All Questions
Tagged with tensor-calculus general-relativity
891
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Notation for vector density in Lagrangian density
Consider a manifold $M$ and a Lagrangian density $\mathcal{L} \equiv \mathcal{L}(\phi, \nabla \phi)$. By varying the action, one obtains the equation
$$\int_M \, dV \; \Big( \frac{\partial \mathcal{L}}...
- 743
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Derivation of Einstein-Cartan (EC) action for parametrized connection $A$ & introduction of torsion
I have some trouble with one missing step when I want to get the teleparallel action from general EC theory, which I am not fully understanding.
The starting form of action (3-Dimensional) is:
$$
S_{...
-1
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1
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Is there an "obvious" reason for why the second derivative of an antisymmetric tensor with respect to coordinates over both of its indices equal to 0?
It was kind of difficult to word the title so I'll restate the question here. My professor took it almost as a given that
$$\frac{\partial T^{\mu\nu}}{\partial X^{\mu}\partial X^\nu} = 0$$
If $T^{\mu\...
- 99
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What is the Lie derivative of Ashtekar connection and its conjugate momentum in LQG?
I am using the reference Black hole entropy from an SU(2)-invariant formulation of Type I isolated horizons for this question. I am trying to understand the two equations (30) that give the variation ...
- 43
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Prove that Levi-Civita tensor density is invariant [closed]
I am struggling with Exercise 7.14 ( the latter part ) of the textbook Supergravity ( by Freedman and Proeyen ). Here is the problem:
We are defining the Levi-Civita form in coordiante basis as:
$$ \...
- 513
4
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1
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292
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Torsion and Compatibility with the Metric
Compatibility with a metric, also referred to as metricity, means, I believe, that the covariant derivative of the metric is zero:
$$g_{ij;k}=g_{ij,k}-\Gamma^m_{ik}g_{mj}-\Gamma^m_{jk}g_{im}=0$$
This ...
- 133
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23
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Variation of nonminimal derivative coupling term
all. Can I request you assistance about the following problem?
How do I vary this action with respect to metric $\delta g_{ab}$
$$
\int d^4x \sqrt{-g} \Big[\kappa R+ G_{ab}\nabla^a \phi \nabla^b \phi \...
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1
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68
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Selecting Indices for the Riemann Tensor
How do I know when computing the Riemann Tensor (in two dimensional) which indices to select? Consider the Riemann Tensor $R^a_{bcd}$ how do I know what values to take for $a$?
As an example, consider ...
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Palatini variation of Ricci tensor
I was looking at Problem 2 of chapter 4 of Sean Carroll's General Relativity book, where you were supposed to demonstrate starting from the Einstein-Hilbert action, and assuming that the connection is ...
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Timelike normal vector becomes null
I have a metric given by
\begin{equation}
ds^2 = \frac{e^{2 A(z)}}{z^2} \left(-g(z) dt^2 + \frac{dz^2}{g(z)} + dx^2 + dx^2_1 + dx^2_2 \right)
\end{equation}
where $A(z) = -a \ln(b z^2 + 1)$ and $g(z)$ ...
- 404
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1
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Books that approach General Relativity via differential forms, without coordinates [duplicate]
Does someone know about some books about differential geometry applied to General Relativity that are written using the language of differential forms, fiber bundles, & spin connection, and not ...
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62
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Equation of motion for gravity in scalar-tensor theory
I'm trying to derive equation of motion in Higgs scalar-tensor theory with the Lagrangian given by
$$\mathcal{L}=[\frac{1}{16\pi}\alpha \phi^{\dagger}\phi R+ \frac{1}{2}\phi^{\dagger}_{;\mu}\phi^{;\mu}...
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Bitensors at three or more space-time points
Bitensors, i.e. tensors at two points that have indices belonging to either of them, have been used in the literature quite a bit and there are many calculations involving them. They are the go-to ...
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1
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Derivation of transformation law for the Hilbert Stress-energy tensor [duplicate]
The Hilbert stress-energy tensor is defined as
$$T_{\mu\nu}=-2 \frac{1}{\sqrt{g}}\frac{\delta S_M}{\delta g^{\mu\nu}}.$$
Given the name one expects that it transform as a tensor, but how to prove this ...
- 327
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Tensor equation
What is a valid tensor equation. In the book by Bernard Schutz, it is often argued that a valid tensor equation will be frame invariant. So the conclusions reached by relatively easy calculation done ...
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