All Questions
Tagged with riemannian-geometry general-relativity
48 questions
6
votes
1
answer
334
views
Tensor component calculation
First of all, this question may be more suited for the Math stack exchange site. If anyone finds this question irrelevant here, please transfer to the relevant site.
Recall that in terms of Weyl and ...
- 319
0
votes
1
answer
128
views
Non-inertial frames of reference in empty space
Imagine that somebody wants to generalize special relativity to non-inertial frames of reference. For example I am going around a point and the metrics of space is non-Euclidean from my point of view. ...
- 125
0
votes
0
answers
47
views
On sub-maximally symmetric Riemannian spaces
Is there a 4-dimensional Riemannian manifold with 8-dimensional isometry group?
Context: Guido Fubini (Annali di Mat., ser. 3, 8 (1903) 54) shows that the dimension $n$ of the isometry group of a $d$-...
- 427
0
votes
1
answer
74
views
Handling degenerate planes in pseudo-Riemannian geometry: impact on sectional curvature and comparison theorems
I've been studying Riemannian and pseudo-Riemannian manifolds and came across an intriguing point regarding the definition of sectional curvature in both geometries.
In pseudo-Riemannian geometry, for ...
- 45
3
votes
0
answers
126
views
On the linearized evolution equations in general relativity
The following puzzles me already for quite some time: In mathematical relativity, especially in the discussion of the Cauchy problem, one usually works in the so-called ADM-Formalism, in which one ...
- 1,429
0
votes
1
answer
117
views
Curvature of an affine system
I find an interesting paper that mentioned the Definition of curvature of an affine optimal control system. It reminded me that many textbooks on Riemannian geometry only tell us about metrics, ...
- 111
1
vote
0
answers
59
views
Number of divergence free symmetric two tensor in dimension 4 [duplicate]
In a $4$ dimensional (semi)-Riemannian manifold $(M^{4}, g)$, both Einstein tensor $G= \operatorname{Ric}(g)- \frac{R(g)}{2}g$ and stress-energy tensor $T$ symmetric and divergence-free. Is there any ...
- 319
3
votes
1
answer
333
views
Definitions fundamental forms and their geometric Intuition
Let $(M^{n+1}, g)$ be a Lorentzian manifold (spacetime) that contains a Riemannian/spacelike hypersurface $(\Sigma ^{n},h).$ Then we can define the second fundamental form of the hypersurface in many ...
- 319
2
votes
1
answer
358
views
Understanding the proof of lemma 1.1 from Fisher, Marsden, and Moncrief's paper
The following lemma is from Fisher, Marsden, and Moncrief's paper: the structure of the space of solutions of Einstein's equations:1
1.1. Lemma.
If Ein( $\left.{ }^{(4)} g\right)=0$, and ${ }^{(4)} h$ ...
- 319
2
votes
1
answer
166
views
Hyperboloids in Minkowski geometry
Let $(\mathbb R^{1+2},\eta)$ be Minkowski with the metric $\eta= -dt^2+(dx^1)^2+(dx^2)^2$. Suppose $\Sigma$ is a smooth timelike hypersurface and denote by $h$ the second fundamental form on $\Sigma$. ...
- 4,135
3
votes
1
answer
239
views
A question on light cones in Lorentzian manifolds with timelike boundary
Suppose $M= \mathbb R \times M_0$ with a Lorentzian metric $g(t,x)=-dt^2+ g_0(t,x)$ where
$M_0$ is a compact manifold with a smooth boundary and $g_0$ is a family of smooth Riemannian metrics on $M_0$ ...
- 4,135
3
votes
3
answers
525
views
Usage/Application of Raychaudhuri equation in Riemann geometry or pure maths
While going through this paper by Witten and seeing a discussion about different aspects of Raychaudhari Equation and Einstein Field Equation. I want to ask if Raychaudhari Equation find any ...
- 139
6
votes
0
answers
129
views
Deriving (Gaussian) curvature bounds from bounds on the metric
I am trying to understand a bound in Christodoulou's 2008 paper on black hole formation.
The paper considers a spacelike surface $S$ diffeomorphic to a sphere, with two metrics:
the induced metric $\...
- 419
3
votes
0
answers
72
views
Lorentzian cobordism through the dominant energy condition
Is the answer to the following problem, or some close variant thereof, known? Briefly:
Given two initial data sets $I_1=(M,g_1,k_1)$ and $I_2=(M,g_2,k_2)$, is there a time-oriented spacetime ...
- 2,247
20
votes
0
answers
2k
views
Schoen and Yau's proof of the higher dimensional positive mass theorem
In April 2017 Schoen and Yau posted on the arxiv their solution of the time-symmetric positive mass theorem in all dimensions, which has been a significant conjecture since the 70s. As of now, July ...
- 2,247
1
vote
0
answers
138
views
Perturbation of a spacetime in general relativity
In general relativity one has the Schwarzchild metric for a non-rotating black hole
$g_{SC} = -\phi^2 \: dt^2 + \Bigg(1 + \frac{m_0}{2r} \Bigg)^4 \delta $
and from this one has the spacelike ...
- 5,146
7
votes
1
answer
247
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 ...
- 1,064
2
votes
2
answers
233
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, ...
- 193
6
votes
0
answers
272
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 ...
- 16.5k
3
votes
1
answer
379
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?
- 159
16
votes
2
answers
1k
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 ...
- 23.3k
2
votes
1
answer
386
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 "...
- 679
1
vote
0
answers
126
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 ...
- 1,379
1
vote
1
answer
142
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 ...
- 679
2
votes
1
answer
492
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 ...
- 679
1
vote
0
answers
156
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 $m_g$...
- 2,533
2
votes
2
answers
403
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 ...
- 272
36
votes
7
answers
5k
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 ...
- 395
2
votes
0
answers
479
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 ...
- 272
3
votes
0
answers
162
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 ...
- 1,163
6
votes
2
answers
196
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 ...
- 1,163
9
votes
2
answers
568
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 ...
- 1,396
2
votes
0
answers
106
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 ...
- 51
6
votes
1
answer
580
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 ...
- 193
3
votes
0
answers
101
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?
...
- 31
2
votes
1
answer
562
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 ...
- 2,792
13
votes
4
answers
3k
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 ...
- 3,574
3
votes
0
answers
96
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 ...
- 2,527
3
votes
0
answers
367
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 ...
- 4,312
4
votes
2
answers
528
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: ...
- 475
2
votes
2
answers
473
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
$$
D^{i}(K_{ij}...
- 21
19
votes
2
answers
7k
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 ...
- 151k
5
votes
1
answer
566
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) := \frac{|\...
- 303
2
votes
0
answers
255
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 ...
- 307
24
votes
4
answers
3k
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 ...
- 639
7
votes
5
answers
4k
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 ...
- 977
4
votes
1
answer
466
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 $R=R_{\mu\nu}dx^\mu\wedge dx^\nu$,...
11
votes
1
answer
1k
views
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 $ (\Sigma,...
- 307