Physics 325
Lecture 22
Four-Vectors
We have discussed treating time, or
ct
, as a fourth dimension.
We have also seen (HW)
that the Lorentz transformations can be thought of as a rotation in this four-dimensional
space.
All that remains is to define the vectors and operations between the vectors in this
space.
We recall that our three-dimensional rotations preserve the length of vectors as
defined by the dot product.
We therefore would like the rotation in four-dimensional
space, give by the Lorentz transformations, to also preserve the length of our “four-
vector”.
In the last lecture we discovered the Lorentz invariant quantity
(
)
(
)
(
)
(
)
(
)
2
2
2
2
2
1
t
x
y
c
τ
2
z
⎡
⎤
∆
= ∆
−
∆
+ ∆
+ ∆
⎣
⎦
(22.1)
and its twin
(22.2)
(
)
(
)
(
)
(
)
(
)
2
2
2
2
2
s
x
y
z
c
⎡
⎤
∆
=
∆
+ ∆
+ ∆
−
∆
⎣
⎦
2
t
These suggest that our four-vector has three spatial components and one time component
and that the dot product involves a sign difference between the space and time
components.
We therefore define the position-time four-vector as
x
x
ict
⎛
⎞
=
⎜
⎟
⎝
⎠
G
G
(22.3)
where I have used the notation with an arrow below the symbol to denote a four-vector.
This definition of the position-time four-vector, with an
i
before the time component,
allows us to take dot products the way we are used to:
(
)(
)
2
2
2
2
2
x x
x x
ict
ict
x
y
z
c t
=
+
=
+
+
−
G G
i
i
G G
Which is the Lorentz invariant
s
2
in (22.2).
This four-vector has a constant length under
Lorentz transformations (rotations).
You will often see the four-vector and the dot product defined slightly differently.
The
four-vector can be defined without the imaginary component as
(
)
,
x
x ct
=
G
G
and the dot
product defined with the help of a “metric tensor”

This
** preview**
has intentionally

**sections.**

*blurred***to view the full version.**

*Sign up*