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## Main Question or Discussion Point

Is a negative zeroth-component of a four-velocity [itex]V^0[/itex] with a curved space metric of signature [itex]-+++[/itex] allowed?

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Is a negative zeroth-component of a four-velocity [itex]V^0[/itex] with a curved space metric of signature [itex]-+++[/itex] allowed?

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I assume we're talking about timelike vectors. A spacelike vector that had a positive timelike component in one frame could have a negative one in a different frame. I think the question is better posed as whether we can prohibit velocity vectors from lying in a certain side of the timelike light cone.

If we restrict ourselves to timelike vectors, then any velocity vector can represent the frame of reference of an observer.

Mathematically either side of the light cone is certainly allowed, because this is a vector space, and one of the axioms of linear algebra is that if v is a vector, there's a vector -v.

Physically, SR is time-reversal invariant, so there is no fundamental physical distinction between the forward and backward directions of time. Therefore there is no fundamental way to state a prohibition on velocity vectors that point a certain way instead of the opposite way. GR inherits all the same local symmetries as SR, so this applies to GR as well. (Spacetimes in GR don't even have to be time-orientable.)

There are reasons that we have an arrow of time, but those reasons can't be found within the fundamental kinematical laws of relativity.

There is at least some kinematical guarantee of sanity, which is that a timelike geodesic always stays timelike. That implies that in SR an inertial observer will never have a velocity vector that wanders from one side of the light cone to the opposite side. (To do that, it would have to pass through the spacelike region.)

For particles, you can interpret this kind of thing in terms of matter/antimatter symmetry, i.e., flipping the velocity four-vector of an electron makes it an antielectron.

If we restrict ourselves to timelike vectors, then any velocity vector can represent the frame of reference of an observer.

Mathematically either side of the light cone is certainly allowed, because this is a vector space, and one of the axioms of linear algebra is that if v is a vector, there's a vector -v.

Physically, SR is time-reversal invariant, so there is no fundamental physical distinction between the forward and backward directions of time. Therefore there is no fundamental way to state a prohibition on velocity vectors that point a certain way instead of the opposite way. GR inherits all the same local symmetries as SR, so this applies to GR as well. (Spacetimes in GR don't even have to be time-orientable.)

There are reasons that we have an arrow of time, but those reasons can't be found within the fundamental kinematical laws of relativity.

There is at least some kinematical guarantee of sanity, which is that a timelike geodesic always stays timelike. That implies that in SR an inertial observer will never have a velocity vector that wanders from one side of the light cone to the opposite side. (To do that, it would have to pass through the spacelike region.)

For particles, you can interpret this kind of thing in terms of matter/antimatter symmetry, i.e., flipping the velocity four-vector of an electron makes it an antielectron.

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Thank you, bcrowell!

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