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Definition of stress tensor

In several of the sources where I’ve found the definition of stress tensor that I’ve found on the internet, it says that
$$T^{\mu
u}=\frac{1}{\sqrt{g}} \sqrt{ -\, g} \, g^{\mu
u} \,.$$
Now, I’ve also seen other definitions of stress-tensor, namely
$$T^{\mu
u}=\frac{1}{\sqrt{g}} \frac{\partial \left( \sqrt{ -\, g} g^{\mu
u} \right)}{\partial x^
u}$$
with $g$ being the determinant of the metric $g$.
Now, these two definitions seem to be equivalent to each other. I can write either one into the form I stated above with a little work, but at the same time, they don’t seem to be equivalent: The first one seems to involve a derivative of the determinant of the metric, while the second doesn’t.
Now, the question I’ve got here is, what is the correct definition of the stress tensor? (Either of the two)?

A:

The most common definition of the stress-energy-momentum tensor is the conservation of energy momentum
$$
abla_\mu T^{\mu
u} =0.\tag{1}$$
This means that the divergence of
$$T^{\mu
u} = \frac{1}{\sqrt{g}}\frac{\partial}{\partial x^\mu} \left(\sqrt{ -g}g^{\mu
u}\right)$$
is the energy density
$$
abla_\mu T^{\mu
u} = \frac{\partial \rho}{\partial x^
u},\tag{2}$$
where $\rho$ is the energy density.
Another, equivalent definition is the local conservation of energy
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