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In Hartle's Gravity we have the covariant derivative (first in an LIF) which is:
##\nabla_{\beta} v^{\alpha} = \frac{\partial v^{\alpha}}{\partial x^{\beta}}##
As the components of the tensor ##\bf{ t = \nabla v}##. But, it's not clear which components they are!
My guess is that ##t^{\alpha}_{\ \beta} = \nabla_{\beta} v^{\alpha} = \frac{\partial v^{\alpha}}{\partial x^{\beta}}##
Rather than: ##t^{\ \ \alpha}_{\beta} = \nabla_{\beta} v^{\alpha} = \frac{\partial v^{\alpha}}{\partial x^{\beta}}##
I'm obliged to anyone who could confirm this one way or the other.
##\nabla_{\beta} v^{\alpha} = \frac{\partial v^{\alpha}}{\partial x^{\beta}}##
As the components of the tensor ##\bf{ t = \nabla v}##. But, it's not clear which components they are!
My guess is that ##t^{\alpha}_{\ \beta} = \nabla_{\beta} v^{\alpha} = \frac{\partial v^{\alpha}}{\partial x^{\beta}}##
Rather than: ##t^{\ \ \alpha}_{\beta} = \nabla_{\beta} v^{\alpha} = \frac{\partial v^{\alpha}}{\partial x^{\beta}}##
I'm obliged to anyone who could confirm this one way or the other.