All Questions
Tagged with spacetime differential-geometry
347
questions
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42
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How is the time defined with paths consistent with the idea that we assign time to frames?
In 49:34 of this lecture by Frederic Schuller, it is explained that time is a derived quantity defined through this integral:
$$\tau = \int_{\lambda_o}^{\lambda_1} \sqrt{ g(v_{\gamma}, v_{\gamma}) }$$...
- 7,980
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1
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59
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Sorts of relativistic effect around black holes [closed]
There are many effects around black holes. In particular it is possible to study the motion of geodesics, calculate tidal tensors, lense-thirring effetcs and so on.
So, beyond tidal effects, geodesic ...
- 315
3
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95
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What does the Einstein-Hilbert action look like in terms of Riemannian metric of positive signature?
For a 4-manifold to admit a Lorentzian metric is equivalent to that manifold having vanishing Euler characteristic. Any spacetime that admits a Lorentzian metric $g^{\mathcal{L}}$ can have that metric ...
- 2,847
2
votes
1
answer
101
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Ultrastatic spacetime and cosmological constant
A spacetime $(\mathcal{M},g)$ is called "ultrastatic", if it admits a set of coordinates such that
$$g=-\mathrm{d}t^{2}+h$$
where $h$ is a Riemannian metric, which does not depend on time. ...
- 854
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2
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121
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What is the formal criteria that the spacetime is curved?
We suppose we have three scenarios.
We are far away from mass and energy in a spot in the universe. We put in free movement a small object $m$, for example, an apple. At the same time, we send a ...
- 123
2
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1
answer
141
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Static Spacetime = no cosmological constant?
I stumbled over a strange result, which cannot be true: In the (3+1)-formulation of general relativity, one considers a metric of the type
$$g_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}=(-\alpha^{2}+\...
- 854
1
vote
1
answer
262
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Gabriel's horn and General Relativity
Is there anything in GR that involves Gabriel's Horn?
This idea came to me when I met Flamm's paraboloid. If we take Schwarzschild metric at constant time and $\theta=\pi/2$, we get
$$ds^2=\left(1-\...
- 2,284
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
2
votes
2
answers
260
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Is a stationary spacetime automatically globally hyperbolic?
Is a stationary spacetime automatically globally hyperbolic? Can one construct a Cauchy-Surface by the existence of a global timelike Killing Vector field?
- 159
7
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3
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1k
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Is spacetime isomorphic to a metric space?
I know that spacetime, as described by General Relativity (GR), is a pseudo-Riemannian manifold. The label "pseudo" is due to the fact that the metric of spacetime entails not only positive ...
- 431
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572
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How to derive the Kerr killing vector?
The Kerr metric have two killing vectors:
$$t^{\mu} \equiv (k_{t})^{\mu} = (1,0,0,0)\hspace{5mm} \mathrm{and}\hspace{5mm} \phi^{\mu} \equiv (k_{\phi})^{\mu} = (0,0,0,1). \tag{1}$$
In general, it is ...
- 3,085
1
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0
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41
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Does a cosmological scale parameter in FLRW split the frame bundle?
Suppose we're looking at an FLRW cosmological metric. Then a set of orthonormal frames on our spacetime in the comoving frame looke like the set:
$$\theta=\left\{ a(x^{0})\theta^{i},\theta^{0}\right\} ...
- 2,847
2
votes
0
answers
95
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Can the Einstein tensor be written as an integral over real spacetime?
Background
We know the Stress-energy tensor can be written as:
$$ T^{\mu \nu} = \int \mathcal{N}(x,p,t) p^\mu \otimes p^\nu \frac{d V_p}{E}$$
where $\mathcal{N}(x,p,t)$ is the distribution function, $...
- 1,389
4
votes
2
answers
619
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Can we just take the underlying set of the spacetime manifold as $\mathbb{R^4}$ for all practical purposes?
In mathematical GR and also in some informal GR presentations (eg: MTW), manifolds are always mentioned before talking about GR... but now I am starting to wonder.. if it even actually neccesary?
In ...
- 7,980
4
votes
1
answer
435
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Foundational considerations in definition of Newtonian Spacetime
My doubt comes from lecture 08 on Theoritische Mechanik by Frederic Schuller.
He gives definition of Newtonian Space time as followg:
A Newtonian Spacetime is a quintuple of structures $(M,\mathcal{O}...
- 7,980