# 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.

3,577 questions
Filter by
Sorted by
Tagged with
1 vote
50 views

### 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 ...
• 11
45 views

### 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 ...
353 views

### 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?
• 335
26 views

### 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 ...
152 views

23 views

### 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 ...
1 vote
62 views

### 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 $\mathrm{d}\tau$ ...
• 782
66 views

### 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 ...
36 views

82 views

### 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 ...
468 views

### 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 ...
• 576
87 views

### 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 ...
• 137
37 views

### 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 $\mathbf{L}$ ...
• 576
60 views

### 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 ...
• 187
76 views

### 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 ...
53 views

• 323
207 views

### 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 $2M$ ...
• 1,171
78 views

### 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 ...
• 1,171
578 views

### 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 ...
• 1,171
79 views

### 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 ...
• 23
1 vote
62 views

• 57
70 views

• 21
1 vote
27 views

### How to derive Feffermann-Graham expansion for AdS Vaidya geometries?

Introduction The Feffermann-Graham expansion for an asymptotically AdS spacetime [0] looks like Poincare AdS but with the flat space replaced by a more general metric i.e. ds^2=\frac{1}{z^2}(g_{\mu \...
• 805
56 views

### Time component of four-velocity

While reading through Spacetime and Geometry by Sean Carroll, I came across the following passage: "Don't get tricked into thinking that the timelike component of the four velocity of a particle ...
• 442
As an exercise, I wanted to apply the tool of taking some "time-evolving" surface embeddable in Euclidean space, defined parametrically as $X_0(u, v, t), X_1(u, v, t), X_2(u, v, t)$, and ...