# 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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### Metric in dilatation transformation of massless scalar field

The lagrangian density of the massless real scalar field is \begin{align} L = \frac{1}{2}\eta^{\mu\nu}\partial_\mu\Phi\partial_\nu\Phi = \frac{1}{2}\partial_\mu\Phi\partial^\mu\Phi. \end{align} I want ...
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### Why is $dt/d\tau=\gamma$? What is $dt/d\tau$ supposed to mean exactly?

I'm a math student trying to learn some physics by reading Susskind's The Theoretical Minimum. In the volume on special relativity he derives that $\frac{dt}{d\tau}=\gamma=1/\sqrt{1-v^2}$ and uses it ...
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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) + ...
1 vote
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### Minkowski space

In Minkowski space, coordinates which satisfy $\Delta s^2 = \Delta t^2 - \Delta ^2 > 0$ are in the region of spacetime that is time-like. If it's $\Delta s^2 = \Delta t^2 - \Delta x^2 < 0$, the ...
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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 ...
1 vote
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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:...
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### Deriving Einstein-Rosen Bridge

I know that an einstein rosen bridge is derived by a coordinate transformation on the schwarzschild metric, but I can't find much on it online, could someone please show how to change the metric into ...
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### Covariant derivative to the metric determinant?

I am reading the paper Alternatives to dark matter and dark energy, but cannot obtain one specific equation no matter how I tried. So I wrote an email to the author, the following is what he replies ...
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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### Is the Birkhoff Theorem valid for the FLRW metric?

The FLRW metric, in the case of positive scalar curvature, is: $ds^2 =- c^2 dt^2+a(t)^2\left(dw^2+\sin^{2}w(d\theta^2+\sin^2\theta d\phi^2)\right)$. The Birkhoff’s theorem states that "any ...
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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 ...
1 vote
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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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### Metric under conformal transformation

I have a question regarding the conformal factor $\Omega(x)$ when dealing with a conformal transformation. We know that under a change of coordinates $x\rightarrow x^{'}=x^{'}(x)$ our metric changes ...
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### What is the gradient of deformation gradient $F$?

Deformation gradient is defined as $$F_{iJ}=\frac{\partial x_i}{\partial X_J},\;\mathbf{F}=\frac{\partial\mathbf{x}}{\partial\mathbf{X}},$$ where $\mathbf{x}$ is spatial coordinates; $\mathbf{X}$ is ...
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### Null vector space in Minkowski space

Let us consider a Minkowski space of the form: $$ds^2 = -dt^2 + dx^2 + dy^2 +dz^2.$$ What would the linearly independent null vectors of this space be? I am aware this is a trivial question but is ...
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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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### 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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### The importance of metric signature in Ricci scalar

I have read this question Different signatures of the metric in Einstein field equations (and related posts) on the invariance of Einstein field equations under metric signature change. However, there ...
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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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### 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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### Diameter of a sphere in the regime of general relativity

Lets start naive: empty space, define the origin somewhere, start putting mirrors in a distance of $r$ in many directions so that they roughly sample the surface of a ball of radius $r$. Someone ...
1 vote
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### Geodesic equation & Four-velocity

I've been studying Kolb & Turner's "The Early Universe", and came across an equation that somehow I can't understand. Given the equation: \frac{1}{u_0} \frac{d |\vec u|}{ds} + \...
1 vote