MOND and holography

These blogs should be sources of provocative ideas that can lead to something new, which is why this article is about MOND and holography. Be sure that I find all MOND theories very unlikely but we should all be aware of interesting speculations that have shown non-trivially looking quantitative results.

Note added later: the readers who find this article interesting should also look at a newer article about the Pioneer anomaly.

Note added in 2006: more direct observations of dark matter have arguably falsified all published versions of MOND, great, and it is a somewhat open question whether more sophisticated versions of MOND like mine - that may involve macroscopic interference - are gone, too. The question can be settled by similar observations of the clusters in the future: if the distribution of the deduced dark matter is going to be random and independent of the visible matter, then any theory without dark matter will be dead.

What is MOND? The acronym MOND stands for "modified Newtonian dynamics", and it is an alternative to the theory of dark matter. Jacob Bekenstein gave a talk about MOND at Harvard when he was visiting us - and he is probably the most famous current advocate of this theory which is viewed as a controversial one: MOND does not quite agree with the usual picture of the Universe that emerges from general relativity. There is a lot of reasons to think that MOND is nonsense, but the goal of this article is just the opposite one. ;-)


Basic argument in favor of dark matter

In order to convince you that there can be something nice about MOND, you need to know the basic reasons why most people believe that there is a lot of dark matter around.




When we observe other galaxies, we can use the usual laws of general relativity (well, Newton's laws are enough) to predict the angular velocity of the stars in that particular galaxy as a function of their distance from the center. We think that we know the mass distribution because we know where the stars are located and what their mass is, and therefore we may calculate the gravitational field, and the motion of the stars in this gravitational field.

What results do we get from Newton's theory if we substitute the masses of the visible stars as the source of the gravitational field? The result says that the angular velocity (the velocity divided by the radius) decreases with the distance from the center (with the radius), much like in the case of planets orbiting the Sun. You know that Pluto's motion around the Sun is much slower than the motion of Mercury, and the ratio of the angular velocities is even more pronounced because Pluto is much further from the Sun and the distance appears in the denominator.

However, the direct observations of the stars' motion lead to a different result: the angular velocity is nearly constant for all stars in a single galaxy, except for a few stars near the galactic center. The rotation of a galaxy resembles the rotation of an LP recording! Although the gravitational laws have been successfully tested for planets and many other celestial bodies, the predictions of the rotation of galaxies seem to fail.

The usual response is that the true gravitational field is very different from the gravitational field of the visible stars: there is a huge amount of invisible matter (dark matter) whose gravitational field is such that the stars in the galaxy rotate like points on an LP recording. One must work with a reasonable distribution of dark matter to achieve this goal, and the theories of dark matter are pretty messy. Most of the dark matter must occupy a "halo" that is pretty far from the galactic center, and one needs to tune many parameters to make the dark matter theory agree with all the observed galaxies. Well, that's what you're forced to do if you believe that general relativity is a good description at very long, cosmological distances.


MOND basic achievements

One can measure the dependence of angular velocity on the distance from the center for different galaxies. Each galaxy gives a slightly different result. The predictions of Newton's theory are pretty good for the stars near the center - but a totally different behavior takes over once the distance from the center is greater than a certain critical distance.

In fact, the point where the angular velocity starts to deviate from Newton's laws - where it starts to be constant - always seems to be at the distance where the acceleration of the stars decreases to a certain universal critical acceleration a_0. For each galaxy, the distance of the first stars for which Newton's laws "break down" is slightly different; but their acceleration is always the same, with a rather good accuracy. That's interesting, is not it?

Moreover, the constant a_0 is a number pretty close to the Hubble constant H. In other words, a_0 is comparable to the inverse age of the Universe (multiplied by the speed of light). This fact will be important at the end of this posting.

The MOND theories, roughly speaking, say that the Newtonian laws are modified in such a way that the inverse square law (1/r^2) for gravity is continously replaced by the inverse distance law (1/r) for all objects whose acceleration is smaller than the critical value a_0. This crazy assumption allows one to reproduce the observed results from a large number of galaxies, using a minimal number of parameters - and without assuming any dark matter. (The agreement is not so great for clusters of galaxies, but this is not what we want to analyze here.)

Many ways how to derive this weird new behavior from "more field-theoretical" laws, closer to GR, have been proposed - but let me say that neither of them looks natural enough. Therefore the phenomenological description with the 1/r^2 force transmuting into 1/r for small accelerations is everything we find valuable about MOND.


Why it may follow from holography?


OK, let me now switch to my speculative explanation why this weird behavior may be derivable from holography. (Jacob Bekenstein told me that a crackpot has already written a paper about this relation MOND-holography, and therefore I apologize if I am not the first one who proposes this idea haha.)

First, imagine the usual holograms in optics, invented by Dennis Gabor in the 1950s. They are two-dimensional pictures. But if you look at them carefully (in some cases, you will need laser - the same laser that is necessary to produce them), you see a three-dimensional object.

The three-dimensional illusion results from a dense network of interference patterns. These patterns reflect that wave character of light. A small piece of the hologram can still be used to reconstruct the whole three-dimensional image, although the quality is reduced. It will be important below to keep in mind that if the piece of the hologram is way too small (compared to the object that we want to see, or even compared to the wavelength of the light), the three-dimensional illusion will disappear.

In quantum gravity, we know that something like holography is important, too. 't Hooft and Susskind were the first to propose the idea (the holographic principle) that in every theory of quantum gravity (which really means in "every solution of string/M-theory"), the information about the "bulk" can be encoded on the surface of this volume, and the density of information is never bigger than roughly one bit per Planck area.

Normally, we think that the "memory" grows with the volume. The bigger volume we have, the more RAM chips we can insert into this volume. However, gravity guarantees that this can't work indefinitely. If you put too many RAM chips into a too small volume, the gravity will be so strong that the chips will collapse into a black hole. It has been realized by Bekenstein and Hawking that the entropy (or information) carried by this black hole only scales like r^2 (in four spacetime dimensions) divided by the Planck area, instead of the usual r^3 behavior. Black hole is, at the same moment, the most entropic system that can fit a given volume. It really means that large volumes can carry much less information than what we would expect from a naive proportionality law (with the volume).

Maldacena's AdS/CFT correspondence is a very rigorous example of holography in string theory: in its most popular version, a four-dimensional (gauge) theory living on the boundary of five-dimensional anti de Sitter space contains all information about quantum gravitational (and stringy) physics inside the five-dimensional "bulk" (multiplied by another five-dimensional sphere, to get the total of ten dimensions).

Forget about AdS/CFT for a while, and think about our real Universe again. OK, I'm now gonna argue that MOND may follow from holography. As I mentioned above, the three-dimensional illusion of a hologram in optics breaks down if the piece of your hologram is too small. It's not unnatural to believe that a similar limitation occurs for holograms in quantum gravity. More generally, I want to argue that holography in quantum gravity implies modifications of dynamics in the "infrared" - and we want to define "infrared" according to the acceleration.

Consider an object whose acceleration is a. The worldline in spacetime is a hyperbola, and the center of curvature of this hyperbola is at distance 1/a. If you associate de Broglie's wave with that object, the lines of constant phase will depend on the velocity, and they will intersect at the center of curvature of that hyperbola. Let's now believe that this self-intersection is necessary for the three-dimensional interpretation of our hologram to be valid. OK?

If you swallowed that, then we're done. It's simply because the 3+1-dimensional physics can only be trusted if the center of the hyperbola fits into the cosmic hologram, i.e. if 1/a is smaller than the radius of the Universe. In other words, the usual physics only occurs if a is greater than the critical acceleration. If the acceleration is smaller, you're not allowed to use the 3+1-dimensional laws to calculate the forces affecting the object. Instead, you should switch to the physics of the hologram which is 2+1-dimensional. If you allow me to make one more leap, I can even say that the usual 1/r^2 force in 3+1 dimensions is replaced by the 1/r force in 2+1 dimensions (of the hologram), which is precisely what we need for MOND to describe the rotation of galaxies without any dark matter.

What do you think about this weird proposal? :-)