76 5. THE PRESENT
And so for straightforward Euclidean geometry, it is clear how to add a fourth dimen-
sion of time; simply add another term of c
2
t
2
. But what Minkowski showed is that the
consequences of special relativity follow from a geometry that is almost straightforward four-
dimensional Euclidean geometry. e distance formula, however, is slightly different, with a
negative sign distinguishing the time coordinate from the space coordinates:
S
2
D d
2
´ c
2
t
2
(5.10)
S
2
D x
2
C y
2
C z
2
´ c
2
t
2
; (5.11)
where S
2
is called the spacetime interval,
3
and it is an example of an invariant—a quantity that
is the same regardless of the frame of reference. In Newton’s laws, both the time between events
and the lengths of objects are invariant. But special relativity shows us that this only seems to be
true at speeds much less than the speed of light. At speeds near the speed of light, both lengths
and intervals of time are relative—this is the meaning of the relativistic phenomena of length
contraction and time dilation. But the spacetime interval, S
2
, is the same for all observers.
It is easy to show that both time dilation and length contraction follow directly from the
invariance of this spacetime interval. And so Minkowski’s geometrical interpretation of Ein-
stein’s special relativity has the advantage of mathematical simplicity and elegance. But there is
more to it than that, as Einstein recognized. He used Minkowski’s idea of a geometrical inter-
pretation of dynamics as the starting point for general relativity, his theory of gravity. We take
up this topic in Chapter 11, Section 11.2.
5.2 RIGHT NOW AND RIGHT HERE
e concept of look-back time (Section 4.3) means that our view of the universe necessarily
connects space to time. To see is to look back in time, and the further away one looks, the more
into the distant past one peers. “A long time ago” really is “in a galaxy far, far away.” But Einstein’s
relativity tells us that this is not simply a consequence of our particular vantage point from Earth.
ere are real, physical consequences to the fact that space and time are ultimately connected by
the speed of light. Since no influence can travel faster than light, spacetime is intertwined with
the very relation between physical causes and their effects.
5.2.1 SIMULTANEITY AND THE MEANING OF LOCAL
Because we look back in time as we look out into space, we see the present only in the near-by.
It makes human sense to want to know what is happening in some distant galaxy “right now,”
but we must be cautious about such notions. One lesson of Einstein’s special relativity is that
time is not absolute. And so why should we expect there to be a universal “present” that applies
equally both to myself and some alien space-rodent in a galaxy millions of light years distant?
3
Some authors, following Minkowski’s original paper, assign the negative signs to the space coordinates and the positive
sign to the time coordinate. e mathematical formalism is slightly different, but the physical consequences are the same.
5.2. RIGHT NOW AND RIGHT HERE 77
An event is a particular point in space and time. We use the term simultaneity to refer
to different events that happen at the same time. For Isaac Newton, any two events are either
simultaneous or they are not. And if Event B happened before Event A, then that is a fact upon
which all agree. But with special relativity, simultaneity is relative. In fact, even the ordering of
two events may depend upon the relative motion of the observers.
When referring to two events in space and time, it is useful to distinguish between two
possibilities.
1. If it is possible for a particle, moving at less than the speed of light, to travel from Event A
to Event B, then the two events are said to be separated by a timelike interval. If the interval
between two events is timelike from the point of view of any one inertial frame of reference,
then it will also be timelike from the point of view of every other inertial frame of reference.
2. If it is not possible even for a light signal to travel from Event A to Event B, then the two
events are said to be separated by a spacelike interval. If the interval between two events is
spacelike from the point of view of any one inertial frame of reference, then it will also be
spacelike from the point of view of every other inertial frame of reference.
And so a spacelike interval is such that the distance, d , measured in light years between the two
events is greater than the number of years, t, that a light signal would require to travel that
distance. A timelike interval is one for which the opposite is true. It should be clear that in terms
of our spacetime interval, as defined in Equation (5.11), S
2
is a positive number for a spacelike
interval and a negative number for a timelike interval. We can also talk about the borderline
case between these two intervals, whereby it requires exactly the speed of light to travel from
one event to the other. Such an interval is called lightlike.
If S
2
can be negative, what about S itself? Wouldn’t that involve the square root of a neg-
ative number, resulting in so-called imaginary numbers? Indeed it would, and therein lies much
of the power of Minkowski’s geometric interpretation of special relativity. e mathematics in-
volving imaginary numbers is called complex analysis, and it lends itself to a richer theoretical
structure than the ordinary real numbers associated with Newton’s laws. And although imagi-
nary numbers and complex analysis can be used in the mathematics of special relativity [Penrose,
2004, p. 414], it always works out that calculated quantities that can be measured are ordinary
real numbers.
e distinction between spacelike and timelike intervals is crucial, because nothing can
travel faster than light. And so if two events are separated by a spacelike interval, then it is impossible
for either event to have caused the other. Events separated by a timelike interval, on the other hand,
may be causally connected; it is physically possible that one of the events might have caused the
other.
In special relativity, simultaneity is relative. If in some particular frame of reference two
events happen at different places but at the same time, then there will always be other frames
of reference for which one of the events happened before the other. But notice that if this is
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