Tensor equation

Question

What is a valid tensor equation. In the book by Bernard Schutz, it is often argued that a valid tensor equation will be frame invariant. So the conclusions reached by relatively easy calculation done on cartesian coordinate is generalized for all other coordinates. eg: $g_{\alpha\beta;\mu}=0$. My question is why shouldn't I take $g_{\alpha\beta,\mu}=0$ to be a valid tensor equation. This equation is also true in cartesian coordinate. Is there some relation with covariant derivatives and valid tensor equation?

Answer 1

A valid tensor equation is simply an equation relating tensors. The partial derivatives of tensor components do not form the components of a tensor simply because they do not obey the tensor transformation rules. If an object is a tensor, its components must obey the tensor transformation rules. However, $$\frac{\partial v'^\rho}{\partial x'^\sigma} = \frac{\partial}{\partial x'^\sigma}\left(\frac{\partial x'^\rho}{\partial x^\mu}v^\mu\right) \\ = \frac{\partial v^\mu}{\partial x'^\sigma}\frac{\partial x'^\rho}{\partial x^\mu} + v^\mu \frac{\partial}{\partial x'^\sigma}\frac{\partial x'^\rho}{\partial x^\mu} \\ = \frac{\partial x^\gamma}{\partial x'^\sigma}\frac{\partial x'^\rho}{\partial x^\mu}\frac{\partial v^\mu}{\partial x^\gamma} + v^\mu \frac{\partial x^\gamma}{\partial x'^\sigma}\frac{\partial^2 x'^\rho}{\partial x^\gamma\partial x^\mu}.$$ Due to the presence of the second term, these cannot be the components of a tensor.