Short answer: we don’t.
Einstein’s relativity has been a successful theory of gravity for over 100 years now. But now most scientists think it is wrong.
In 1919, the world celebrated the measurement of light bending around the Sun by the exact amount Einstein’s theory predicted.
Einstein became a superstar.
Over the next several decades, Einstein’s theory was confirmed over and over again.
When problems did crop up, they turned out to be attributed to things we couldn’t see. As early as the 1930s, we discovered dark matter. Dark energy was discovered much later in the 1990s with the confirmation of the accelerating expansion of the universe.
Both of these explanations worked well in models and were essentially placeholders but neither pointed to Einstein’s theory being wrong.
Meanwhile, quantum field theorists showed that the other three forces of nature could be combined with matter into what is called the Standard Model of quantum physics. All of these forces were what is called renormalizable.
All the models for these forces make predictions that are infinite, but they can be fixed so that they do produce sensible results if we renormalize them.
Renormalization is a two step process. First you assume that your model breaks down at some small scale or high energy. (You can think of the correspondence of length and energy in terms of frequency of waves. Smaller wavelengths, larger energy.) This scale is outside what your particle accelerators can measure. You use that to cut off your equations so they only apply down to a certain length scale. This makes the results finite but you don’t know at what scale your math stops working!
But there is a solution. You assume that the strength of interactions between particles and forces, called a coupling, is dependent on the ratio of the length scale you are measuring and the length scale where the theory breaks down.
An example of a coupling constant is the fine structure constant for electromagnetism which is about 1/137 at normal, human length scales, but actually becomes larger at smaller scales. So we know that this happens.
Since you can measure couplings and you assume that your cutoff is contained inside the coupling, you don’t need to know the actual cutoff. Therefore, you now have a working theory that predicts sensible results.
It is actually pretty surprising that this trick works on three known forces and matter because it doesn’t always work. It works because all these forces have coupling constants that are independent of units, what physicists call non-dimensional.
If you had a coupling that had units, then you could scale it all you wanted with a change of units from say pounds to kilograms.
That doesn’t seem like a problem on the face of it, and it isn’t unless your coupling has units of inverse energy or mass (we know from Einstein mass and energy are the same). In that case, your theory blows up because smaller and smaller interactions become more important and not less in the theory.
This problem cropped up when we tried to understand the weak force, which causes radioactive decay. In that case, it turned out that at very high energies, the weak force combined with electromagnetism to become another theory, the electroweak theory! That did have a non-dimensional coupling and so, by explaining that transition, we saved quantum field theory.
Gravity also has such a coupling constant called Newton’s universal constant of gravitation, G. Einstein uses the same constant. And G has units of inverse mass squared to cancel out the mass units of the two objects it is coupling to one another, so, like the weak force, it fails renormalization.
A simple thought experiment shows why. At smaller and smaller scales you have higher and higher energies. At some energy scale, gravity will have so much energy that it will spontaneously form black holes from itself (with no matter present). That would mean that at the tiniest scales the universe is entirely made of black holes popping in and out of existence. Their behavior would dominate any large scale measurements we make, but it doesn’t. So something else must be going on.
There are a number of research programs to try to solve the problem of renormalization. The most prominent attempts solve it the way that the weak force problem was solved.
String theory attempts to show that gravity is part of a much larger theory that combines all forces and matter into strings vibrating in multi-dimensional spaces. String theory is trivially renormalizable under certain assumptions.
Others have suggested that Einstein’s theory can be put into a quantum framework by looking at what is called lattice gauge theory. Lattice gauge theory is a mechanism for modeling forces as loops on a four dimensional grid called a lattice.
We know that these loops become the correct equations when we take the lattice size to zero in a limit. With gravity it is a little different, however, because gravity determines length scales itself, so you can’t define a four dimensional lattice with gravity on it. Gravity is the lattice. You can’t take the lattice size to zero because there is no length scale to scale.
What becomes important in that theory is the intersections between loops, not the loops themselves and so it has a deep connection to graph theory, the theory of interconnections. These are called spin networks.
That theory is unsurprisingly called Loop Quantum Gravity.
(I have to throw in the obligatory caution that LQG isn’t a workable theory purely because it introduces a smallest length scale. It depends heavily on a reformulation of Einstein’s theory into what are called Ashtekar variables which allow it to be expressed as a lattice quantum theory in the first place.)
Another theory assumes that gravity actually is renormalizable but is just more difficult to renormalize than other theories. When we renormalize other forces, we are doing something called “perturbative” renormalization, which I like to call “super” renormalization.
Physically, perturbation theory breaks a theory apart into individual particle interactions — two particles, three particles, combinations of different types of particles, and so on. Each perturbation is a separate interaction. When we renormalize most theories, this is what we are attempting to fix. We want to stop these individual particle interactions from giving infinite results.
That seems to make sense until you realize that there are an infinite number of infinitely complex particle interactions and that it is possible that we are throwing away higher level interactions that might actually solve the lower renormalization problems or maybe those higher level interactions just don’t exist.
When it comes to gravity, all those black holes popping in and out of existence might never happen because, perhaps, there is a limit to how energetic the theory can become at zero scale. More complex, nonlinear behavior is stopping it from happening.
This is called asymptotic safety and it is my favorite answer to this problem. The reason is because this tells us that the problem we are trying to solve with new theories doesn’t exist. It is a math problem that was too hard for us to solve at one time.
We know of other theories that are not perturbatively renormalizable but are non-perturbatively renormalizable, so it is within the realm of possibility. There is no reason that gravity has to be solved the same way as weak theory.
When it comes to science, the first question you have to ask before replacing a theory is: what is the evidence?
Right now, at the quantum level, there is zero evidence that Einstein’s theory needs to be replaced. The reasoning is purely mathematical. So, we may just be trying to come up with a physics answer to a math problem.
That doesn’t address the bigger problems of dark matter and dark energy, but so far no observation requires a modification to Einstein’s theory. While some modifications can account for what we observe, these could easily be something else.
A first principles analysis of this problem, therefore, is as follows:
Einstein’s relativity is not perturbatively renormalizable.
Renormalization can be solved in basically three ways: (1) find a new renormalizable theory that has the old theory as a “low energy” version, (2) find a way to eliminate infinite energy scales from the theory entirely, or (3) accept the lack of perturbative renormalizability and show that the theory is non-perturbatively renormalizable.
(1) Is an example of reasoning by analogy. We want the analogy between electroweak theory and gravity to hold. Not first principle thinking!
(2) That there is a minimum length scale has no basis in evidence or basic principles other than analogy with atomic theory (that there is a smallest scale or atom making up space and time)
(3) Requires no new evidence. It says you need better math.
Without any physical evidence that the theory is wrong at small scales and no reason to accept a new theory at large, we should assume the problem is a mathematical one.
For those of you familiar with or curious about my 5-D theory of quantum gravity, you might wonder why I favor asymptotic safety instead of my own. The reason is because my theory requires it, albeit in a different form than the standard one.
My theory has a smallest scale built into it but it is not a smallest possible scale like LQG. Rather it is a smallest scale at which gravity behaves randomly.
This is also true of molecular motion. At a certain size and time scale molecular motion in a gas for example stops appearing random and becomes smooth.
A cartoon picture is the billiard ball model of gases. The billiard balls on a table, if encouraged to move continuously, experience smooth, straight line motion interspersed with rapid and chaotic changes in direction. There is a characteristic scale of time and distance between collisions. You can see this by playing with this simulator (requires java) and changing the number and speed of the balls.
My theory proposes that the same is true of gravity. That helps resolve part of the renormalization issue. The other part relates to how to take averages over the fifth dimension which is a separate renormalization problem. You have to show that the theory behaves well over infinite distances.
This is essentially an asymptotic safety style solution but perhaps more easily solved than the standard one. The motivation for the theory, however, is to solve the quantum measurement problem as well as quantum gravity.
Becker, Katrin, Melanie Becker, and John H. Schwarz. String theory and M-theory: A modern introduction. Cambridge university press, 2006.
Ashtekar, Abhay. “Ashtekar variables.” Scholarpedia 10.6 (2015): 32900.
Rovelli, Carlo. “Loop quantum gravity.” Living reviews in relativity 11.1 (2008): 1–69.
Wilson, Kenneth G., and John Kogut. “The renormalization group and the ϵ expansion.” Physics reports 12.2 (1974): 75–199.
Niedermaier, Max, and Martin Reuter. “The asymptotic safety scenario in quantum gravity.” Living Reviews in Relativity 9.1 (2006): 1–173.