The interplay between internal and external accelerations

Center of Mass Motion and The External Field Effect

There are two questions that often come up in the context of MOND, regarding the behavior of a small gravitating system in the field of a larger, mother system:

(See also the MOND laws of galactic dynamics.)

An explanation by Moti Milgrom:

MOND has to be described by a nonlinear theory. This basically means that the acceleration endowed by two bodies to a third is not the (vectorial) sum of the individual accelerations produced by each separately. For example, if we double the mass of a body, the acceleration it produces at a given position is not doubled (as happens in Newtonian dynamics); it is increased only by sqrt(2). This nonlinearity leads to some results that do not conform with the intuition we have gained from long experience with linear theories (such as Newtonian dynamics and Maxwell's electrodynamics). The above mentioned questions show up in the following contexts: Suppose we want to describe the dynamics of a self gravitating composite system made of many masses. Now suppose that the system can be divided into two subsystems, one of a low mass and small extent (the ``subsystem''), the other of a large mass and large extent (the ``mother system''). The subsystem could be a star, a binary star, a star cluster, or a gas cloud in a mother galaxy; or the subsystem could be a galaxy in a mother galaxy cluster. Then, if we are only interested in what happens to, and inside, the subsystem, we can save ourselves the complications of accounting for the full field of the mother system, by approximately encapsulating its effects by an external acceleration field in which the subsystem is thought of as being embedded. This is a common practice in many situations in physics.

In the linear, Newtonian dynamics, the internal dynamics of the subsystem, and its center-of-mass motion in the field of the mother system decouple. Namely, the internal dynamics is always the same no matter what the external field is, and the COM motion is oblivious to what happens internally.

Not so in a nonlinear theory, such as MOND.

In Maxwel'll electrostatics, which is linear, the external field can be felt internally--e.g., as polarization--if particles in the subsystem have different charge-to-mass ratios. But if they all have the same ratio, the situation is similar to that in Newtonian gravity.

It is good to remember that physics is rife with nonlinear theories. General relativity is, of course, nonlinear (see below). I can also mention the Born-Infeld generalization of Maxwell's electrodynamics, propounded in the 1930s, which has regained much popularity recently, as it appears as an effective theory in the context of string theory, and which has much in common with certain versions of MOND. And there are many other examples of nonlinear behavior in physical systems.

So the short ``explanation'' of the above two effects would be that they somehow follow from a more detailed account of how exactly the nonlinearity acts. In fact, the EFE is easier to understand, because it is clear that the nonlinearity does not permit us to ignore the external field (the field of the mother system) when dealing with the internal dynamics. For example, in the framework of ``modified inertia'' MOND--namely modification of the second law--the acceleration that enters the argument of mu, namely, the acceleration that tells you whether you are in the MOND or high-acceleration regime, is the total acceleration of the particle with respect to some absolute inertial frame. This could be the quantum vacuum, for example. Then, when you are performing an experiment on earth, you have to recon not just with the small increment of force and acceleration internal to your experiment, but with the fact that each mass in your experiment is also undergoing the acceleration due to the rotation of the earth, its revolution around the sun, of the sun's around the galaxy, and even the accelerations due to micro-earthquakes are larger than a0. Since this absolute acceleration is very large, MOND effects are erased.

General relativity is not linear (for example if you double the mass of a black hole you do not double the acceleration at a given radius). But in GR, the strong equivalence principle tells you that IN A SMALL ENOUGH system you can forget about the external acceleration. This principle is, however, rather unique to GR.

I will now try to give a somewhat more detailed (if still incomplete) explanation.

As a caricature of the more general configuration, look at a simple three-body system: two small bodies b1 and b2, and a massive body B, all three interacting through gravity. Suppose that b1 and b2 are at a small distance from each other compared with their distance from B (as said above, this could model a table top experiment made of b1 and b2 in the field of the earth, or a globular cluster in the field of a galaxy). The EFE is a result of the fact that in MOND we cannot treat the dynamics in the b1-b2 system ignoring the presence of B? We know, for example, that if the acceleration of the b1-b2 system due to B is much larger than a0, then, even if the mutual acceleration in the b1-b2 system is small, its dynamics is Newtonian, while, if the b1-b2 system is isolated, its behavior is MONDian. In Newtonian dynamics we can ignore body B (barring tidal effects). So what makes MOND so different?

The reason the presence of B affects so much the internal, b1-b2 dynamics (the external field effect), is that MOND is nonlinear. You cannot calculate the combined effects of B and b1, on b2, by first calculating separately the acceleration produced by B alone and by b1 alone, then add them (vectorially). Newtonian dynamics is linear (and the superposition principle does apply); so is Maxwell's electromagnetism. But, as I said, these are exceptions. General Relativity, for example, is non-linear (MOND is nonlinear both in the relativistic an nonrelativistic regime). In Newtonian dynamics you can, for example write the acceleration of b2 as a2=a2(B)+a2(b1), where a2(B) is the acceleration due to B alone, and a2(b1) is due to b1 alone. Similarly, a1=a1(B)+a1(b2). Now subtract. Because b1 and b2 are much nearer each other than to B, we have a2(B)=a1(B) ; so subtracting the above two you have a1-a2=a1(b2)-a1(b1), which means that B is out of the problem. This cannot be done in MOND. Now take the case where the internal accelerations in the b1-b2 system are high compared with a0, but the acceleration of them due to B is small. This could model a star (with b1, b2 standing for atoms in the star), or a close binary of two stars, moving in the outskirts of a galaxy (body B), where the galactic acceleration is small. Since the total acceleration of b1 and b2 separately are high, and should thus be Newtonian, not MONDian, how come the combined, center-of-mass acceleration in the field of B is MONDian?

I don't actually have a non-technical, intuitive, or heuristic answer. This, in fact, happens differently in different MOND theories. In the modified gravity theories, such as the one Bekenstein and I wrote in 1984 (modified, nonlinear Poisson equation), or in the more recent QUMOND (quasi-linear formulation of non-relativistic MOND), this happens because the gravitational force on any body made of a distribution of masses can be written as an integral over any surface that surrounds that body (excluding all other masses). You can take the integral on a surface surrounding the b1-b2 system far from the system itself, and show that that integral depends only on the total mass (in fact is proportional to the total mass) of the b1-b2 system, as long as this mass is small compared to that of B. This is an exact result in the limit where the mass of the b1/b2 system (the star) is small compared with that of B (the galaxy).

(Seen differently, but still on a technical level, the solution of Poisson-like equations depends not only of the mass distribution, but also on the boundary conditions at infinity. In the present case, this reflects the acceleration field produced by body B at the position of the small b1-b2 system. This is imprinted on the solution even inside the system, where the internal accelerations are high.)

I can also add that a similar "paradox" occurs in GR: Suppose b1 and b2 are two black holes in a close binary system, not many Schwarzschild radii away from each other. B is far from the b1/b2 system, and produces a Newtonin field at their position. The motion of say b1 in the combined field of the system is highly relativistic and very complicated (you would not even be able to calculate it in general). The same is true of b2. So you wouldn't even know to calculate their joint, center-of-mass motion by calculating each separately and combining. Yet, you know well that the center-of-mass motion of the b1/b2 system in the field of B will be simple and Newtonian. There too this follows from general arguments of the above type.

One can summarize the situation as follows: Write the total (vectorial) acceleration of b1 as a1=a(B)+q1, and that of b2 as a2=a(B)+q2, where a(B) is the acceleration produced by B alone, at the position of the b1/b2 system. The theory then tells us that despite the nonlinearity, and the fact that q1 is NOT the acceleration produced on b1 by b2 alone (and vice versa), we still have m1*q1+m2*q2=0. This means that the COM acceleration of the b1/b2 system is always a(B), no matter what the internal accelerations are. However, q1 and q2, which describe the internal dynamics, do depend also on the external field. When they are much larger than a(B), they do not depend on the external field, ARE the mutual b1-b2 accelerations, and so the internal dynamics are not affected by the external field. But when q1, q2 are much smaller than a(B) they are affected by B, leading to the EFE..

In modified-inertia theories the explanation of the COM motion is different.