Spacetime Foam (Script) (Patreon)

Spacetime on its smallest scales is a seething ocean of black holes and wormholes flickering into and out of existence—or so many physicists think has to be the case. But why should we take this spacetime foam seriously if we’ve never seen any evidence of it?

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A while ago we started talking about the fundamentalness of space—is it the most elementary stage that holds all of the complexity of physics? Or is it just the way our brains make sense of something that’s nothing like the space of our experience. Well, we have good reason to at least doubt that the fundamental building block of space is just more space. And that’s because our best theory of the nature of space—Einstein’s general theory of relativity—is incomplete. It comes into severe conflict with the other great pillar of modern physics: quantum mechanics.

We’ve spoken in the past about the source of this conflict, and about some of the speculative solutions such as string theory and loop quantum gravity. In both of these theories, space gives way to something rather different—something more fundamental at the tiniest scales. But these theories have proved damnably difficult to test, and so we can’t say what the most elementary building block of space really is. But that doesn’t mean we can’t say something about what space looks like at close to the smallest scales. By using a little logical deduction, we can combine general relativity and quantum mechanics just well enough to trick Nature into giving us an answer. And that answer is that spacetime is … foamy.

It was John Archibald Wheeler who worked this one out. Wheeler was a central figure in the development of both general relativity and quantum mechanics, and adviser to Richard Feynman and Hugh Everett and Kip Thorne and many many others. Here’s the analogy that Wheeler came up with to describe his conclusions about the minutest scales of the fabric of spacetime.

Imagine you're flying in a plane over the Atlantic ocean. From a regular cruising altitude, looking down, the surface of the ocean appears smooth and relatively featureless. But, if your plane lowered its altitude and got closer you may start to see the waves—from here just tiny disturbances in the surface of the water. But if you got even close, perhaps on a large ship sailing through the waters you'd see the foam and white caps of waves forming and breaking, and you’d feel their effect on your motion. And if you’re on a rowboat the wave would be everything—you’d be moving through a space that was far from flat, but rather dominated by everchanging geometries.

In this analogy, the 2-D surface of the ocean is 3-dimensional space. For large creatures like us humans, we’re flying high above and see smooth, flat, and perfectly continuous spatial dimensions. But “descending” closer to the Planck scale, we would see ubiquitous tiny fluctuations as distances and geometries became warped. And at the Planck scale, spacetime would become so curved that black holes and wormholes would be popping into existence, only to vanish again—as if our ocean were now violently boiling.

This is Wheeler’s “spacetime foam’’, and there’s good reason to think that spacetime really does look like this when you’re close to the smallest scales—regardless of your preferred theory of quantum gravity. We need the latter to say what spacetime is made of—to understand the analogous level of the water molecules in our ocean. However, the behaviour of a quantum spacetime just above that level can perhaps be understood regardless of its underlying theory, just as we can describe the behaviour of water without knowing its chemistry.

Today we’re going to see where the idea of spacetime foam came from, and also how it might be tested and used to progress further towards the unification of quantum mechanics and general relativity. We’ll do that by combining two of the fundamental ideas in each theory - the uncertainty of quantum mechanics with the geometry of GR. Because spacetime foam is what you get when geometry itself becomes uncertain.

Let’s start with the bit about uncertainty. The Heisenberg uncertainty principle is one of the foundational concepts in quantum mechanics. One version tells us that we can’t simultaneously know both the position AND the momentum of a particle to arbitrary precision. The product of their uncertainties will always be larger than a particular very small number.

\Delta x\Delta p \geq \frac{h}{4\pi}

This is a very deep relationship that we get into in detail in this episode. But let me also remind you of a simpler way to think about it that we covered more recently—the Heisenberg microscope. Imagine you try to measure the position of an object by bouncing a photon off it. You’ll get a more precise position measurement with a shorter  wavelength photon. That photon will transfer some of its momentum to the object, as will any measurement attempt. The shorter the wavelength, the more momentum the photon can transfer. So the more precisely you measure position, the less certain you become about the final momentum of the object.

But there’s a limit to the precision of our position measurement, even if we’re happy with absolute uncertainty in momentum. That limit is a consequence of us bringing general relativity into the picture. GR tells us that mass and energy changes the geometry of space. This is described by the Einstein field equations, with the mass-energy content of space completely determines the geometry of that space.

So as our measuring photon gains momentum and therefore energy, it introduces uncertainty into the geometry of space between you and the object, which increases our uncertainty in the distance to the object. So as the photon gets more energetic, the Heisenberg uncertainty goes down but the uncertainty in the geometry goes up. They equal each other when the uncertainty is equal to the Planck length, which is a tiny 1.6x10^-35 meters. So the absolute limit of our ability to pinpoint objects in space is the Planck length. But what does that mean for the structure of space on that scale?

Let’s say we now want to measure the size of a little block of space. The measuring photon needs to have a wavelength at least smaller than the width of that block. That photon also warps our chunk of space. The Einstein field equations can tell us roughly when the introduced curvature is equal to the size of the space we’re trying to measure. It turns out that if we try to measure the size of a 1-Planck-length block of space, the curvature introduced is equal to a Planck length. We can express our curvature uncertainty with a new version of the uncertainty principle that combines quantum mechanics with general relativity.

\Delta x\Delta r \geq \ell_{P}^{2}

Here we have our first glimpse of the spacetime foam. As we peer closer and closer at the fabric of spacetime, we see that its curvature at any point becomes more and more variable; its geometry more uncertain, until the curvature dominates. What does that mean? Well, geometries that have a radius of curvature similar to their size include spheres, cylinders, and black holes. So those are the geometries that emerge at the Planck scale. Space at that scale curves back completely on itself, infinitesimal black holes appear, and wormholes connect nearby regions.  But such Planck scale black holes would likely rapidly evaporate due to Hawking radiation and the wormholes would be unstable and collapse in a very short amount of time. The quantum foam is as transitory as any quantum fluctuation.

To better understand the dynamics of the spacetime foam, let’s think about quantum fluctuations. For that it’s helpful to turn to the other popular version of the Heisenberg uncertainty principle, which tells us that we can’t simultaneously know both the energy AND the time or duration of an event.

So if you try to measure the energy of a system at a very precise time, or over a very short duration, then your energy measurement becomes highly uncertain. But if energy can be fundamentally uncertain, that means there can be a fundamental uncertainty in the physical contents of the universe. To understand that, let me give you a quick refresher on our best theory of stuff in the universe. Quantum field theory and in particular the standard model of particle physics nicely describes all matter and 3 of the 4 forces as excitations in different quantum fields that permeate all of space.

But because of the time-energy uncertainty principle, we can never know the exact energy present in a particular patch of space. We can’t even know that the energy is precisely zero in the complete absence of real particles. What should be a complete vacuum might be observed to have non-zero energy, and that becomes more likely the shorter the timescale of our observation. This manifests as an underlying buzz of rapid energy fluctuations in the quantum vacuum. It’s sometimes described as an ocean of virtual particles constantly appearing and vanishing, although it’s more complicated than that. Now, this isn’t quite our spacetime foam, because it says nothing about the shape of the fabric of spacetime on its own.

But let’s bring general relativity back into it. This fluctuating energy of the vacuum can be thought of as an uncertainty in whatever’s sitting on the right side of the Einstein equations. The mass-energy content of a tiny chunk of space is uncertain, then the corresponding geometry should be uncertain also. If we think about this in terms of virtual particles, then every time one appears, we should see an accompanying tiny gravitational field that lasts just for an instant. But because these quantum fluctuations are extremely complex and constantly changing, the effect on the geometry of space is also complex and ever-changing. Virtual particles aren’t even restricted to the normal rules that real particles have—for example, negative masses are possible—which means exotic objects like wormholes can appear and vanish amid the general roiling mess of spacetime at the Planck scale.

And because these quantum fluctuations are actually in a superposition of many different states, we need to think of the geometry of spacetime as also being in a quantum superposition at that scale. Many, and perhaps every possible geometry exists simultaneously. What we see on the macroscopic scale is the washed out sum of all Planck-scale configurations—and it ends up conveniently nice and flat.

We can think of the quantum foam as being due to an intrinsic uncertainty in either the stuff space contains or in the geometry of spacetime itself. These are two ways to get at the same conclusion: as long as the uncertainty principle holds in some reasonable way for gravity, then on the Planck scale the fabric of spacetime should be foamy.

So now we have a picture of what space might look like on scales well below our experience, or even the experience of atoms or nuclei. How do we test this? If the standard picture of is right, quantum fluctuations of spacetime only become significant near the Planck scale, which is far, far smaller than any of our direct probes of spacetime structure—and that’ll probably be the case until we’re a super-advanced galactic civilization.

But Indirect tests are possible now. Remember Wheeler’s analogy of spacetime foam as a choppy ocean surface. A gigantic ship doesn’t notice the waves, but a rowboat does. A vessel in between may have a relatively smooth ride, but might be deflected or impeded over time on a wavy ocean versus a still one. If spacetime foam is real, then we might expect a particle traveling very long distances to experience a slight shift in its course.

And we can test this with a surprisingly simple experiment. A distant point of light can produce diffraction patterns, for example these diffraction spikes if we observe through a telescope aperture or diffraction fringes through a pair of slits. Those patterns are due to individual photons interfering with themselves after interacting with the gap. Photons that reach us traveling in the same direction will experience the same interference. By catching many such photons, we slowly build up a single interference pattern. But two photons coming in at even slightly different angles will land on our detector according to different interference patterns. If we look at many such photons, those different patterns will be overlaid on each other and result in just a blur.

That’s why we see diffraction spikes around stars in Hubble or JWST images, but only a blurry blob from even our best ground-based telescopes. In the latter case, light from the star is bounced around by our atmosphere so that it arrives traveling in slightly different directions. Well, if spacetime is foamy then we might expect a similar effect for light arriving from very great distances. Stars in our galaxy are far enough away, but some objects like quasars and gamma ray bursts can be billions of light years distant.

A number of studies have tried to detect the effect of spacetime foam on the diffraction patterns of distant objects in Hubble Space Telescope data. Hubble is particularly good because it’s very sensitive to ultraviolet light—a short wavelength light that should be more influenced by a turbulent spacetime fabric compared to, say, the long-wavelength infrared light that JWST is sensitive to. There’s nothing conclusive yet. So far the best these studies have been able to do is to rule out certain models which produce a particularly strong spacetime foam. But some of the measurements are right on the edge of the needed sensitivity, so perhaps a bigger ultra-violet-sensitive space telescope will be able to detect this effect. Only then can we really believe that a tempestuous ocean of uncertain geometries underlies our deceptively tranquil spacetime.