Monday, November 13, 2023

How not to detect MOND

 You might have heard about recent efforts to inspect lots of "wide binaries", double stars that orbit each other at very large distances, which is one of the tasks the Gaia mission was built for, to determine if their dynamics follows Newtonian gravity or rather MOND, the modified Newtonian dynamics (Einstein theory plays no role at such weak fields). 

You can learn about the latest update from this video by Dr. Betty (spoiler: Newton's just fine).

MOND is an alternative theory of gravity that was originally proposed as an alternative to dark matter to explain galactic rotation curves (which it does quite well, some argue better than dark matter). Since, it has been investigated in other weak gravity situations as well. In short, it introduces an additional scale \(a_0\) of dimension acceleration and posits that gravitational acceleration (either in Newton's law of gravity or in Newton's second law) are weakened by a factor


where a is the acceleration without the correction.

In the recent studies reported on in the video, people measure the stars' velocities and have to do statistics because they don't know about the orbital parameters and the orientation of the orbit relative to the line of sight.

That gave me an idea of what else one could try: When the law of gravity gets modified from its \(1/r^2\) form for large separations and correspondingly small gravitational accelerations, the orbits will no longer be Keppler ellipses. What happens for example if this modified dynamics would result for example in eccentricities growing or shrinking systematically? Then we might observe too many binaries with large/small eccentricities and that would be in indication of a modified gravitational law.

The only question is: What does the modification result in? A quick internet search did not reveal anything useful combining celestial mechanics and MOND, so I had to figure out myself. Inspection shows that you can put the modification into a modification of \(1/r^2\) into 

$$\mu(1/r^2) \frac{\vec r}{r^3}$$

and thus into a corresponding new gravitational potential. Thus much of the usual analysis carries over: Energy and angular momentum would still be conserved and one can go into the center of mass system and work with the reduced mass of the system. And I will use units in which \(GM=1\) to simplify calculations.

The only thing that will no longer be conserved is the Runge-Lenz-vector

$$\vec A= \vec p\times\vec L - \vec e_r.$$

\(\vec A\) points in the direction of the major semi-axis and its length equals the eccentricity of the ellipse.

Just recall that in Newton gravity, this is an additional constant of motion (which made the system \(SO(4,2)\) rather than \(SO(3)\) symmetric and is responsible for states with different \(\ell\) being degenerate in energy for the hydrogen atom), as one can easily check

$$\dot{\vec A} = \{H, \vec A\}= \dot{\vec p}\times \vec L-\dot{\vec e_r}=\dots=0$$

using the equations of motion in the first term. 

To test this idea I started Mathematica and used the numerical ODE solver to solve the modified equations of motion and plot the resulting orbit. I used initial data that implies a large eccentricity (so one can easily see the orientation of the ellipse) and an \(a_0\) that kicks in for about the further away half of the orbit.

Clearly, the orbit is no longer elliptic but precesses around the center of the potential. On the other hand, it does not look like the instantaneous ellipses would get rounder or narrower. So let's plot the orbit of the would be Runge Lenz vector:
Orbit of would be Runge Lenz vector \(\vec A\)

What a disappointment! Even if it is no longer conserved it seems to move on a circle with some additional wiggles on it (Did anybody mention epicycles?). So it is only the orientation of the orbit that changes with time but there is no general trend toward smaller or larger eccentricities that one might look out for in real data.

On the other hand the eccentricity \(\|\vec A\|\) is not exactly conserved but wiggles a bit with the orbit but comes back to its original value after one full rotation. Can we understand that analytically?

To this end, we make use the fact that the equation of motion is only used in the first term when computing the time derivative of \(\vec A\):

 $$\dot{\vec A}=\left(1-\mu(1/r^2)\right) \dot{\vec e_r}.$$

\(\mu\) differs from 1 far away from the center, where the acceleration is weakest. On the other hand, since \(\vec e_r\) is a unit vector, its time derivative has to be orthogonal to it. But in the far away part of the the ellipse, \(\vec e_r\) is almost parallel to the major semi axis and thus \(\vec A\) and thus \(\dot{\vec a}\) is almost orthogonal to \(\vec A\). Furthermore, due to the reflection symmetry of the ellipse, the parts of \(\dot{\vec e_r}\) that are not orthogonal to \(\vec A\) will cancel each other on both sides and thus the wiggling around the average \(\|\vec a\|\) is periodic with the period of the orbit. q.e.d.

There is only a tiny net effect since the ellipse is not exactly symmetric but precesses a little bit. This can be seen when plotting \(\|\vec A\|\) as a function of time:

\(\|\vec A\|\) as a function of time for the first 1000 units of time (brown) and from time 9000 to 10,000 (red)

The same plot zoomed in. One can see that the brown line's minimum is slightly below the red one.
If one looks very carefully, one sees a tiny trend towards larger values of eccentricity.

If one looks very carefully, one sees a tiny trend towards larger values of eccentricity.

This is probably far too weak to have any observable consequence (in particular since there are a million other perturbing effects), but these numerics suggests that binaries whose orbits probe the MOND regime for a long time should show slightly larger eccentricities on average.

So Gaia people, go out an check this!

Friday, April 28, 2023

Can you create a black hole in AdS?

 Here is a little puzzle I just came up with when in today's hep-th serving I found 

  arXiv:2304.14351 [pdfother]
Operator growth and black hole formation
Comments: 20+9 pages, 10 figures. arXiv admin note: text overlap with arXiv:2104.02736
Subjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc); Quantum Physics (quant-ph)

When two particles collide in an asymptotically AdS spacetime with high enough energy and small enough impact parameter, they can form a black hole. 

But to explain it, I should probably say one or two things about thermal states in the algebraic QFT language: There (as we teach for example in our "Mathematical Statistical Physics" class) you take take to distinguish (quasi-local) observables which form a C*-algebra and representations of these on a Hilbert space. In particular, like for example for Lie algebras, there can be inequivalent representations that is different Hilbert spaces where the observables act as operators but there are no (quasi-local) operators that you can use to act on a vector state in one Hilbert space that brings you to the other Hilbert space. The different Hilbert space representations are different super-selection sectors of the theory.

A typical example are states of different density in infinite volume: The difference in particle number is infinite but any finite product of creation and annihilation operators cannot change the particle number by an infinite amount. Or said differently: In Fock space, there are only states with arbitrary but finite particle number, trying to change that you run into IR divergent operators.

Similarly, assuming that the (weak closure) of the representation on one Hilbert space if a type III factor as it should be for a good QFT, states of different temperatures (KMS states in that language) are disjoint, meaning they live in different Hilbert spaces and you cannot go from one to the other by acting with a quasi-local operator. This is proven as Theorem 5.3.35 in volume 2 of the Bratelli/Robinson textbook.

Now to the AdS black holes: Start with empty AdS space also encoded by the vacuum in the holographic boundary theory. Now, at t=0 you act with two boundary operators (obviously quasi-local) to create two strong gravitational wave packets heading towards each other with very small impact parameter. Assuming the hoop conjecture, they will create a black hole when they collide (probably plus some outgoing gravitational radiation). 

Then we wait long enough for things to settle (but not so long as the black hole starts to evaporate in a significant amount). We should be left with some AdS-Kerr black hole. From the boundary perspective, this should now be a thermal state (of the black hole temperature) according to the usual dictionary.

So, from the point of the boundary, we started from the vacuum, acted with local operators and ended up in a thermal state. But this is exactly what the abstract reasoning above says is impossible.

How can this be? Comments are open!