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.
98 questions
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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 \...
0
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1answer
82 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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162 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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0answers
116 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
186 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
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 ...
6
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1answer
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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0answers
85 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 ...
1
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1answer
132 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 ...
27
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5answers
2k 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
321 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
141 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
148 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 ...
1
vote
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 ...
6
votes
2answers
117 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 ...
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2answers
379 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
84 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
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 ...
3
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0answers
65 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/...
10
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0answers
135 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 ...
2
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0answers
69 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 ...
3
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1answer
198 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 $\...
43
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2answers
1k 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
200 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
499 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
204 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
votes
2answers
280 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 ...
3
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0answers
87 views
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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1answer
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 ...
1
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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 ...
5
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2answers
407 views
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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1answer
85 views
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}\...
13
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2answers
406 views
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 ...
2
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0answers
128 views
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 ...
4
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0answers
214 views
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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4answers
1k views
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 ...
1
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1answer
95 views
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,...
2
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1answer
225 views
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=...
3
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0answers
84 views
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 ...
6
votes
2answers
699 views
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 ...
4
votes
2answers
315 views
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 ...
51
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2answers
4k views
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
...
1
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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 ...
0
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0answers
101 views
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 ...
3
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1answer
1k views
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 ...
3
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0answers
239 views
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 ...
1
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1answer
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 ...
4
votes
1answer
289 views
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 ...
4
votes
2answers
311 views
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: ...
6
votes
0answers
417 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 I^-(p)...
