THE OTHER WAY in which gravity is different from the forces of the Standard Model is related to the way it is understood within Einstein’s theory of General Relativity. In this theory, it is questionable whether we should even call gravity a force at all.
When getting to grips with the Dirac equation for our journeys across the map, we saw that the ‘special’ theory of relativity arises when you insist that the laws of physics are the same for all ‘inertial observers’ – observers travelling with constant speed in a constant direction. The laws of physics include electromagnetism, so that means that the speed of light must be the same for all those observers, and once you insist on that, all the weird effects on how time passes and how space contracts follow. As does the famous equivalence equation between energy and mass, E = mc2. To get such astounding new physics from such a general principle is remarkable.
But what about other observers? Most of the time we are not ‘inertial observers’. Surely the laws of physics should not change just because we are accelerating or decelerating? It is certainly tempting to insist that physics should be the same for all observers, even ‘non-inertial’ ones, if only because we got such a lot of new and correct physical understanding from doing that for the inertial ones. Anyway, it seems intuitively right, somehow.
That is the challenge Einstein addressed with his general theory of relativity – ‘general’ because it applies to all observers, not just to the special case of inertial ones considered in his previous masterpiece, the Special Theory of Relativity.
What are the effects of acceleration? There is an airport not so far away from us – we will visit this soon on our travels. In an aeroplane accelerating down the runway for take-off, the passengers feel themselves pressed back into their seats. From their point of view this feels like a force, and that’s why they are not inertial observers. Imagine yourself in their place. The water bottle you left on the floor beneath your seat rolls backwards as though someone is pushing it. Objects in your frame of reference do not move with constant speeds, as they should for an inertial observer. From your point of view, objects accelerate backwards. From my point of view, watching from the park, you are accelerating forwards and leaving them behind.
How does this connect to gravity? Well, imagine it is night. It is pitch black outside. You are pressed back in your seat as the plane accelerates horizontally. But could you tell the difference between this situation and the possibility that the aeroplane might be flying upwards at some angle and a constant speed?
In both cases the bottle (or carelessly stowed laptop) would accelerate backwards until it hits something, maybe the back of the cabin. Einstein’s key observation was to notice that it is difficult, perhaps impossible, to tell the difference between a frame which is non-inertial because it is experiencing a gravitational force (the climbing aeroplane) and a frame which is non-inertial because it is accelerating (the horizontal aeroplane speeding up).
Similarly, a frame which is falling freely under gravity looks very much like an inertial frame, in which no gravity acts. In fact, the only frames we’re familiar with which really look inertial are those in free fall; most obviously the frame moving along with the International Space Station (ISS), continually freefalling along its orbit round the Earth.
It is worth thinking about the situation of something in orbit like the ISS, compared to someone or something on, for example, the roundabout in the children’s playground we’ve just noticed across the park from us.
From the point of view of someone on Earth, the ISS is moving quickly, and it would move in a straight line, according to the conservation of momentum, but for the gravitational attraction between it and the Earth. The gravitational force plays the same role as the arms of a child holding on to the roundabout, spinning in the playground we can see. By holding on, the child’s arms exert a force that pulls them continually towards the centre of the roundabout and so keeps them turning around it. Likewise, if gravity suddenly stopped working, the ISS would fly off into outer space. But gravity keeps it circling in orbit. Gravity provides a centripetal force, directed towards the centre of the Earth.
But there are some big differences between the situation of the child and the ISS, apart from the obvious ones involving spacesuits and stuff.
On the roundabout, the child experiences a centrifugal ‘pseudoforce’. They definitely are not in an inertial frame. Anything they drop will fly off the roundabout, away from the axis of rotation in the middle. Yet on the ISS, the astronauts seem to experience fewer forces acting. Not only do they not experience a centrifugal pseudoforce, but they seem to be weightless.
The reason is that in orbit, your weight and the centrifugal pseudoforce completely cancel, leaving you in free fall. This is a very different experience from being flung around a roundabout, for two main reasons.
Firstly, gravity acts on your whole body, and indeed all matter, equally and simultaneously, so you don’t have to grab on to a handle and then have your arms pull the rest of you round the circle. Your arms, legs and the rest are all being acted on by gravity.
The second reason goes to the heart of General Relativity and the reason you are, on the ISS, in fact in an inertial frame.
Mass appears in two important equations – force is mass multiplied by acceleration,fn1 and the force of gravity is proportional to mass. General Relativity works because the mass in those two relations is identical. On the face of it there is no reason that this should be so, but in General Relativity it is built in. That was another of Einstein’s great insights.
This means that in your ISS reference frame, the centrifugal pseudoforce can be cancelled by the force of gravity, not just for one particular mass approximately, but for all masses, exactly, all the time, leaving you in an inertial frame. In fact, because it cancels a pseudoforce in this way, it is legitimate to say that the General Relativity reduces the ‘force of gravity’ to the status of just another pseudoforce.
The principle of conservation of momentum, used to define an inertial frame, still applies and still defines such a frame, but inertial frames now include any frame falling freely under gravity. Bodies falling freely like this are said to be travelling along a ‘geodesic’, which is a redefinition in space and time of what constitutes a straight line, the shortest route between two points.
In the absence of a gravitational field, a geodesic is a straight line in the ‘Euclidian’ sense. Euclid was the first person we know of to define rules for geometry. Amongst these rules, or axioms, is the statement that the maximum number of times a pair of straight lines can cross each other is once, and parallel straight lines never meet each other. If there is no gravitational field around, geodesics are straight lines, and General Relativity, Special Relativity and Newton’s laws of motion all agree that freely moving bodies will travel along a straight line at a constant speed.
But near a large mass, General Relativity states that geodesics bend into curves, or even into the closed ellipse of an orbiting ISS. Space and time, defined by geodesics, are no longer Euclidian. The very meaning of a ‘straight line’ has changed.
Maybe the easiest way to get an idea of what is going on is to imagine two people at the equator a few miles apart, setting off due north, parallel to each other. Though they set off parallel and keep travelling in the same direction – due north – they will eventually meet, at the North Pole, because the surface of the Earth is curved, and defines a non-Euclidian two-dimensional geometry.
Near a large mass, space is curved in three dimensions, and ‘straight lines’ – geodesics – can become orbits. This curvature is what we, and all other masses, experience as gravitational force. This is why gravity is, in some sense, a pseudoforce. It is an effect generated by curves in the geometry of space–time.
This is a very different way of thinking about a ‘force’ from how we think about the forces in the Standard Model. If we think of particles and forces as actors on the stage of space–time, gravity isn’t just another actor, it is something which bends the stage on which the others perform. Also, as is already being discovered in our exploration, the other forces are fully developed in quantum field theories, while gravity is definitely not.