All Questions
23
questions
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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
0
votes
1
answer
110
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Geodesic beeing inextendible and incomplete
We say that a geodesic is (future ) inextendible if there exist no (future) endpoint for example. Wouldnt that imply the domain of the geodesic is $[p, \infty)$ with some beginning point $p$? And it ...
user390166
3
votes
0
answers
131
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I need help with a proof in Hawking & Ellis [closed]
Here's a proof in Hawking and Ellis (1973) of proposition 6.4.6:
The definition of "strong causality" used in the book is that for every point $p$ and every neighborhood $U$ of $p$, there ...
- 863
0
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2
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158
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Does a straight line in flat space become a geodesic in curved space when the space becomes curved?
Flexible foam has shortest path from Point-A to Point-B. When the foam is not curved (space-time is not curved), the shortest path is Path-1 (straight line - before curving the foam). But if the foam ...
- 1
1
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3
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2k
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What is timelike geodesic?
I have searched the internet for the definition of timelike geodesic curves. But I am not getting a consistent definition. In some places I saw the geodesic maximises the proper time and in some ...
- 399
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98
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Geodesically complete spacetime
By definition, a spacetime is geodesically complete if all inextendible curve are complete but is this equivalent to «if all geodesic of finite length has endpoints»?
My situation is:
I have a set of ...
- 357
1
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1
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472
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Geodesic deviation equation in flat spacetime $\sim$ divergence of geodesics
Consider the above two neighbouring geodesics $\mathcal{Y}$ given by $x^{\alpha}(\sigma)$ and $\mathcal{\tilde{Y}}$ by $\tilde{x}^{\alpha}(\sigma)$ for top and bottom curves respectively. Vector ...
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After conformal compactification, do null geodesics intersect future null infinity nonasymptotically?
I was going through this paper and was worried about an assumption in the main proof (Theorem 3.1), where they assume null geodesics intersect the boundary extension after conformal compactification ...
1
vote
1
answer
144
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Geodesic incompleteness of static spherically symmetric solution
Static spherically symmetric solution of Einstein equations is given by the metric
$$
ds^2=f(r)dt^2-\frac{dr^2}{f(r)}-r^2d\Omega^2,
$$
where $f(r)=1-(kr)^2$, $d\Omega^2$ is the metric of unit sphere.
...
- 31
1
vote
1
answer
304
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Combining the Einstein field equations with the geodesic equation
I've seen this question How does Einstein field equations interact with geodesic equation?, but it doesn't make any sense to me. If spacetime is a Lorentzian manifold, then surely one thing general ...
- 2,475
3
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1
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754
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Geodesics and constraints on the parameterization
I am following an introductory course on General Relativity based on the work of Sean Carroll in: Spacetime and Geometry.
After a lot of trouble we get to the following differential equation:
$$\frac{...
- 4,577
0
votes
1
answer
95
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Geodesic with decreasing value of time coordinate
Does there exists an example of geodesic for an exotic space-time manifold in which even though the proper time on the geodesic is increasing but still the time coordinate of the geodesic in global ...
- 3,043
0
votes
0
answers
50
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Trajectory of photons in Schwarzschild coordinates 2D + t
I'm french and my english is awfull, excuse me for this. I hope you could however understand my question.
I'm looking for a formula of trajectory of photons in Schwarzschild coordinates. For example ...
- 3
0
votes
2
answers
110
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GR visualization
I'm watching some GR lectures by Schuller (more or less rushing through them so bear with my ignorance here please) in Lecture 10: Metric Manifolds.
He's talking about geodesics in a manifold with a ...
- 165
3
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184
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Doubt about energy conditions: the Time-like Convergence Condition
First of all, consider a congruence of smooth time-like geodesics parametrized by proper time $\tau$. So, a tangent vector to a time-like geodesic is indeed a four-velocity up to a factor constant; ...
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