How to Detect Extra Dimensions - Script (Patreon)

The hunt for extra dimensions sounds like science fiction. Fortunately, with the discovery of gravitational waves, we’re now living in a science fiction future. So, how many dimensions are there?

We may have mentioned once or twice that the new era of gravitational wave astronomy is going to open new windows to the universe and unlock many mysteries. When does all that start to happen? Oh, it has. We’re already making definitive statements about hypotheses that were previously untestable. Today on Space Time Journal Club I want to tell you about one in particular, described in a new paper “Limits on the number of spacetime dimensions from GW170817”, by Pardo, Fishbach, Holz and Spergel.

The key to this breakthrough was the gravitational wave event observed in August of 2017- GW170817. A pair of neutron stars spiraled together and merged. These super dense remnants of dead stars churned the fabric of space and time in their death spiral, and the LIGO and VIRGO gravitational wave observatories detected the resulting ripples. Unlike merging black holes, which are invisible, merging neutron stars explode spectacularly. The resulting kilonova is first observed as a gamma-ray burst. In GW170817 the flash of gamma radiation arrived 1.7 seconds after the gravitational waves. This was followed by a glow across the electromagnetic spectrum, and ultimately with discovery of the distant galaxy in which the explosion happened.

Among other things, this optical identification, gave a completely independent measurement of the distance traveled by the gravitational waves. This allowed us to make some really important conclusions about how gravity travels through space. In the case of today’s paper, it allows us to measure how many dimensions that space actually has. Yeah, that’s actually a serious question! We think of space as 3-dimensional. Add the one dimension of time to give us 4-D spacetime – which we’ll also refer to as 3+1 dimensional spacetime. But adding extra spatial dimensions beyond the usual three could actually explain a lot – from the difference between gravity and the other forces of nature to the nature of dark energy.

But before we get all hyperdimensional, let’s think a bit more about 3+1-D spacetime, and

how gravity, light, and matter behave there. Imagine a pulse of light traveling from some distant source. We can think of light rays spreading evenly over an expanding spherical shell. If we see that pulse it means our eye or our telescope intercepts some of those light rays. The brightness of the pulse is determined by now many rays we intercept. So as this shell expands, the light rays become more spread out. Intensity drops proportional to the surface area of the shell – which is proportional to the square of its radius – the square of the distance to the source. This is the famous inverse square law.

If we instead lived in 2-D space then the same pulse would be spread over an expanding circle, not a sphere – it would diminish in intensity proportional to the circumference, and also proportional to the radius – so the distance to the pulse, not the distance squared. The way pulses fade in brightness depends on the number of dimensions – typically proportional to 1-over-distance to the power of the number of dimensions minus 1. 

So with 4 dimensions of space objects should dim proportional to distance cubed.

This relationship also applies to the force felt in a gravitational field. In our universe it appears to diminish according to the inverse square law, as reflected in Newton’s Law of Universal Gravitation. We do see slight deviations in very strong gravitational fields, like close to the Sun. But even there, Einstein’s general relativity describes gravity perfectly with 3 spatial dimensions. In general, general relativity in 3+1 spacetime does a great job at describing gravity in the large-scale universe. But there are some things about this version of gravity that seem peculiar. For example its pathetic strength. While the electromagnetic, strong, and weak forces are in the same ballpark in terms of strength, gravity is vastly weaker – 10^32 times weaker than even the weak nuclear force. The only reason we see so much of it is that its range is infinite – unlike the nuclear forces - and it doesn’t cancel out like the electromagnetic force.

This mismatch in strength might be because gravity is really fundamentally different to the other forces, but that idea makes some physicists sad. Many would like to find a “theory of everything” which merges the forces of nature into the same uber-force. That means gravity has to look just like the other forces at very high energies. It needs to be intrinsically strong, but then become weakened in the low-energy, large-scale regime of our familiar universe. One fun way to do that is to throw in an extra spatial dimension. If you recall, intensity drops off more quickly the more dimensions you have. So you drain gravity into an extra dimension, but you restrict all of the other stuff in the universe – matter, radiation, astronomers – to only 3 spatial dimensions. All of that stuff will behave relatively normally while gravity is weakened.

Let’s get a bit more technical: there are these theoretical objects called branes. We can think of them as geometrical structures of potentially any number of dimensions on which the quantum fields and their corresponding particles can live. They’re used in string theory, where they typically have a large number of dimensions – 11 is popular. But in string theory all but 3 spatial dimensions of the brane are inaccessible. They’re finite and coiled up on themselves – compactified – allowing us to cram them into our 3 spatial dimensions.

But you can also flip this idea around. You can imagine a 3-dimensional brane – a 3-brane – embedded in a spacetime with 4 spatial dimensions, where the extra dimension is extended rather than compact. Most of the stuff in such a universe, including all the fundamental forces besides gravity – would be restricted to the 3-brane. Tune your theory just right and you get normal physics for matter and radiation in 3 spatial dimensions – for example, the usual inverse square law for light. On some spatial scales you even get the inverse square law for gravity. But on other spatial scales gravity can behave very differently. If gravity spreads out in 4 dimensions rather than three, then it should become much weaker.

This can be used to explain the general weakness of the gravitational force on all but the tiniest scales. It can also be used to explain another mysterious phenomenon – dark energy. This is something we’ve gone into in great depth, but in short: the expansion of the universe seems to be accelerating. This is usually thought of as coming from the action of the energy of the vacuum. But there’s another way to get this type of acceleration. In our hypothetical universe with 4 spatial dimensions, gravity is already weak on the scale of the solar system and the galaxy, but can become even weaker on larger scales. Depending on how you tweak the theory, gravity can obey an inverse square law on galactic scales where it’s sort of coupled to the 3 spatial dimensions of the 3-brane. But it starts to obey the inverse cubed law on much larger scales. In fact the 3-brane itself, which defines the 3-dimensional structure on which our observable universe exists – can actually expand into the extra 4th spatial dimension. To us, that would look like an accelerating expansion of the universe. It would look like dark energy.

So how do you test a wild idea like this? Here’s where we finally get back to our gravitational waves. If the gravitational field can extend into extra spatial dimensions, then gravitational waves should lose energy to that extra dimension as they travel through space. Here’s where I have to complicate things a tiny bit more, but I promise we’re nearly there! While light and the force of gravity appear to obey the inverse square law, in regular 3-D space gravitational waves drop in intensity proportional to just distance, not distance squared. But it’s the same general trend. If space has four or more dimensions then gravitational waves should to drop off faster than you would expect in 3 dimensions.

So that gives us a simple test: just observe a gravitational wave and figure out how much its intensity dropped over the distance it traveled to us. Does that match what you expect in a universe with 3 spatial dimensions? If the drop in intensity is higher, then you have evidence for extra dimensions. Basic stuff, right? All you need is a billion dollar network of gravitational wave detectors and a way to independently measure the distance the wave traveled. Fortunately we have both. We have LIGO and VIRGO, and now we also have GW170817. The electromagnetic signal from these merging-exploding neutron stars allowed us to measure its distance completely independently to the gravitational wave signal – something that’s impossible with black hole mergers.

One other important factor here: in order to determine how much intensity was lost by the gravitational wave, we need to know how intense this was when it started its journey. A super convenient property of gravitational waves is that you can figure this out by looking at other properties of the merger event – namely masses of the merging objects and the frequency of the wave, combined with our independent distance measurement.

OK, so what’s the conclusion? How many extra dimensions did we discover? Uh, zero – precisely zero. The gravitational wave lost the right amount of intensity for a 3+1 dimensional universe. There was no observable leakage of gravity into extra spatial dimensions, pretty much ruling this out as an explanation of dark energy. There still might be compactified extra dimensions, so string theorists are okay – for now. By the way, comparison of the electromagnetic and gravitational wave arrival times also allowed us to verify that gravity really does travel at pretty much exactly the speed of light. This ruled out or constrained various alternative theories to general relativity.

This sort of null result might sound like the less interesting outcome. I mean, how cool would it have been to discover extra dimensions? But don’t be disappointed – it’s completely mind-blowing that we can even test these crazy ideas. Ruling them out narrows the vast scope of possible theoretical models of our universe, bringing us closer and closer to the truth. And apparently that truth doesn’t include a 3-brane embedded in an extended 4+1-dimensional spacetime.