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Can anyone explain what metric signature is?

I have a basic knowledge regarding tensors, btw.

Also, how is it related to fundamental understanding of general relativity?

Thanks.

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Do you know what the metric tensor is, or should we explain that before explain what it's signature means? –  John Rennie Apr 27 '12 at 15:00
    
@JohnRennie I sort of know what the metric tensor is, but if metric tensor is explained at least briefly, I will appreciate :) –  user27515 Apr 27 '12 at 15:03
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1 Answer

up vote 3 down vote accepted

In relativity there is an invariant called the proper time, $\tau$. It's an invariant in the sense that all observers will agree on it's value. In special relativity the proper time is defined as:

$$d\tau^2 = ds^2 = c^2dt^2 - dx^2 - dy^2 - dz^2$$ or $$d\tau^2 = -ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2$$

You see both sign conventions and I've never been sure which is more generally accepted. Anyhow, you can write the equation for $ds^2$ as a matrix equation using:

$$ds^2 = g_{\alpha\beta}x^\alpha x^\beta$$

where $x$ is the vector $(t, x, y, z)$ and $g$ is the matrix:

$$\left( \begin{matrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}\right)$$

The matrix is called the metric tensor (or a representation of it) and the signature is the number of positive and negative values on the leading diagonal. In this case it's (1,3) or you often just add together the negative and positive numbers to give, in this case, just 2.

Exactly the same equation is used in general relativity, but the matrix representing the metric tensor is more complicated and generally not diagonal, so you have to diagonalise it to calculate the signature. The Wikipedia article goes into this in more detail.

The reason we're interested in the signature is that we expect spacetime to have one timelike co-ordinate and three spacelike co-ordinates, so we expect the signature to be always (1,3).

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You missed a $d$ in $c^2dt^2$. But more importantly: I think it's pretty universal that the sign of proper time matches the sign of coordinate time. $c^2d\tau^2 = c^2dt^2-dx^2-dy^2-dz^2$. The issue with the metric signature is just about whether $c^2d\tau^2 = \pm ds^2$. In one signature, the Minkowski metric tensor has diagonal elements $-1,1,1,1$ and in the other signature it has diagonal elements $1,-1,-1,-1$. The latter is more common in particle physics, the former in GR and fundamental QFT, I think. –  David Z Apr 27 '12 at 23:12
    
You say often just add the numbers to signature of 2 but I have never before heard the signature of Minkowski spacetime referred to as 2. I have always seen the (1,3) notation (or -,+,+,+ or +,-,-,-). It doesn't make sense to add the diagonal elements and say the signature is 2. A flat 2 dimensional plane space would also have signature 2 by that method which certainly is not the same as (1,3). –  FrankH Mar 19 '13 at 19:17
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