VISUALIZING THE IMPOSSIBLE

In this occasional series on science, which started a year ago (2/06) with The Search for Dark Matter, I discussed Einstein's concept of special relativity last August in A Personal Journey Through Relativity.  Jack Wheeler provided an engaging account of it two years ago (2/05) in Aristotle, Einstein, and Ayn Rand.

Einstein though did not have a Theory of Relativity  — ‘a' as in one.  He had two.  His Special Theory is about the relativity of time.  His General Theory is about the relativity of gravity.  The puzzles and paradoxes of the first pale in comparison to those of the second.

So much so that most anyone can visualize clocks telling time at different rates and astronauts coming back from another galaxy much younger than the people they left on earth.  The math and explanations may be hard but one can at least visualize the effects of special relativity.

It seems well-nigh impossible to visualize gravity not as a force but as a manifestation of curved space.

I had picked up some of the oddments about the general theory while still in high school.

I tried without success to picture space curving. I could readily imagine the geometry of a block of rubber being distorted, because rubber consists of stuff – but  space is just emptiness.

How can ‘nothing' be curved? And precisely where does it curve?  A block of rubber can curve ‘in' space, but space isn't ‘in' anything!

At this stage I formed the impression that the curvature of space manifested itself by causing the path of the planets to curve around the Sun. The  Earth moved in its elliptical orbit, I believed, not because it was pulled by a gravitational force from the Sun, but because the Sun curved the space in its vicinity.

The Earth merely followed the straightest possible path through this curved space. This seemed to make sense to me, because after all the Earth's orbit is curved, and I knew that even light rays were curved by the Sun.

That must be it, I thought.  Curved space simply means that bodies followed curved paths.  Easy!

Now came a new puzzle:  the Earth's orbit is a closed path.  According to the mental image I had constructed, this implied that space was somehow folded around completely, engulfing the Solar System in a way that would cut it off from the rest of the Universe.

Clearly that could not be right.  The curvature of Earth's orbit was obviously far too great for it to be due to curved apace.

Finally I learned the mistake I made – and it was a subtle one.  The curvature involved was  not one of space all by its lonesome, but of space plus time, what Einstein called spacetime. The difference is crucial.  

Viewed in spacetime, the Earth's orbit is not a closed ellipse, but is shaped like a coiled spring, known as a helix.  Visualize a stretched-out Slinky.

The curvature involves both space and time, and whenever the time part appears in the picture, Einstein's theory requires one to multiply it by the speed of light.  This is a very big number, and it has the effect of enormously affecting the helix.  So although the orbit is tightly curved in space (a closed Slinky),  it is a very shallow curve in spacetime (a stretched-out Slinky).

Spacetime Slinkies:  that's the way the curvature of space can be viewed in terms of the paths of moving bodies, I thought.

I hoped I was now making progress in understanding relativity.  I soon learned greater difficulties were to come, such as Einstein's concept of a ‘closed-but-unbounded' universe.

This concept defeated all my attempts as visualization.  I had still not really got used to the idea that spacetime could be curved without being curved in something.  Now I expected to believe that the whole of space might curve right around so that it joined again on the far side of the Universe.

The pictures didn't really help. Showing the surface of a sphere and saying this was a closed but unbounded surface in two dimensions was all very well, but going from two to three dimensions wasn't the simple extension that the writers who presented this analogy seemed to imagine.

After all, a two dimensional surface can be curved in three dimensions of space, but what could three dimensions curve into? I was stumped by the same old problem.

Then I stumbled on the solution to visualizing the impossible:  reading science fiction  — good science fiction where the writer understands the basic science involved and entertainingly describes its effects.

By reading good science fiction, I got used to picture myself through the characters, seeing an unfamiliar world through their eyes and sharing their experiences.

Even when reading the impossible, you can still imagine what it would be like for certain things to happen.  That really helped me grasp concepts that math and experiments showed to be true but are so elusive to picture.

And anyway, it's a great rationale for my reading habit.

Dennis Turner