line element proper time proper distance differential contravariant four-vector position
The coordinate differentials transform also contravariantly:




d

X


μ
′



=

Λ


μ
′







ν


d

X

ν




{\displaystyle dX^{\mu '}=\Lambda ^{\mu '}{}_{\nu }dX^{\nu }}


so the squared length of the differential of the position four-vector dX constructed using




d


X


2


=
d

X

μ



d

X

μ


=

η

μ
ν



d

X

μ



d

X

ν


=
−
(
c
d
t

)

2


+
(
d
x

)

2


+
(
d
y

)

2


+
(
d
z

)

2




{\displaystyle d\mathbf {X} ^{2}=dX^{\mu }\,dX_{\mu }=\eta _{\mu \nu }\,dX^{\mu }\,dX^{\nu }=-(cdt)^{2}+(dx)^{2}+(dy)^{2}+(dz)^{2}}


is an invariant. Notice that when the line element dX is negative that √−dX is the differential of proper time, while when dX is positive, √dX is differential of the proper distance.
