Chapter 5.3 Spacetime

Upon giving up the hypothesis of the invariant and absolute character of time, particularly that of simultaneity, the four- dimensionality of the time–space concept was immediately recognised. It is neither the point in space, nor the instant in time, at which something happens that has physical reality, but only the event itself. There is no absolute relation in space, and no absolute relation in time between two events, but there is an absolute relation in space and time . . . Upon this depends the great advance in method which the theory of relativity owes to Minkovski. (Albert Einstein, Meaning of Relativity)

Before relativity, it was thought that the world and objects in it were all three-dimensional. This meant everything had a length, breadth and height. It is hard to say what a dimension is. We picture a dimension as a long, straight line. If three straight lines can be drawn at right angles to each other, the space they are drawn in is three- dimensional. On a piece of paper, only two straight lines can be drawn at right angles, so the paper is two-dimensional.

Lines through space can be used to name the locations of objects.

On the two-dimensional surface of the earth, places can be located by their latitude and longitude. In three-dimensional space, we can name places using the three coordinates (x, y, z). What would it mean to say that the world is really four-dimensional? It is easy to locate all events in space and time because each has a place and a date. Each event could therefore be given four coordinates (x, y, z, t). In effect, we routinely recognize that events have four dimensions when we agree to meet someone for coffee at a certain place and time. Thus it is true but trivial to say that events have four coordinates. When some physicists say the world is four-dimensional they are making a different and much stronger statement. To understand their claim, we begin with a fact, and then consider its interpretation.

Suppose there are two events, A and B, each at a different place and time. Using rulers and clocks we could measure the distance between the two places and the duration between the two times.
These two numbers indicate how separated the events are in space and time. But, as we have seen, these distances and durations are not invariant and therefore are not real properties of anything. Now something magical happens. Although neither the distance nor the duration is invariant, together they do form an invariant number.

They are combined using a peculiar recipe. The distance and duration are treated as if they were two sides of a right-angled triangle. Using a formula very similar to Pythagoras’ theorem, we calculate the length of the “third” side of the triangle. This new number is invariant. Since it is some kind of combination of space and time, it is called the spacetime interval.

Measurements of distances and durations depend on speeds, since they will contract or dilate, but the spacetime interval between two particular events is always the same: everyone who measures and calculates will find the same number. This is surprising. How do two variable numbers combine into a constant? One way of thinking of it is that, at higher speeds, the lengths shrink and the times elongate, and these variations compensate for each other. All physicists agree that the invariance of spacetime intervals is a fact. That is just a statement about numbers that we calculate from measurements. But what does this imply about reality? How should we interpret that invariance?

As we have seen, the mainstream interpretation of relativity denies the existence of real distances and durations. They are not real properties of individuals. This is, however, a purely negative doctrine. But clearly things are separated from each other in space as well as in time. What are they separated by? The mainstream physicist answers that, while neither space nor time exists in its own right, the combination of them does.

Things are separated from each other by stretches of spacetime.

The central argument for the reality of the spacetime interval was made by Hermann Minkovski. In essence, he argued that if all measurements give the same value for a property, then the property must be real. If there were a painting so beautiful that everyone fell down instantly babbling about its beauty, then we would conclude that beauty really was a property of the painting. Likewise, Minkovski argues that if the spacetime interval appears the same in all measurements, then it must be real.

The positive doctrine of the mainstream interpretation is thus found in the following argument:

Reality of the spacetime interval

• A. The apparent (measured) spacetime interval is invariant. (P)
• B. If an apparent property is invariant, then it corresponds to a real property. (P)
• C. Therefore, the spacetime interval is a real property. (from A,B)

It is now clear that the mainstream interpretation denies three- dimensional distances and durations, which are not real properties of individual things, but affirms the existence of four-dimensional spacetime intervals, which are real properties of events.

Minkovski helped clarify the meaning of the spacetime interval with his well-known rotation analogy. Consider some three-dimen- sional object such as a sculpture of Venus. As we view it from different angles, its width may change. It may seem wide when viewed from the front, but seem narrow when viewed from the side. Minkovski said that spacetime is real, but that different sets of rulers and clocks are all “viewing it from different perspectives”. According to one set, a spacetime interval may appear short in space and long in time, but another set may find it long in space and short in time. More crudely, it might be said that when we treat distances and durations independently, we are arbitrarily chopping up a spacetime interval into so much space and so much time. Another observer may choose to chop it up differently, into less space and more time.

This rotation analogy also explains what it means to say that distances and durations are relations. Suppose that the sculpture of Venus sits in a space, and that we choose three lines in the space to be the mutually perpendicular x-axis, y-axis and z-axis. Given these lines, we can say the sculpture has, say, a length of two metres along the x-axis and three metres along the y-axis. But these lengths are relations between the three-dimensional shape of the statue and certain lines. If we chose different lines to be our axes, then the “length along the x-axis” would change. In short, “length along the x- axis” is not a property that depends only on the individual statue; it is a relation between the statue and a direction.

Likewise, the mainstream interpretation asserts that four- dimensional “shapes” and intervals are real. Choosing an x-axis in space and a time axis defines the distance and duration of a four- dimensional shape (an event). But if the directions of these axes change, then the distance and duration change. They are relations between the four-dimensional shape and certain directions.

Spacetime intervals may also be helpfully interpreted as “sizes” of events. In three-dimensional space, volume is length multiplied by breadth multiplied by height. Likewise, in four-dimensional space, the four-dimensional volume is duration multiplied by length multiplied by breadth multiplied by height. Thus a tennis match may take three hours and fill a tennis court, and we can calculate the four-dimensional volume of this event. This region of spacetime has an invariant volume, even though the length of the court and the duration of the match are relative to the clocks and rulers used to measure them.

To summarize, the mainstream interpretation makes several claims:

• distances and durations are not real properties of individuals

• nor are they mere appearances

• spacetime intervals are invariant and therefore real

• distances and durations are relations between spacetime intervals and directions in spacetime.

Einstein always insisted on the first two ideas, and later accepted Minkovski’s interpretations of the spacetime interval.

The minority interpretation accepts, of course, the fact that the spacetime interval is invariant, but it interprets it as a mathematical accident. Movement through the ether causes lengths to contract and clocks to slow. Since these two processes have “opposite” effects, we should not be surprised that, if we combine both in a calculation, they cancel and leave a constant. Lorentz never thought that the invariance of the spacetime interval was important.