All Questions
Tagged with conventions general-relativity
86
questions
0
votes
1
answer
146
views
What's the difference? $\nabla_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~\text{ and }~\partial_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~?$
What's the difference? $$\nabla_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~\text{ and }~\partial_\mu e_\nu=\Gamma_{\mu \nu}^\rho e_\rho~?$$
In John Dirk Walecka's book 'Introduction to General Relativity',...
- 51
0
votes
1
answer
66
views
Name of metric used by Friedmann
In his original paper, Friedmann used the following dynamic and symmetrical metric:
$$\mathrm{d}s^2=a(t)^2\left(\mathrm{d}\chi^2+\sin (\chi)^2\left(\mathrm{d}\theta^2+\sin (\theta)^2 \mathrm{d}\phi^2\...
- 181
1
vote
1
answer
69
views
Sign convention for the Lagrangian of a free massive point particle in general relativity
As far as I understand, the Lagrangian of a massive free particle in the context of general relativity is the following:
$$L=-mc\sqrt{g_{\mu\nu}\dfrac{dx^\mu}{dt}\dfrac{dx^\nu}{dt}}.$$
But is this the ...
- 293
0
votes
0
answers
57
views
The energy-momentum tensor of a scalar field
In Carroll's Introduction to General Relativity: Spacetime and Geometry, I am going to express the components of the energy-momentum tensor of scalar field in three-
vector notation, using the ...
- 157
0
votes
0
answers
56
views
Embedding diagram in west coast signature
Assume you have a metric in (+,-,-,-) signature,
$$d s^2=e^{2 \Phi(r)} d t^2-\frac{d r^2}{1-\frac{b(r)}{r}}-r^2 d \Omega^2.$$
To embed it, we take $t=$Constant, $\theta=\pi/2$ slice,
$$d s^2=-\frac{d ...
- 546
1
vote
0
answers
207
views
Christoffel symbol with third index up
Generally the Christoffel symbol of the first kind is defined as
$$\Gamma_{\lambda\mu\nu}=\frac12\,(\partial_\nu g_{\lambda\mu}+\partial_\mu g_{\lambda\nu}-\partial_\lambda g_{\mu\nu}) \tag{1}$$
and ...
- 392
1
vote
1
answer
194
views
The importance of metric signature in Ricci scalar
I have read this question Different signatures of the metric in Einstein field equations (and related posts) on the invariance of Einstein field equations under metric signature change.
However, there ...
- 169
2
votes
0
answers
108
views
Convention of notation $(p,q)$ for the metric signature, which number is first?
This is a really simple question that I fail to quickly find and answer for.
The metric signature for Minkowski space or more general Pseudo-Riemannian manifolds is usually denoted by a list of signs ...
- 96
9
votes
4
answers
1k
views
Is the four-velocity always normalized?
In the book i have reading defines the four-velocity like $$U^\mu=dx^\mu/d\tau.$$ The metric used is $\eta_{\mu\nu}=diag(-1,1,1,1)$. It is straightforward to show that the norm of the four-velocity is ...
- 171
1
vote
1
answer
110
views
Position of an index when raising and lowering indices
I'm reading Carroll's book on GR, page 25, and have a question about raising and lowering indices in 1.72:
For the first equation, why do we have (sorry I don't know how to leave spaces for the lower ...
- 1,783
0
votes
1
answer
126
views
What is the symbol to differentiate between 3D and 4D tensors?
I am writing a computer program and in there I need to differentiate 3D tensors (metric tensor, Riemann tensor, Ricci scalar, Christoffel Symbols, etc.) from 4D ones.
I wanted to write something like $...
- 3,122
1
vote
0
answers
79
views
Clarification of Notation in MTW's Gravitation, Section 6.4
Three difficulties concerning notation in MTW's Gravitation Section 6.4, which introduces a coordinate system for an accelerated frame:
the time axis e0' is the observer's 4-velocity, so he is always ...
0
votes
1
answer
82
views
Elementary question about Einstein notation
I have encountered, in a physics textbook, the following Lagrangian:
$$L=\dfrac{m}{2}g_{ij}(x^k)\dot{x}^i\dot{x}^j.$$
I understand that Einstein notation is being used, and therefore there is an ...
- 293
1
vote
1
answer
124
views
Confusion about convention for curvature tensor
I am a little bit confused about the convention of the curvature tensor. The books of Wald and Misner/Deser/Wheeler seem to have the same conventions, i.e. the indices of the Riemann curvature tensor ...
- 854
0
votes
0
answers
75
views
Question about placing of indices in tetrads/vierbein
I'm going to use boldface to denote the metric tensor as a geometric rank 2 tensor and I'll expand it in different basis. First, let's define the coordinate vector and 1-form basis
\begin{equation}
\...
- 4,827