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Lecture 14 Notes:
07 / 25
4vectors
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
4vector
X = (ct, x, y, z
).
A 4vector is defined as a 4component 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 4vectors,
∆
X
is a 4vector as well, and transforms according to the
Lorentz transformation.
Moreover, it has an invariant length, which is a
4scalar
,
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.
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View Full DocumentThus, the invariant interval
∆
s
is equal to
t
0
, the proper time that elapses between the
two events.
This is the time from one event to another according to an observer who is
present at one event and moves uniformly in a straight line, in such a way as to be
present at the next.
Defined this way, this quantity is clearly a 4scalar:
different
observers might measure different times between these events, but they all agree that an
observer moving from one event to another would measure a length of time equal to
s
.
Note that
s
2
can be negative, implying an imaginary “proper time” between the two
events.
This happens when
Thus if is
s
2
negative, the observer would have to move faster than the speed of light to
get from event 1 to event 2.
This is not possible; thus there is no proper time between
the two events, and
s
2
does not have this physical interpretation.
However, it can be
shown that in this case, there exists an observer for which the two events are
simultaneous.

s
2
then gives the square of the distance between the two events
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 Spring '07
 Smith

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