- #1

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## Main Question or Discussion Point

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:

\begin{equation}

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

\end{equation}

?

Thanks a lot!

If the previous condition was true doesn't this simply mean that ##g_{\mu\nu}^\prime## isA metric ##g_{\mu\nu}## is said to beform-invariantunder 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 ofitsargument ##x^\mu##, that is,

\begin{equation}

g^\prime_{\mu\nu}(y)=g_{\mu\nu}(y) \; \text{ for all }y.

\end{equation}

*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!