Loop Quantum Gravity Explained! (Script) (Patreon)

It’s time we talked about loop quantum gravity. What exactly is it? What are the loops? And can it really defeat string theory in our quest for a Theory of Everything?

The holy grail of physics is to connect our understanding of the tiny scales of atoms and subatomic particles with that of the vast scales of planets, galaxies, and the entire universe. To connect quantum physics with Einstein’s general theory of relativity. Our search for a theory of quantum gravity is a century old, and we’ve talked quite a bit about it already, including what’s probably the lead contender - string theory. But string theory isn’t the only game in town - or so some physicists believe. There may be another way to reconcile the physics of the tiny and the gigantic - another way to a theory of quantum gravity that avoids a lot of conceptual baggage like tiny wiggling strings made of coiled up extra dimensions. That other way would be loop quantum gravity, and today we’re going to learn exactly what it is. 

Back in the day we talked about why combining quantum mechanics with general relativity was so hard. For example, there’s the fact that general relativity - or perhaps quantum mechanics breaks down when we think about the extreme densities of the black hole or big bang singularities. But there are more fundamental conflicts: namely, background independence and the problem of time. Attempts to resolve these conflicts ultimately led to to the invention of loop quantum gravity. I’ll mention the problem of time briefly, but the real focus is going to be on this background independence thing. So what is it?

Quantum mechanics, and indeed most theories in physics, involve a set of equations describing how stuff moves around, exerts force, etc. on some background coordinate system. Like actors on a stage, where the actors are particles and wavefunctions and fields and the stage is the coordinates of space and time. In quantum mechanics that stage is flat and static and isn’t influenced by the actors. It requires some giant hacks to even attempt regular quantum calculations on a non-flat geometry. In short: quantum mechanics is NOT background independent.

General relativity, on the other hand, has to be background independent - because that’s what its equations do - they change the background. They describe how the presence of mass and energy warps the fabric of spacetime. Our background coordinate system itself becomes a dynamic entity. More precisely, the metric - the object encapsulating the geometry and causal structure of spacetime - evolves in the equations of GR ,  So those equations need to work regardless of that background. 

In string theory, a type of background independence emerges in an abstract space of moving strings and with that a gravitational field. But for that to work first you need those strings to exist - and we don’t know if they do. Loop quantum gravity tries to quantize general relativity with no strings attached, while preserving the background independence already inherent to GR.

But why is quantizing general relativity so difficult in the first place? The challenge really gets us to the fundamentals of what a quantum theory actually is. So just quickly, let’s review all of quantum mechanics. In classical physics, we have variables like position, time, momentum, energy - mathematical expressions that represent the observable properties of the object or system that you’re trying to describe. Some of these - say, position and time - also form our background coordinate system. But in quantum mechanics, things aren’t so straightforward. Certain properties have an in-built uncertainty and only take concrete values after measurement. Absent measurement, they exist in a fuzzy space of possibilities called a wavefunction. 

In the first formulations of quantum mechanics, that wavefunction describes the distribution of possible positions or momenta of, say, a particle. These can then be resolved into concrete, measured values by acting on the wavefunction with so-called position and momentum operators. The wavefunction and operators are fundamentally tied to the coordinate system. After all, the position and momentum of quantum mechanics literally describes location on a spatial coordinate system and the change in that location over time. That makes it highly background dependent. There are other ways to formulate quantum mechanics, like quantum field theory, but these ultimately have the same issue

It gets worse actually. In quantum mechanics, time is treated completely separately to other variables - there’s no “time wavefunction” or “time operator”. This is completely at odds with general relativity, in which time is treated as just another dimension. This is the “problem of time” that I mentioned, and it’s strongly connected to background independence. A quantum theory of gravity needs to fix both of these issues - but we’re going to focus on background independence for now – it’s the main motivation behind quantum loop gravity.

The equations of quantum mechanics let you calculate changing properties of a particle- - like its position or momentum - relative to the background coordinate system. The equations of general relativity let you calculate the changing shape of the coordinate system itself, encapsulated in the metric. So maybe instead of thinking about the quantum fuzziness of position and momentum we can think about the quantum fuzziness of the metric itself. And instead of an equation that describes the quantum evolution of the properties of an object IN spacetime, maybe there’s an equation that describes the quantum evolution of the geometry of space. 

There is - or at least an attempt at one. It’s called the Wheeler-deWitt equation, based on something called the ADM formalism. ADM starts by defining this abstract space of spaces - 3-D spatial metrics, 3-D space slices cut out of 4-D spacetime. It then gives a sort of equation of motion for how these metrics evolve through time. You can imagine a funky coordinate system describing where you are in this abstract space of metrics. As you move through this coordinate system, the geometry of space changes. You can also imagine analogies of the position and momentum in this space of metrics.  So the Wheeler-deWitt quantizes these - turns them into quantum operators. The result is a quantum equation for the fabric of space. A contender for a theory of quantum gravity.

The Wheeler-deWitt equation was promising, but turned out to be … unsolvable. Which makes it not so useful, and impossible to verify as correct. So perhaps this whole path of using abstract coordinates is a dead end, or perhaps we just haven’t gone down it far enough. That’s what loop quantum gravity does - it takes us down the abstraction rabbit hole - past our space of metrics into a space of something called connections. And these connections are going to give us our loops.

Connections are mathematical functions that tell you how something, like a vector, changes as it moves between two points in a space. We saw an example way back in the day when we looked at parallel transport. As you move the base of a vector along a path in curved space, the vector rotates. And the amount of rotation encodes information about the changes in geometry along the path. If connections contain all the information about spacetime, them maybe we can represent spacetime with these connections instead of with regular coordinates. In the 1950s Einstein himself tried to rewrite general relativity in terms of these parallel transport vector connections, but the result was a mess. 

The breakthrough came in the 80s when Abey Ashketar tried a different type of connection: one in which you parallel transport not a vector but something called a spinor - a vector-like thing that also represents a quantum of angular momentum - or spin. Ashketar rewrote general relativity in terms of these spin-connections - now known as Ashketar variables. In this formalism, the “space of metrics” looks just like a space of fields in quantum field theory. Quantized gravity suddenly looked to be in reach.

And now we get to the loops of loop quantum gravity. Lee Smolin and Carlo Rovelli realized they could fully solve the Wheeler-deWitt equation representing spatial metrics using Ashketar's spin connections. But they needed one more trick - one layer deeper in abstraction. They evaluated these connections over closed loops – so each point connected back to itself. They realized it was possible to define any geometry of 3-D space out of a sort of weave of these closed loops, with each loop like an elementary closed circuit of gravitational field. So now you have a space of loops with with to construct the fabric of space - and THAT space of loops can be quantized rather neatly, AND in a background independent way. After all, there IS no background until these now-quantum loop states build it.

The result, of course, is loop quantum gravity. It’s general relativity – our modern theory of gravity – cast in terms of very abstract building blocks. Not with chunks of spacetime but with quantum circuits of gravitational field. 3-D space can be sort of woven from these loops into something called a spin-network - which is a concept too abstract for even this episode. But the resulting 3-D space  looks normal on large scales - it looks like space. But on the tiniest of scales – the Planck scale - it’s sort of pixelated. At the nexuses of this weave you have quantized volume elements – irreducible grains of space - connected by quantized area faces like facets. Even that description is too space-like - probably the underlying weave of the fabric of space doesn't resemble anything intuitive at all.

The big success of loop quantum gravity is that it manages to combine general relativity and quantum mechanics in their accepted forms, without taking away their most important foundational principles. And without adding big assumptions – like the existence of strings or extra dimensions or supersymmetry. The theory has some other successes, for example, the theory seems to predict Hawking radiation and black hole entropy consistent with Hawking and Bekenstein’s equations.

However there are also many who identify serious, fundamental issues the theory. Also, while LQG has background independence in terms of different 3-D spatial geometries, it’s not clear that this independence extends to 4-D spacetime. And connected to this, LQG doesn’t solve the problem of time. More generally, for this theory to be successful it needs give you the equations of good old general relativity on large, non-quantum scales - in the so-called classical limit. But it’s not clear that it can do that. 

This is still a hotly debated topic. Some researchers think that the method and foundations are sound and that the current criticisms and shortcomings can be resolved with more research and extensions to the existing formalism. Others argue that the problems are fundamental and that no amount of tinkering and extending will resolve them. A lot more exploration is needed on the theoretical side. But what about experiments?  Surprisingly, some experiments have actually been proposed. Quantum loop gravity seems to predict that the speed of light should depend very slightly on the energy of the photon, with, for example, high-energy gamma rays travelling a wee bit slower than low energy radio waves due to the way they propagate through the graininess of an LQG spacetime. This was tested in 2009 by looking for differences in the arrival time of light from a gamma ray burst nearly a billion light years away. If there was any difference it was barely measurable, and that doesn't look great for loop quantum gravity.

Loop quantum gravity is an intriguing alternative to the more popular string theory. Both currently live deep in their respective theoretical rabbit-holes, not yet able to make experimental contact with the real universe. But the mathematics have yielded intriguing clues to the nature of the fabric of the universe – and that nature is very weird. One way or another, we live in a seriously loopy space time.

As always, a huge thanks to our Patreon contributors. Your help makes an enormous difference. And today I want to give an extra special thanks to David Barnholdt, a new Big Bang level supporter. David, we already spent all of your money ... on aspirin, after loop quantum gravity broke our brains. Honestly, we can't thank you enough. Still hurts a little though. Last time we talked about Black Hole Harmonics, let's see what you had to say.

David Bennack likes the idea of Gravitational lensing of gravitational waves. First the context: in August LIGO detected two black hole merger signals within 20 minutes of each other and in similar patches of the sky. It's really REALLy hard to come up with a scenario for two actual black hole pairs to collide at the same time in the same general location. An alternative possibility is that it was just one black hole merger, but the gravitational wave from it was deflected by a galaxy or something on its way to us - it was gravitationally lensed so as to arrive via two separate paths through the universe. If those paths were different lengths we'd see the same signal separated by a small time delay. We do see this effect in the light from gravitationally lensed quasars and supernovae. Gravitational waves should be lensed in the same way as light, so it's all plausible. In the case of this particular example it's not really the favored explanation - the signals don't look like they were from quite the same location after all, and they were different enough. Still, we'll probably see a lensed gravitational wave at some point.

I talked about the maximum rate of spin of a back hole, and so Lucas Przybyla rightly asks how fast is as fast as possible? So there's a limit to the rotation of a black hole which comes from the fact that the outward centrifugal force of rotation partially counters the gravitatinally attraction - to use crude Newtonian speak. If the black hole rotates more than a certain amount then the event horizon evaporates, exposing the singularity to the universe - or in the case of a rotating black hole, an infinite density ring. Such naked singularities are expected to be impossible, and so we expect a maximum rotation rate for black holes.f The details of all this need their own episode, so let's leave it at that.

A few of you asked a really on-point question: if the fabric of space and time can be stretched and if can have waves, that means it must have a sort of elasticity and resistance to stretching. So just how resilient is the fabric of spacetime? The answer lies in the Einstein field equation of general relativity - in a remarkably simple way. That equation says that the amount stretching of spacetime is proportional to the mass and energy contained by that spacetime. The constant of proportionality can be thought of as the tensile strength of space - the resistance to stretching. The smaller the number, the more energy is needed to stretch spacetime. And that constant is very, very small - 2 by 10^-43. Spacetime is a very, very stiff fabric.

A few of you noticed we missed a huge opportunity by not calling quasinormal modes quasimodos. Especially given I kept saying a struck black hole rings like a bell. I had a hunch someone would notice that. Notre Damn!