# 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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### On the Background Independence condition

In General Relativity, one has that the equations of motion for any matter distribution are given by extremizing the following action: S[g] = \int\left[\frac{1}{8\pi}(R - 2\Lambda) + ...
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### In a Spatially One-Dimensional Universe, is a Minkowski Space-Time Diagram accurately graphable if we include the effects of "gravity"?

I've been working on studying Special Relativity and General Relativity for the past few years. As I think we all know, GR gets a lot more complicated than SR, and my knowledge is limited. I am very ...
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### Importance of orthogonality in Minkowski space [closed]

I am currently studying Minkowski space. Orthogonality in this space is new to me. I have seen in a blog post, in 1 that states that, orthogonality is important in this space. It will be helpful, if ...
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### Discrepancy with a result from Peskin & Schroeder's QFT

I'm trying to understand a result in Peskin and Schroeders chapter 3 on the dirac equation, during the free particle calculation for the case $p^0=E>0$. Peskin and schroeder derived this expression:...
1 vote
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### Why cant I measure proper length on spacetime curvature with the following formula? [closed]

I'm struggling right now from the definition of proper length along spacetime curvature, it is said as I found online the length that object covered on his spacetime rest frame , so why cant I use the ...
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### What metric to use for this dark matter simulation?

I am reading this paper https://arxiv.org/abs/1901.08064 which uses the GR version of euler equations in fluid dynamic to simulate the evolution of a perfect fluid system. (PDF) and this is the paper ...
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### Inverse problem for geodesic

If I know the expressions for geodesic distance between any points $x$ and $y$: $$L=L(x^\mu,y^\nu) \ .$$ How do I find the metric of the corresponding space?
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### When we write down the FLRW metric,what are the basis vector or coordiante lines of the coordiante system?

When we consider the coordinate system,it seems we can always ask for how the curvlinear coordinate lines looks like. So if the universe started evluting from a point,then whether the coordinate ...
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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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### 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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### Photonic black holes

"Can a photon turn into a black hole?" - usually the answer to this question is - it can't, because it has zero rest mass. However, when we derive the Schwarzchild Metric initially the $2M$ ...
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### Homogeneous and Isotropic But not Maximally Symmetric Space

Is this statement correct: "In a homogeneous and Isotropic space the sectional curvature is constant, while in a maximally symmetric space the Riemann Curvature Tensor is covariantly constant in ...
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### Constant curvature on a sphere?

$ds^2 = \frac{1}{1- r^2}dr^2 + r^2d\theta^2$ denotes a 2d spherical surface and it should have a constant curvature. The Riemann curvature tensor components are linear in their all 3 inputs. Since the ...
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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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### Conformal equivalent to Schwarzschild metric

Consider Schwarzschild spacetime in Eddington-Finkelstein coordinates $(v,r,\theta,\phi)$ g \enspace = \enspace -f(r) \, dv^2 + 2 \, dv \, dr + r^2 \, d\Omega^2 \quad , \qquad f(r) = 1 - \frac{2m}{r}...
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### Are the quantity $\frac{\delta \sqrt{-g}}{\delta g_{\mu \nu}}$, $\frac{\delta \sqrt{-g}}{\delta g^{\mu \nu}}$ are computable? [duplicate]

From $\delta \sqrt{-g} = -\frac{1}{2} \sqrt{-g} g_{\alpha \beta}\delta g^{\alpha \beta} = \frac{1}{2}\sqrt{-g} g^{\alpha \beta} \delta g_{\alpha \beta}$, can we compute \begin{align} &\frac{\...
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### Can a vector tangent to a spacelike surface be null?

I'm studying the peeling-off behaviour of zero rest-mass fields, as described in Penrose's paper. In it, he talks about the boundary $\mathscr{I}$ of the conformal completion of an asymptotically ...
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### What is the topology of a spacelike cross-section of a null hypersurface?

I'm studying the peeling-off behaviour of zero rest-mass fields, as described in Penrose's paper. In it, he talks about the boundary $\mathscr{I}$ of the conformal completion of an asymptotically ...
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### Graviton metric interaction

In most discussions on quantum-gravity, graviton is considered as a perturbation that is being added linearly to a flat metric (the $h_{\mu\nu}$ term in $g_{\mu\nu} = \eta_{\mu\nu} + h_{\mu\nu}$). ...
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