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!

 

[sardar]

Hello,

Thank you for your question. I can provide you with a response to help clarify the definition of a form-invariant function.

Firstly, a form-invariant function is a mathematical concept that is used in the study of general relativity. In this context, a form-invariant function refers to a function that remains unchanged under a given coordinate transformation.

In the definition provided, the transformed metric $g^\prime_{\mu\nu}(x^\prime)$ is said to be the same function of its argument $x^{\prime\mu}$ as the original metric $g_{\mu\nu}(x)$ was of its argument $x^\mu$. This means that the two metrics, the original and the transformed, have the same functional form. However, the arguments of the two metrics are different, as one is in the original coordinate system and the other is in the transformed coordinate system.

To answer your question, yes, the condition $g^\prime_{\mu\nu}(y)=g_{\mu\nu}(y)$ does mean that the two metrics are the same function. However, to be more precise, we should ask for the condition $g^\prime_{\mu\nu}(x^\prime)=g_{\mu\nu}(x)$, as this takes into account the fact that the arguments of the two metrics are different due to the coordinate transformation.

I hope this helps clarify the definition of a form-invariant function for you. If you have any further questions, please don't hesitate to ask.