Particle physics signature versus relativity signature of metric tensor

Question

In almost every book that has a introductory notion on relativity, the author usually says the signature that he uses: $(+---)$ or $(-+++)$. The book I'm reading says:

Note that the convention on the metric signature is not unique and in several textbooks it is used the other one; the physics, of course, is left unchanged.

Why does the physics not change?

They say that physics cannot depend on a special coordinate system and it is quite simple why but it is not completely obvious (to me) that changing signatures will not lead to change the physics of the problem. There is some full explanation on why the physics is left unchanged or there is some study that proved that this is true?

In this Physics.SE question the two different conventions are explained. This question is more in the last part of this usual statement.

Answer 1

The physics doesn't change because both conventions give the same definition for all physical quantities. For instance, proper time in the mostly negative convention is given by $\mathrm{d}\tau^2=\eta_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}$ and in the mostly positive convention is given by $\mathrm{d}\tau^2=-\eta_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}$. Since in both conventions $\mathrm{d}\tau^2 $ is given by $ \mathrm{d}t^2-\mathrm{d}\textbf{x}^2$ and every physical quantity in a relativistic theory is dependent only on the definition of $\tau$, both methods make the same predictions.

Another justification lies in the fact that every time a particle physicist uses $\eta_{\mu\nu}$, a gravitational physicist would use $-\eta_{\mu\nu}$, which are the exact same metric.

I hope this helped!

Answer 2

You're right that the metric convention choice is slightly different from others.

For example, in the case of the left-hand and right-hand conventions for the cross product, no equations need to be changed at all, because all physically observable quantities come from combinations of two cross products. Similarly, in theories that are invariant under coordinate changes, equations should look essentially the same in all coordinate systems.

However, switching the metric signature to $(+---)$ from $(-+++)$ isn't the same, as there are physically observable quantities with one factor of the metric, such as the rest mass $m^2 = p^\mu p_\mu$. If we just changed the signature and nothing else, we'd end up in a world with three timelike dimensions instead of one. The fix is that equations with the 'flipped' convention need an extra minus sign for every factor of the metric.