
#19
Mar1312, 04:19 AM

P: 71

Depends on your definition of the dot product, but I see what you mean. But I don't see why [Ux/Ut,Uy/Ut,Uz/Ut] would correspond to a spatial velocity.




#20
Mar1312, 08:08 AM

Sci Advisor
PF Gold
P: 4,862

If U is some tangent vector, and x,y,z,t are unit vectors for some frame, then the dot product of U with such unit vectors expresses U in that frame basis. Then Ux/Ut gives the x speed (well, actually, xspeed/c , but that is just as good). Look at the tangent vector itself expressed in your starting coordinates (cnormed, canonic metric; works the same in any other convention): U = gamma(c,u) Ux = gamma * ux is not the x speed; but note Ux/Ut = ux/c. This feature will be true in any other basis. In particular, in an orthonormal basis with U itself taken as the time unit vector, you get spatial speed of zero  the particle has no spatial speed in its own basis. 


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