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Results tagged with general-relativity
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user 5626
A theory that describes how matter interacts dynamically with the geometry of space and time. It was first published by Einstein in 1915 and is currently used to study the structure and evolution of the universe, as well as having practical applications like GPS.
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Given symmetry of metric tensor, can't all the partial derivatives in the Christoffel symbol...
The Christoffel symbols of the first kind:
$$\Gamma_{mij}=\frac{1}{2}\sum_i \sum_j\left( \frac{\partial g_{jm}}{\partial x_i}+\frac{\partial g_{im}}{\partial x_j}- \frac{\partial g_{ij}}{\partial x_m …
- 1,527
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2
answers
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How does the friedmon solution to Einstein's equations resolve paradox of bounded infinities?
This article talks about a potential explanation of dark matter based on something called the "friedmon." I have no interest in the dark matter question, but the article has made me curious about thi …
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What is the "momentum" referred to in the energy-momentum tensor
What is the "momentum" referred to in the energy momentum tensor from GR?
Is it $m\dot{x}$ or is it the canonical momentum $\frac{d}{dt} \left(\frac{\partial L}{\partial \dot{x}}\right)$
Also, I fin …
- 1,527
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Physical meaning of the Rindler hyperbola vertex and the Rindler lines
Two questions regarding the Rindler diagram:
1) Does the vertex of a given hyperbola in the diagram have physical meaning? I know it is the inverse of the constant proper acceleration ($\alpha$) ass …
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Are there any conditions under which the Christoffel symbols can be treated as a damping ter...
(Mathjax did not seem to be working as I composed this question. Hopefully it will kick into action once I post.)
Note I am a novice at tensor notation.
I am working with the following Lagrangian ( …
- 1,527
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3
answers
2k
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Need some basic help with notation and the Christoffel symbols
Apologies in advance if some of the questions below seem overly simple.
In an introductory GR book, I find the following expression for the autoparallel of the affine connection (the upper bound of t …
- 1,527
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3
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429
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Having trouble seeing the similarity between these two energy-momentum tensors
Leonard Suskind gives the following formulation of the energy-momentum tensor in his Stanford lectures on GR (#10, I believe):
$$T_{\mu \nu}=\partial_{\mu}\phi \partial_{\nu}\phi-\frac{1}{2}g_{\mu \n …
- 1,527
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2
answers
2k
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How do we know the geodesic is a minimum?
The geodesic equation is derived from the Euler-Lagrange equation, which (as I understand it) is a necessary but not sufficient condition to ensure that the geodesic is a minimum.
The introductory GR …
- 1,527
5
votes
1
answer
549
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Confused about indices of the Ricci tensor
In an intro to GR book the Ricci tensor is given as:
$$R_{\mu\nu}=\partial_{\lambda}\Gamma_{\mu \nu}^{\lambda}-\Gamma_{\lambda \sigma}^{\lambda}\Gamma_{\mu \nu}^{\sigma}-[\partial_{\nu}\Gamma_{\mu \l …
- 1,527
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2
answers
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How to think of the harmonic oscillator equation in terms of "acceleration = gradient"
This is related to another question I just asked where I learned that the equation of motion of a harmonic oscillator is expressed as:
$$\ddot{x}+kx=0$$
What little physics I grasp centers on geodes …
- 1,527
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2
answers
130
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Is the gravitational potential a measurable physical quantity or an artifact of warped measu...
The Euler-Lagrange conditions for stationary points of $$L=m/2 v(\mathbf{\dot{x}})^2-U(\mathbf{x})$$
($m$ is mass, $v()$ is velocity, $U()$ is the scalar potential, and the boldfaced arguments of the …
- 1,527
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3
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Clarifying what metric counts as flat space
In (2D) Cartesian coordinates, the Euclidean metric...
$$\begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}$$
...is flat space. If the diagonal elements are exchanged for other real numbers greater or l …
- 1,527
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2
answers
464
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Using the area element in derivation of geodesic
In the derivation of the geodesic, one starts with the integral of the line element (arclength):
$$L(C)=\int_{\tau_1}^{\tau_2}d\tau\sqrt{g_{\mu \nu}\dot{x}^{\mu} \dot{x}^{\nu}}$$
The integrand is th …
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