Spacetime Symmetry: Find Quick Answer to Simple Question

[ZirkMan]

I've got a simple question I can't find any quick answer to.

I understand that if various observers with different relative kinetic energy (velocity) are to measure the speed of light of the same event the same (c), time and space values must be different for them.
But how do we know which of the variables changes and by what fraction? I believe that the Lorentz transformation take the fraction to be 50:50 for both time and space values. Why not that time slows by 1/3 and space shortens by 2/3 or any other arbitrary fraction?

 

[atyy]

I don't know the proportion of change given by the Lorentz transformations (you can look that up). I do know that the Lorentz transornations are meant to change space and time so if you know the laws of physics in an inertial coordinate system A, then the laws of physics will look the same in another orthonormal coordinate system B moving with constant velocity relative to inertial coordinate system A, ie. the moving orthonormal coordinate system B is also an inertial coordinate system.

 

[Mentz114]

I've got a simple question I can't find any quick answer to.

I understand that if various observers with different relative kinetic energy (velocity) are to measure the speed of light of the same event the same (c), time and space values must be different for them.
But how do we know which of the variables changes and by what fraction? I believe that the Lorentz transformation take the fraction to be 50:50 for both time and space values. Why not that time slows by 1/3 and space shortens by 2/3 or any other arbitrary fraction?

If I understand the question correctly, the fraction of space swapping with time in a Lorentz boost depends on the relative velocity. It's not a constant 50:50.

 

[bcrowell]

I believe that the Lorentz transformation take the fraction to be 50:50 for both time and space values. Why not that time slows by 1/3 and space shortens by 2/3 or any other arbitrary fraction?

In other words, why are time dilation and length contraction both given by the same factor $\gamma$, rather than having $\gamma_{time}\ne\gamma_{length}$?

The answer to any question like this is going to depend on what axioms you choose for special relativity. If you use Einstein's 1905 axiomatization, then $\gamma_{time}\ne\gamma_{length}$ would violate the axiom that the speed of light is frame-independent. If c has a certain value in one frame, and $\gamma_{time}\ne\gamma_{length}$, then in another frame, the ratio of the distance traveled by a ray of light to the time elapsed will have some other value.

 

[ZirkMan]

In other words, why are time dilation and length contraction both given by the same factor $\gamma$, rather than having $\gamma_{time}\ne\gamma_{length}$?

Yes, although I didn't state it like that this is the question I ask.

The answer to any question like this is going to depend on what axioms you choose for special relativity. If you use Einstein's 1905 axiomatization, then $\gamma_{time}\ne\gamma_{length}$ would violate the axiom that the speed of light is frame-independent. If c has a certain value in one frame, and $\gamma_{time}\ne\gamma_{length}$, then in another frame, the ratio of the distance traveled by a ray of light to the time elapsed will have some other value.

Ok, so that means that it is mathematically impossible to have different gammas for time and space and at the same time satisfy the condition of constant c in all frames? If yes, then this is the answer I needed to hear.