Friday, May 19, 2017

Can we use gravitational waves to rule out extra dimensions – and string theory with it?

Gravitational Waves,
Computer simulation.

Credits: Henze, NASA
Tl;dr: Probably not.

Last week I learned from New Scientist that “Gravitational waves could show hints of extra dimensions.” The article is about a paper which recently appeared on the arxiv:

The claim in this paper is nothing but stunning. Authors Andriot and Gómez argue that if our universe has additional dimensions, no matter how small, then we could find out using gravitational waves in the frequency regime accessible by LIGO.

While LIGO alone cannot do it because the measurement requires three independent detectors, soon upcoming experiments could either confirm or forever rule out extra dimensions – and kill string theory along the way. That, ladies and gentlemen, would be the discovery of the millennium. And, almost equally stunning, you heard it first from New Scientist.

Additional dimensions are today primarily associated with string theory, but the idea is much older. In the context of general relativity, it dates back to the work of Kaluza and Klein the 1920s. I came across their papers as an undergraduate and was fascinated. Kaluza and Klein showed that if you add a fourth space-like coordinate to our universe and curl it up to a tiny circle, you don’t get back general relativity – you get back general relativity plus electrodynamics.

In the presently most widely used variants of string theory one has not one, but six additional dimensions and they can be curled up – or ‘compactified,’ as they say – to complicated shapes. But a key feature of the original idea survives: Waves which extend into the extra dimension must have wavelengths in integer fractions of the extra dimension’s radius. This gives rise to an infinite number of higher harmonics – the “Kaluza-Klein tower” – that appear like massive excitations of any particle that can travel into the extra dimensions.

The mass of these excitations is inversely proportional to the radius (in natural units). This means if the radius is small, one needs a lot of energy to create an excitation, and this explains why he haven’t yet noticed the additional dimensions.

In the most commonly used model, one further assumes that the only particle that experiences the extra-dimensions is the graviton – the hypothetical quantum of the gravitational interaction. Since we have not measured the gravitational interaction on short distances as precisely as the other interactions, such gravity-only extra-dimensions allow for larger radii than all-particle extra-dimensions (known as “universal extra-dimensions”.) In the new paper, the authors deal with gravity-only extra-dimensions.

From the current lack of observation, one can then derive bounds on the size of the extra-dimension. These bounds depend on the number of extra-dimensions and on their intrinsic curvature. For the simplest case – the flat extra-dimensions used in the paper – the bounds range from a few micrometers (for two extra-dimensions) to a few inverse MeV for six extra dimensions (natural units again).

Such extra-dimensions do more, however, than giving rise to a tower of massive graviton excitations. Gravitational waves have spin two regardless of the number of spacelike dimensions, but the number of possible polarizations depends on the number of dimensions. More dimensions, more possible polarizations. And the number of polarizations, importantly, doesn’t depend on the size of the extra-dimensions at all.

In the new paper, the authors point out that the additional polarization of the graviton affects the propagation even of the non-excited gravitational waves, ie the ones that we can measure. The modified geometry of general relativity gives rise to a “breathing mode,” that is a gravitational wave which expands and contracts synchronously in the two (large) dimensions perpendicular to the direction of the wave. Such a breathing mode does not exist in normal general relativity, but it is not specific to extra-dimensions; other modifications of general relativity also have a breathing mode. Still, its non-observation would indicate no extra-dimensions.

But an old problem of Kaluza-Klein theories stands in the way of drawing this conclusion. The radii of the additional dimensions (also known as “moduli”) are unstable. You can assume that they have particular initial values, but there is no reason for the radii to stay at these values. If you shake an extra-dimension, its radius tends to run away. That’s a problem because then it becomes very difficult to explain why we haven’t yet noticed the extra-dimensions.

To deal with the unstable radius of an extra-dimension, theoretical physicists hence introduce a potential with a minimum at which the value of the radius is stuck. This isn’t optional – it’s necessary to prevent conflict with observation. One can debate how well-motivated that is, but it’s certainly possible, and it removes the stability problem.

Fixing the radius of an extra-dimension, however, will also make it more difficult to wiggle it – after all, that’s exactly what the potential was made to do. Unfortunately, in the above mentioned paper the authors don’t have stabilizing potentials.

I do not know for sure what stabilizing the extra-dimensions would do to their analysis. This would depend not only on the type and number of extra-dimension but also on the potential. Maybe there is a range in parameter-space where the effect they speak of survives. But from the analysis provided so far it’s not clear, and I am – as always – skeptical.

In summary: I don’t think we’ll rule out string theory any time soon.

[Updated to clarify breathing mode also appears in other modifications of general relativity.]


  1. Hi Sabine,

    very good post, thank you! I have two remarks on what you say they claim "While LIGO alone cannot do it because the measurement requires three independent detectors, soon upcoming experiments could either confirm or forever rule out extra dimensions – and kill string theory along the way."

    1/ Since one cannot predict the amplitude of the effects they describe, there is no way one can rule out extra-dimensions from a search of them : one can only put limits on their amplitudes. At the same time, extra-polarization modes are generic in non-GR theories of gravitation. Therefore, observing them wouldn't be a confirmation of extra-dimension either. Although observing a discrete set of high frequency signals in all polarization modes may be a smoking gun.

    2/ It turns out that the LIGO detectors alone can detect extra-polarization modes. More, the two LIGO and the Virgo detectors can distinguish all the different extra-polarization modes. Not for transient events indeed, but from the stochastic gravitational wave background: (see figure 14 and 15).



  2. Hi Olivier,

    Thanks for pointing out, I have added a clarification on the first point. 2.) Interesting, I'll have a look at this!

  3. Great post on a topic of much interest today, Dr. Bee. I don't read New Scientist any more, and I am far to skeptical of string theory to buy into it (but mainly because I can't imagine Nature being so complicated).

    Question: might the extra dimensions in string theory just be a bunch of parameters needed to make it work, kind of like curve fitting data to a polynomial?

  4. "... extra dimensions add another way for gravitational waves to make space shape-shift, called a breathing mode. expands and contracts as gravitational waves pass through, in addition to stretching and squishing." Expect added degrees of freedom in gravitational radiation orbital decay and elsewhere, yes?

    Pulsar binaries lack orbital decay anomalies. PSR B1913+16 Hulse–Taylor, pulsar-neutron star, 7.75-hr orbit. PSR J0348+0432, pulsar-white dwarf, 2.46-hr orbit. PSR J1903+0327, pulsar- solar star, 95.17-day orbit.

    Hydrogen Lyman-alpha, 21-cm hyperfine transition, Lamb shift.
    ...H-like (91+) and He-like (90+) uranium ions
    ...Table 5.22; Section 5.9 Muonic atom decay
    ... Mössbauer spectroscopy. Fe-57, 14.4 KeV gamma-ray, 5×10^(-9) eV linewidth, 10^(-12) resolution.
    ...Casimir effect, 29.5 - 86 nm interaction range.

    Dark, sterile, see-saw, selective extra dimensions?

  5. Bill,

    The number of dimensions is a parameter and a specific number is necessary to 'make it work' and I think I really don't understand the question.

  6. With the exception that this is not KK but a warped compactification.

    KK ansatz is a direct product (e.g. M₄ x S¹) for the vacuum but you always get moduli in the uncompactified directions.

    To stabilize e.g. via flux compactification you have to use Warped ansatz (due to the flux you can't have a direct product anymore) where you have a warp factor in front of the line element of the 4d metric that depends on the compactified coordinates.

    But forget all these subtleties the big news is that String theory is falsifiable after all and It can finally get the legit scientific theory stump by the popperian bureaucrats!!


  7. As the second author of the paper in question I fully agree with Olivier's comment (first comment here above), particularly with his first point. Moreover, I believe the whole point of vue of your post is a little forced. Talking about falsifying string theory on the basis of our analysis is quite a stretch, to say the least, and I don't think this viewpoint conveys the message very well or fits with the study at all. While ruling out string theory is a legitimate question to ask, I do not think one can answer it based on our simple study, which focuses on the propagation (ignoring sources and emission) and furthermore is not directly related to string theory but more genetically to extra dimensions.

  8. Gustavo,

    Ruling out extra-dimensions is pretty much the only way to rule out string theory (at low energies). I agree with you of course that it wouldn't be as easy as my blogpost might have made it sound because many other possible explanations for the absence of a signal would have to be ruled out.


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