All Questions
8 questions
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What is the relation between gauge field and Levi-Civita connection?
In field theory, covariant derivative is something like
$$D_{\mu}\phi=(\partial_{\mu}-igA_{\mu})\phi$$
while in differential geometry, covariant derivative is something like
$$D_{\mu}V^{\nu}=\partial_{...
- 101
1
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0
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57
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If there is a spin-0 structure and i differentiate it with respect to a space dimension, does it become a spin-1 structure?
It might be a naive question but i was wondering what a derivative can do regarding spin. If there is a Riemann scalar it is clear that its an invariant object under tensor transformation and it does ...
- 153
3
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0
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153
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d'Alembertian operator in presence of torsion
Consider a Riemann-Cartan 4-dimensional spacetime with torsion. In such a spacetime, I have been asked to compute the d'Alembertian operator acting on a scalar field. Here's what I tried:
$$ g^{\mu\nu}...
- 702
0
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1
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153
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Differentiating the index notation
I am always confused with the algebra of differentiating the index notation, and have browsed many other posts but still confused. There must be details I have been missing. It would be really ...
- 41
1
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0
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170
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What is the meaning of $\nabla _{\mu}\nabla _{\nu}\phi(r)$ in general relativity?
I know the covariant derivative of a tensor is
$$\nabla_{\mu} V_{\nu}=\partial_\mu V_\nu-\Gamma_{\mu\nu}^{\lambda}V_{\lambda}$$
Now I want to obtain $\nabla_{\mu}\nabla_{\nu}\Phi(x)$ where $\Phi(x)$...
- 67
3
votes
2
answers
455
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Lie derivatives and the tetrad formalism
I have been trying to learn about the tetrad formalism in general relativity and I understand the basic idea, but there is one issue that I can't seem to figure out: Is there a definition of a Lie ...
- 1,113
2
votes
1
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807
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The equation of motion for a scalar field in curved spacetime in terms of the covariant derivative
The equation of motion for a scalar field in curved spacetime $$\frac{\partial\mathcal{L}}{\partial\phi}=\frac{1}{\sqrt{-g}}\partial_{\mu}\left[\sqrt{-g}\frac{\partial\mathcal{L}}{\partial\left(\...
- 12.3k
6
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1k
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Covariant derivative of a Dirac spinor and Kosmann lift
In [1] I have found a definition of the covariant derivative of a Dirac field with a general connection $\omega_{\mu a}{}^{b}$ (with torsion and non-metricity) [see eq. (29)]:
$$\nabla_{\mu}\psi=\...
- 567