- #1

Jianbing_Shao

- 102

- 2

- TL;DR Summary
- metric compatibility equation, connection, constraint

We all know that The gradient of a scalar-valued function ##f(x)## in ##IR^n## is a vector field ##V_\mu(x)=\partial_\mu f(x)##, Such a vector field is said to be conservative. Not all vector fields are conservative. A conservative vector field should meet certain constraints ##curlV_\mu(x)=0##.

In the discussion of a vector field ##V(x)## , If we take the partial derivative of this vector field we can get ##n## matrix function ##A_\mu(x)##.

$$

\partial_\mu V(x)=A_\mu(x)V(x)

$$If matrix functions ##A_\mu(x)## are continuously differentiable with respect to all variables. then we will find that only when$$

R_{\mu\nu}= \partial_\mu A_\nu-\partial_\nu A_\mu+[A_\mu , A_\nu]=0

$$ then there exist a matrix function ##G(x)## which satisfy: ##A_\mu(x)=\partial_\mu G(x)G^{-1}(x)##.(we can find the proof of this conclusion in the book 'Product Integration, Its History And Applications'P41) also we can get: ##V(x')=G(x')G^{-1}(x)V(x)##.

In general relativity, we calculate a Christoffel connection ##\Gamma(x)##(torsion free)from a metric ##g(x)## using the metric compatibility equation ##\nabla_\rho g_{\mu\nu}=0##. the metric compatible connection ##\gamma(x)## is not necessarily torsion free. From the discussion above, we can say that if the matric ##g(x)## is a matrix function, the metric compatible connection ##\gamma(x)## we get should also satisfy certain constraints. then what is it?

In the discussion of a vector field ##V(x)## , If we take the partial derivative of this vector field we can get ##n## matrix function ##A_\mu(x)##.

$$

\partial_\mu V(x)=A_\mu(x)V(x)

$$If matrix functions ##A_\mu(x)## are continuously differentiable with respect to all variables. then we will find that only when$$

R_{\mu\nu}= \partial_\mu A_\nu-\partial_\nu A_\mu+[A_\mu , A_\nu]=0

$$ then there exist a matrix function ##G(x)## which satisfy: ##A_\mu(x)=\partial_\mu G(x)G^{-1}(x)##.(we can find the proof of this conclusion in the book 'Product Integration, Its History And Applications'P41) also we can get: ##V(x')=G(x')G^{-1}(x)V(x)##.

In general relativity, we calculate a Christoffel connection ##\Gamma(x)##(torsion free)from a metric ##g(x)## using the metric compatibility equation ##\nabla_\rho g_{\mu\nu}=0##. the metric compatible connection ##\gamma(x)## is not necessarily torsion free. From the discussion above, we can say that if the matric ##g(x)## is a matrix function, the metric compatible connection ##\gamma(x)## we get should also satisfy certain constraints. then what is it?

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