Lecture 14 Notes:
07 / 25
4-vectors
Interval between events
Last time, we implicitly made use of the concept of an
event
, which is described by the
coordinates (
x, y, z
) and the time
t
.
The coordinates and the time vary depending on the
observer, and can be translated from one observer's frame to another via the Lorentz
transformation.
The coordinates and the time together make up a
4-vector
X = (ct, x, y, z
).
A 4-vector is defined as a 4-component object that takes on different
values for different observers according to the Lorentz transformation.
Suppose we have two events at
X
1
= (
ct
1
, x
1
, y
1
, z
1
) and
X
2
= (
ct
2
, x
2
, y
2
, z
2
).
The
separation
between these two events is the difference between their coordinates in time
and space:
Since
X
1
and
X
2
are 4-vectors,
∆
X
is a 4-vector as well, and transforms according to the
Lorentz transformation.
Moreover, it has an invariant length, which is a
4-scalar
,
meaning that it is the same according to all observers.
The invariant length of the
interval is known as the
invariant interval
.
Let us consider the meaning of this invariant interval.
Suppose that there exists an
observer
B
who is moving in such a way that he is at position (
x
1
, y
1
, z
1
) at
t
1
and at
position (
x
2
, y
2
, z
2
) at
t
2
.
In this case, according to this observer, both events occur at his
own position, that is, at the origin:
x
1B
= x
2B
= y
1B
= y
2B
= z
1B
= z
2B
=
0.
The observer
measures the invariant interval to be
Thus
∆
s
is the
proper time
between the two events, multiplied by a factor of
c
.
Recall
from last lecture that the proper time for a process was the time elapsed for that process
according to the observer who is at rest with respect to it.
In this case, the positions of
the two events are the same according to the observer, so if the events were caused by
the same process, then the observer would be at rest with respect to it.