# Why does matter curve space and time?

This is one of those questions that has always bugged me.

Suppose that, somewhere in the universe, there is a very large closed box made out of some kind of heavy, neutral matter. Inside this box a civilisation of intelligent creatures have evolved. They are made out of normal matter like you and me, except that for some reason they are very light — their bodies do not contain much matter at all. What’s more, there are no other heavy bodies or planets inside this large box aside from the population of aliens, whose total mass is too small to have any noticeable effect on the gravitational field. Thus, the only gravitational field that the aliens are aware of is the field created by the box itself (I’m assuming there are no other massive bodies near to the box).

Setting aside the obvious questions about how these aliens came to exist without an energy source like the sun, and where the heck the giant box came from, I want to examine the following question: in principle, is there any way that these aliens could figure out that matter is the source of gravitational fields?

Now, to make it interesting, let us assume the density of the box is not uniform, so there are some parts of its walls that have a stronger gravitational pull than others. Our aliens can walk around on these parts of the walls, and in some parts the aliens even become too heavy to support their own weight and get stuck until someone rescues them. Elsewhere, the walls of the box are low density and so the gravitational attraction to them is very weak. Here, the aliens can easily jump off and float away from the wall. Indeed, the aliens spend much of their time floating freely near the center of the box where the gravitational fields are weak. Apart from that, the composition of the box itself does not change with time and the box is not rotating, so the aliens are quickly able to map out the constant gravitational field that surrounds them inside the box, with its strong and weak points.

Like us, the aliens have developed technology to manipulate the electromagnetic field, and they know that it is the electromagnetic forces that keeps their bodies intact and stops matter from passing through itself. More importantly, they can accelerate objects of different masses by pushing on them, or applying an electric force to charged test bodies, so they quickly discover that matter has inertia, measured by its mass. In this way, they are able to discover Newton’s laws of mechanics. In addition, their experiments with electromagnetism and light eventually lead them to upgrade their picture of space-time, and their Newtonian mechanics is replaced by special relativistic mechanics and Maxwell’s equations for the electromagnetic field.

So far, so good! Except that, because they do not observe any orbiting planets or moving gravitating bodies (their own bodies being too light to produce any noticible attractive forces), they still have not reproduced Newtonian gravity. They know that there is a static field permeating space-time, called the gravitational field, that seems to be fixed to the frame of the box — but they have no reason to think that this gravitational force originates from matter. Indeed, there are two philosophical schools of thought on this. The first group holds that the gravitational field is to be thought of analogously to the electromagnetic field, and is therefore sourced by special “gravitational charges”. It was originally claimed that the material of the box itself carries gravitational charge, but scrapings of the material from the box revealed it to be the same kind of matter from which the aliens themselves were composed (let’s say Carbon) and the scrapings themselves seemed not to produce any gravitational fields, even when collected together in large amounts of several kilograms (a truly humungous weight to the minds of the aliens, whose entire population combined would only weigh ten kilograms). Some aliens pointed out that the gravitational charge of Carbon might be extremely weak, and since the mass of the entire box was likely to be many orders of magnitude larger than anything they had experienced before, it is possible that its cumulative charge would be enough to produce the field. However, these aliens were criticised for making ad-hoc modifications to their theory to avoid its obvious refutation by the kilograms-of-Carbon experiments. If gravity is analogous to the electromagnetic force — they were asked with a sneer — then why should it be so much weaker than electromagnetism? It seemed rather too convenient.

Some people suggested that the true gravitational charge was not Carbon, but some other material that coated the outside of the box. However, these people were derided even more severely than were the Carbon Gravitists (as they had become known). Instead, the popular scientific consensus shifted to a modern idea in which the gravitational force was considered to be a special kind of force field that simply had no source charges. It was a God-given field whose origin and patterns were not to be questioned but simply accepted, much like the very existence of the Great Box itself. This following gained great support when someone made a great discovery: the gravitational force could be regarded as the very geometry of spacetime itself.

The motivation for this was the peculiar observation, long known but never explained, that massive bodies always had the same acceleration in the gravitational field regardless of their different masses. A single alien falling towards one of the gravitating walls of the box would keep speed perfectly with a group of a hundred Aliens tied together, despite their clearly different masses. This dealt a crushing blow to the remnants of the Carbon Gravitists, for it implied that the gravitational charge of matter was exactly proportional to its inertial mass. This coincidence had no precedent in electromagnetism, where it was known that bodies of the same mass could have very different electric charges.

Under the new school of thought, the gravitational force was reinterpreted as the background geometry of space-time inside the box, which specified the inertial trajectories of all massive bodies. Hence, the gravitational force was not a force at all, so it was meaningless to ascribe a “gravitational charge” to matter. Tensor calculus was developed as a natural extension of special relativity, and the aliens derived the geodesic equation describing the motion of matter in a fixed curved space-time metric. The metric of the box was mapped out with high precision, and all questions about the universe seemed to have been settled.

Well, almost all. Some troublesome philosophers continued to insist that there should be some kind of connection between space-time geometry and matter. They wanted more than just the well-known description of how geometry caused matter to move: they tried to argue that matter should also tell space-time how to curve.

“Our entire population combined only weighs a fraction of the mass of the box. What would happen if there were more matter available to us? What if we did the Carbon-kilogram experiment again, but with 100 kilograms? Or a million? Surely the presence of such a large amount of matter would have an effect on space-time itself?”

But these philosophers were just laughed at. Why should any amount of matter affect the eternal and never-changing space-time geometry? Even if the Great Box itself were removed, the prevailing thought was that the gravitational field would remain, fixed as it was in space-time and not to any material source. So they all lived happily ever after, in blissful ignorance of the gravitational constant G, planetary orbits, and other such fantasies.

***

Did you find this fairytale disturbing? I did. It illustrates what I think is an under-appreciated uncomfortable feature of our best theories of gravity: they all take the fact that matter generates gravity as a premise, without justification apart from empirical observation. There’s nothing strictly wrong with this — we do essentially the same thing in special relativity when we take the speed of light to be constant regardless of the motion of its source, historically an empirically determined fact (and one that was found quite surprising).

However, there is a slight difference: one can in principle argue that the speed of light should be reference-frame independent from philosophical grounds, without appealing to empirical observations. Roughly, the relativity principle states that the laws of physics should be the same in all frames of motion, and from among the laws of physics we can include the non-relativistic equations of the electromagnetic field, from which the constant speed of light can be derived from the electric and magnetic constants of the vacuum. As far as I know, there is no similar philosophical grounding for the connection between matter and geometry as embodied by the gravitational constant, and hence no compelling reason for our hypothetical aliens to ever believe that matter is the source of space-time geometry.

Could it be that there is an essential piece missing from our accounts of the connection between matter and space-time? Or are our aliens are doomed by their unfortunately contrived situation, never to deduce the complete laws of the universe?