi Identity- $R_{abcd;e} + R_{abde;c} + R_{abec;d} = 0$ It is said that ...
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Invariance of spin coefficients

I have a question about how spin coefficients (Newman Penrose formalism) transform. I know that if we perform a tetrad rotation, say of Class III: $(l,n,m,\overline{m})\mapsto \left(\frac{1}{A}l, An,...
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Choosing a coordinate transformation

I was reading the following paper http://scitation.aip.org/docserver/fulltext/aip/journal/jmp/4/7/1.1704018.pdf?expires=1460721373&id=id&accname=2112043&checksum=...
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Invariant Lagrangians of a connection and its derivatives: how do they look like?

Let $$ L=L(\Gamma,\partial\Gamma,\ldots,\partial^n\Gamma) $$ be a Lagrangian depending on a linear symmetric connection $\Gamma$ on the tangent space of a manifold $M$ together with its derivatives up ...
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Principal bundle approach to general relativity

I am curious if there is any literature (texbooks, mainly, but articles will do too, though I don't have easy access to any paid journal) that deals with general relativity by using Ehresmann ...
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Holonomy of a Ricci-flat affine connection

There is some link between Ricci-flatness and reduction of holonomy. For example a Kahler manifold is Ricci-flat if and only if it has at most $SU(n)$ holonomy rather than $U(n)$, and it's apparently ...
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Recent observation of gravitational waves

It was exciting to hear that LIGO detected the merging of two black holes one billion light-years away. One of the black holes had 36 times the mass of the sun, and the other 29. After the merging the ...
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1answer
162 views

Marginally Trapped surfaces with constant Gaussian curvature

By marginally trapped surface I mean a spacelike surface in a 4-dimensional Lorentzian manifold such that the mean curvature vector is lightlike. In my research I have stumbled across marginally ...
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Proper time and asymptotic flatness

I have asked this question at physics stackexchange but got no response. I thought I could try my luck here: I'm trying to understand the concept of asymptotic flatness in general relativity, and ...
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1answer
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Intuition behind the “Lapse Function”

I came across the following definite of the Lapse Function: $N=\sqrt{\frac{1}{2}g(L,\overline{L})}$ where $L,\overline{L}$ are the null geodesic vector fields. Further, I have been looking at this ...
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Obtaining the metric from the mixed Ricci tensor $R^i{}_j$

In chapter 5 of the book "Einstein Manifolds", Arthur Besse discusses the possibility to find the metric $g$ when knowing the Ricci curvature tensor $Ric(g)$ ($=R_{ij}$). But what do we know about ...
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188 views

Eikonal equation and double null coordinates

I"m trying to understand the exact/technical link between the Eikonal equation and a double-null form of the metric (if such a direct link even exists). R. Wald, in his "General Relativity", doesn't ...
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Conformal compactification of Kerr spacetime

I'm looking for a book/paper where the conformal compactification of Kerr spacetime is calculated. I've seen plenty of reference for the Minkowski, but none (explicitly calculated) for Kerr. Thank ...
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2answers
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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: ...
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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 I^-(p)...