All Questions
113
questions
2
votes
0
answers
60
views
Under what circumstances can a 4D singularity occur in General Relativity?
I've tried to find on the literature about 4D (single point) singularities, but most of the theorems about singularities pertain to either space-like or time-like singularities, which always have some ...
- 159
0
votes
1
answer
83
views
What objects are solutions to the Einstein Field Equations?
The usual way the solutions of the Einstien Field Equations are introduced is by saying they are (pseudo-) riemannian metrics that satiafy the diff equations for a given EM Tensor. My question is: ...
0
votes
1
answer
76
views
How to motivate that in presence of gravity the spacetime metric must be modified to $ds^2=g_{ab}(x)dx^adx^b$?
In the presence of a gravitational field, the spacetime metric, $$ds^2=\eta_{ab}dx^a dx^b,$$ should be changed to, $$ds^2=g_{ab}(x)dx^adx^b.$$ What are the convincing physical arguments that motivate ...
- 11.8k
1
vote
0
answers
32
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Example of lightlike curve that's not a geodesic in Lorentz spacetime [duplicate]
Let $(M,g)$ be a 4 dimensional Lorentz spacetime. A smooth curve $\alpha:\ I\to M$ is called lightlike if $\alpha'(s)\in TM_{\alpha(s)}$ is lightlike for all $s\in I$, which means
$$g_{\alpha(s)}\big(\...
- 143
1
vote
0
answers
62
views
Confused about spherically symmetric spacetimes
I'm following Schutz's General Relativity book and I am confused about his description and derivations of a spherically symmetric spacetime. I searched online and found that using Killing vectors is a ...
- 151
0
votes
0
answers
56
views
Why is spacetime pseudo-Riemannian manifold?
Forgive me for asking, This is a relatively naïve question, though, one i've had for a while now. I know that a pseudo-Riemannian manifold is a differentiable manifold with a metric tensor that is ...
2
votes
2
answers
189
views
Can $\mathbb{R}^4$ be globally equipped with a non-trivial non-singular Ricci-flat metric?
I'm self-studying general relativity. I just learned the Schwarzschild metric, which is defined on $\mathbb{R}\times (E^3-O)$. So I got a natural question: does there exist a nontrivial solution (...
- 107
7
votes
2
answers
2k
views
Do the Einstein Field Equations force the metric to be Lorentzian?
In GR, we are working with Lorentzian metrics, which are examples of a pseudo-Riemannian metrics. That is, we are trying to find pseudo-Riemannian $g_{\mu\nu}$ that are solutions to the field equation ...
- 453
2
votes
2
answers
156
views
Relationship between spacelike and timelike distances in General Relativity vs. Special Relativity
In Minkowski spacetime, the distance $d_S$ between two space-like separated events $x$ and $y$ can (up to constant) be given by a distance between the two time-like separated events $z$ and $w$ where $...
- 125
3
votes
2
answers
165
views
What is difference between an infinitesimal displacement $dx$ and a basis one-form given by the gradient of a coordinate function?
In general relativity, we introduce the line element as $$ds^2=g_{\mu \nu}dx^{\mu}dx^{\nu}\tag{1}$$ which is used to get the length of a path and $dx$ is an infinitesimal displacement But for a ...
- 374
2
votes
2
answers
305
views
Derivation of the Schwarzschild metric: why are $g_{22}$ and $g_{33}$ the same as for flat spacetime?
I'm trying to understand the derivation of the Schwarzschild metric from Wikipedia, but I simply do not understand why, therein, $g_{22}$ and $g_{33}$ must be those of the flat spacetime.
Couldn't $g_{...
- 93
5
votes
0
answers
127
views
Is it possible to create a Nil geometry in real spacetime according to general relativity? (What metrics are possible in the real world?)
Background
I've heard that it is possible to construct a Penrose triangle in the 3D geometry Nil. And I wondered: Can we build a Penrose triangle in the real world if spacetime is appropriately ...
0
votes
1
answer
353
views
Angular Deficit of a Conical Singularity
I'm currently studying the Bonnor solution starting with this paper on Black Diholes. The metric is given by :
$$ ds^2 = \left(1-\frac{2Mr}\Sigma\right)^2 \left[-dt^2 + \frac{\Sigma^4}{(\Delta + (M^2 +...
- 55
3
votes
1
answer
530
views
Confusion regarding Geodesics
Suppose we have a causal curve and we can cover the causal curve by convex normal neighborhoods. We also know that, in convex normal neighborhood there will exist a unique geodesic inside the ...
- 27
2
votes
3
answers
220
views
What is Dirac's reasoning when showing the curvature vanishing implies we can choose rectilinear coordinates?
In section 12 of Dirac's book "General Theory of Relativity" he is justifying the name of the curvature tensor, which he has just defined as the difference between taking the covariant ...
- 43