Four-position

A point in Minkowski space is a time and spatial position, called an "event", or sometimes the position four-vector or four-position or 4-position, described in some reference frame by a set of four coordinates:

R

=

(

c
t
,

r

)

=

[

c
t

x
(
t
)

y
(
t
)

z
(
t
)

]

{\displaystyle
\mathbf{R}= \left(ct, \mathbf{r}\right) =
\begin{bmatrix}
ct \\ x(t) \\ y(t) \\ z(t)
\end{bmatrix}
}

where **r** is the three-dimensional space position vector. If **r** is a function of coordinate time *t* in the same frame, i.e. **r** = **r**(*t*), this corresponds to a sequence of events as *t* varies. The definition *R*0 = *ct* ensures that all the coordinates have the same units (of distance). These coordinates are the components of the *position four-vector* for the event.
The *displacement four-vector* is defined to be an "arrow" linking two events:

Δ

R

=

(

c
Δ
t
,
Δ

r

)

{\displaystyle \Delta \mathbf{R} = \left(c\Delta t, \Delta \mathbf{r} \right) }

For the differential four-position on a world line we have, using a norm notation:

‖
d

R

‖
2

=

d
R
⋅
d
R

=
d

R
μ

d

R
μ

=

c
2

d

τ
2

=
d

s
2

,

{\displaystyle \|d\mathbf{R}\|^2 = \mathbf{dR \cdot dR} = dR^\mu dR\_\mu=c^2d\tau^2=ds^2 \,,}

defining the differential line element d*s* and differential proper time increment d*τ*, but this "norm" is also:

‖
d

R

‖
2

=
(
c
d
t

)
2

−
d

r

⋅
d

r

,

{\displaystyle \|d\mathbf{R}\|^2 = (cdt)^2 - d\mathbf{r}\cdot d\mathbf{r} \,,}

so that:

(
c
d
τ

)
2

=
(
c
d
t

)
2

−
d

r

⋅
d

r

.

{\displaystyle (c d\tau)^2 = (cdt)^2 - d\mathbf{r}\cdot d\mathbf{r} \,.}

When considering physical phenomena, differential equations arise naturally; however, when considering space and time derivatives of functions, it is unclear which reference frame these derivatives are taken with respect to. It is agreed that time derivatives are taken with respect to the proper time

τ

{\displaystyle \tau}

. As proper time is an invariant, this guarantees that the proper-time-derivative of any four-vector is itself a four-vector. It is then important to find a relation between this proper-time-derivative and another time derivative (using the coordinate time *t* of an inertial reference frame). This relation is provided by taking the above differential invariant spacetime interval, then dividing by (*cdt*)2 to obtain:

(

c
d
τ

c
d
t

)

2

=
1
−

(

d

r

c
d
t

⋅

d

r

c
d
t

)

=
1
−

u

⋅

u

c
2

=

1

γ
(

u

)
2

,

{\displaystyle \left(\frac{cd\tau}{cdt}\right)^2 = 1 - \left(\frac{d\mathbf{r}}{cdt}\cdot \frac{d\mathbf{r}}{cdt}\right) = 1 - \frac{\mathbf{u}\cdot\mathbf{u}}{c^2} = \frac{1}{\gamma(\mathbf{u})^2} \,,}

where **u** = *d***r**/*dt* is the coordinate 3-velocity of an object measured in the same frame as the coordinates *x*, *y*, *z*, and coordinate time *t*, and

γ
(

u

# )

1

1
−

u

⋅

u

c
2

{\displaystyle \gamma(\mathbf{u}) = \frac{1}{\sqrt{1- \frac{\mathbf{u}\cdot\mathbf{u}}{c^2}}}}

is the Lorentz factor. This provides a useful relation between the differentials in coordinate time and proper time:

d
# t
γ
(

u

)
d
τ

.

{\displaystyle dt = \gamma(\mathbf{u})d\tau \,.}

This relation can also be found from the time transformation in the Lorentz transformations.

Important four-vectors in relativity theory can be defined by applying this differential

d

d
τ

{\displaystyle \frac{d}{d\tau}}

.