All Questions
Tagged with differentiation metric-tensor
125
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Derivative of the induced metric
2-metric $\gamma_{AB}$ induced on the world sheet by the spacetime metric $g_{\mu\nu}$ is $$\gamma_{AB}=g_{\mu\nu}X^{\mu},_A X^{\nu},_B$$
$$\gamma^{AB},_B=-\gamma^{AC}\gamma^{BD}\gamma_{CD},_B$$
How ...
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Covariant derivative contracted with a metric
I would like to calculate $\nabla_\mu(g^{\mu\alpha}g^{\nu\beta}\nabla_\alpha \kappa_\beta)$. How would this expand?
Where $\nabla$ is the covariant derivative, g the metric and $\kappa_\beta$ a 1-...
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Differentiation of the determinant $g$
Let $g$ be the determinant of the metric tensor.
I want to derive the following equation $g_{,\nu}=gg^{\lambda \mu}g_{\lambda \mu,\nu}$. It is said that $gg^{\lambda \mu}$ is a cofactor, but I can't ...
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Regd. derivation of some equations in “Bertrand Spacetimes” by Pelick
We are going through "Bertrand Spacetimes" by Dr Perlick, in which he first gave the idea of a new class of spacetimes named as Bertrand spacetimes after the well-known Bertrand's Theorem in Classical ...
2
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Divergence of a tensor
On pg.70 of Dalarsson's "Tensors, Relativity and Cosmology"
For a mixed tensor of contravariant order 2 and covariant order 1 $(T^{mn}_{p,m})$, the divergence with respect to m is defined as:$$T^{...
- 1,047
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Box operator in FLRW metric
Definition of box operator in curved space time is $g^{\mu \nu}\partial_{\mu}\partial_{\nu}$ and in FLRW metric $g_{\mu \nu}$ is $diag(1 ,-a^2(t)$ $,-a^2(t),-a^2(t) )$ so the box operator should be $\...
- 3,043
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Index (Einstein summation) notation question
Question #1:
Lost as to how the second equality in the following equation holds —
$$\frac{\partial}{\partial \tau} (A^2) = \frac{\partial}{\partial \tau} (\eta_{\mu\nu}A^\mu A^\nu) = 2\eta_{\mu\nu}...
- 1,031
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What is the covariant derivative of a metric tensor $\nabla_{\mu} g^{\mu\nu}$ =?
What is the covariant derivative of a metric tensor, this particular one to be specific $\nabla_{\mu} g^{\mu\nu}$? Notice we've got repetitive indices here. Is it zero and has it got to do anything ...
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Why does nature favour the Laplacian?
The three-dimensional Laplacian can be defined as $$\nabla^2=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+\frac{\partial^2}{\partial z^2}.$$ Expressed in spherical coordinates, it ...
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Question about Lie derivative of connection [closed]
This is from Hughston and Tod, Ex 8.5. Please give me an idea how to start with this.
If $\mathcal{L}_V g_{ij} = 2\phi g_{ij}$ then prove that
$$
\mathcal{L}_V\Gamma^{i}_{jk} = \phi_{,j}\delta^{...
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General Relativity: Exchanging a field with its infinitesimal components on metric tensor
On a youtube video about Einstein's field equations, the author writes the following equation (https://youtu.be/foRPKAKZWx8?t=1078):
$$d\phi=\sum_{n} \frac{\partial \phi}{\partial x^n} dx^n\tag{1}$$
...
- 212
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When can we raise lower indices on "nontensors" as described in Dirac's book *General Theory of Relativity*?
This is a follow on to my previous question: Dirac's book *General Theory of Relativity*: Doesn't this show the partial derivative of the metric tensor is zero?
I should not have made that ...
- 2,015
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Dirac's book *General Theory of Relativity*: Doesn't this show the partial derivative of the metric tensor is zero?
See the bold text for my question.
This question regards Dirac's General Theory of Relativity page 13. In his demonstration that the length of a vector is unchanged by parallel displacement Dirac ...
- 2,015
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What's the variation of a product of two metrics? [closed]
I was trying variate an action in General Relativity, and I come to the next calculus:
$\delta(g^{\alpha\beta}g^{\mu\nu})$
And I did:
$\delta(g^{\alpha\beta}g^{\mu\nu})=g^{\alpha\beta}\delta g^{\mu\...
3
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Why is it natural to impose the condition that the metric remains unchanged under parallel transport?
In Wald’s General Relativity, given a metric tensor $g_{ab}$, and two vectors $v^a,w^b$, the author said it is “natural” to impose the condition that the $g_{ab}v^aw^b$ is invariant under parallel ...
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