To talk about Einstein and general relativity, I had to try to explain it. That inevitably meant trying to explain the curvature of space and time. Having categorically poo-poohed the standard curved-space-motion example of a heavy ball on a trampoline, I was obliged to come up with something else. I could have instead declared the effort futile -- a stance with which I have also flirted -- but, as difficult and risky as it is to explain complex scientific concepts to non-experts, I'm not quite ready to concede that it is fundamentally impossible. That would be a cop-out.

So here is my attempt to explain how curved space and time can produce an effect that looks like the force of gravity. I look forward to comments, criticism, and suggestions for improvement, which I will ignore or dismiss at my leisure.

The purpose isn't to explain exactly how the curvature of space-time produces the effect that Newton interpreted as a gravitational force. To see how that effect falls out of Einstein's equations, you have to do the calculations -- and that's important, because, while you might be convinced after my explanation that it's quite reasonable that space-time curvature might lead to a force like gravity, that doesn't mean that it really

*does*predict precisely the force that we observe. A heuristic argument about curvature does not prove that Einstein's theory is correct. For that you need careful calculations and precise experiments. The purpose of the explanation is merely to give you some idea of how it works.

Enough of excuses and weaselling.

The first thing to get out of the way is what it means for space to be curved. Isn't space just emptiness? How do you bend emptiness? How do you stretch and curve

*nothing*?

It's not the emptiness that's distorted. It's the distances between objects that are stretched. It's the rate of flow of time that is slowed down. The presence of the Earth and Sun makes the distance between them larger. If you were to replace each with a peanut, the distance between the two peanuts would be less than the distance between the Earth and the Sun. Similarly, a clock on Earth ticks more slowly than the same clock out in empty space. It is the

*geometry*that is distorted by masses -- distances and times.

Near a massive object, parallel lines can meet, the interior angles of a triangle do not necessarily add up to 180 degrees, and the ratio of a circle's circumference to its diameter may not be quite pi. The geometry has been altered; space-time has been curved.

That doesn't explain

*how*any of this happens. It's just what Einstein's theory says will happen.

If we accept that that's what happens, we can then ask: how is it that stretching distances and slowing down clocks produces a gravitational force? Just because the Sun causes a slight stretching of distances -- and, let's face it, it's an extremely slight stretch -- it doesn't obviously follow that the planets are now going to orbit the Sun.

One way to see how motion on a curved surface can look like a force is as follows. Let's get back to good old Earth, and away from warped space and slow clocks. Let's imagine two people standing at the equator. The map shows (approximately) the equator through Africa; the line passes through Lake Victoria. Now imagine that two people go to different points along the equator, several hundred miles apart, and decide to hike North. They are very serious, and have compasses, and are determined to hike directly North, along perfectly straight lines. And being well versed in Euclid, they know that, since they are travelling in parallel, they will remain exactly the same distance apart.

What will happen? Well, we know that the Earth is a ball, and that if they head directly North, they will eventually reach the North pole. They will have to cross some water, and they may find themselves in a few political hotspots, but with enough determination, ingenuity, diplomacy, and a sufficient set of valid visas in their passports, they can in principle travel directly North and meet at the North Pole.

They certainly don't remain the same distance apart. Their attempt to travel on parallel paths has been thwarted -- eventually their paths meet!

That's not so surprising when we look at a globe of the Earth. We understand this perfectly: the Earth is curved, and so parallel lines don't work the same way as they do on a flat surface.

But now imagine that these intrepid hikers, in addition to having a very particular and uncompromising idea of a fun way to spend their time, are also dedicated flat-Earthers. They believe that the Earth is flat. As they travel they keep track of the distance between them. For example, they could hold tight a very long piece of thread.

One of them crosses the Mediterranean, and the other hikes directly through the Holy Land. Somehow they not only manage to negotiate a daunting series of obstacles, geographic and political, but they also manage to keep holding their thread pulled tight. By the time one has reached Poland and the other Russia, they realise that something has gone wrong. They have been very slowly reeling in their thread. They have moved closer.

There is only one sensible conclusion: there is a mysterious force at work. As much as they endeavour to travel along straight lines, this mysterious force pulls them towards each other. It is like swimming directly across a flowing river. You can at all times be swimming directly towards the opposite bank, but when you get there you find that you've gone downstream.

We know that the explanation is a curved surface. But if they don't realise that, and are convinced that the Earth is flat, then they must conclude that they were subject to some invisible force.

That's how Newton (and two hundred years worth of successors) saw gravity. He assumed that the geometry of space is as described by Euclid, and therefore concluded that planets orbit stars and moons orbit planets due to a mysterious, invisible force. And let's face it, that was a perfectly reasonable assumption to make, so reasonable that he probably wasn't aware that he was making it. It was only after that explanation no longer worked -- in particular, Newton's force leaped across space at infinite speed, and Einstein knew (for other reasons) that it had to be limited by the speed of light -- that the real explanation could be found.

Does my little hiking example explain how gravity works? Of course not. Planets don't hike through space. The motion they follow depends on their mass and speed, while the hikers would follow the same paths irrespective of whether they travelled by foot, amphibious tank, or jet aircraft. All the example does is show how motion in a curved geometry can look like motion that is affected by a force.

A more important question: does this explanation help anyone to understand anything? Unfortunately, I was not allowed to quiz the audience members before the talk and ask them whether they understood curved space-time, and identify those who did not and then afterwards ask whether they had (a) paid close attention, and (b) now understood. Even if I had done those things, I would also have had to wait until the next day and eavesdrop on them repeating the explanation to someone else, to see if they really understood it. Having failed to do any of that, I have no idea whether this is a useful explanation or not.

One of my in-laws is a science groupie (of the smart kind) and he told me he was frustrated by the usual popular explanation of general relativity (small ball rolling on a trampoline curved by a big bowling ball in the middle) for the following reasons:

ReplyDelete1) The trampoline is curved, he could see *that*. But it's curved because there is an extra dimension in which to curve!

I explained to him that space can be curved even without being embedded in a bigger flat space and you can test this by triangulations.

2) The small ball seems to rotate around the bowling ball, but a child could tell you that that's because there is an even bigger ball (the Earth) pulling both of them down. You use Earth's gravity to explain gravity! I told him to relax and poured him and myself a glass of wine.