Tuesday, 15 March 2016

Is spacetime really curved?

(Spoiler: probably)

In the interests of my relativist’s street cred, I want to expand on my one-line explanation of distortions in spacetime. It is possible that the result will be quite the reverse — I will only reveal my ignorance and attract the scorn of my colleagues. A typical day at the office, really.
I wrote last time that “gravity makes the distance between two objects larger, even if they do not move”. Serious thinkers may legitimately respond, “What does `do not move’ mean? If they are now further apart, isn’t that the very definition of moving?”

This kind of argument comes up all the time when you talk about relativity. In fact, it’s usually even worse. If I say, “The distance between you and someone on the other side of the solar system would be smaller if the Sun were not there,” as a “simple” illustration of how gravity stretches space, then I’ve just done the intellectual equivalent of nose-diving into a bathtub of crocodiles.

How, in the vast wastes of empty space, do I determine that myself and my distant friend are fixed in position? What does “fixed in position” even mean? And if I’ve fought off that one, the next question is, “How can you compare with the case when the Sun is there? Because then they certainly won’t be fixed in position — they’ll be attracted towards the Sun!” What a mess!

Here’s an example I cooked up to try to get around this problem.

Imagine that someone (somehow) goes to the North Pole and drills a shaft down through the Earth all the way to the South Pole. Then they use a laser to measure the distance from pole to pole, through the Earth. They get some number.

Now imagine that someone goes out into space and pumps up an inflatable copy of the Earth. It’s exactly the same size, and exactly the same shape on the surface. I’ve no idea how you would do that, but what matters is that the surface is indistinguishable from the real Earth, but it is almost weightless (i.e., some sort of lightweight plastic, instead of rock and earth and water and so on), and it is hollow, except for the tiny amount of gas needed to keep it inflated in empty space.

Ok, we now have two Earths. The real Earth, packed with rock and magma and whatever else, with a mass of 10^24 kg, and the fake inflatable Earth, which looks identical from outside, but is essentially weightless.

If you measure the distance from pole to pole through the centre of the fake Earth, it will be smaller than for the real Earth. You should have guessed that: the mass of real Earth stretches space and so the distance from pole to pole is enlarged. However, the size of the two Earths is the same, by which I mean that the surface area of both Earths is identical, and the circumference of the two Earths is identical.

We can push this a bit further. Now let’s pump matter into the fake Earth. We replace the air with some heavy stuff. (We leave free our little tube/shaft between the poles, so that we can keep making our laser measurements.) As the Earth gets more massive, the distance between the poles gets larger. But so long as the matter is distributed uniformly, the area of the Earth stays the same. It doesn’t get any bigger.

The two poles have got further apart, but they have not moved!

We can pump in increasingly dense matter. Each time we pump in more matter, our fake Earth gets more massive and the distance between the poles gets larger. But, again, the Earth’s size (it’s surface area) stays the same. We could even make the interior so dense that it was at the very limit of collapsing to a black hole, and now the distance would be very large indeed. (I haven’t tried to work out what the effect would be, so I don’t know just how large we could make this distance, or even if there would be a limit. If someone wants to work it out, that would be great.)

The nice thing in this example is that we have very clearly kept the circumference and area of the Earth fixed, while increasing its diameter. We don’t have any ambiguities about whether anything is moving apart — if you want to say that the two poles have moved further apart, you also have to accept that any group of toy penguins gathered around the pole (remember, this is a fake Earth) have not moved further apart. Space has stretched, and there’s no getting away from it.

There are trickier problems with talking about how gravitational-wave detectors work. There are mirrors 4km apart. A laser beam is sent back and forth to measure the distance between them. When  a gravitational-wave passes, the space is stretched, the distance gets alternately larger and shorter, and the laser measures it. (There is a great sadistic pleasure in making it sound so easy, but only if all experimenters are beyond punching distance.) But does the space stretch, or do the mirrors move?

Anyone foolish enough to pretend to be wise based on my last post will announce, “Of course the space has stretched. That’s what Einstein’s theory says.” But Einstein’s theory also says that we are free to choose coordinates as we wish, i.e., we can imagine a cosmic tape measure where we draw on markings however we like — and can let them move around as well. It is equally valid to interpret what happens as the mirrors moving back and forth, and space staying just the way it is.

The problem with this picture is that it only describes the mirrors. The theory says that the distance changes between everything that the wave passes through -- the entire 4km-long tunnel stretches, and so does the Earth around it. Surely it is more sensible to gently stretch the space, than to apply a very particular force to every single particle in the relevant chunk of the universe?

Certainly the theory is written in terms of the geometry, and I would have thought that was the “standard” way to view what happens. But perhaps not — the LIGO animation illustrating how the detectors work involves mirrors that move back and forth when a gravitational-wave passes through. Behind the mirrors is a nice clean grid, to make it very clear that the mirrors are moving, and not that the mirrors are sitting in place and the space stretching. Was this done just to make a simpler illustration for the public announcement, or was this a careful philosophical choice?

It was only when I tried to argue to colleagues that the animation was wrong that I opened up the interpretation can of worms. I’m going to stick with my view that space stretches and the animation is dubious, on the grounds of an irrefutable metaphysical position known as bloody-mindedness. But I do have one argument for why I think it is more reasonable to think in terms of space stretching.

Imagine some sort of localised gravitational wave, a sort of gravitational-wave laser beam, which is only a few metres across. This beam is fired across the detector arm. For example, it passes through the exact middle of the arm. Imagine that this beam is so intense that it seriously stretches the space that it passes through. We could imagine that the distance between the ends of the detector arm are changed from 4km to 6km. (Ok, I’ve just stretched a few-metres-across section of the detector arm by a factor of a thousand, and perhaps such an insane gravitational wave would induce gravitational collapse, but we can easily scale down this example to less ridiculous magnitudes if we wish.)

Now the distance between the end mirrors is greatly increased. Have either of them moved? In their vicinity nothing has changed. There is no gravitational wave there. They have experienced no acceleration. The mirrors have stayed fixed. They haven’t noticed a thing. All the apparatus near the mirrors have stayed fixed, and haven’t noticed anything, either. No-one sitting at the end-station has the slightest hint that any gravitational wave has passed, no matter how sensitive their local equipment. The only way they can know that something has happened is to read off the measurements from the laser beams, and discover to their amazement that the travel distance has mysteriously changed.

I just made up that example, and probably it is full of holes. If I can’t plug up the holes, then maybe I have to live with the other interpretation.

The point here is that this stuff really is tricky. But the other point, which fluffy thinkers will forget in all this talk of “interpretation”, is that the predictions of measurements are unambiguous. That is what matters. If you are strictly wedded to one interpretation — even to the view that space and time are curved — then maybe you are blinding yourself to the next big discovery. There is a conceptual framework that motivates a theory, and then grows up further around it if the theory is successful, but in the end it is the observations that matter. If you can find another interpretation that is completely consistent with all of the results, that’s perfectly fine, and perhaps it will even lead you to some new predictions and a whole new theory.

One day we might have a completely different picture, and curved spacetime will be a historical curiosity. There might be a whole new crazy picture every 150 years. Fine. Each time our ability to describe the universe will become more refined, more precise. The key observations won’t change. Gravity will still be real, and gravitational waves will still be real. This is a point that I suspect a lot of people do not appreciate.


We detected gravitational waves!

What it feels like to detect gravitational waves.

How to decode gravitational waves from black holes.

Why bother trying to explain gravitational waves?


  1. Pulsar timing makes you think even harder about this. Or it should do.

  2. I think I almost understand this, which is great. And I like your last paragraph - remember phlogiston!

  3. Polite cough: http://i.stack.imgur.com/IlSrh.jpg

  4. All the visualizations of a GW show them as a wave moving freely floating particles in space – so this shows how the GW interacts with freely falling particles in space (not time). If GW are ripples in the fabric of spacetime – is it possible to represent the time part “visually”? http://www.einstein-online.info/spotlights/gw_waves

  5. That's a good point, and it's often forgotten that both space and time are stretching. I imagine one *could* represent the distortions in time as well (a row of clocks displayed along the LIGO arm, for example), but I've never seen it done.

  6. And if one were to (hypothically) compare 2 interferometers – one with the mirrors freely suspended, and another with the mirrors anchored – would both interferometers detect the signal - but at different levels of intensity?

    The question is whether the GW actually stretches and compresses space(time?) more in a vacuum as compared to a solid mass. Recalling Feynman’s sticky bead argument - "Feynman’s gravitational wave detector: It is simply two beads sliding freely (but with a small amount of friction) on a rigid rod. As the wave passes over the rod, atomic forces hold the length of the rod fixed, but the proper distance between the two beads oscillates. Thus, the beads rub against the rod, dissipating heat." (from Wikipedia)

    I must admit that understanding how the interferometers conceptually work is not easy but I do appreciate the explanations you have presented on this topic.


[Note: comments do not seem to work from Facebook.]