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We have seen that the quantity transforms as a 4vector under
a general Lorentz transformation (see Eq. (2.47)). Since
it follows that

(114) 
also transforms as a 4vector. This quantity is known as the 4velocity.
Likewise, the quantity

(115) 
is a 4vector, and is called the 4acceleration.
For events along the worldline of a particle traveling with 3velocity
we have

(116) 
where use has been made of Eq. (2.78). This gives the relationship between
a particle's 3velocity and its 4velocity. The relationship between the
3acceleration and the 4acceleration is less straightforward. We
have

(117) 
where
is the 3acceleration.
In the rest frame of the particle
and
. It follows that

(118) 
(note that is an invariant quantity).
In other words, the 4acceleration of a particle is always orthogonal
to its 4velocity.
Next: The current density 4vector
Up: Relativity and electromagnetism
Previous: Proper time
Richard Fitzpatrick
20020518