I posted a question on the canonical for of the metric to physics forums here. It is about equation (2.46) in the book. he writes:

Quote: A useful characterisation of the metric is obtained by putting

There were two good answers and I would rewrite (2.46) as

Sadly Orodruin wrote "

I then wrote:

Orodruin:

martinbn:

I was tempted to reply

So the canonical form of the R² metric is

and the canonical form of the Minkowski metric (which is Lorentzian) is

Quote: A useful characterisation of the metric is obtained by putting

*g*into its canonical form. In this form the metric components become_{μν}
$$ g_{\mu\nu} = \rm{diag} (-1, -1,...-1,+1,+1, ... +1,0,0, ... ,0) $$

where "diag" means a diagonal matrix with the given elements. 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. That's too vague!There were two good answers and I would rewrite (2.46) as

$$ g_{\mu\nu} = \rm{diag} (-1, 0,1...-1, 0,1 ... -1, 0,1 ... -1, 0,1) $$

i.e. each diagonal element can be -1,0 or 1.

Sadly Orodruin wrote "

*That would have been wrong and very unclear.*" He did not elucidate.I then wrote:

*Does this work?*

*As a matrix, a non-degenerate metric in canonical form is diagonal with ±1 in each component.*

*Technically, that would be a pseudo-metric unless you have +1 in all diagonals, but in physics we just call it metric anyway.*martinbn:

*This*
$$ g_{\mu\nu} = \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} $$

*and this*
$$ g_{\mu\nu} = \rm{diag} (-1, 1, 1,1) $$

*are the same.*I was tempted to reply

*Are we discussing how many angels can fit on a pinhead?*But religion is not allowed.So the canonical form of the R² metric is

$$ \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ \end{pmatrix} $$

and the canonical form of the Minkowski metric (which is Lorentzian) is

$$ \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} $$

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