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George Keeling

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I am reading Spacetime and Geometry : An Introduction to General Relativity – by Sean M Carroll and he writes:

Quote: A useful characterisation of the metric is obtained by putting ##g_{\mu\nu}## into its

Wikipedia tells me to "Write ## \rm{diag} (a_1, ..., a_n)## for a diagonal matrix whose diagonal entries starting in the upper left corner are ##a_1, ..., a_n##." So Carroll's expression seems to imply a diagonal matrix with with a minimum size 9x9 and one extra row and column wherever one of his .'s takes a value. Or something like $$\begin{pmatrix}

-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & . & 0 \\

0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & . & 0 \\

. & . & . \\

0 & 0 & 0 & 0 & -1& 0 & 0 & 0 & . & 0 \\

0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & . & 0 \\

. & . & .

\end{pmatrix}$$ In general Carroll does not assume that ##\mu,\nu## have a range such as 1,2,3 or 0,1,2,3 until he gets to examples which are much simpler.

I don't think my interpretation is correct. Can anybody cast any light for me?

Quote: A useful characterisation of the metric is obtained by putting ##g_{\mu\nu}## into its

**canonical form**. In this form the metric components become $$ g_{\mu\nu} = \rm{diag} (-1, -1,...-1,+1,+1, ... +1,0,0, ... ,0) $$where "diag" means a diagonal matrix with the given elements. End quote.Wikipedia tells me to "Write ## \rm{diag} (a_1, ..., a_n)## for a diagonal matrix whose diagonal entries starting in the upper left corner are ##a_1, ..., a_n##." So Carroll's expression seems to imply a diagonal matrix with with a minimum size 9x9 and one extra row and column wherever one of his .'s takes a value. Or something like $$\begin{pmatrix}

-1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & . & 0 \\

0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & . & 0 \\

. & . & . \\

0 & 0 & 0 & 0 & -1& 0 & 0 & 0 & . & 0 \\

0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & . & 0 \\

. & . & .

\end{pmatrix}$$ In general Carroll does not assume that ##\mu,\nu## have a range such as 1,2,3 or 0,1,2,3 until he gets to examples which are much simpler.

I don't think my interpretation is correct. Can anybody cast any light for me?

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