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I have friend who has a friend who is having trouble visualising relativity.
My friend’s friend complains that she cannot see that curved ‘world lines’
are shorter than straight lines: ‘They don’t look like that,’ she says.
And of course they don’t. Nor are they. Take a ruler and see. The straight
lines are shorter than the curved ones.

The problem arises because the diagrams are drawn on a page in a book.
Such pages are, unless you drop them in the bath, two-dimensional. Space-time
is four-dimensional. Two-dimensional drawings of four-dimensional scenes
leave a lot to be desired.

I’ve been thinking hard about this problem of curved world lines. Even
though physicists insist such lines are shorter than straight ones, I believe
the straight ones are shorter. We are told to draw analogies with arcs of
a great circle, the shortest distances on the surface of a sphere, such
as Earth, which, when placed on a Mercator projection, are longer than a
straight line between the two points. World lines have to include a fourth
dimension, time, but the example is similar, we are told. Curved world lines
are the shortest distance between two points when the surface involved is
curved. Sounds simple. We should stop thinking about the lines, we are directed,
and think instead of the kind of surface they traverse. The trouble is,
the shortest distance between two points on the surface of a sphere is a
straight line – not the arc of a great circle – that makes use of the third
dimension and tunnels through the sphere from one point to the other. You
can make a lot of money by asking intellectuals what is the shortest distance
between two points that are on the surface of a sphere, and when they smugly
answer, ‘The arc of a great circle, peasant,’ bet them they’re wrong, then
go out and get an orange and a long pin, reminding them that you have said
nothing about the route to be taken, that is, you did not say that that
had to be on the sphere’s surface, only that the points to be connected
were.

A curved world line is the fourth-dimensional analogue of that long
pin skewering a three-dimensional orange. This should please my friend’s
friend because it shows that straight lines are shorter than curved ones,
point to point. It all depends on the route’s topography and what is meant
by ‘straight’, that is to say, how many dimensions you have in mind. If
you are excited by all this you can start thinking about shortest distances
in four or more spatial dimensions; for example, what is the shortest distance
between two points in a hypersphere? In a Klein-Kaluza 10-dimensional space?
And so on. In all cases, the answer is ‘a straight line’ (or geodesic).
It all comes down to just what is meant by ‘a straight line’ in the particular
dimension you are dealing with.

The answer to lots of ‘intractable’ questions often comes down to what
we mean by the words we use. Two men, for separate and unrelated reasons,
decide to go to the market at the same time. They meet in the market. Is
their meeting determined or indeterminate? I have not made up this conundrum;
Aristotle did in his Physics, Book II, Chapter 5. The answer is, of course,
what do you mean by those words? Depending on what we mean by them, we
can then easily go on to decide whether their unplanned meeting is ‘determined’
or not. A lot of quantum theorists would make things a lot clearer to themselves
and the rest of us poor souls if they took the trouble to define what they
mean by ‘determined’ and ‘indeterminate’, by ’cause’ and ‘effect’. The two
men’s meeting in the market seems to be the effect of a cause, or a number
of causes. But has the meeting been determined, is it the effect of causes(s),
or are there no causes that brought about this effect and is the meeting
indeterminate? It’s all a terrible dilemma – until we say how we are using
the words, then the dilemma disappears.

Other words like ‘chance’ and ‘random’ need defining too. This may take
time but in the long run it will save us a good deal of unnecessary confusion.
Radioactive atoms decay in ‘random’ order that is unpredictable. Does this
mean that decay is ‘uncaused’? Light exhibits either particle – or wave-like
characteristics, depending on the instrument we use to observe it. Does
this mean that light is both a particle and wave alternately? I bounce a
tennis ball on a firm, hard surface. The ball reacts a certain way. I bounce
the same ball on a sticky, soft surface. The ball reacts differently. I
drop it down a 12-metre well. It doesn’t seem to react at all. The ball
possesses a ‘complementarity of bounceabilities’? Well, that’s one way of
putting it. But is it the explanation that accounts for the most facts most
simply? I suspect it isn’t. I suspect our problem with the ‘dual’ nature
of light (and particles) is a problem we have constructed by asking the
wrong questions, using the wrong words, or rather, not really defining how
we are using the words.

I suspect that, at the nitty-gritty level, many of our ‘insoluble’ dilemmas
in physics and elsewhere are the result of sloppy words. Physicists are
rarely much good with words, so they prefer talking in mathematics, forgetting
that maths is just another, more concentrated, form of language and no better,
no more accurate and precise, than our definitions of the forms and principles
involved. Things may happen in such-and-such a way in mathematical models,
but that does not necessarily mean it happens that way in the real world
of time and space, the one ‘out there’.

We are all amused by the story of the paranoiac-schizophrenic: a passer-by
says ‘Hello, how are you?’ and the paranoiac-schizophrenic eyes him beadingly
and asks ‘What do you mean by that?’ Do we ever wonder what we do mean
by that? Perhaps it is time we started wondering exactly what physicists
do mean by all the wonderful things they say.

Ralph Estling is a retired schoolteacher living in Somerset.