# Can the overall sign of the Minkowski metric be changed?

If we take the Minkowski metric, $\eta_{\mu\nu}=(1,-1,-1,-1)$, instead of the usual $(-1,1,1,1)$, does this change the form of the Lorentz Transform? I think the standard Lorentz Transform looks like: $$\left( \begin{matrix} \gamma & -\gamma\beta & 0 & 0\\ -\gamma\beta & \gamma & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 &0 & 1 \end{matrix} \right)$$

• The conventional way to indicate which one you are using is by writing $\mathrm{Tr}(g) = \pm 1$ or saying "using the trace equals (minus) one metric". – dmckee Jan 13 '13 at 3:56

## 1 Answer

An interesting question indeed :-) Yes, you can flip the overall sign of the Minkowski metric, and in fact a lot of physicists do this! The sign choice $\operatorname{diag}(-1, 1, 1, 1)$ is conventional in fundamental quantum field theory and in quantum gravity, if I remember correctly, whereas $\operatorname{diag}(1, -1, -1, -1)$ is conventional in particle physics.

This doesn't affect the Lorentz transform, though. If you apply the Lorentz transform to a metric tensor, it computes as $g'_{\alpha\beta} = \Lambda_\alpha^\mu \Lambda_\beta^\nu g_{\mu\nu}$, and so you will automatically come out with the same sign convention that you put in.

• However, four-vector norms change sign. A timelike one (inside the light-cone) has a negative norm when using (-1+1+1+1) and positive otherwise. Einstein used (-1 -1 -1 +1) in the Princeton Lectures, for GR. Some of the equations result also formally different. – Eduardo Guerras Valera Jan 13 '13 at 7:44
• There is an annoying subtlety to this: The possible spinor representations are not always the same for the flipped overall sign, for instance, in this case one would have to distinguish between Majorana and pseudo-Majorana spinors. In the end, this is irrelevant to the physics in all cases known to me, but formally, the two choices are not fully equivalent. – ACuriousMind Oct 3 '16 at 8:02
• Wald is making an interesting comment on this and spinors in his book on general relativity. He's using the $(-1, +1, +1, +1)$ convention in his book, then change to $(+1, -1, -1, -1)$ in the chapter about spinors (same book). Also, Misner-Thorne-Wheeler are giving a nice summary of all conventions used in the world, in their Big Black Book "Gravitation". – Cham Oct 3 '16 at 13:21