All Questions
Tagged with differentiation stress-energy-momentum-tensor
10
questions
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Does (covariant) divergence-freeness of the stress-energy tensor ${T^{\mu\nu}}_{;\nu}=0$ follow from the Bianchi identity?
I'm working through Chap. $30$ of Dirac's "GTR" where he develops the "comprehensive action principle". He makes a very slick and mathematically elegant argument to show that the ...
- 366
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How do you differentiate $F^{αβ}$ with respect to $g_{μν}$?
I want to experiment with this relation (from Dirac's "General Theory of Relativity"):
$$T^{μν} = -\left(2 \frac{∂L}{∂g_{μν}} + g^{μν} L \right)$$
using the electromagnetic Lagrangian $L = -(...
- 366
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Confusion regarding 4-Velocity Derivative Identity (for conservation of energy momentum tensor) in Carroll's Spacetime and Geometry
During Carroll's discussion of the energy-momentum tensor for a perfect fluid (page 36), he writes out that its divergence should be zero. He then expands this as follows:
$$\partial_\mu T^{\mu\nu} = \...
4
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How to derive $\nabla_{\mu} T^{\mu\nu}=\frac{1}{\sqrt {-g}}\partial_{\mu}(\sqrt{-g}\,T^{\mu\nu})$?
This $$\nabla_{\mu} T^{\mu\nu}=\frac{1}{\sqrt {-g}}\partial_{\mu}(\sqrt{-g}\,T^{\mu\nu})\tag{3.39}$$ is from a textbook on general relativity on black hole Vaidya metric, where only non-zero term of ...
- 175
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Covariant derivative in a basis
Reading through this paper, I saw that the energy momentum conservation:
$$\nabla_\mu T^{\mu\nu}=0$$
can be evaluated as:
$$\partial_t(\sqrt{-g}T^{t}_\nu)=-\partial_i(\sqrt{-g}T^{i}_\nu)+\sqrt{-g}T^...
- 43
1
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Double divergence of stress tensor for migration flux
I am looking to calculate migration as a function of time using equation in Image 1. SigmaP is the total particle stress tensor in the cylindrical coordinates (r, theta, z). I am only interested in ...
- 11
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Schwarzian derivative from conformal factor
Suppose I have a 2D Lorentzian conformally flat metric
$$ ds^2 = -\Omega(u, v) du dv.$$
I consider a conformal field theory whose stress-energy tensor $T_{ab}$ is known on the flat metric
$$ds^2 = -...
- 4,251
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Regarding $T^{\mu\nu};_{\mu}=0$ in general relativity
In Physics we say we have a conserved entity $\vec{P}$ (to use a common example) if we can write:
$$\frac{\partial}{\partial x^{\mu}}P^{\mu}=P^{\mu},_{\mu}=0\tag{1}$$
Where "," denotes the partial ...
- 2,847
4
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Computation of $T^{\mu\nu}$ from Lagrangian density $\mathscr{L} $
I am trying to understand how upper and lower indices are connected when computing the energy-momentum tensor. In particular, I found the simple problem where the Lagrangian density is given as
$$\...
- 143
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Covariant derivative of stress-energy tensor for a scalar field [closed]
In order to prove that $$\nabla ^\mu T_{\mu\nu} =0$$ I want to find the
covariant derivative of $$T_{\mu\nu} = \partial_\mu\phi \partial_\nu \phi -\frac{1}{2}g_{\mu\nu}(g ^{\lambda\sigma}\partial_\...
- 1,529