# Tagged Questions

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.

1answer
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### Would this be a metric?

would a matrix $M$ with diagonal entries not necessarily equal 1, i.e. diag $M = (a,1,1,1)$ be a metric if $a \neq 1$ or $\neq 0$? I.e. in this case would this be like some sort of more general ...
1answer
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### 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 (-+++) ...
1answer
59 views

### Evaluating the Ricci tensor effectively

If given a metric of the form $$ds^2=\alpha^2(dr^2+r^2d\theta^2)$$ where $\alpha=\alpha(r)$, then can one immediately conclude that $$R_{\theta\theta}=r^2R_{rr}$$ where $R_{ab}$ is the Ricci tensor, ...
0answers
33 views

### metric extension outside the light cone

Could anyone explain what "extending the solution" beyond the past light cone means? Say, for example, if I have a metric (no coordinate singularities), how can I extend it to the outside of the past ...
1answer
85 views

### When a variation of a tensor is not a tensor?

In a comment about variation of metric tensor it was shown that $$\delta g_{\mu\nu}=-g_{\mu\rho}g_{\nu\,\sigma}\delta g^{\rho\,\sigma}$$ which is contrary to the usual rule of lowering indeces of a ...
1answer
32 views

### What is the Lorentz tensor with a superscript and subscript index?

I have been reading about symmetries of systems' actions, e.g. the Polyakov action, and I have encountered Lorentz transformations of the form: $\Lambda^{\mu}_{\nu} X^{\nu}$. I am moderately familiar ...
1answer
110 views

### Christoffel symbol for Schwarzschild metric

I know that the christoffel (second kind) can be defined like this: \Gamma^m_{ij} = \frac{1}{2} g^{mk}(\frac{\partial g_{ki}}{\partial U^j}+\frac{\partial g_{jk}}{\partial U^i}-\frac{\partial ...
1answer
64 views

### Derivation of the volume element (which uses the metric tensor)?

I have often seen $\sqrt{-g}$ in integrals, especially actions, where $g=\mathrm{det}(g_{\mu \nu})$. Does anyone know of a derivation that shows that this is indeed the volume element which must be ...
1answer
51 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 ...
0answers
35 views

### Null vector fields given Bondi metric

I'm trying to understand how to compute the null future-directed vector fields if I have a given (Bondi) metric $g=-e^{2\nu}du^{2}-2e^{\nu+\lambda}dudr+r^{2}d\Omega$ with $d\Omega$-standard metric ...
1answer
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### Parallel transport of a vector along a closed curve in curvilinear coordinates

There is an expression indicating the change of the vector parallel translation along a closed infinitesimal curve in curvilinear coordinates (one way of introducing curvature tensor):  \Delta A_{k} ...
1answer
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### Plane waves in QFT

Suppose we work in the metric $(-1,+1)$. How do we describe an incoming particle with a plane wave; $\exp(-\mathrm ikx)$ or $\exp(+\mathrm ikx)$? What's the difference? Does it change if we work in ...
2answers
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### 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 ...
0answers
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### Singularities in Schwarzchild space-time

Can anyone explain when a co-ordinate and geometric singularity arise in Schwarzschild space-time with the element  ...
0answers
98 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 ...
0answers
25 views

### How to prove the derive the expression for space part of Riemann tensor for homogeneous and isotropic space-time?

It's not a homework!! For spheric, hyperbolic and flat case $$dl^{2} = R^{2}\left(d \psi^{2} + sin^{2}(\psi )(d \theta^{2} + sin^{2}(\theta )d \varphi^{2})\right),$$  dl^{2} = R^{2}\left(d ...
1answer
38 views

### Contraction of the metric tensor

This is perhaps a simple tensor calculus problem -- but I just can't see why... I have notes (in GR) that contains a proof of the statement In space of constant sectional curvature, $K$ is ...
0answers
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### The interior of a cylinder as an Einstein manifold

The interior of a curved cylinder is an Einstein manifold (the Ricci Curvature Tensor is proportional to the Metric $R_{\mu\nu}=kg_{\mu\nu}$) since it has a constant curvature. However, I was unable ...
3answers
585 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 ...
3answers
113 views

### How scalar curvature of following spacetime can be equal to zero?

For an interval of this spacetime, $$ds^{2} = c^{2}dt^{2} - c^{2}t^{2}(d \psi^{2} + sh^{2}(\psi )(d \theta^{2} + sin^{2}(\theta )d \varphi^{2})),$$ scalar curvature is equal to zero. Also, Ricci ...
1answer
45 views

### Change of variables in an interval expression

This question is a continuation of How to calculate a scalar curvature fast? . Let's have Lorentz-Fock spacetime with an interval  d \hat {s}^{2} = \frac{t_{0}^{2}R^{2}}{\hat {t}^{4}}\left( d \hat ...
3answers
63 views

### Combining metric tensors/curvature tensors

I was thinking about the following scenario: Consider a particle which causes a metric $g_{\mu\nu}$ on an otherwise Minkowski spacetime (or any manifold). Now, consider another particle, somewhere in ...
0answers
83 views

### How to calculate a scalar curvature fast? [closed]

Let's have a metric tensor  g^{\alpha \beta} = \frac{1}{\left( 1 + \frac{ct}{R} \right)^{2}}\begin{bmatrix} \frac{1 - \frac{r^{2}}{R^{2}}}{\left(1 + \frac{ct}{R}\right)^{2}} & ...
2answers
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### What is metric of spherical coordinates $(t,r,\theta,\phi)$?

In spherical coordinates the flat space-time metric takes: $$ds^2=-c^2dt^2+dr^2+r^2d\Omega^2$$ where $r^2d\Omega^2$ come from when the signature of metric $g_{\mu\nu}$ is (-,+,+,+)? what is ...
2answers
149 views

### Null geodesic given metric

I (desperately) need help with the following: What is the null geodesic for the space time $$ds^2=-x^2 dt^2 +dx^2?$$ I don't know how to transform a metric into a geodesic...! There is no need to ...
3answers
104 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 ...
0answers
46 views

### When is spacetime homogenous and isotropic?

When is spacetime homogenous and isotropic? For example, some metric $g_{\mu \nu}$ is homogeneous and isotropic. We now construct effective metric n_{\mu \nu} ~\rightarrow~ g_{\mu \nu} + ...
2answers
56 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.
2answers
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### Coordinate and conformal transformations of the FRW metric

I'm considering a metric of the following form (signature $(+,-,-,-)$): $$ds^2 = (F(r,t)-G(r,t))dt^2 - (F(r,t)+G(r,t))dr^2 - r^2(d\Omega)^2$$ where $F(r,t)$ and $G(r,t)$ are arbitrary scalar ...
3answers
138 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 - ...
0answers
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### Switching from an accelerated frame of reference to a locally inertial reference system

Using the equivalence principle, show that the interval for an accelerated observer ($\textbf{g}$ uniform and constant) has the form  ds^2|_{\text{first order in ...
2answers
106 views

### Difference between slanted indices on a tensor

In my class, there is no distinction made between, $$C_{ab}{}^{b}$$ and $$C^{b}{}_{ab}.$$ All I know, and read about so far, is the distinction of covariant and contravariant, form/vector, etc. ...
2answers
136 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 + ...
2answers
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### metric tensor of expanding universe

Why is the metric tensor of a expanding universe a function of time? Why is it not a function of distance between the galaxies? I heard this from a lecture. Can anyone help me understand?
1answer
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### Non-diagonal elements when switching metric signature?

Considering a metric tensor with the signature $(-,+,+,+)$: \$g_{\mu\nu}= \begin{pmatrix} -c^2 & g_{01} & g_{02} & g_{03}\\ g_{10} & a^2 & g_{12} & g_{13}\\ g_{20} & g_{21} ...
2answers
87 views

### What is the link between the metric signature of spacetime and fundamental field equations?

The signature of Minkowski spacetime is 2, as is explained here: metric signature explanation. The signature is related to the form the fundamental equations take, but I'm not totally clear on the ...
2answers
85 views

### Sign convention for basic Dirac equation

The dirac equation;$$(i\gamma^\mu\partial_{\mu} - m)\psi=0$$ is just; $$(i\gamma^{0}\partial_{0} - i\gamma^{i}\partial_{i} - m)\psi=0$$ in a (+,---) metric right?
1answer
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### Spacelike slicing of Schwarzschild geometry

I am having trouble understanding how to obtain a spacelike slicing of the Schwarchild black hole. I understand there is not a globally well defined timelike killing vector, so we can define t=cte ...
3answers
114 views

### Relation between the determinants of metric tensors

Recently I have started to study the classical theory of gravity. In Landau, Classical Theory of Field, paragraph 84 ("Distances and time intervals") , it is written We also state that the ...
2answers
71 views

### Why can certain functions be absorbed into the Schwarzschild metric, while others can't?

Another question about the Schwarzschild solution of General Relativity: In the derivation (shown below) of the Schwarzschild metric from the vacuum Einstein Equation, at the step marked "HERE," we ...
1answer
170 views

### Polyakov action: difference induced metric and dynamical metric

The Polyakov action is given by:  S_p ~=~ -\frac{T}{2}\int d^2\sigma \sqrt{-g}g^{\alpha\beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X^{\nu}\eta_{\mu\nu} ~=~ -\frac{T}{2}\int d^2\sigma ...
1answer
117 views

### Material strain from spacetime curvature

Let's say that you moved an object made of rigid materials into a place with extreme tidal forces. Materials have a modulus of elasticity and a yield strength. Does the corresponding 3D geometric ...
1answer
51 views

### Constraint on a metric

Given a metric of the form $$ds^2=dr^2+a^2\tanh^2(r/b)d\theta^2$$ why does it follow that $a=b$? I can't quite spot a constraint condition...
1answer
187 views

### Covariant derivative

I would very much appreciate some help in The following: What is 2nd order covariant derivative $$\nabla_i\nabla_jf(r)$$ in terms of $r,\theta, g(r)$ and partial derivative, given that the metric ...
2answers
245 views

### Question about proper time in general relativity

I think I may have some fundamental misunderstanding about what $dt, dx$ are in general relativity. As I understand it, in special relativity, $ds^2=dt^2-dx^2$, we call this the length because it is ...
2answers
171 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} ...
1answer
113 views

### Lorentz transformation problem

In the equation (1.18) they omitted the translation vector $a^\mu$, but why?
2answers
125 views

### Metric coefficients in rotating coordinates

Let $(t,x,y,z)$ be the standard coordinates on $\mathbb{R}^4$ and consider the Minkowski metric $$ds^2 = -dt^2+dx^2+dy^2+dz^2.$$ I am trying to compute the metric coefficients under the change of ...
1answer
87 views

### Question from Schutz's

In q. 22 in page 141, I am asked to show that if $U^{\alpha}\nabla_{\alpha} V^{\beta} = W^{\beta}$, then $U^{\alpha}\nabla_{\alpha}V_{\beta}=W_{\beta}$. Here's what I have done: \$V_{\beta}=g_{\beta ...
1answer
104 views

### Covariant derivative with upper index

I just need clarification, that is, to see that I'm doing the right thing. When calculating central charge for certain metric, I need to solve an integral that contains Lie brackets etc. And I have ...
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