## Thursday, 20 December 2018

### Question on canonical for of the metric

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 gμν 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. 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.
Orodruin:
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

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