four-vector four-vectors
The 4-velocity U has an invariant form:






U


2


=

η

ν
μ



U

ν



U

μ


=
−

c

2



,


{\displaystyle \mathbf {U} ^{2}=\eta _{\nu \mu }U^{\nu }U^{\mu }=-c^{2}\,,}


which means all velocity four-vectors have a magnitude of c. This is an expression of the fact that there is no such thing as being at coordinate rest in relativity: at the least, you are always moving forward through time. Differentiating the above equation by τ produces:




2

η

μ
ν



A

μ



U

ν


=
0.


{\displaystyle 2\eta _{\mu \nu }A^{\mu }U^{\nu }=0.}


So in special relativity, the acceleration four-vector and the velocity four-vector are orthogonal.
