All Questions
Tagged with conventions differential-geometry
38
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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
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207
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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
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108
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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
1
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580
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Scalar curvature of a 2-sphere via the Ricci tensor
Using the usual coordinates on a 2-sphere of radius $r$, I get the metric tensor $g_{\mu\nu}=\text{diag}(r^2, r^2\sin^2\theta)$ and so $g^{\mu\nu}=\text{diag}(1/r^2,1/r^2\sin^2\theta)$.
Hence the only ...
- 366
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124
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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
3
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120
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The inhomogeneous Maxwell equation ${*}\mathrm d{*}F=J^\flat$ is only true for the signature $(+---)$?
In short, we would expect that considering the Riemannian manifold $(M,g)$ and the Riemannian manifold $(M,-g)$ leads to the same Maxwell equations, i.e. we would expect that
$\newcommand{\imult}{\...
- 1,801
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75
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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
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354
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Different definitions of Riemann tensor components
Hobson (General Relativity 2006 ed. page 158 eq. 7.13) and Ryder (Introduction to General Relativity 2009 ed. page 124 eq. 4.31) define Riemann tensor components as
$$ {R^\alpha}_{\beta\gamma\delta} = ...
- 882
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205
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Spin connection raise and lower flat indices
The spin connection $\omega^a_{b\nu}$ is used to define the covariant derivative of a spinor in curved spacetime. I want to explicitly calculate the covariant derivative:
$$\nabla_\nu\Psi=(\partial_\...
- 505
2
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246
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Alternative expression for Riemann curvature tensor
There is the usual expression for the Riemann tensor
$$R_{abcd}=\partial_c\Gamma_{adb}-\partial_d\Gamma_{acb}+\Gamma_{ace}{\Gamma^e}_{db}-\Gamma_{ade}{\Gamma^e}_{cb}.$$
However, in the last page of ...
- 3,915
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378
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Are differential geometric and physics conventions for covariant derivatives consistent?
In a differential geometric setting, the covariant derivative can be defined as a map $\nabla_X:\Gamma(TM)\to\Gamma(TM)$, for any vector field $X\in\Gamma(TM)$, satisfying certain conditions. In other ...
- 14.8k
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248
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Annoying sign in geometric interpretation of curvature
In this video Susskind gives a heuristic derivation of the curvature formula which I summarize as follows:
In a coordinate system, start with a vector $v_A$ at the origin, and extend it to a parallel ...
- 592
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149
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Curvature 2-forms: missing factor 1/2 in (14.21) and (14.22) of Misner, Thorne and Wheeler?
It seems that the correct version of (14.21) is
$$
\frac{1}{2}\,\langle d\alpha,u\wedge v\rangle=\partial_u\langle\alpha,v\rangle-\partial_v\langle\alpha,u\rangle-\langle\alpha,[u,v]\rangle
$$
where $\...
- 1,839
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Conventions for graded wedge product in supergeometry
There are two conventions for the graded exterior product on superspace (see https://ncatlab.org/nlab/show/signs+in+supergeometry):
$$\alpha \wedge \beta = (-1)^{pq+|\alpha||\beta|}\beta\wedge\alpha \;...
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Einstein notation and writing down the geodesic equation - a misunderstanding?
if one wants to write down the geodesic equation to describe the movement of the planets (for i=1 in the following context) one uses the metric tensor $g_{ik}$ for spherical symmetric coordinates.
One ...
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