Definition of Form-Invariant Function: Q&A

[Einj]

Hello everyone. I'm reading Weinberg's 'Gravitation and Cosmology' and I'm having some problems understanding the definition of a 'form-invariat function'. He says:

A metric $g_{\mu\nu}$ is said to be form-invariant under a given coordinate transformation $x\to x^\prime$, when the transformed metric $g^\prime_{\mu\nu}(x^\prime)$ is the same function of its argument $x^{\prime\mu}$ as the original metric $g_{\mu\nu}(x)$ was of its argument $x^\mu$, that is,
\begin{equation}
g^\prime_{\mu\nu}(y)=g_{\mu\nu}(y) \; \text{ for all }y.
\end{equation}

If the previous condition was true doesn't this simply mean that $g_{\mu\nu}^\prime$ is the same function as $g_{\mu\nu}$? Should we ask for:
\begin{equation}
g^\prime_{\mu\nu}(x^\prime)=g_{\mu\nu}(x)
\end{equation}
?

Thanks a lot!

 

[Cruz Martinez]

If the previous condition was true doesn't this simply mean that $g_{\mu\nu}^\prime$ is the same function as $g_{\mu\nu}$?

That is precisely what he means, that they are the same function.
I think that's why he used the symbol "y" instead of x's, so that it wouldn't be confused as having something to do with the coordinates x and x'.

 

[ShayanJ]

The problem with g'_{\mu\nu}(x')=g_{\mu\nu}(x) is that it means the two functions have same value in a single point. But because x' associates different numbers to that point compared to x, g'_{\mu\nu}(x')=g_{\mu\nu}(x) necessarily means g and g' are different functions and don't have the same form.
But g'_{\mu\nu}(y)=g_{\mu\nu}(y) means that if we give the two functions, the same numbers, they will give equal results which means they're the same function. Note that y in the right refers to a different point than y in the left because they are same numbers in different coordinate systems.

 

[Einj]

Note that y in the right refers to a different point than y in the left because they are same numbers in different coordinate systems.

Oh I see. That is clear. Thanks!