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## Main Question or Discussion Point

General four-dimensional (symmetric) metric tensor has 10 algebraic independent components.

But transformation of coordinates allows choose four components of metric tensor almost arbitrarily.

My question is how much freedom is in choose this components?

Do exist for most general metric any coordinates transformation

[tex] g_{\mu\nu}(x)\rightarrow\tilde{g}_{\mu\nu}(\tilde{x})[/tex] which transforming matric to new form?

[tex]

\[ \left( \begin{array}{cccc}

-1 & 0 & 0 & 0 \\

0 & \tilde{g}11 & \tilde{g}12 & \tilde{g}13\\

0 & \tilde{g}12 & \tilde{g}22 & \tilde{g}23\\

0 & \tilde{g}13 & \tilde{g}23 & \tilde{g}33 \end{array} \right)\]

[/tex]

[tex]

\[ \left( \begin{array}{cccc}

-1 & \tilde{g}01 & \tilde{g}02 & \tilde{g}02 \\

\tilde{g}01 & \tilde{g}11 & 0 & 0\\

\tilde{g}02 & 0 & \tilde{g}22 & 0\\

\tilde{g}03 & 0 & 0 & \tilde{g}33 \end{array} \right)\]

[/tex]

But transformation of coordinates allows choose four components of metric tensor almost arbitrarily.

My question is how much freedom is in choose this components?

Do exist for most general metric any coordinates transformation

[tex] g_{\mu\nu}(x)\rightarrow\tilde{g}_{\mu\nu}(\tilde{x})[/tex] which transforming matric to new form?

[tex]

\[ \left( \begin{array}{cccc}

-1 & 0 & 0 & 0 \\

0 & \tilde{g}11 & \tilde{g}12 & \tilde{g}13\\

0 & \tilde{g}12 & \tilde{g}22 & \tilde{g}23\\

0 & \tilde{g}13 & \tilde{g}23 & \tilde{g}33 \end{array} \right)\]

[/tex]

Or

[tex]

\[ \left( \begin{array}{cccc}

-1 & \tilde{g}01 & \tilde{g}02 & \tilde{g}02 \\

\tilde{g}01 & \tilde{g}11 & 0 & 0\\

\tilde{g}02 & 0 & \tilde{g}22 & 0\\

\tilde{g}03 & 0 & 0 & \tilde{g}33 \end{array} \right)\]

[/tex]