Gauge Theoretic Gravity: Why space and time might be illusions
By: Tim Andersen
Tags: Time, Space, Physics, Gravity, Philosophy

Einstein’s theory presented the universe as a four dimensional “manifold”, which is a word for a shape with certain properties. People and things in that manifold had to move around according to a strict rule: no traveling faster than the speed of light. Traveling faster than light is, unfortunately, the only apparent way to go back in time in Minkowski’s universe. But, in Einstein’s, there was a loophole. Spacetime itself could be shaped in such a way that some paths between points were shorter than others and so, even if you couldn’t go faster than light, you could shorten the journey between two points. You could also connect a point in the future to a point in the past by a bridge. You can’t turn around on the highway and go back, but, like a cloverleaf, you can take an exit and backtrack by another route.

The other property of Einstein’s manifold is that there is no such thing as universal time. Everything has its own “proper” time that it experiences as it slices its way through the manifold. Objects that are near one other and in a similar state of motion tend to experience similar flows of time as well, which is why we all on Earth experience nearly the same flow of time. The variations for ordinary speeds on Earth and small differences in gravitational fields are tiny to the point where only super precise atomic clocks can measure the differences.

It seems like all this discussion of bridges between times and places and no universal time means that time works like a highway. We are all moving along in our cars, experiencing our own flow. There are exit ramps and on ramps. You can go back in time if you can find an exit to take you by another route. All of this supports the idea that those dinos we want to see are just “back there” 65 million exits behind us. A time traveler just needs to come “unstuck” in this flow of time, take a shortcut back to the past, and viola, instant Cretaceous vacation (watch out for meteors!)

But wait, things aren’t so cut and dried.

The truth is that General Relativity is formulated according to Einstein’s philosophy. It proposes that space and time have a geometry, and physicists and mathematicians usually present it this way, the way he did. If something has a geometry, it must have every point existing somewhere or somewhen. There is no evidence for this part of the theory because all tests of the theory have been carried out over a very small patch of the universe, on the Earth and its immediate environs. Even observations of deep space have been made by looking at “old light” that has been traveling for millions or billions of years to reach us. There is a problem with the Einstein formula.

The problem is that Einstein takes a leap from spacetime having a geometry to anything in “free fall”, that is something not experiencing an external force, following a “geodesic” in that spacetime. A geodesic is a shortest line path in a curved geometry. It comes from the Greek geodaisia, ge-, the Earth, and daio-, “to divide”. On the Earth, a geodesic is a “great circle” path between two points; hence, it always divides the Earth into two equal halves.

We all know that if you were to go out on the ocean and follow a constant heading, you would not follow a great circle path except under contrived circumstances. Instead, you would follow a “rhumb line” path which is not a shortest line path at all. In order to follow a great circle path, you need to make periodic course corrections. In Einstein’s curved spacetime, there is a mechanism for making these course corrections, it is gravity. But now it seems we have two separate pieces to the theory: a curved spacetime, like the Earth, and a force that makes us obey that curvature, gravity, like the boat captain.

Ockham’s razor tells us that we shouldn’t invent things that serve no purpose, so it seems reasonable that if we want to explain gravity, we could just get rid of the curved spacetime and keep the gravity. But, then we are left with everything happening all at the same time in the same place. We need a coordinate system to keep things separated, but we don’t need Einstein’s ideas about geometry as much.

In 1950, Utiyama created a version of General Relativity that matched this perspective more or less, called gauge theoretic gravity. Gauge theory then took off and explained all the other forces: weak, strong, and electromagnetic, becoming what is now known as the Standard Model of quantum physics. All gauge theories have a geometric interpretation, but for other forces it is not seen as fundamental. (The geometries of the other forces exist in spaces made of complex numbers, so they aren’t that intuitive.) Still, most presentations of gravity used the geometric approach. Gauge theory gravity has the advantage that it presents gravity as a force, which is what it is, and not as a geometry that needs gravity to order phenomena according to geodesics.

Let’s go back to our highway analogy now. In our Einsteinean, curved manifold theory, we travel on a highway and the past is “back there” and the future is “up ahead”. In a gauge theoretic version, however, we could be on a treadmill with screens around us showing scenery like some sort of racing simulator. The forces exerted on us are the same in both cases. (They are mathematically equivalent.) A spacetime coordinate, rather than representing a point on a manifold, is an artificial construct that helps us label events as distinct from one another. When I travel from one place to another, the gauge theory tells me what will happen as I traverse that “distance”. It also tells me how different people (“observers” in relativity parlance) will have different experiences depending on how they interact with the gauge force of gravity.

Suppose that the universe is not a manifold. Suppose you are, in reality, experiencing neither motion in space nor in time according to Einstein’s ideas but interacting with a gauge force that makes it appear as if you are moving. This means that, not only is the past not “back there” somewhere, but other places are not exactly “over there” either. Rather, to move from place to place or time to time you interact with the gauge force of gravity which translates, rotates, twists, and accelerates you like the treadmill. As in the movie, The Matrix, other people experience their own versions of reality which they reconcile by communicating with one another and sharing observations through forces.

The universe does not need to be a computer simulation for it to be like The Matrix. (That would just invoke another universe that is “real” like the manifold we just got rid of and send Willem of Ockham into a tizzy.) Gauge theory just works that way. It is not like elephants standing on the turtle’s back from Hindu myth. As there is no spoon in The Matrix, there is no turtle.

While gauge theory has few implications for physical measurements right now, since it is mathematically equivalent to the geometric form of General Relativity, it has philosophical implications. If the past and the future do not exist “out there”, then they could be created out of the interaction between ourselves and the universe, especially when we bring quantum uncertainty into the mix. Our choices could, in fact, profoundly impact what we experience because the future has not been written and the past has been erased. I say could because we just don’t know.

It is dangerous to draw philosophical conclusions from physics, more dangerous still to present one interpretation of a physical theory as the only one. If the history of philosophy is any guide, people will choose sides depending on what they want to be true. I, for one, like the idea that my choices matter and that the future is mine to choose. Those like Einstein will wish for a universe where they bear no responsibility for their actions. If the history of physics is any guide, just when we think we’ve made sense out of the universe, it will reveal that we still haven’t a clue.