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.
183
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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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2
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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 ...
- 1,406
3
votes
0
answers
98
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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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1
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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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3
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0
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73
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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/...
- 31
10
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0
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167
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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 ...
user21349
2
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0
answers
101
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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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4
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1
answer
439
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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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3
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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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1
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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 ...
- 31
6
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1
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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 ...
- 183
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1
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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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2
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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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1
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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 ...
2
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1
answer
517
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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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2
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545
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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 ...
- 3,323
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1
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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}\...
- 475
14
votes
2
answers
611
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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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0
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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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4
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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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1
answer
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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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1
answer
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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=...
- 475
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0
answers
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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 ...
- 2,517
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2
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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 ...
- 2,612
4
votes
2
answers
481
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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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1
vote
1
answer
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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 ...
- 403
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0
answers
115
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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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votes
1
answer
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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 ...
- 475
3
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0
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362
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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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1
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389
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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 ...
- 475
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2
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472
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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 ...
- 475
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votes
2
answers
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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: ...
- 475
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0
answers
516
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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)...
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8
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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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1
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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 ...
user21349
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1
answer
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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 "...
- 475
2
votes
2
answers
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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
$$
D^{i}(K_{ij}...
- 21
4
votes
2
answers
403
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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 ...
- 411
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1
answer
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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, n,m,\overline{m}\...
- 475
1
vote
1
answer
362
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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) ...
- 475
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votes
2
answers
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*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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4
votes
1
answer
365
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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 ...
- 475
10
votes
1
answer
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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 ...
- 1,025
1
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1
answer
286
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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 ...
- 405
3
votes
3
answers
472
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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 $\beta(1)$....
- 1,341
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1
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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$$
Is $u$ necessarily ...
2
votes
0
answers
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Differential equation related to the Schwarzschild metric
How can one find solutions of the following second-order differential equation
$$\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 ...
- 16.4k