# Question on definition of form-invariant

1. Jan 3, 2015

### 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:
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:

g^\prime_{\mu\nu}(x^\prime)=g_{\mu\nu}(x)

?

Thanks a lot!

2. Jan 3, 2015

### Cruz Martinez

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'.

Last edited: Jan 3, 2015
3. Jan 3, 2015

### 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.

4. Jan 3, 2015

### Einj

Oh I see. That is clear. Thanks!