When doing physics there are many conventions that are left up to the choice of the practitioner. Many conventions are very standard, but some still have no general consensus. In relativity there is a sign convention for the metric that is somewhat not agreed on, but I will argue should be +---. First I will give you the ONLY reason that it is sometimes taken to be -+++ and then show why this is a bad choice. In non-relativistic Euclidean physics the line element describing distance along a path is given by \(d\sigma \) in \(d\sigma ^{2} = dx^{2} + dy^{2} + dz^{2}\) In special relativity there is an invariant interval between events which can be described in terms of an invariant line element ds, which looks much like this with the exception that it must include a time differential with the opposite sign of the other terms in order to be invariant. In other words \(– dct^{2} + d\sigma ^{2}\) is invariant, but also \(d\sigma ^{2} – dct^{2}\) is invariant. So the question is then which way should we define \(ds^{2}\). Minkowski unfortunately chose the former merely for the only reason that the interval would look more like the Euclidean line element that he was used to dealing with. Which led him to the erroneous(which I will demonstrate) conclusion that time is an imaginary spatial coordinate. This was the wrong thing to do and is unfortunately carried on in too many papers and texts involving relativistic metrics. Here’s the first reason. Squared lengths of vectors for real physical quantities become negative. The inner product of four-vectors A and B involves the metric \(A \cdot B = g_{\mu \nu }A^{\mu }B^{\nu }\) So for example the squared length of the momentum four-vector in special relativity is \(p \cdot p = \eta _{\mu \nu}p^{\mu }p^{\nu }\) and with the +--- convention that yields a positive result \( \eta _{\mu \nu}p^{\mu }p^{\nu } = (\frac{E}{c})^{2} – (p)^{2} = (mc)^{2}\) It is nice that the mass is the length of the momentum four-vector and not imaginary times mass, which would have been the case if we chose -+++. As another example consider the velocity four-vector. \(U \cdot U = g_{\mu \nu }U^{\mu }U^{\nu }\) With the +--- convention the length of the velocity four-vector is the speed of light. \(g_{\mu \nu }U^{\mu }U^{\nu } = c^{2}\) This yields an elegant interpretation of relativity that all things travel in 4d spacetime at the speed of light, merely rotated in direction in spacetime by four-forces. Thats why four-force is always perpendicular to four-velocity. \(g_{\mu \nu }F^{\mu }U^{\nu } = 0\) It would be awfully inconvenient if the interpretation had to be that all things traveled through 4d spacetime with imaginary velocity because we chose -+++ for no good reason. Heres the second reason. It leads to the erroneous interpretation of time as an imaginary spatial coordinate instead of associated imaginary with the spatial coordinates with which it now obviously belongs. Consider for a moment the 2x2 identity and the three Pauli matrices. The square of the 2x2 identity is identity is \(\sigma _{0}^{2} = \sigma _{0}\) The square of any other Pauli matrix is minus one times identity. \(\sigma _{i}^{2} = – \sigma _{0}\) Lets see how these would affect physics if we let them serve as a component of an orthonormal basis with the 2x2 identity associated with time. It is not imaginary that is associated with time. It is imaginary in the Pauli matrices that is associated with space. A displacement vector ds in space and time with these associations is \(\mathbf{ds} = \sigma_{0}dct\mathbf{e_0} + \sigma_{x}dx\mathbf{e_x} + \sigma_{y}dy\mathbf{e_y} + \sigma_{z}dz\mathbf{e_z}\) and the square of that displacement vector without using the metric this time as its effect is imbedded in the Pauli matrices is \((\mathbf{ds}^{2}) = \mathbf{1}ds^{2} = \mathbf{1}(dct^{2} – dx^{2} – dy^{2} – dz^{2})\) Minkowski associated imaginary with the wrong coordinate, BECAUSE he chose the wrong sign convention.
Sorry, but SO(3,1) is isomorphic to SO(1,3), so that taking the metric to be -+++ is totally equivalent to +---. Taking spacelike intervals to be positive and timelike interval to be negative is not a better or worst choice than taking the opposite.
My point totally went over your head. I never argued against the equivalence. I was discussing what interpretations and roads of discovery the different choices would lead a person into. Its a matter of pedagogy.
I'm inclined to agree with you, Trilairian. But then, I was taught with +---, and I think it's quite hard to escape from such biases. Please Register or Log in to view the hidden image!
