How to compute the variation of two covariant derivatives?

In summary, the conversation discusses the perturbation of field equations in modified gravity models. The individual explains their approach and asks for confirmation on the correctness of their reasoning, which is deemed to be correct with the addition of the product rule for derivatives.
  • #1
balaustrada
1
0
Homework Statement
I need to compute the variation of two covariant derivatives applied to a tensor, and I'm not sure how it should be computed.
Relevant Equations
$$ \delta ( \nabla_\rho \nabla_\nu \left[ F'(G) R_{\mu}^{\hphantom{\mu} \rho} \right])$$
I'm working with modfied gravity models and I need to consider the perturbation of field equations. I have problems with the term were I have two covariant derivatives, I'm not sure if I'm doing it right.

I have:
$$\delta(\nabla_\rho \nabla_\nu \left[F'(G)R_{\mu}^{\hphantom{\mu} \rho}\right])$$
where F(G) a function of the Gauss-Bonet comination.

What I consider, is the following:
$$ \delta (\nabla_\rho V^\mu) = \delta ( \partial_\rho V^\mu + \Gamma^\mu_{\rho \xi} V^\xi) = \partial_\rho \delta(V^\mu) + \Gamma^\mu_{\rho \xi} \delta(V^\xi) + \delta (\Gamma^{\mu}_{\rho \xi}) V^\xi \\
= \nabla_\rho \delta(V^{\mu}) + \delta(\Gamma^\mu_{\rho \xi} V^\xi) $$

Applied with two covariant derivatives gives:
$$\delta(\nabla_\rho \nabla_\nu \left[ F'(G) R_{\mu}^{\hphantom{\mu} \rho} \right]) = \nabla_\rho \delta(\nabla_\nu \left[ F'(G) R_{\mu}^{\hphantom{\mu} \rho} \right]) - \delta(\Gamma^\xi_{\rho \nu}) \left(\nabla_\xi \left[ F'(G) R_{\mu}^{\hphantom{\mu} \rho} \right] \right) - \delta(\Gamma^\xi_{\rho \mu})\left(\nabla_\nu \left[ F'(G) R_{\xi}^{\hphantom{\mu} \rho} \right] \right)
+ \delta(\Gamma^\rho_{\nu \xi})\left(\nabla_\nu \left[ F'(G) R_{\mu}^{\hphantom{\mu} \xi} \right] \right)$$

and finally:

$$ \delta(\nabla_\rho \nabla_\nu \left[F'(G)R_{\mu}^{\hphantom{\mu} \rho}\right]) = \nabla_\rho \left[ \nabla_\nu \delta( \left[ F'(G) R_{\mu}^{\hphantom{\mu} \rho} \right] ) - \delta(\Gamma^\xi_{\nu \mu}) \left[F'(G)R_{\xi}^{\hphantom{\xi} \rho} \right] + \delta(\Gamma^\rho_{\nu \xi}) \left[F'(G) R_{\mu}^{\hphantom{\xi} \xi} \right] \right] \\
- \delta(\Gamma^\xi_{\rho \nu}) \left(\nabla_\xi \left[ F'(G) R_{\mu}^{\hphantom{\mu} \rho} \right] \right) - \delta(\Gamma^\xi_{\rho \mu})\left(\nabla_\nu \left[ F'(G) R_{\xi}^{\hphantom{\mu} \rho} \right] \right) + \delta(\Gamma^\rho_{\nu \xi})\left(\nabla_\nu \left[ F'(G) R_{\mu}^{\hphantom{\mu} \xi} \right] \right)$$

Is this reasoning true?
 
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  • #2
Am I missing something? The answer to your question is yes, your reasoning is correct. The only thing you may be missing is the product rule for derivatives, which states that $$\delta(\nabla_\rho \nabla_\nu F(G)) = \nabla_\rho \delta(\nabla_\nu F(G)) + \nabla_\nu \delta (\nabla_\rho F(G))$$This rule can be used to simplify some of the terms in your expression.
 

FAQ: How to compute the variation of two covariant derivatives?

What is the formula for computing the variation of two covariant derivatives?

The formula for computing the variation of two covariant derivatives is: Δ(∇ₐ∇ᵇT) = ∇ᵇ(∇ₐT) - ∇ₐ(∇ᵇT)

How do you interpret the result of computing the variation of two covariant derivatives?

The result of computing the variation of two covariant derivatives represents the difference between the two covariant derivatives when applied to a tensor field. It measures the non-commutativity of covariant derivatives and is a key concept in differential geometry.

Can the variation of two covariant derivatives be zero?

Yes, the variation of two covariant derivatives can be zero if the covariant derivatives commute. This is the case when the tensor field is constant or if the Christoffel symbols are symmetric.

How is the variation of two covariant derivatives related to curvature?

The variation of two covariant derivatives is related to curvature through the Riemann curvature tensor. Specifically, the Riemann tensor can be expressed in terms of the variation of two covariant derivatives of the metric tensor. This illustrates the geometric significance of the variation of two covariant derivatives.

Are there any applications of computing the variation of two covariant derivatives?

The variation of two covariant derivatives has various applications in physics, particularly in general relativity and gauge theories. It is also used in geometric analysis to study the curvature and geometry of manifolds. Additionally, it has applications in optimization and numerical methods for solving differential equations.

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