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This is the so called "principle of uniformity". Basically, it stipulates that the laws of physics are the same everywhere in space and time. Now, why should we believe in such principle ? I have thought a bit about that, and here is my reasoning. Let's define two "types" (in the sense of logic) of "objects". Call pobject, any physical object you can see, ...

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You should always work with $$\nabla_\rho K_\sigma=\partial_\rho K_\sigma-\Gamma^\mu_{\rho\sigma} K_\mu$$ even if $K_\mu$ is a constant vector. Here $\partial_t K_t=0$ but you must consider the Christoffel symbols. For example: $$\nabla_t K_t=\partial_t K_t-\Gamma^\mu_{tt} K_\mu=-\Gamma^t_{tt} K_t$$ ...

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You could make an argument that the four-current is most naturally defined as a vector density. This is because a vector density uniquely defines a three-form (i.e. a totally antisymmetric tensor $J_{\alpha\beta\gamma}$), and three-forms can be integrated on hypersurfaces without any reference to a metric. So if you think of a current as an object which, ...

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Clearly MTW's definition of $J^\alpha$ is a vector field due to the argument given. Note however that in the Wikipedia article you linked, under Summary, $J^\alpha$ is defined with an additional factor $\sqrt{-g}$, making it a vector density. You can easily see this by replacing the ordinary derivative in the definition of $J^\mu$ by a covariant one (which ...

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Wald is a first rate relativist, and as such he is phrasing the concept of general covariance in terms of purely geometrical quantities, rather than resorting to the somewhat imprecise notion of coordinate transformations. In the discussion on pg. 57, he goes on to give an example of what it means to violate the principle of general covariance. In his ...

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General covariance basically means you can change your coordinate system arbitrarily and express the laws of physics in the new coordinates. Because of this freedom, the relationship between coordinate distances, angles, etc. and physical distances, angles, etc. is variable and is expressed by the metric. So the quoted statement is basically saying that ...

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