Gravity is not a force

Nor is it a gauge theory, and this ought to motivate us to consider alternatives to quantum gravity.

The debate about whether gravity is a force has just flared up yet again so I thought now would be a good time to give Substack a start, and wade in. The debate is partly semantics, and sometime a bit of physics-trolling but it’s weirdly heated, I think because it hinges on differing attitudes to geometry and field theory.

So, while all the arguments in this debate have been well rehearsed across the internet, I think it’s worth outlining here, since I think it gets to the heart of Einstein's theory of gravity, General Relativity, and what makes it different to the quantum forces of nature. It’s also a good starting point for this Substack, because in a subsequent post, I will explain that not only is gravity not a force, it is not a gauge theory (like electromagnetism). I'll argue that this makes it important to question whether we should be attempting to reconcile quantum theory and gravity by quantising spacetime. The interactions we know how to quantise are gauge theories, but gravity is not a gauge theory.  This is what forms part of my motivation for considering an alternative to quantum gravity, the postquantum theory of classical gravity.

In this post, I’ll just consider the question of whether we should call gravity a force. Newton certainly did, and Newton's law of gravity superficially looks like Coulomb's force law for electrostatics, so it would be natural to call gravity a force. But we've learned a lot since Newton’s time, and we can now be more precise about what constitutes a force.  To explain this, I’ll start with Einstein’s equivalence principle, but mostly to say that I think we talk too much about the equivalence principle! Instead, we can jump straight to the conclusion Einstein drew from the equivalence principle — that we can describe gravity purely in terms of spacetime geometry. If you accept that we live in a spacetime which can have curvature, then I argue you should conclude that no force is acting on a test particle moving freely in this curved spacetime. Hence gravity is not a force. To call gravity a force, is to privilege flat space as somehow being special.

To put it all together in advance: I will argue that gravity is special – it is not a force; it is not a gauge theory; and only gravity is describable in terms of the curvature of spacetime. If we take this geometric picture seriously, I will argue that this should lead us to question whether we should be seeking a quantum theory of gravity in which we describe gravity through gauge bosons evolving in flat spacetime rather than as a theory of spacetime.

The equivalence principle is a consequence of a principle

Let’s first consider Einstein’s famous thought experiment with Alice trapped in an elevator that is hurtling towards the ground because some fool has cut the rope which held it up. The equivalence principle states that from inside the elevator, Alice can’t tell if she’s freely falling towards the Earth, or instead floating gently in the outer regions of space. Or the equivalence principle can be stated as the equivalence of inertial and gravitational mass. Einstein’s elevator pitch of the equivalence principle is partly true, since if Alice were a point particle, she couldn’t tell if she was falling or floating. But it’s not strictly true, since in the case where she is falling, her feet are falling faster than her head, and she could use this information to tell whether she is falling or floating. This stretching of Alice is what we sometimes refer to as "tidal forces", (or in the extreme, case, "spaghettification"), but as we will see, this tidal force is not really a force.

The elevator version of the equivalence principle doesn't apply to extended objects or test particles which are themselves very heavy, and I think many statements of the equivalence principle often create confusion. We can gain more insight by instead focusing on the underlying reason for the equivalence principle. Namely, that gravity can be described purely in terms of spacetime curvature. This is powerful, because general relativity then becomes almost inevitable. We start by recognizing that spacetime can have curvature. If spacetime could have curvature there is no reason to insist it  must be flat. What would  constrain spacetime to be flat? And if we embrace the dictum that what is not forbidden is required, then we live in spacetime which can have curvature. We are then led to consider geometric terms in the action such as the Einstein-Hilbert term, and this then gives us Einstein’s equations which tells us how matter causes spacetime to bend. So once you accept that we live in spacetime, and it can be curved, you are led to General Relativity which describes gravity in terms of geometry.

If you accept that we live in spacetime, and it can be curved, then I think you should accept that gravity cannot be a force. If no forces acts on a test particle, how should test particles in a curved spacetime move? Newton taught us that in flat space, a test particle moves in a straight line if no forces act on it. We want to generalise Newton’s observation to the case where spacetime is curved. Since in flat space, inertial motion is the shortest distance between two points, I think the best way to generalise it, is to say that when no force acts on a test particle in curved space, it should move along a geodesic. Inertial motion corresponds to taking the shortest distance between spacetime points.

In other words, in the absence of being pushed or pulled, test particles in a curved spacetime will free fall. In a metric theory of gravity, that is what happens. Heavy objects like the sun, the earth and 1 kg masses can cause spacetime to bend, but have no other gravitational effect. The Earth orbits the Sun because the Sun bends spacetime, and the Earth freely falls along this curvature. Since the Earth is following a geodesic in curved space, it is following an inertial trajectory -- no force is acting on it. This is usually illustrated through a diagram in which space bends and forms a funnel. But it is the stretching of time which is the dominant effect resulting in Newton’s inverse square law for gravity, so I’ve used a clock to illustrate this. Newton’s law of gravity simply follows from the fact that time flows at a different rate at different points in space

Time dilation causes the Earth to go around the Sun, which we imagine to be at the centre of this disk. Each cell represents ticks of the clock. As the minute hand sweeps the disk, there are more ticks of the clock further away from the Sun. The Earth moves in an orbit around the Sun because it takes a path which has the fewest clicks of a clock on Earth.

Calling gravity a force, requires flat spacetime to be privileged.

Those who argue that we should call gravity a force are essentially arguing that inertial motion requires spacetime to be flat, and they want to define a force to be any deviation from that. I think that's a defensible position, made by reasonable people. But I think this approach privileges flat space as being special.  And flat space can't be that special, because we don't live in a flat spacetime, we live in a spacetime which is closer to an expanding space time (Friedman-Robertson-Walker). And in general relativity, or quantum field theory in curved space we can take any spacetime as our background spacetime, and consider perturbations around it. In a background independent theory like General Relativity, here is nothing special about flat space.

So if you want to claim that test particles are subject to the force of gravity, you won't be able to define inertial motion without saying that inertial motion can only happen in a flat spacetime. And that privileges flatness, which seems antithetical the General Relativity. I think it’s for this reason that in this debate, it’s mostly General Relativists who tend to argue that gravity is not a force, and high energy theorists who claim that it is.

There is another reason people argue that gravity is a force, which is that an extended freely falling object experiences "tidal forces". Tidal forces occur because Alice's toes are falling slightly faster than her head. She can therefore tell whether she is freely falling near the Earth vs floating in outer space. But I don't think this is relevant, since she is still freely falling in both cases  -- in the falling elevator, all the atoms in her body try to follow a geodesic. Both her toes and her head want to free fall -- the only force that is acting on them is the force that her body exerts to keep her feet and head together. Likewise, in the falling elevator, the speed at which she falls changes slowly with time, so even looking at one point, Alice can tell whether she is freely falling near the Earth, or floating in outer space. But in both cases, we should conclude that she is freely falling because all her individual atoms follow geodesics unless a force acts on it.

Is this just semantics?

Spacetime is just three dimensional space and time taken together as a manifold, and with a notion of distance. The key point is that if we take this concept seriously, then there is no reason for spacetime to be flat. And if it can be curved then we need to generalize what we mean by inertial motion to geodesics. If that’s the case, then I think we’re led to the idea that gravity is not a force.

Is this purely a semantic difference? You could argue that it doesn't really matter whether we describe gravity as a force or through geometry, and we should conflate these two concepts.  But I think this distinction is important to make because it has predictive power. If you believe that gravity is manifest through spacetime bending, then you will never find two different test particles that follow different geodesics. It was Galileo at the tower of Pisa who first demonstrated that two test particles of different masses follow the same geodesic. Experimentalists continue to test this, but if you believe that gravity is a metric theory, then you never expect these experiments to demonstrate a deviation from this.

Gravity is unique in being describable geometrically. Other forces can be partially recast as geometry, but not in a universal way. You can’t recast electromagnetism as geometry because you would need a different geometry depending on the charge of the test particle. Sources alone could not determine the geometry. This is what makes gravity special. Test particles in gravity have no gravitational charge. This is the essence of the equivalence principle, which allows us to describe gravity through geometry. No other interaction, including fictitious ones we invent, or any we may discover in the future can have this property.

I’m sympathetic to the idea that spacetime is not “a thing”, and merely a convenient way to describe gravity. Especially as the way we move through time, and the concept of the present, do not seem to be encoded in our notion of spacetime. Indeed, geometry and quantum fields are mathematical constructs we use to describe the world around us, and their ontological significance will change as our understanding of nature changes. But history has taught as that these mathematical objects have predictive power, and so we should take the concept of spacetime seriously, just as we treat quantum fields as having ontological significance. And if we take the concept of spacetime seriously, and generalise our notion of inertial motion to hold in any spacetime, then I think you are compelled to say that gravity is not a force.

Gravity is not a gauge theory....

I'll next write a Substack post explaining why gravity is not a gauge, and why that should lead us to question whether spacetime has a quantum nature.