All Questions
Tagged with spacetime differential-geometry
347
questions
161
votes
6
answers
55k
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Why would spacetime curvature cause gravity?
It is fine to say that for an object flying past a massive object, the spacetime is curved by the massive object, and so the object flying past follows the curved path of the geodesic, so it "appears" ...
- 1,926
125
votes
6
answers
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What is known about the topological structure of spacetime?
General relativity says that spacetime is a Lorentzian 4-manifold $M$ whose metric satisfies Einstein's field equations. I have two questions:
What topological restrictions do Einstein's equations ...
- 1,734
98
votes
9
answers
23k
views
What is a manifold? [closed]
For complete dummies when it comes to space-time, what is a manifold and how can space-time be modelled using these concepts?
- 1,095
59
votes
11
answers
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Is spacetime wholly a mathematical construct and not a real thing? [closed]
Speaking of what I understood, spacetime is three dimensions of space and one of time. Now, if we look at general relativity, spacetime is generally reckoned as a 'fabric'. So my question is, whether ...
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37
votes
8
answers
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Interval preserving transformations are linear in special relativity
In almost all proofs I've seen of the Lorentz transformations one starts on the assumption that the required transformations are linear. I'm wondering if there is a way to prove the linearity:
Prove ...
- 3,732
33
votes
10
answers
10k
views
Why do objects follow geodesics in spacetime?
Trying to teach myself general relativity. I sort of understand the derivation of the geodesic equation $$\frac{d^{2}x^{\alpha}}{d\tau^{2}}+\Gamma_{\gamma\beta}^{\alpha}\frac{dx^{\beta}}{d\tau}\frac{...
- 3,059
29
votes
4
answers
5k
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Can spacetime be non-orientable?
This question asks what constraints there are on the global topology of spacetime from the Einstein equations. It seems to me the quotient of any global solution can in turn be a global solution. In ...
user1532
29
votes
5
answers
6k
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Does curved spacetime change the volume of the space?
Mass (which can here be considered equivalent to energy) curves spacetime, so a body with mass makes the spacetime around it curved. But we live in 3 spatial dimensions, so this curving could only be ...
- 421
25
votes
1
answer
8k
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How to prove that a spacetime is maximally symmetric?
In Carroll's book on general relativity ("Spacetime and Geometry"), I found the following remark:
In two dimensions, finding that $R$ is a constant suffices to prove that the space is maximally ...
- 16.4k
20
votes
2
answers
3k
views
Is spacetime simply connected?
As I've stated in a prior question of mine, I am a mathematician with very little knowledge of Physics, and I ask here things I'm curious about/things that will help me learn.
This falls into the ...
- 847
18
votes
5
answers
2k
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Where is the Lorentz signature enforced in general relativity?
I'm trying to understand general relativity. Where in the field equations is it enforced that the metric will take on the (+---) form in some basis at each point?
Some thoughts I've had:
It's baked ...
- 183
18
votes
2
answers
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Can a non-Euclidean space be descripted through an Euclidean space of higher dimension? So why use non-Euclidean?
If you draw a big triangle in Earth 2D surface you will have an approximated spherical triangle, this will be a non euclidean geometry.
but from a 3D perspective, for example the same triangle from ...
- 2,909
16
votes
5
answers
5k
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Does space curvature automatically imply extra dimensions?
Total newbie with basically no physics knowledge here :) I would welcome any correction to the steps of my reasoning that lead to my question, which could easily turn out to be invalid :)
My current ...
- 271
16
votes
5
answers
2k
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Is energy "equal" to the curvature of spacetime?
When you are solving the Einstein field equations (EFE), you basically have to input a stress–energy tensor and solve for the metric.
$$
R_{\mu \nu} - \frac{1}{2}R g_{\mu \nu} = 8 \pi T_{\mu \nu}
$$
...
- 1,525
15
votes
1
answer
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How to measure Torsion and Non-metricity?
In General Relativity, we most often work with the Levi-Civita connection (metric and torsion-free). What kind of experiment can we make to be sure that our physical space-time indeed is torsion-free ...
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