What is the rationale for the sign convention in the space-time 4-vector? How is it related to the sign convention in the energy-momentum 4-vector, if at all?
Relativists seem to prefer the latter, while field theorists seem to prefer the former. Which sucks when you're both a relativist and a field theorist =)I assume you mean, why do we write [itex]ds^2 = (cdt)^2 - dx^2 - dy^2 - dz^2[/itex] rather than [itex]ds^2 = dx^2 + dy^2 + dz^2 - (cdt)^2[/itex]?
For one thing, with this choice, we get a positive number for a "causally connectable" spacetime interval. Also, if we use the same convention for all four-vectors, then for energy-momentum [itex]E^2 - (p_x c)^2 - (p_y c)^2 - (p_z c)^2 = (m_0 c^2)^2[/itex] which is also a positive number.
We could do it the other way, but it seems to me that this way, we get fewer minus signs associated with the invariant quantities that we're usually interested in.
Because that's not what nature chose to give us. Also, you'll notice that there's no cross-over with that metric. In other words, there is no limiting speed like the speed of light.My real question is the following: why we don't use the standard convention for scalar products of vectors (+,+,+,+) in GR?
That's not actually a "convention".My real question is the following: why we don't use the standard convention for scalar products of vectors (+,+,+,+) in GR?