Momentary Co-Moving Reference Frame in SR

[Vitani1]

TL;DR Summary

What are the basis vectors?

In SR, for the momentary co-moving reference frame, U (the velocity four vector) takes the form (1,0,0,0). I'm wondering whether the basis vectors associated with this velocity are zero or if the coefficients in front of the basis vectors are zero. In classical mechanics we would say that the coefficients in front of the basis vectors are zero. I ask because the four-acceleration is defined as the proper-time derivative of this velocity and has components (0,x,y,z) hence the dot product of these two vectors is zero like in classical mechanics which makes sense. I also ask this because I'm trying to develop a lorentz transformation that takes a vector U in some frame to the MCRF frame using the lorentz matrix and if I know which it is (components or basis vectors) that cause the last 3 components of the four-velocity to be zero it would save me a lot of work.

 

[Dale]

Summary:: What are the basis vectors?

I'm wondering whether the basis vectors associated with this velocity are zero or if the coefficients in front of the basis vectors are zero.

Basis vectors must be linearly independent. So they cannot be zero.

 

[Vitani1]

Thank you... this is what I thought.

 

[pervect]

There are technically an infinite number of possible basis vectors - as other posters have mentioned, the only real requirement for the basis vecotrs is that they are linearly independent.

There are also an infinite number of possible spatial vectors that have unit length and are orthogonal to the timelike vector <1,0,0,0>. However, for the Minkowskii metric -dt^2 + dx^2 + dy^2 + dz^2, a convenient set of orthonormal spatial basis vectors ortohgonal to the vectors with components <1,0,0,0> are vectors with components <0,1,0,0>, <0,0,1,0>, <0,0,0,1>. Those are probably the vectors you are thinking of when you talk about "the" co-moving reference frame.

Note that I'm using geometrized units where c=1 for simplicity - otherwise <1,0,0,0> wouldn't be a unit length timelike vector.