Questions tagged [metric-tensor]
The variables used in general relativity to describe the shape of spacetime. If your question is about metric units, use the tag "units", and/or "si-units" if it is about the SI system specifically.
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A question about Friedmann cosmological expansion equation
A recent paper, arXiv:2403.01555, gives the equations for homogeneity and isotropy of an expanding 3-space as expressed in the following
distance interval as $x^i = (t, \chi, \theta, \phi)$ and $x^i + ...
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Torsion and Compatibility with the Metric
Compatibility with a metric, also referred to as metricity, means, I believe, that the covariant derivative of the metric is zero:
$$g_{ij;k}=g_{ij,k}-\Gamma^m_{ik}g_{mj}-\Gamma^m_{jk}g_{im}=0$$
This ...
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Turning a Lagrangian contains superscript and subscript indices into energy
I'm recently reading the book "Solitons and Instantons" written by R. RAJARAMAN. However, for lacking of ability, I couldn't figure out how to derivate the static solution for energy with ...
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The Asymmetric Metric [closed]
Update1: closed because the metric has to be symmetric? Ok, but I'm not the first to study an asymmetric metric, which is neither symmetric nor anti-symmetric, but has components of both. In my ...
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Alternative definition of the Ricci Scalar
I came across this definition of the Ricci Scalar on its Spanish Wikipedia page:
$$R=-g^{\mu\nu}\left(\Gamma_{\mu\nu}^{\lambda} \Gamma_{\lambda\sigma}^{\sigma} - \Gamma_{\mu\sigma}^{\lambda}\Gamma_{\...
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Variation of action of non-critical string under Weyl transformation (worldsheet cosmological constant term)
In David Tong's lecture notes on string theory, section 5.3.2 An Aside: Non-Critial Strings, page 121, he describes the non-critical string with the following action:
$$S_{\text{non-critical}} = \frac{...
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Partial derivatives of Christoffel symbols to Covariant derivatives
I wanted to express this thing: $g^{ab}\partial_c\Gamma^c_{ab} - g^{ab}\partial_a\Gamma^c_{cb}$, in terms of a covariant derivative. I figured out that if you swap $a$ and $c$ in the $\partial \Gamma$ ...
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Understanding differentials in an equation in general relativity
I have not studied physics but I was browsing Carroll's relativity book and randomly stumbled upon the following which I would like to understand mathematically. It says
$$ds^{2} = 0 = - \left( 1 - \...
Esoog
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Complex coordinates $ds^2 = dzdz̄$ in 2d
I have a very elementary question about complex coordinates in two dimensions. When we have a 2D Euclidean space,
$$ds^2 = dx^2 +dy^2$$
and we go to complex coordinates:
$$z = x + iy$$ $$z̄ = x - iy$$
...
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Length near the black hole
One meter rod at long distance is thrown to the Schwarzschild black hole. How its length near the black hole seems to distant observer?
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Orbit description in Schwarzschild metric
Suppose to have a restricted 2-body system (BH + star with $M_{BH}\gg M_{\mathrm{star}}$) and you want to describe the orbit of the star relative to the BH, i.e. in the Schwarzschild metric.
Usually, ...
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Derivation of transformation law for the Hilbert Stress-energy tensor [duplicate]
The Hilbert stress-energy tensor is defined as
$$T_{\mu\nu}=-2 \frac{1}{\sqrt{g}}\frac{\delta S_M}{\delta g^{\mu\nu}}.$$
Given the name one expects that it transform as a tensor, but how to prove this ...
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How does the light from distant stars change for an observer at the center of the collapsing or falling sphere?
At the center of a spherically symmetric thin solid static shell lies a point observer. For this observer, distant stars appear violet shifted slightly more $\frac{{G \cdot M}}{{{c^2} \cdot r}}$ ($\...
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How to motivate that in presence of gravity the spacetime metric must be modified to $ds^2=g_{ab}(x)dx^adx^b$?
In the presence of a gravitational field, the spacetime metric, $$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 ...
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Geodesic in flat space in spherical coordinates
let's consider the expression, where $u^\mu$ is the tangent vector to the geodesic
$\theta = \nabla_\mu u^\mu$....scalar $\Rightarrow$ valid in every coordinate system
So in flat space in Cartesian ...
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