From Ancient Egypt to Gauge Theory, the story of the groma
When Albert Einstein conceived his theory of General Relativity, he described the universe to be much like the surface of the Earth but in…
When Albert Einstein conceived his theory of General Relativity, he described the universe to be much like the surface of the Earth but in four dimensions instead of two. This connection is more than analogy but hints at a deeper reality that both the Earth and the universe are, at large scales, “differentiable manifolds”, that is, shapes where change and rate of change in their curvatures in all directions are continuous. Without this property, both our planet and the universe would be impossible to understand or navigate. And because of this property, a simple surveyors tool, developed over 5000 years ago, can tell us a lot about the structure of space and time.
It is common knowledge that the Ancient Egyptians built the pyramids without the benefit of the wheel. One piece of technology they did have, however, allowed them to build numerous monumental buildings including their massive pyramids.
Later in history, this same device, vastly improved over the Egyptian model to be a large self-supporting instrument, enabled the Roman military to build a vast, continent spanning road network, construct aqueducts to bring water to growing cities, build city walls to keep out invaders, and enrich their cities with stone temples, palaces, forums, and theatres. Without that one small tool, the Roman Empire would never have built the lasting civilization it did.
The device is easy to make. All you need are two rods set at right angles, four pieces of string, four weights (preferably plum bobs), and either a staff with an arm at the top for the Roman style or a loop of rope for the Egyptian. You connect the strings to the ends of the rods, and, to the bottoms of the strings, you attach the weights. To the place where the rods meet you either attach the arm of the staff to the bottom or the loop of rope to the top. What this simple arrangement makes is one of the most powerful inventions of all time and one that has a profound relationship to our understanding of space and time.
It’s name is the groma.
The groma solves a problem that anyone who wants to build a large project with straight lines and right angles needs to solve. Egyptians wanted large buildings like pyramids. The Romans wanted even more: long, straight roads and aqueducts that spanned miles.
Here’s how it works:
Suppose you are a Roman surveyor, measuring for a new road. Your road must be long and straight, and it must precisely line up with its route. Roman roads were extremely costly and labor intensive to build, after all. You couldn’t afford to get it wrong. You know the route it has to take already. But how do you make sure that the actual road follows that route when it gets built? For small jobs, a taut rope as in the Egyptian relief above might do the trick as many DIY home improvers snap the chalk line to get a straight line. For long distances over hills, through deserts and forests, that will not do. You need to create a straight line by sight alone and that’s what the groma does.
The setup is simple. You set up the groma at a point on the ground. The Roman one above has a spike at one end so you can secure it in the ground to make sure the bottom end doesn’t move and attach a line so it won’t twist. Then you line up the plum bobs in each direction. This is the critical step because once the plum bobs are all lined up you know the groma is level. Each pair of plum bobs creates a sight, like the sight of a gun, along which you can then survey a straight line.
Here is a demonstration:
The groma is a beautiful tool because it creates, without the aid of lasers, satellites, or even telescopes, an imaginary coordinate system on the surface of the Earth at the point where the surveyor is standing. It does this without the surveyor having to leave the spot where he is. If you imagined a flat plane formed by the four points of the plum bobs extending outward infinitely in all directions, what you would get is the tangent plane to the Earth at that point. A coordinate system on a tangent plane is an example of a “local coordinate system” on a “manifold”, where the manifold in this case is the Earth.
If you imagine that the surveyor is at position (0,0) in this coordinate system, based on how he rotates the groma, his coordinate system also rotates, but the tangent plane stays the same. If he aligns it according to the compass points in a East-North system, one pace to East is position (1,0). One pace North is (0,1). South is (0,-1). And West is (-1,0).
He can walk in any direction as long as he likes of course, but eventually his tangent plane will no longer conform to the surface of the Earth. That is why it is only a local coordinate system. It is only good for distances that are short, say, no more than a few miles. Once he has traveled a few miles, he can set up his groma again and resurvey with a new local coordinate system, and so on, traveling over the surface of the Earth as the road proceeds on to its destination.
The groma works because the surface of the Earth, at the scales that matter to surveyors, is approximately differentiable, meaning that it is continuous and its rate of change is continuous in all directions from every point. Because it is continuous, every point on the Earth has a local coordinate system that is a decent approximation of the surface at that point for some small distance. Because it is differentiable, no matter what direction you go, the change in your local coordinate system is also continuous.
According to Einstein’s equations for General Relativity, the universe is also differentiable. There are no pits where you suddenly disappear. Even black holes are continuous and differentiable right up to the singularity. It also means that, like on the Earth, wherever you are in the universe it “looks” flat in a four dimensional sense. This means that, since you can use a groma to define a local coordinate system at any point on Earth, you can use a similar tool to define a local coordinate system anywhere in the universe. You just need to add some dimensions.
Enter the hypergroma.
My new invention, the hypergroma, is a surveyor’s tool, like the groma. I call it a hypergroma since it is like a groma but creates a four dimensional local coordinate system instead of two. If an alien or future human civilization wanted to survey roads through an Empire spanning a large proportion of the universe, they might need such a device. Like the regular groma, however, one is easy to make.
To make a hypergroma, all you need are three straight rods of fixed length, say, 1 meter each, and a clock. You can use any kind of clock. I recommend using something called a “light clock” which is a clock that measures time by bouncing a pulse of light between two mirrored surfaces. Each time the light bounces off the mirror and comes back it counts one tick.
You build it by setting all the rods at right angles to one another and then set the clock in the middle. That’s it. There are no plum bobs because there is no “down” in space. Duh.
Let’s do a little role play:
You are a Hyper-roman surveyor. Her Majesty, the Hyper-empress, wants a new Transgalactic Imperial Highway so that her Hyper-imperial fleet of Hyper-warships can quickly mobilize to defeat invading barbarian aliens from the frozen outer fringes of her domain. Your task is to survey the route. You need to survey through a particularly knotty cluster of galaxies to make sure that the route is straight and clear of gravitational anomalies. You set up your hypergroma at the outskirts of the Imperial Capital intending to survey the route to a distant border province some hundred million light years away. Your next waypoint, in a backwater local cluster, is a dull spiral galaxy called the Milky Way where, on its way to blast aliens, the Hyper-imperial fleet will gather a few young blue giants for fuel and maybe use a yellow dwarf star or two for target practice (nobody cares about those anyway, do they?).
You can’t survey the whole route at once because space and time warp as you pass by and between galaxies. (This is why you get phenomena like gravitational lensing.) Instead, you need to do it little by little, taking more care in places where matter is more dense and the curvature greater. You move your hypergroma periodically a few thousand lightyears at a time to survey your route as the roman surveyor did, jettisoning marking probes as you go so the builders to come later can find the route. The hypergroma will ensure that the legs of your path are straight as can be in space and time. Route complete, the Transgalactic Highway can be built. Your Hyper-emperess won’t have any trouble with those alien baddies thanks to you, not to mention any uppity provincial governors. Watch out for black holes!
The hypergroma defines a local coordinate system for whoever is holding it by pointing in three spatial directions and one temporal. Like a regular groma, you can rotate it, not only in one axis but in three axes (these are the familiar pitch, yaw, and roll of an airplane). You can also accelerate it in three directions. These changes in velocity, called Lorentz “boosts” in the parlance of relativity, in addition to the three rotations, give you six “degrees of freedom”. Every time you change one or more of these degrees of freedom by rotation or change of velocity, you get a new local coordinate system or “frame”. All these frames are different coordinate systems on the same tangent hyperplane (plane is to hyperplane as groma is to hypergroma) in space and time.
As with the groma, the hypergroma defines a coordinate system that, if extended outward infinitely, will describe the space and time near you pretty well, but, the farther away from you into the tangent hyperplane, which is, of course, imaginary, you go, the worse it will match the real universe. Because the universe is differentiable, you know that the change will be continuous and you won’t hit any discontinuous pits where your approximation is suddenly really bad. But you can’t assume that the universe matches your local coordinate system for too many lightyears, and near massive bodies like stars and planets, much less.
Gauge theory gravity, which I wrote about in this article, is one way of breaking down gravitation into the observations of a Hyper-roman surveyor traveling on a route through the universe. At each point, you define a local coordinate system also called a Local Inertial Frame (LIF). As you string together many of the LIFs you get a picture of a Global Inertial Frame (GIF), which describes how your hypergroma changes orientation and velocity as it moves along its path. The GIF is analogous to your complete road with each LIF being a segment on it. If your route is a loop and you come back to where you started, even if you don’t change the orientation of your hypergroma as you move, you could find that it doesn’t match a similar hypergroma that you left behind. Even your clocks might not agree! That is called “parallel transport”, and we see it on Earth too, but you only see it if you travel over very long distances. If you combine many of these GIFs, you get a more complete picture of the universe you inhabit (pun intended).
Should the human species rise to the level of a multi-galaxy spanning civilization, we will certainly use tools like the hypergroma to survey it. They will likely be more like modern surveyors tools with lasers and telescopes or the future equivalent, perhaps even gravitational wave-based. Nevertheless, geometry is geometry and the facts about the Earth and the universe won’t change, at least on the larger scales. On the small scales, the size of atoms, we know that Earth doesn’t exactly behave in a differentiable way, at least not a way that is useful to a surveyor. Perhaps our universe is the same way. If so, any theory of quantum gravity would have to do away with Einstein’s differential geometry. So far, we haven’t detected any violations of Einstein’s ideas, however, but, to the future scientists who make those discoveries, our best particle accelerators, wave detectors, and telescopes may seem as primitive as the groma.