Relativistic Vectors: Opposite Direction Assumption

[anantchowdhary]

In special relativity we assume relative velocity of B wrt A is the same in magnitude and opposite in direction as of velocity of A wrt B. Now doesn't this demand that the position vectors in both frames(A's and B's) are the just opposite in direction...?

I mean to ask...how can we take for granted that the position vectors will just be the same in magnitude and opposite in direction ??

Thanks!

 

[JesseM]

Position vectors at what moment in each frame? Because of the relativity of simultaneity, you can't just compare the position vector of B in A's frame with the position vector of A in B's frame "at the same moment" as you could in classical physics.

 

[anantchowdhary]

What i meant to ask is that how is v(ba)=-v(ab) ??

In Newtonian mechanics we do this simply as we consider space to be absolute,so the vector as seen from A's origin to B's origin will be just the opposite of what it is from B's origin to A's origin(we take that for granted i guess)

So isn;t a vector frame specific??

According to SR it should be,shouldn't it?

So as you said,we ca't compare position vectors in both frames just like that,how can we say that the relative velocitues are same in magnitude ??

Thanks

 

[JesseM]

What i meant to ask is that how is v(ba)=-v(ab) ??

In Newtonian mechanics we do this simply as we consider space to be absolute,so the vector as seen from A's origin to B's origin will be just the opposite of what it is from B's origin to A's origin(we take that for granted i guess)

So isn;t a vector frame specific??

According to SR it should be,shouldn't it?

So as you said,we ca't compare position vectors in both frames just like that,how can we say that the relative velocitues are same in magnitude ??

Well, when you talk about the symmetry of velocities in SR you're talking about the velocities of inertial objects which never accelerate, so whatever the value of their velocity is at one moment in a given frame, it should be the same at every other moment in that frame (unlike with position vectors which are constantly changing). The fact that v(ba) = -v(ab) can be derived from the Lorentz transformation, which itself is derived from the two basic postulates of SR.

 

[anantchowdhary]

The fact that v(ba) = -v(ab) can be derived from the Lorentz transformation, which itself is derived from the two basic postulates of SR.

Could you please illustrate this

Thanks!