Negative Velocities in Lorentz Transformations

[TheConfusedOne]

Undergrad studying engineering here, and my physics class has been doing a unit about intro to special relativity. Essentially, all of our problems and studies concern themselves with velocities which are in the +x direction relative to a "home frame" (I think physicists call this standard orientation). However, I'm curious as to how relativistic effects occur when velocities are allowed to be negative.

To begin I'll summarize an issue:

For example, does a Lorentz Elongation exist? For example, letting Frame A move in the -x direction relative to Frame B is the same as letting Frame B move in the +x direction relative to Frame A. Therefore isn't it equally valid to say that Frame A measures lengths in Frame B along x to be LONGER than they are as resting in B? Of course, one could note that any length has a "proper length", which is measured in a frame which is not moving relative to this length (I suppose by length I mean the distance between two independent "events". In this notation the "rest frame" is defined for which the events occur at the simultaneously). Here is where my troubles begin, however. Suppose there is a defined proper length L. Thus, in a frame with +x velocity, this L is measured as L' which is shorter than L. But, in a frame with -x velocity, according to L'=Lsqrt(1-v^2/c^2) this L' is still shorter, even though I have this weird gut feeling it might be longer. I am concurrently in multivariable calculus and I suspect the equation my textbook gave me fails for -x direction velocities because it has neglected to treat the matter as vectors and instead has opted for simpler scalar equations.

 

[WannabeNewton]

Nope. $\vec{v} \rightarrow -\vec{v}$ just represents an inverse Lorentz boost.

 

[Ibix]

(v)²=(-v)²

Event ordering depends on the direction of motion, but length contraction and time dilation both depend on the square of the velocity, which is independent of the sign.

 

[yuiop]

How would you choose between East and West to determine which is the length expansion direction?

Lets say you choose East and between North and South you choose North as the length expansion direction. This means that an object going North West or South East relative to you, would have no length contraction, an object going North East would have length expansion squared and an object going or South West would have length contraction squared. Since time dilation is intimately connected to length contraction, we would have to have directional time dilation and relative time speeding up in some directions.

 

[PeroK]

I suspect the equation my textbook gave me fails for -x direction velocities because it has neglected to treat the matter as vectors and instead has opted for simpler scalar equations.

Note that a scalar is a 1-dimensional vector. The "V" in the standard Lorentz tranformations is a scalar, but can be +ve or -ve. Although the equations may have been derived using diagrams that show the S' frame moving to the right (+ve), they are valid for the S' frame moving to the left (where V is -ve).

It might be a useful exercise to check this out for yourself.

(I'm just learning this stuff myself, so apologies to the experts!)