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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Second-order perturbations of gauge field in GR
When expanding a Lagrangian $\mathcal{L}[g_{\mu\nu},A_\mu,\chi]$ to second order in perturbations, the metric is expanded like
$$g_{\mu\nu}\to g_{\mu\nu}+\delta g_{\mu\nu}+\frac{1}{2}\delta g_{\mu}^{\,...
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Finding coordinate transfromations by line element
I’m confused on how to generally approach these coordinate transformations: I initially thought we can set $dT^2=(dt-b/2dx)^2$ and $a^2dX^2= (a^2+\frac{b^2}{4})dx^2$. This way, when carrying out the ...
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In which direction is the relation between the time-component of celerity and the Lorentz factor defined?
Celerity (a.k.a. proper velocity) is defined as $w^\alpha=\frac{\mathrm{d}X^\alpha}{\mathrm{d}\tau}$, where $\mathrm{d}X^\alpha=(\mathrm{d}t,\mathrm{d}x,\mathrm{d}y,\mathrm{d}z)$ and $\mathrm{d}\tau$ ...
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Does Mass Actually Displace Space-Time, or does Mass only Distort it?
1. Question
Given the plethora of space-time illustrations, there is a sense that space-time is actually being displaced by mass, (planets). But on its face, this doesn't really make sense because ...
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Is the tensor product involved in the metric a symmetric product?
The expression of the FRW metric in Cosmology in usually written as:
$$ds^2=-dt^2+a^2(t)d\vec{x}^2$$
where $c=1$. However, $dt^2$ is a shortening of $dt\otimes dt$, that is, of the tensor product of $...
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Question coming from Cosmological Perturbation
We consider the following scalar perturbation on the FRW metric:
$$ ds^2 = -(1 + 2\phi)dt^2 +2a\partial_i B dx^i dt + a^2 \left( (1 - 2\psi)\delta_{ij} + 2\partial_{ij}E\right) dx^i dx^j $$
where $\...
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Why is $(\partial_\mu F_{\alpha\beta})F^{\alpha\beta}=F_{\alpha\beta}\partial_\mu(F^{\alpha\beta})$?
I'm trying to prove that the divergence of the energy-momentum-tensor is zero by expressing it in terms of the field strength tensor: $\partial_\mu T^{\mu\nu}=0$.
In doing this, letting the derivative ...
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Question on special relativity
I am trying to learn special relativity. If we consider two inertial reference frames with spacetime co-ordinates $(t,x,y,z)$ and $(t',x',y',z')$ and let there be 2 observers who measure the speed of ...
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Is there a metric, a solution to Einstein's field equations, for a single body in a space of uniform non-zero density?
The Swarzschild metric describes a single body in an empty space with zero density, while the FLRW metric is presumably for a space with uniform non-zero density but no single body. But is there a ...
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Derive Minkowski metric from Lorentz transformation
I am trying to learn special relativity. My goal is to prove that given the fact that a 4-vector $\mathbf{x}$ is transformed as $\mathbf{Lx}$, between two inertial reference frames where $\mathbf{L}$ ...
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Under what circumstances can a 4D singularity occur in General Relativity?
I've tried to find on the literature about 4D (single point) singularities, but most of the theorems about singularities pertain to either space-like or time-like singularities, which always have some ...
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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 ...
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Extrinsic Curvature Calculation on the Sphere
Given the following 2+1 dimensional metric:
$$ds^{2}=2k\left(dr^{2}+\left(1-\frac{2\sin\left(\chi\right)\sin\left(\chi-\psi\right)}{\Delta}\right)d\theta^{2}\right)-\frac{2\cos\left(\chi\right)\cos\...
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Interpretation of degenerate metrics
I was studying the metric tensor and saw all about degenerate metrics. I would like what is the physical or geometrical intuition of a degenerate metric.
What is the meaning of $g(v,w) = 0$ for a ...
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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 $\vec{a}$ ...
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