Velocity difference not preserved in special relativity

[Malvia]

Consider single line motion. If an observer sees two objects, and one is seen moving say 50 m/s faster than the other, will all other observers measure the same velocity difference? The answer seems to be 'No' from the velocity addition formula of special relativity?

Thus same velocity difference (i.e. same relative motion) is not preserved in special relativity. Does that not seem counter-intuitive? I understand no observed individual velocity can exceed c so we must give up simple classical velocity addition, but is there any intuitive reason why this must extend to the same velocity difference not being preserved? Intuitively - should relativity not preserve this relative value ?

It would also seem from the velocity addition formula that the 50 m/s difference can be made to vary the full range between 0 and c, depending on the speed assigned to new observer. So there is no sense of any preservation of this relative value.

 

[Orodruin]

No, the velocity difference between two objects is not frame independent in relativity. However, you need to be careful with your nomenclature here, "velocity difference" will often be used to specifically mean the velocity of one of the objects in the rest frame of the other.

I also do not understand why you find this any more counter intuitive than the other relativistic effects. In an object's rest frame, the magnitude of its velocity difference to a light signal is clearly given by c. However, this will not be the case in most frames where the object is moving since the object will have a non-zero velocity and the light still travels at c.

It would also seem from the velocity addition formula that the 50 m/s difference can be made to vary the full range between 0 and c, depending on the speed assigned to new observer.

This is not correct. The maximal velocity difference will be observed in a frame where the objects' velocities have the same magnitude, but opposite directions.

 

[Dale]

is there any intuitive reason why this must extend to the same velocity difference not being preserved?

Certainly. If the velocity difference were 50 m/s in the frame where one object were at rest then in the frame where that object is moving at c - 1m/s if the difference were preserved the other object would exceed c.

 

[Dragon27]

Thus same velocity difference (i.e. same relative motion)

Intuitively - should relativity not preserve this relative value ?

You're clinging to the word "relative" too much. In classical mechanics relative motion means the difference in motion between two objects (you simply subtract one position vector from the other - a simple mathematical procedure). This fits our everyday intuition quite nicely, but our everyday intuition doesn't carry over to special relativity. There the motion of one object relative to the other usually means the motion of one object in the frame of reference of the other, which isn't given by the simple mathematical subtraction of position vectors from each another, like in classical mechanics. So it's better not to call the difference in position vectors (and, by extension, in velocity vectors - which is a simple difference of speeds in the case of a one dimensional motion) relative motion in SR. This quantity is not very useful anyway.

 

[Ibix]

is there any intuitive reason why this must extend to the same velocity difference not being preserved?

How do you measure velocity? Basically you measure the time taken for an object to travel a measured distance.

Length contraction means frames don't agree on the measured distance. Time dilation and the relativity of simultaneity mean that frames don't agree on the time taken. Why would you expect the frames to agree on the velocity?

This is not correct. The maximal velocity difference will be observed in a frame where the objects' velocities have the same magnitude, but opposite directions.

I think this is an illustration of your point about the distinction between relative velocity and separation rate. If I am at rest and an object is moving at 50m/s in my rest frame then there exists a frame (with velocity almost c) where our separation rate is almost zero. Conversely, if there is a frame where our separation rate is almost 2c then there is a frame where our separation rate is 50m/s or lower.

 

[PeroK]

So it's better not to call the difference in position vectors (and, by extension, in velocity vectors - which is a simple difference of speeds in the case of a one dimensional motion) relative motion in SR. This quantity is not very useful anyway.

It seems pretty useful to me for studying the kinematics of a scenario, once you have chosen your frame of reference.

 

[Malvia]

Certainly. If the velocity difference were 50 m/s in the frame where one object were at rest then in the frame where that object is moving at c - 1m/s if the difference were preserved the other object would exceed c.

Thanks, nice answer.