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.
12
votes
3answers
115 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 ...
8
votes
2answers
268 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 ...
7
votes
3answers
712 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
1answer
124 views
Uniqueness of Riemann curvature tensor
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 ...
6
votes
2answers
210 views
Einstein tensor in Friedmann equations : where is the missing $c^2$?
I would like to demonstrate the several forms of the Friedmann equations WITH the $c^2$ factors. Everything is fine ... apart that I have a missing $c^2$ factor somewhere.
In all the following $\rho$ ...
6
votes
2answers
146 views
What's the basic premise of General Relativity?
What is the basic assumption(s) required to explore general relativity?
For example, if one merely assumes that the speed of light $c$ is the same for all observers, and the laws of physics are the ...
5
votes
0answers
169 views
Penrose Conformal diagram for flat 2-dim Lorentz space-time
I have the following metric
$$ds^2 ~=~ Tdv^2 + 2dTdv,$$
defined for
$$(v,T)~\in~ S^1\times \mathbb{R},$$
e.g. $v$ is periodic.
This is the according Penrose diagram:
Question 1) Is the ...
4
votes
2answers
117 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. ...
4
votes
3answers
144 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 - ...
4
votes
2answers
193 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:
$$\tilde{g}_{AB}=\begin{pmatrix}
...
3
votes
1answer
196 views
Can the overall sign of the Minkowski metric be changed?
If we take the Minkowski metric, $\eta_{\mu\nu}=(1,-1,-1,-1)$, instead of the usual $(-1,1,1,1)$, does this change the form of the Lorentz Transform? I think the standard Lorentz Transform looks like: ...
3
votes
1answer
94 views
Would this be a metric?
would a matrix $M$ with diagonal entries not necessarily equal 1, i.e. diag $M = (a,1,1,1)$ be a metric if $a \neq 1$ or $\neq 0$? I.e. in this case would this be like some sort of more general ...
3
votes
3answers
125 views
How scalar curvature of following spacetime can be equal to zero?
For an interval of this spacetime,
$$
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})),
$$
scalar curvature is equal to zero. Also, Ricci ...
3
votes
2answers
131 views
Metric coefficients in rotating coordinates
Let $(t,x,y,z)$ be the standard coordinates on $\mathbb{R}^4$ and consider the Minkowski metric
$$ds^2 = -dt^2+dx^2+dy^2+dz^2.$$
I am trying to compute the metric coefficients under the change of ...
3
votes
1answer
100 views
When a variation of a tensor is not a tensor?
In a comment about variation of metric tensor it was shown that
$$\delta g_{\mu\nu}=-g_{\mu\rho}g_{\nu\,\sigma}\delta g^{\rho\,\sigma}$$
which is contrary to the usual rule of lowering indeces of a ...
3
votes
3answers
129 views
Relation between the determinants of metric tensors
Recently I have started to study the classical theory of gravity. In Landau, Classical Theory of Field, paragraph 84 ("Distances and time intervals") , it is written
We also state that the ...
3
votes
1answer
82 views
Derivation of the volume element (which uses the metric tensor)?
I have often seen $\sqrt{-g}$ in integrals, especially actions, where $g=\mathrm{det}(g_{\mu \nu})$. Does anyone know of a derivation that shows that this is indeed the volume element which must be ...
3
votes
1answer
92 views
Intervals as infinitesimals of same order (Landau & Lifshitz)
I don't understand the following statement in Landau & Lifshitz, Classical Theory of Fields, p.5:
$ds$ and $ds'$ are infinitesimals of same order. [...] It follows that $ds^2$ and $ds'^2$ must ...
3
votes
1answer
91 views
The most general form of the metric for a homogeneous, isotropic and static space-time
What is the most general form of the metric for a homogeneous, isotropic and static space-time?
For the first 2 criteria, the Robertson-Walker metric springs to mind. (I shall adopt the (-+++) ...
3
votes
1answer
123 views
Material strain from spacetime curvature
Let's say that you moved an object made of rigid materials into a place with extreme tidal forces. Materials have a modulus of elasticity and a yield strength. Does the corresponding 3D geometric ...
2
votes
2answers
90 views
Sign convention for basic Dirac equation
The dirac equation;$$(i\gamma^\mu\partial_{\mu} - m)\psi=0 $$ is just; $$(i\gamma^{0}\partial_{0} - i\gamma^{i}\partial_{i} - m)\psi=0 $$ in a (+,---) metric right?
2
votes
1answer
96 views
Question from Schutz's
In q. 22 in page 141, I am asked to show that if $U^{\alpha}\nabla_{\alpha} V^{\beta} = W^{\beta}$, then $U^{\alpha}\nabla_{\alpha}V_{\beta}=W_{\beta}$.
Here's what I have done:
$V_{\beta}=g_{\beta ...
2
votes
3answers
128 views
Clarifying what metric counts as flat space
In (2D) Cartesian coordinates, the Euclidean metric...
$$\begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix}$$
...is flat space. If the diagonal elements are exchanged for other real numbers ...
2
votes
1answer
195 views
Polyakov action: difference induced metric and dynamical metric
The Polyakov action is given by:
$$
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 ...
2
votes
1answer
209 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 ...
2
votes
1answer
133 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
44 views
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} ...
2
votes
2answers
187 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 ...
2
votes
1answer
45 views
General expression of the redshift: explanation?
In some papers, authors put the following formula for the cosmological redshift $z$ :
$1+z=\frac{\left(g_{\mu\nu}k^{\mu}u^{\nu}\right)_{S}}{\left(g_{\mu\nu}k^{\mu}u^{\nu}\right)_{O}}$
where :
$S$ ...
2
votes
1answer
77 views
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 ...
2
votes
2answers
113 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
67 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 ...
1
vote
3answers
81 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 ...
1
vote
2answers
86 views
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 ...
1
vote
2answers
55 views
Kronecker delta confusion
I'm confused about the Kronecker delta. In the context of four-dimensional spacetime, multiplying the metric tensor by its inverse, I've seen (where the upstairs and downstairs indices are the same):
...
1
vote
2answers
139 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 + ...
1
vote
2answers
74 views
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?
1
vote
2answers
193 views
Visualization of de Sitter spaces
When I was reading about de Sitter it said that the we looked in 5-dimensional Minkowski metric and that we had immersed in it a hyperboloid, and with some coordinate transformation we get some metric ...
1
vote
2answers
120 views
Most suitable metric for the Solar system?
If I wanted to solve the Einstein equations for the solar system, which choice of $g_{\mu\nu}$ and $T_{\mu\nu}$ is more suitable?
I thought about using a Schwarzschild metric near each planet, but ...
1
vote
1answer
51 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 ...
1
vote
2answers
74 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 ...
1
vote
1answer
113 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 ...
1
vote
1answer
33 views
Vector analysis in curvilinear coordinates using the metric tensor
In Weinberg's Gravitation,
the formula for the volume element in curviliniar coordinates is given by
$$dV=h_1 h_2 h_3 dx^1 dx^2 dx^3.$$
The metric is given by $ds^2=h_1^2 dx_1^2+h_2^2 ...
1
vote
1answer
68 views
Evaluating the Ricci tensor effectively
If given a metric of the form $$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}$$ where $R_{ab}$ is the Ricci tensor, ...
1
vote
1answer
79 views
Spacelike slicing of Schwarzschild geometry
I am having trouble understanding how to obtain a spacelike slicing of the Schwarchild black hole. I understand there is not a globally well defined timelike killing vector, so we can define t=cte ...
1
vote
2answers
113 views
What should I call an n>4 dimensional Minkowski metric?
I am manipulating an $nxn$ metric where $n$ is often $> 4$, depending on the model. The $00$ component is always tau*constant, as in the Minkowski metric, but the signs on all components might be ...
1
vote
1answer
58 views
Finding the Basis vectors of a Killing field vector space
I have solved the Killing vector equations for a 2-sphere and got the following answer. $A,B,C$ are three integration constants as expected.
$$\xi_{\theta}=A \sin{\phi}+B\cos{\phi}$$
...
1
vote
0answers
38 views
Null vector fields given Bondi metric
I'm trying to understand how to compute the null future-directed vector fields if I have a given (Bondi) metric
$g=-e^{2\nu}du^{2}-2e^{\nu+\lambda}dudr+r^{2}d\Omega$
with $d\Omega$-standard metric ...
1
vote
0answers
50 views
Singularities in Schwarzchild space-time
Can anyone explain when a co-ordinate and geometric singularity arise in Schwarzschild space-time with the element
$$ ...
1
vote
0answers
130 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 ...
