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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1answer
90 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 ...
6
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1answer
62 views

Why does the $L_2$ norm give the shortest path between 2 points?

Why not the $L_1$ or $L_3$ distances? Is there some deep reason why the universe (at least at human scales) looks pretty much Euclidean? Could we imagine a different universe where a different ...
2
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3answers
476 views

D'Alembertian for a scalar field

I have read that the D'Alembertian for a scalar field is $$ \Box = g^{\nu\mu}\nabla_\nu\nabla_\mu = \frac{1}{\sqrt{-g}}\partial_\mu (\sqrt{-g}\partial^\mu). $$ Exactly when is this correct? Only for ...
2
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1answer
85 views

Can these two terms cancel out?

In trying to prove that $$\Gamma_{\mu\nu\lambda} = \eta_{ab}J^a_bJ^b_{\nu\lambda}.$$ The author canceled out while expanding the first equation $$J^a_{\mu\lambda}J^b_\nu$$ with ...
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1answer
57 views

Timelike curves in Special Relativity

I have a question that probably might sound silly to most of you. We know that a natural Lorentz-invariant parametrization of a timelike curve is provided by: $$\tau$$ the Lorentz-invariant proper ...
4
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2answers
96 views

Why is the phase space a symplectic manifold rather than a manifold with a metric?

Why does phase space require a symplectic geometry rather than a metric? Is there some scenario where a metric is unable to describe the notion of length in phase space, specifically in relation to ...
7
votes
1answer
210 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 ...
2
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1answer
65 views

Is it okay to Wick rotate to give the negative of the Euclidean metric? Also, could we make the space-like coordinates imaginary instead?

There are 2 parts to my question: 1) Say we choose the metric signature to be (-+++), as in the Wikipedia page. Then the invariant interval in Minkowski space is written: $ds^{2} = -(dt^{2}) + ...
1
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0answers
64 views

Help with parametrization of surface if I'm given the metric [closed]

I've got a homework question. Consider a 2 dimensional space with metric $$ ds^{2} = \frac{dr^{2}}{1 -\frac{2}{r} } + r^{2}d\theta^{2} .$$ I need to show that this is the induced metric ...
4
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2answers
117 views

Getting the Lagrangian from the action in curved spacetime

Suppose I have this action: $$ S = \int \mathrm d^4 x\sqrt{-g}\times \text{something}$$ where $g$ is the determinant of the metric. Should I take the Lagrangian to be: $$ \mathcal L = \sqrt{-g} ...
3
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0answers
88 views

Question about derivation of tensor in Di Francesco's CFT

This is a question for anyone who is familiar with Di Francesco's book on Conformal Field theory. In particular, on P.108 when he is deriving the general form of the 2-point Schwinger function in two ...
1
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1answer
57 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
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0answers
28 views

Supergravity solution, metric for the total space, and connection

In supergravity solutions, one sometimes encounters the case where the manifold may be a bundle over some base space, and one has to write down the explicit metric regarding such bundle. I would like ...
1
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1answer
92 views

Geodesic curvature and Weyl transformations

The geodesic curvature is given by $$k=\pm t^a n_b\nabla_a t^b,$$ where $t^a$ is a unit vector tangent to the boundary of the string worldsheet and $n_a$ is an outward vector orthogonal to $t^a$. I ...
0
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1answer
44 views

Why does product of Moduli and Diff x Weyl Variation vanish?

According to equation 5.2.5 in Polchinski :- $$\int d^2 \sigma~ \delta^{'}g_{ab} \times [-2(P_1 \delta \sigma)_{ab} +(2\delta w - \Delta \cdot \delta \sigma)g^{ab}]=0$$ The assumption here is that " ...
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6answers
838 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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0answers
20 views

Using Polyakov-Alvarez Anomaly Formula [closed]

Take $\Sigma=\mathbb{D}$ to be the unit disk with metric $g=\frac{4}{(1+|z|^2)^2}\,|dz|^2$. If $\phi$ is a nice enough function on $\mathbb{D}$, then I want to compute $$\int_{\partial \Sigma} k_g ...
4
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1answer
136 views

What is the Schwarzschild metric with proper radial distance?

Reading the marvellous book "The Membrane Paradigm" I stumbled upon a suggested change of variable that I'm not able to deal with. Starting with the usual Schwarzschild metric for the spatial ...
1
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0answers
66 views

How to test that a flat metric represents a global three-torus geometry

When introducing Robertson-Walker metrics, Carroll's suggests that we consider our spacetime to be $R \times \Sigma$, where $R$ represents the time direction and $\Sigma$ is a maximally symmetric ...
-2
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1answer
62 views

Length in polar coordinates

Say we are in 3 dimensions and use $(-++)$. If we have the metric $$ds^2=-dt^2+dr^2+r^2df^2(t),$$ then what is the third coordinate if the first two were $t$ and $r$? $$X^iX_i=-t^2+r^2+?$$
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0answers
72 views

Minkowski to Euclidean

When dealing with solutions to Einstein's equations given by a 4d metric with signature $(-,+,+,+)$, we're able to move to Euclidean space using some transformation so that our signature is now ...
2
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0answers
45 views

In KK theory, is proper time defined using the 5 dimensional or the 4 dimensional line element?

Let's consider five dimensional KK theory. This is Klein's metric $\hat{g}_{AB}= \begin{pmatrix} g_{00}+A_{0}A_{0}&g_{01}+A_{0}A_{1}&g_{02}+A_{0}A_{2}&g_{03}+A_{0}A_{3}&A_ 0\\ ...
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0answers
51 views

Radial Null Geodesics in Static Maximally Symmetric DeSitter Space

Given a DeSitter-space metric from the line element: $$ ds^2=\left(1-\frac{r^2}{R^2}\right)dt^2-\left(1-\frac{r^2}{R^2}\right)^{-1}dr^2-r^2d\Omega^2 $$ Where $R=\sqrt{\frac{3}{\Lambda}}$, and ...
2
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2answers
323 views

Weight of a tensor density

Is there any freedom in choosing the weight of a tensor density? I have seen in some papers that they introduce a tensor density made from metric with a special weight. There is a tensor density with ...
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 ...
1
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0answers
43 views

Unknown Function in the Tolman-Bondi-de Sitter Metric

I've been working with some dust solutions in General Relativity, practicing calculating the Riemann curvature tensor, and I came across an odd metric: the Tolman-Bondi-de Sitter metric. A quick ...
0
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0answers
41 views

Metric convention conversions between expressions

I'm sure this has caused many people headaches. First, is there a metric $(-+++)\leftrightarrow(+---)$ convention conversion chart where many common expressions are listed? Thank you in advance for ...
1
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1answer
34 views

CFT Entanglement Entropy - relation between translations and the stress-energy tensor

In a recent paper on CFT entanglement entropy, I want to understand the defintion of a certain partition function. They consider a metric space $S^1 \times \mathbb{H}^{d-1}_q$ with metric: $$ ...
0
votes
1answer
62 views

Showing a fourth rank tensor in $\epsilon$'s reduces to one in the metric $g$

Consider the fourth rank tensor $$S_{\mu \nu \rho \sigma} = a(\epsilon_{\mu \sigma}\epsilon_{\nu \rho} + \epsilon_{\mu \rho}\epsilon_{\nu \sigma})f(x^2),$$ in 2D where $a$ is a constant and $f(x^2)$ ...
1
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0answers
60 views

Higher-Dimensional Metrics in (Hyper)-Spherical Coordinates

I want to compute the components of the Riemann curvature tensor (for a case similar to the Schwarzschild solution) in 4 + 1 dimensions, but I want to use a higher-dimensional analogue of spherical ...
0
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1answer
48 views

The commutator of Killing vectors

I'm going over an assignment for my general relativity course. My solution to the question below strikes me as too short, considering that it appeared in the "longer questions" section of the ...
1
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0answers
55 views

Lagrangian for FRW metric

For the metric $$ds^2=-dt^2+a^2(t)(dx^2+dy^2+dz^2),$$ $$L= \sqrt{-g_{\alpha\beta}\frac{dx^\alpha}{dt}\frac{dx^\beta}{dt}}$$ How does this become $$L= \sqrt{1-a^2 (\frac{dx}{dt})^2}~? $$ I guess ...
0
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0answers
42 views

Compactification and off-diagonal terms of the metric tensor

In standard 3+1 dimensional spacetime, the metric tensor is of order 4 and had ten independent coefficients, hence there are 6 terms off the diagonal in the corresponding $4\times 4$ real symmetric ...
4
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2answers
201 views

Metric tensor in special and general relativity

I'm having trouble understanding the metric tensor in general relativity. What I've understood so far has come from my course lecture notes used in conjunction with "The Road to Reality" by Roger ...
2
votes
2answers
212 views

Why “light cones” have different shapes near black holes?

There is theory that light cone shape does not depend on the reference frame in which it is viewed. So why we draw light cones near black hole differently? I thought that if I am observing (from the ...
1
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0answers
44 views

Laplacian in tensor [closed]

Find $\vec \nabla^2\phi $ when $$ds^{2}=-dt^{2}+a^{2}(t)[dx^{2}+dy^{2}+dz^{2}] $$ or $$g_{ij}=\begin{bmatrix} -1 & 0 &0 &0 \\ 0 &a^{2}(t) &0 &0 \\ 0&0 ...
3
votes
1answer
83 views

Ricci flat compact manifold with $U(1)\times{}SU(2)\times{}SU(3)$ isometry group?

As the title says, is it possible to have a Riemannian Ricci flat compact manifold with $U(1)\times{}SU(2)\times{}SU(3) $ isometry group?
2
votes
1answer
209 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 ...
3
votes
1answer
255 views

The most general form of the metric for a homogeneous, isotropic and static space-time

What is the most general form of the metric for a homogeneous, isotropic and static space-time? For the first 2 criteria, the Robertson-Walker metric springs to mind. (I shall adopt the (-+++) ...
8
votes
2answers
429 views

Does Kaluza-Klein Theory Require an Additional Scalar Field?

I've seen the Kaluza-Klein metric presented in two different ways., cf. Refs. 1 and 2. In one, there is a constant as well as an additional scalar field introduced: $$\tilde{g}_{AB}=\begin{pmatrix} ...
2
votes
1answer
104 views

Computing the Christoffel symbols with the geodesic equation

I would like to compute the Christoffel symbols of the second kind using the geodesic equation. To practice, I have tried the Schwarzschild Ansatz $$ g_{00} = \mathrm e^\nu,\quad g_{11} = - \mathrm ...
1
vote
1answer
40 views

Homogeneity and isotropy of stress energy tensor

Given the energy momentum tensor in E&M: $T_{\mu\nu} = -F_{\mu\alpha} g^{\alpha \beta} F_{\beta \nu} +\frac{1}{4} g_{\mu \nu} F_{\sigma \alpha} g^{\alpha \beta} F_{\beta \rho} g^{\rho \sigma}$ I ...
0
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4answers
153 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 ...
1
vote
1answer
73 views

Has a metric formulation of electromagnetism ever been attempted? [duplicate]

I understand that electromagnetic fields carry energy, and this energy curves spacetime gravitationally. That's not my question. I'm asking if anyone has tried to formulate electromagnetism in such ...
1
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1answer
57 views

Interpretation of space time Minkowski diagram [closed]

How to interpret the following space-time diagram in the image. I know how to interpret euclidean distance from Euclidean space diagram omit the line "whereas for Euclidean space".
1
vote
1answer
71 views

Tensors in special relativity [duplicate]

I'm trying to understand tensors, but I've come across the following question: Let $T^{\mu\nu}$ by a $(2,0)$ tensor. Give the definitions of $T_\mu^{\,\nu}$, $T_{\mu\nu}$, and ...
4
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4answers
180 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 ...
3
votes
0answers
37 views

Sign convention with the $AdS$ metric

One would say that $AdS_n$ satisfies the equations for the scalar curvature (R) and Ricci tensor ($R_{\mu \nu}$), $R = - \frac{n(n-1)}{L^2}$ and $R_{ab} = - \frac{n-1}{L^2}g_{ab}$. But do the signs ...
3
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0answers
50 views

Why is the Taub-NUT instanton singular at $\theta=\pi$?

Consider the following metric $$ds^2=V(dx+4m(1-\cos\theta)d\phi)^2+\frac{1}{V}(dr+r^2d\theta^2+r^2\sin^2\theta{}d\phi^2),$$ where $$V=1+\frac{4m}{r}.$$ That is the Taub-NUT instanton. I have been ...
1
vote
3answers
126 views

Why do we need a metric to define gradient?

For me, the gradient of a scalar field (say, in three dimensions) is simply (formally) $\nabla f = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y},\frac{\partial f}{\partial z} ...