# Geometry: space, spacetime, gravity and gravitational waves

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Clearly, what the world needs is another explanation of where gravity comes from. This one comes at it from a slightly unusual direction – geometry all the way. We cover quite a lot of ground, here, and some bits are rather hand-waving, because you can pick up further details elsewhere: we could potentially have book-length footnotes all the way through this; the point I want to make is that there is a very direct story which can be uncovered.

This arose from a conversation with a smart but non-techie friend, which seemed to make sense at the time.

## Distances

FIrst we start with distances; that gives us geometry and straight lines, and then we can go to movement. The theme here is that sometimes the obvious is special and surprising.

Look at the clumps of stones in this picture. We can ask how far apart they are from each other, and get an answer like 10 metres. Or we might get an answer like 30 feet or 21.83 cubits, depending on taste and accuracy.

Stones, 10 metres apart

The distance between these flags is a real thing. If I have a length of string, then it may or may not stretch between them; if I employ someone to dig a trench between the two flags, at so many UK£ per metre of digging, then there's nothing mathematically sophisticated about the invoice.

### Geometry

We can measure the distances between lots of things, and we discover in doing so that the distances between things are not random, and that patterns emerge:

• If I measure the distance to one of the other rocks near the right-hand pole, I'll find it to be quite similar to the distance between the poles.
• If I walk from one flag to the other, and then walk for the same time again, I'll cover something like 20 metres.
• That is, the distances between things aren't just random numbers, but systematically related to the world.

Further, if we look at the distance across the field of view, and then measure how much the left flag is further from the right one...

Stones, slanted

...then we discover that there's a pattern here, too: six metres, squared, plus eight metres, squared, is 100, or 10 metres squared. This is of course just Pythagoras's theorem.

We can keep going at this game, perhaps tie a 10m string to one of the flags, walk in a circle keeping the string taut, and discover that the distance we've walked is 2𝛑 times 10m. And so on and on. If we kept going, we'd discover that we'd reinvented Eucliean geometry – that whole edifice of the sizes of circles, the internal angles of triangles, parallel lines, and so on. For Euclid, Pythagoras's theorem was a consequence of Euclid's geometrical axioms, but we can turn this round and discover that Euclid's geometry is a consequence of Pythagoras's theorem: if you ask the question ‘what do you get if you insist Pythagoras's theorem is true?’, the answer is ‘Euclidean geometry’.

Going down a mathematical layer or two, it is Pythagoras's theorem that gives a meaning to the notion of ‘distance’ on a plane surface. Without this notion, all we have is a bunch of things sitting in a space, with some of them bumping into one another.

### Other geometries

All of this sounds rather trivial. Of course the circumference of a circle is 2𝛑 times its radius; of course parallel lines never meet. I learned this in school! Why are we even talking about this?

It doesn't have to be this way. If you tied a 10m string to the North Pole, and walked in a circle, the circle you'd trace out, before you succumbed to frostbite, would have a length which is pretty much 2𝛑 times 10m. If however the string was 10000km long, and you walked south until the string was taut, then you'd be at the equator (and no longer at risk of frostbite); and the circle you'd draw out if you walked while keeping the string taut would be a little over 40000km, which is a lot less than 2𝛑 times 10000km.

Parallel lines from the equator

Similarly, if you and a friend started at the equator and walked north along parallel paths, you'd end up colliding at the North Pole (in the way that Euclid says you can't if you trace out parallel paths), and if you look at the triangle in the image above, you'll see it has two right angles at the equator, and a non-zero one at the top, meaning that its internal angles add up to more than 180 degrees, which is not what you learned about triangles in school, and which is again contra Euclid.

The rules for distances are different on the surface of a sphere, as well. If I know the latitude and longitude of London and New York, then I can work out the direct distance between them. I don't do that using Pythagoras's theorem, because this is the surface of a sphere, where the rules are different, but using the corresponding distance-theorem found in so-called spherical trigonometry and the mathematical techniques of navigation flogged into midshipmen from the age of sail onwards.

There's nothing freaky about this, and Euclid isn't wrong. It's just that Euclid's geometry doesn't apply to a space which is curved as the surface of the earth is curved.

### Dimensions

A bit of parenthetical terminology:

Euclid describes the geometry of a flat plane. The word ‘flat’ here is a technical term, and means specifically this geometry, where the perimeters of circles are 𝛑 times their diameters, and the angles of triangles add up to 180 degrees, and so on. This plane is also two dimensional: we can go north, or east, or a combination of the two, but that's it. We can also think of a three-dimensional space, where we add the notion of ‘up’. That familiar space is also referred to (by mathematicians, somewhat counter-intuitively) as ‘flat’, since we have familiar calculations such as the surface area of a sphere being 4𝛑 times its radius squared, and its volume being 4/3𝛑 times its radius cubed.

When we're talking about the geometry of the earth's surface, we're talking about the surface of a sphere. Since we're confined to that surface, we can go in the direction of increasing latitude, or increasing longitude (or combinations of the two), but no other: that is, this is also a two-dimensional space, but on this occasion not flat (because Euclid's geometry doesn't work); it's a ‘curved’ space.

## Movement

Classical mechanics is based, ultimately, on Newton's laws. Of those, the second law is best known: you push an object – exert a force on it – and it moves; you push it twice as hard and it accelerates twice as much; you apply the same force to an object twice as massive, and it accelerates half as much.

You can calculate with this law. You can build an industrial revolution on the back of it, predict the motions of the planets well enough to navigate across oceans with it, and use it to build a rocket to carry you, and navigate you, to the moon and back.

The first law is apparently very simple:

Newton's first law (conventional version): Objects move in straight lines at a constant speed, unless acted on by an external force.

In contrast to the second law, the first law always seemed to me rather redundant. Of course things more in straight lines; what else are they going to do?

### Isn't that obvious?

This wasn't obvious, however, to physicists before Newton, and it's not obvious to us before we learn a little bit of physics at school. It goes contrary to our experience, since we see real things around us slow to a stop if they're not actively pushed, since real objects experience friction, always acting as a force against their motion. The physics of Aristotle and his successors, did they but know it, was concerned with explaining motion-in-the-presence-of-friction, and it involved what we now see as mistaken ideas about elements and their natural motions: earth and water naturally moved towards the centre of the universe (at the centre of the earth); air and fire naturally moved away; and the fifth element, aether, which formed the universe above the sphere of the moon, naturally moved in circles (this, incidentally, is one reason why Ptolemy and his successors were so fixated on describing the motions of the planets using circles-within-circles, to the exclusion of any other geometrical figures).

This is why Newton's declaration that a straight line is the natural motion of all matter, above and below the moon, is such a physically significant statement. You could imagine the world being different from this – clearly Aristotle and co imagined this, and we imagine this until we're taught otherwise – but it's not. Just to emphasise this remark: this is a contentful physical statement about a contingent feature of our universe.

### Straight lines

But why a straight line? What's special about a straight line?

What is a ‘straight line’ anyway? Answer: it's the path you trace out if you keep walking straight ahead. That sounds stupidly tautologous, but in fact it's an important operational definition.

You can imagine standing up and walking straight ahead. Perhaps you have a good sense of direction, or perhaps you use a contraption like the parallel ruler used by navigators, and use it to keep your next step parallel to, and in line with, your last one. If you do this, you will trace out what you will in retrospect recognise as a ‘straight line’.

Parallel ruler (image by Rodhullandemu – CC BY-SA 3.0, [wikimedia](https://commons.wikimedia.org/w/index.php?curid=7106686))

But why define this in such a bizarrely indirect fashion? The normal way you'd think to draw a straight line might be using a straight edge such as a ruler to join a start and end point (but that is genuinely begging the question), or else you might be smart, remember that ‘a straight line is the shortest distance between two points’, and stretch a bit of string between the start and end. But both of these techniques depend on you knowing the end-point of your line, whereas the ‘keep walking forwards’ idea allows you to discover the straight line as you go, without any precognition. The importance of the distinction might become evident later.

If you do the same thing on the surface of a sphere – for example, you stand at the equator, face north, and keep walking forwards – you will trace out what counts as a ‘straight line’ in that space: it's known as a great circle to folk who care about naming such things. That great circle isn't a straight line in the three-dimensional space in which the sphere sits (that straight line tunnels through the sphere), but it's the straight-ahead line in the two dimensional space of the sphere's surface.

The usual technical term for one of these ‘straight lines’ is geodesic.

### Newton's mechanics in geometrical terms

All this allows us to rearticulate Newton's first law in geometrical terms:

Newton's first law (geometrical version): The natural motion of objects is to move, at a constant speed, along straight lines (ie, ‘geodesics’) in flat/Euclidean three-dimensional space.

Newton's theory of Gravity hypothesises that gravity is a force emitted by massive objects, which reaches out and tugs on nearby masses, accelerating them in accordance with Newton's second law, and thus pulling them away from the path Newton's first law says they would otherwise follow.

It's important to note that Newton didn't – and couldn't – have anything to say about where this gravitational force came from, and indeed was uncomfortably aware that, in involving action at a distance, it was contradicting what was and is a deep physical principle, that forces and influences can be transmitted only through direct contact (the apparent violation of this principle is why the effects of quantum coherence are regarded with such discomfort). Part of the General Scholium attached to the end of Newton's Principia says (paraphrasing wildly) that this action-at-a-distance gravitational force makes no sense and can't be justified, but that it works, and that's good enough for now.

Hold on to that thought.

## Special Relativity

Three-dimensional space is what you get when you take ‘north’ and ‘east’ and add ‘up’, and with distance defined essentially as with Pythagoras's theorem. Newtonian ‘space-time’, if we can call it that, adds in time as a sort of auxiliary dimension: it views the universe as a sequence of three-dimensional snapshots strung out along the washing-line of time.

In contrast, Einstein's special relativity says that ‘space-time’ is more accurately described in a way which couples space dimensions and a time dimension more intimately than that. It would take us too far out of our way to go into the details of this (though you can find extensive details elsewhere), but if we jump straight to the relevant conclusion, we find that we can fairly naturally discuss motion in terms of a four dimensional space with three space and one time direction, and with a definition of distance which involves all four dimensions which looks similar to, but not the same as Pythagoras's theorem. The time dilation and length contraction which are so exotic in special relativity are, in this point of view, ‘merely’ perspective effects caused by rotation in this space.

Hold on to that thought, too.

### A couple of parentheses

Parenthesis number 1: In the Newtonian framework, a trajectory, such as that of a thrown ball, is a line in three-dimensional space; in the framework of Special Relativity, in contrast, a trajectory is a line drawn through space and time, joining the sequence of positions and corresponding times. In the Newtonian framework, the thrown ball moves along the trajectory like a ball on a wire, and further bits of Newtonian mechanics work out how quickly it moves along that wire; in the SR framework, the movement, through space and time, is ‘frozen’ into the four-dimensional line (this is quite a big deal in SR, but the discussion below doesn't really depend on it, so if it seems too odd, don't worry about it).

Parenthesis number 2: Although it's irrelevant to this discusison, it's curious to note that although a straight line turns out to be the shortest distance between two points in Euclidean space, and on the surface of a sphere, in the space of Special Relativity, a straight line ends up as the longest distance between two points. This is another reason for the ‘one foot in front of the other’ definition of a straight line, and another example of the wide variety of ways that the geometrical question ‘what's the distance from A to B?’ can be answered.

## Another theory of gravity

Here's where we've got to:

• Although we're not used to even noticing it as a Thing, the geometry of a (two-, three- or even four-dimensional) space is an important property of the space. We're intuitively, from the cradle, used to one particular (Euclidean) geometry, but it's not the only possibility.

• The idea of ‘movement’ becomes a geometrical one, and it's natural to express a law of physics – Newton's first law – in geometrical terms.

• In particular, that law links the idea of free or ‘natural’ motion (a physical idea) with the idea of a straight line (a mathematical idea).

Let's first have a final further version of Newton's first law:

Newton’s first law (geometrical version, in spacetime): The natural motion of objects is represented by straight lines (ie, ‘geodesics’) in four-dimensional spacetime.

The difference from the previous version is firstly that we're talking about the four-dimensional space+time of special relativity rather than the three-dimensional-space plus time of the previous version; and secondly that we've omitted ‘at constant speed’, since that's bundled into the idea of the ‘straight line’ in spacetime.

If we're not talking about gravity, then we're in the world of special relativity, and the ‘natural motion’ of objects is exactly the same as the one that we (and Newton) expect – we've changed the ‘why’ but not the ‘what’. In this case, the four-dimensional spacetime in question is the direct analogue of flat Euclidean space, called ‘Minkowski space’.

If, however, we are talking about gravity, because we're talking about the space nearby something heavy, like a planet or a star, then the space we're talking about is not Minkowski space, but something very similar to it, systematically slightly distorted. This is where we get to the crucial point of this essay.

### Einstein's theory of gravity

Newton said (despite his disquiet) that the effect of mass was to create a mysterious force which reached out and pulled things away from their natural motion. Einstein, in his General Theory of Relativity, says that objects continue to follow their natural (Newton's first law) motions, but in a spacetime which is distorted, by the presence of masses, slightly away from Minkowski space.

That is, objects continue to follow their natural motion through that spacetime, according to the final version of Newton's first law, above. That is, Newton's first law doesn't change in General Relativity. The particles don't know and don't care that the spacetime has changed shape, but simply trace out a ‘keep walking forwards’ straight line in that spacetime. The distortion that the mass creates is such that the set of points in space and time, that that straight line traces out, are very, very, nearly the same as the orbits that Newton's gravitational law says will be traced out by a planet moving around the sun. That is, the straight line in the curved (four dimensional) spacetime corresponds to a curved line when projected into the three-dimensional fragment of spacetime, that we think of as ‘space’.

The two orbits are not exactly they same (ie, this is a testable prediction), but the tiny difference between the predicted orbits is just the right amount to account for the long-known defect between the Newtonian predictions of the orbit of Mercury, and the actual orbit.

### Thus...

Thus gravity, in the current picture, is not a force, in the same way that electricity is a force, or being prodded with a stick is a force, or the force that holds together atomic nuclei is a force. And the space(time) we live in is not just the stage on which physics (and chemistry, biology, psychology, sociology and politics) happens, but instead a participant, distorted by the presence of masses. That is, spacetime is a Thing, which has properties and shape of its own, and is not just an abstract set of gaps between objects.

And what that also means is that Newton's cinderella first law (‘why bother even saying this?’) is a contentful physical statement about the way that objects – us – move through the universe.

## And so, finally, to the point

The point of this is to pass on the thought that GR is, from a suitable vantage point, a simplification, and that it is continuous with the physics that preceded it, starting from Newton.

Now, ‘simple’ isn't the same as ‘easy’, and the mathematics which allows you to perform actual calculations with GR is ferociously hard, but conceptually ‘General Relativity’ consists of taking ‘Special Relativity’ – which we already believe is necessary, and correct, to describe dynamics in the special case of no acceleration and no gravity, and which we already believe to be thoroughly supported by experiment – and generalising it to remove the special case. So GR (i) is continuous with what we already believe ot be true, (ii) allows us to get away from the conceptually mysterious gravitational force of Newton, and most importantly (iii) is correct in cases where Newton's theory turns out to be wrong.

Of course, the development of GR hasn't solved physics. Of the four forces in the Standard Model, three of them are plausibly unifiable into a more fundamental force, leaving gravity still special. A quantum theory of gravity would be needed to describe the sort of ultra-exotic circumstance where spacetime is so tightly furled that a ‘gravitational’ force is as strong as the other forces. There are ideas about that, but they're a very separate problem.

## A coda: gravitational waves

It's an additional thought, but with that point above, we can go on to get a slightly deeper picture of what gravitational waves are.

### Waves on a drum-head: Pythagoras the Ant

Look at the image here, which schematically shows the surface of a vibrating drum-head. Let's freeze the motion at two points in time, shown top and bottom, and ask: What's the diameter of this drum?

Waves on a drum

One answer, of course, is that it's the diameter of the ring around the outside. Another (if you're being fancy), is the circumference of the drum-head divided by 𝛑. If you're an ant on the surface of this drum, however, then you'll have a different and very pragmatic answer, which is the answer to the substitute question: how long does it take to walk across the drum? As we've seen from the discussion about the distances between flags, at the top of this essay, there's an argument to be made that this is the most fundamental answer to the question.

In the top case, you can see that, however long it takes to walk across the drum-head from left to right, it'll take slightly longer to walk front to back, because of the briefly distorted shape of the drum-head. That is, the front and back points on the circumference are further apart, as far as the walking ant is concerned, then the left and right points. In the second image, fractionally later, the left-right diameter is longer than the front-back one, for the same reason. Thus two ants standing at opposite ends of a diameter of this drum-head will find, as the drum-head vibrates, that at successive times there is a greater and lesser distance between them, even though the ants are not themselves moving.

There's nothing ‘subjective’ or ‘relative’ about this question: the ant really would take longer to walk back and forth across the drum-head if they were going front-to-back in the top image, than if they were going left-to-right. In the sense illustrated at te top of this essay, distance is a real thing.

### Waves in spacetime: gravitational waves

There is an analogy between this drum-head and gravitational waves.

We can carefully hang two mirrors some distance apart, and use a light beam to measure the distance between them very carefully (we use a laser light-beam to do this because Special Relativity tells us that the time-of-flight of a light beam is a reliable measure of distance, evading a set of puzzles that we needn't go into just here). This is the setup that represents the real 3km-long detectors of the LIGO and Virgo collaborations.

If a gravitational wave travels between those mirrors, then the distance between them changes in a way analogous to the change of the drum-head, even though the mirrors don't move; and this change in distance (which is a fraction of the diameter of an atom) can be detected, just, by the LIGO and Virgo instruments.

Gravitational waves are one of the last tests of GR. But apart from their astronomical significance, they vividly illustrate the way that spacetime is not just a network of separations between objects – a mathematical contrivance – but an experimentally accessible actor in the universe. You have to bang a couple of black holes together to generate waves we can measure, but the fact that they are measurable brings about one of the biggest changes in our understanding of the material inventory of our universe, in the last few hundred years.

Norman, 2021 May 1