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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9
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1answer
317 views

How does the Hubble parameter change with the age of the universe?

How does the Hubble parameter change with the age of the universe? This question was posted recently, and I had almost finished writing an answer when the question was deleted. Since it's a shame to ...
15
votes
6answers
906 views

Proving that interval preserving transformations are linear

In almost all proofs I've seen of the Lorentz transformations one starts on the assumption that the required transformations are linear. I'm wondering if there is a way to prove the linearity: Prove ...
2
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3answers
2k views

Ricci scalar for a diagonal metric tensor

I was wondering if there is a general formula for calculating Ricci scalar for any diagonal $n\times n$ metric tensor?
8
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4answers
1k views

Minkowski spacetime: Is there a signature (+,+,+,+)?

In history there was an attempt to reach (+, +, +, +) by replacing "ct" with "ict", still employed today in form of the "Wick rotation". Wick rotation supposes that time is imaginary. I wonder if ...
14
votes
3answers
284 views

What is meant when it is said that the universe is homogeneous and isotropic?

It is sometimes said that the universe is homogeneous and isotropic. What is meant by each of these descriptions? Are they mutually exclusive, or does one require the other? And what implications rise ...
2
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3answers
245 views

Clarifying what metric counts as flat space

In (2D) Cartesian coordinates, the Euclidean metric... $$\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$ ...is flat space. If the diagonal elements are exchanged for other real numbers ...
13
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1answer
463 views

What are the local covariant tensors one can form from the metric?

Normally in differential geometry, we assume that the only way to produce a tensorial quantity by differentiation is to (1) start with a tensor, and then (2) apply a covariant derivative (not a plain ...
9
votes
5answers
931 views

Minkowski Metric Signature

When I learned about the Minkowski Space and it's coordinates, it was explained such that the metric turns out to be $$ ds^{2} = -(cdx^{0})^{2} +(dx^{1})^{2} + (dx^{2})^{2} + (dx^{3})^{2} $$ where $ ...
4
votes
1answer
207 views

Computing Curvature via Cartan Formalism

Given a metric $g_{\mu \nu}$, one can select an orthonormal basis $\omega^{\hat{a}}$ such that, $$ds^2= \omega^{\hat{t}}\otimes\omega^{\hat{t}} - \omega^{\hat{x}} \otimes \omega^{\hat{x}} - ...$$ By ...
9
votes
1answer
325 views

Contracting Indices

Does anyone know how to get from (1) to (2) in the system $$ \begin{align} \mathrm{g}^{\mu\nu}_{,\rho}+ \mathrm{g}^{\sigma\nu}{{\Gamma}}^{\mu}_{\sigma\rho}+ ...
14
votes
4answers
544 views

The Lagrangian as a metric

My question is, can the (classical) Lagrangian be thought of as a metric? That is, is there a meaningful sense in which we can think of the least-action path from the initial to the final ...
8
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3answers
4k views

Why is the covariant derivative of the metric tensor zero?

I've consulted several books for the explanation of why $$\nabla _{\mu}g_{\alpha \beta} = 0,$$ and hence derive the relation between metric tensor and affine connection $\Gamma ^{\sigma}_{\mu ...
7
votes
4answers
416 views

How do you tell if a metric is curved?

I was reading up on the Kerr metric (from Sean Carroll's book) and something that he said confused me. To start with, the Kerr metric is pretty messy, but importantly, it contains two constants - ...
5
votes
4answers
659 views

How did “no prior geometry” father 50 years of confusion?

I've come across this quote attributed to Misner, Thorne & Wheeler from their book, Gravitation: Mathematics was not sufficiently refined in 1917 to cleave apart the demands for "no prior ...
2
votes
1answer
251 views

How to properly construct the electromagnetic tensor in curved space-time?

How do I properly construct the electromagnetic tensor in curved space-time? I have my curved spacetime metric $(+,-,-,-)$ and my magnetic vector potential $A$. I tried two ways but not sure which is ...
2
votes
1answer
343 views

Variation of modified Einstein Hilbert Action

In general relativity one can derive the Einstein Field Equations by the principle of least action through variations with respect to the inverse of the metric tensor. In some modified theories of ...
1
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2answers
65 views

Proper time in general relativity

For general relativity, Wald's GR states that timelike curves, with the norm $g_{ab}T^{a}T^{b} < 0$, can be parameterized by the "proper time" $$\tau = \int (-g_{ab}T^{a}T^{b})^{1/2} dt.$$ This ...
6
votes
2answers
879 views

Infinitesimal Lorentz transformation is antisymmetric

The Minkowski metric transforms under Lorentz transformations as \begin{align*}\eta_{\rho\sigma} = \eta_{\mu\nu}\Lambda^\mu_{\ \ \ \rho} \Lambda^\nu_{\ \ \ \sigma} \end{align*} I want to show that ...
4
votes
4answers
190 views

What makes a coordinate curved?

Bear with me while I try to explain exactly what the question is. The question Can a curvature in time (and not space) cause acceleration? is imagining a coordinate system in which the curvature is ...
4
votes
1answer
138 views

Some hints for special case of metric tensor in GR

Let's have metric $$ ds^2 = dt^2 - dx^2 - dy^2 - dz^2 - 2f(t - z, x, y)(dt - dz)^2. $$ I need to prove that it is an exact solution for Einstein equations in vacuum for $\partial_{x}^{2}f + ...
3
votes
1answer
153 views

Index raising and lowering - how does it work?

In the context of four-dimensional spacetime, how does the metric turn a tangent vector into a gradient, and vice versa? By this I mean that I know the metric can be used to raise and lower indices: ...
1
vote
1answer
285 views

Finding the Basis vectors of a Killing field vector space

I have solved the Killing vector equations for a 2-sphere and got the following answer. $A,B,C$ are three integration constants as expected. $$\xi_{\theta}=A \sin{\phi}+B\cos{\phi}$$ ...
0
votes
3answers
149 views

Calculating the Riemann tensor for a 3-Sphere

I have worked out all the connection symbols for the 3-sphere using calculus of variations, cf. this Phys.SE post. So to find the Riemann tensor I am trying to find all the nonzero components of: ...
0
votes
5answers
210 views

How to determine “timelike”-ness without using a coordinate system?

It has been stated here that: we can say, without introducing a coordinate system, that the interval associated with two events is timelike, lightlike, or spacelike. This assertion appears at ...
3
votes
1answer
292 views

Can the overall sign of the Minkowski metric be changed?

If we take the Minkowski metric, $\eta_{\mu\nu}=(1,-1,-1,-1)$, instead of the usual $(-1,1,1,1)$, does this change the form of the Lorentz Transform? I think the standard Lorentz Transform looks like: ...
2
votes
1answer
465 views

Why vary the action with respect to the inverse metric?

Whenever I have read texts which employ actions that contain metric tensors, such as the Nambu-Goto, Polyakov or Einstein-Hilbert action, the equations of motion are derived by varying with respect to ...
2
votes
2answers
454 views

Ricci tensor for a 3-sphere without Math packets

Let's have the metric for a 3-sphere: $$ dl^{2} = R^{2}\left(d\psi ^{2} + sin^{2}(\psi )(d \theta ^{2} + sin^{2}(\theta ) d \varphi^{2})\right). $$ I tried to calculate Riemann or Ricci tensor's ...
1
vote
1answer
60 views

Demostrating possible equivalence of two tensors

Is there anyway to see by inspection that a form like $$a(x^2 )^{-3} (g _{μσ} x_{\rho} x_{ ν} + g_{μρ} x_{σ} x_{ ν} +g_{νσ} x_{ρ} x_{ μ} + g_{ νρ} x_{ σ} x_{ μ} ) $$ may be equivalent to (i.e ...
1
vote
0answers
573 views

How to calculate Riemann and Ricci tensors for a sphere? [closed]

Let's have the metric for a sphere: $$ dl^{2} = R^{2}\left(d\psi ^{2} + sin^{2}(\psi )(d \theta ^{2} + sin^{2}(\theta ) d \varphi^{2})\right). $$ I tried to calculate Riemann or Ricci tensor's ...
1
vote
2answers
193 views

How to find a curvature of the space-time by having $g^{\alpha \beta}$ in the following case without cumbersome calculations?

The metric tensor for Fock-Lorentz space-time, $$ \mathbf r_{||}{'} = \frac{\gamma (u)(\mathbf r_{||} - \mathbf u t)}{\lambda \gamma (u) (\mathbf u \cdot \mathbf r) + \lambda c^{2} (1 - \gamma (u))t + ...
0
votes
1answer
145 views

Geodesic deviation on a unit sphere

Very little interest in the original version of this question so I've rejigged it hoping for a more positive response. I'm trying to use the geodesic deviation ...
0
votes
2answers
86 views

Changing the scalar curvature (k = 0,+1,-1) with coordinate transformations?

I would like to prove that I can (or can't) change curvature of space, k = 0,+1,-1, via general coordinate transformations, which in principle can mix space and time coordinates together.