Tagged Questions
1
vote
1answer
62 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, ...
3
votes
1answer
66 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 ...
0
votes
1answer
48 views
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):
$$
\Delta A_{k} ...
3
votes
1answer
77 views
The most general form of the metric for a homogeneous, isotropic and static space-time
What is the most general form of the metric for a homogeneous, isotropic and static space-time?
For the first 2 criteria, the Robertson-Walker metric springs to mind. (I shall adopt the (-+++) ...
1
vote
2answers
94 views
Ricci tensor for a 3-sphere without Math packets
Let's have the metric for a 3-sphere:
$$
dl^{2} = R^{2}\left(d\psi ^{2} + sin^{2}(\psi )(d \theta ^{2} + sin^{2}(\theta ) d \varphi^{2})\right).
$$
I tried to calculate Riemann or Ricci tensor's ...
0
votes
1answer
41 views
Contraction of the metric tensor
This is perhaps a simple tensor calculus problem -- but I just can't see why...
I have notes (in GR) that contains a proof of the statement
In space of constant sectional curvature, $K$ is ...
1
vote
3answers
68 views
Combining metric tensors/curvature tensors
I was thinking about the following scenario:
Consider a particle which causes a metric $g_{\mu\nu}$ on an otherwise Minkowski spacetime (or any manifold). Now, consider another particle, somewhere in ...
0
votes
2answers
128 views
What is metric of spherical coordinates $(t,r,\theta,\phi)$?
In spherical coordinates the flat space-time metric takes:
$$ds^2=-c^2dt^2+dr^2+r^2d\Omega^2$$
where $r^2d\Omega^2$ come from when the signature of metric $g_{\mu\nu}$ is (-,+,+,+)?
what is ...
2
votes
1answer
176 views
Polyakov action: difference induced metric and dynamical metric
The Polyakov action is given by:
$$
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 ...
0
votes
1answer
52 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...
2
votes
1answer
198 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 ...
1
vote
2answers
157 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 ...
4
votes
2answers
180 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:
$$\tilde{g}_{AB}=\begin{pmatrix}
...
3
votes
2answers
125 views
Metric coefficients in rotating coordinates
Let $(t,x,y,z)$ be the standard coordinates on $\mathbb{R}^4$ and consider the Minkowski metric
$$ds^2 = -dt^2+dx^2+dy^2+dz^2.$$
I am trying to compute the metric coefficients under the change of ...
1
vote
1answer
110 views
Covariant derivative with upper index
I just need clarification, that is, to see that I'm doing the right thing.
When calculating central charge for certain metric, I need to solve an integral that contains Lie brackets etc. And I have ...
7
votes
3answers
622 views
Why is the covariant derivative of the metric tensor zero?
I've consulted several books for the explanation of why
$$\nabla _{\mu}g_{\alpha \beta} = 0,$$
and hence derive the relation between metric tensor and affine connection $\Gamma ^{\sigma}_{\mu ...
2
votes
1answer
126 views
Cosmology with a negative cosmological constant
Based on the Friedmann equation for a universe with only cosmological constant,
$$\left(\frac{\dot{a}}{a}\right)^2 \sim \Lambda$$
I would expect the scale factor $a(t) \sim e^{-it}$ if $\Lambda < ...
2
votes
1answer
92 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 ...
