All Questions
Tagged with spacetime metric-tensor
567
questions
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What happens if we differentiate spacetime with respect to time? [closed]
Essentially, what would differentiating space-time with respect to time provide us with? What are the constraints associated with such operations? Is it possible to obtain a useful physical quantity ...
1
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2
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123
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Is the FRW metric, based on spatial homogeneity and isotropy, rotationally and translationally invariant? If so, how?
The spatial part of the Minkowski metric, written in the Cartesian coordinates, $$d\vec{ x}^2=dx^2+dy^2+dz^2,$$ is invariant under spatial translations: $\vec{x}\to \vec{x}+\vec{a}$, where $\vec{a}$ ...
- 11.8k
2
votes
1
answer
72
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A few doubts regarding the geometry and representations of spacetime diagrams [closed]
I had a couple questions regarding the geometry of space-time diagrams, and I believe that this specific example in Hartle's book will help me understand.
However, I am unable to wrap my head around ...
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69
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Confusion about local Minkowski frames
This is sort of a follow-up to the question I asked here:
Confusion about timelike spatial coordinates
The important context is that we imagine a metric that, as $t\rightarrow\infty$, approaches the ...
- 180
2
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1
answer
88
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Confusion about timelike spatial coordinates
I'm pretty new to general relativity, and I'm self-studying it using Sean M. Carroll's text on the subject. In Section 2.7, he introduces the notion of closed timelike curves. He gives the example of ...
- 180
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0
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How to derive Feffermann-Graham expansion for AdS Vaidya geometries?
Introduction
The Feffermann-Graham expansion for an asymptotically AdS spacetime [0] looks like Poincare AdS but with the flat space replaced by a more general metric i.e.
$$ds^2=\frac{1}{z^2}(g_{\mu \...
- 785
3
votes
1
answer
55
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Time component of four-velocity
While reading through Spacetime and Geometry by Sean Carroll, I came across the following passage:
"Don't get tricked into thinking that the timelike component of the four velocity of a particle ...
- 442
4
votes
3
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197
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Change of variables from FRW metric to Newtonian gauge
My question arises from a physics paper, where they state that if we take the FRW metric as follows, where $t_c$ and $\vec{x}$ are the FRW comoving coordinates:
$$ds^2=-dt_c^2+a^2(t_c)d\vec{x}_c^2$$
...
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1
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82
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What objects are solutions to the Einstein Field Equations?
The usual way the solutions of the Einstien Field Equations are introduced is by saying they are (pseudo-) riemannian metrics that satiafy the diff equations for a given EM Tensor. My question is: ...
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2
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64
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What is $r$ in a metric signature in general relativity? If $v$ and $p$ are the time and spatial coordinates?
The Wikipedia article on metric signatures says that the signature of a metric can be written $(v,p,r)$, where $v$ is the number of positive eigenvalues, $p$ is the number of negative eigenvalues, and ...
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How to motivate that in presence of gravity the spacetime metric must be modified to $ds^2=g_{ab}(x)dx^adx^b$?
In the presence of a gravitational field, the spacetime metric, $$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
3
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140
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Clarification on Representing Distances and Trajectories in Minkowski Spacetime
In the context of Minkowski spacetime, where the metric has a signature of (-, +, +, +), the $x-t$ plane (spacetime diagram) is commonly used to visualize events and their evolution in both space and ...
- 487
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4
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364
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Regarding the signature of special relativity
in special relativity we add time as a dimension and replace euclidean space $ \mathbb{R}^4 $ with a pseudo-euclidean space $ \mathbb{R}^{1,3} $ of signature $ (1,3) $ by defining a quadratic form $\...
- 309
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Proof of the invariance of $c$ using the Lorentz group
Apologies if this question was already asked a few times but i could only find proofs of the invariance of $ ds^2 $.
Is there any way of proving the 2nd postulate (that $c$ is invariant in all ...
- 309
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2
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Do we have notion of a proper time for any two timelike separated arbitrary events?
Consider two infinitesimally close, timelike separated but otherwise arbitrary events $P$ and $Q$ with coordinates $(t,\vec{x})$ and $(t+dt,\vec{x}+d\vec{x})$. For example, imagine event $P$ is "...
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