All Questions
Tagged with riemannian-geometry general-relativity
16 questions with no upvoted or accepted answers
20
votes
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answers
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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
6
votes
0
answers
129
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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
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
0
answers
126
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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
3
votes
0
answers
72
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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
3
votes
0
answers
162
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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 ...
- 1,163
3
votes
0
answers
101
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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?
...
- 31
3
votes
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96
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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 ...
- 2,527
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
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
2
votes
0
answers
106
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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 ...
- 51
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
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
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
0
answers
156
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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 $m_g$...
- 2,533
0
votes
0
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47
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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