Joseph over at Entsophy has posted about some theoretical physics, namely the problem of what constitutes a Newtonian inertial reference frame. I have some comments over there that aren't terribly well developed, but I thought I'd ruminate here on this topic with a little bit of Nonstandard Analysis (NSA) flavor as well.

Suppose that we are in a spaceship somewhere in the solar system. Do we experience centrifugal forces? The answer seems to depend on our state of rotation, but rotation with respect to what? Newton more or less says to consider the distant stars as fixed, create a coordinate system relative to them, and determine if we are rotating relative to that coordinate system. In practice this produces a frame in which if we are not rotating, we experience essentially no centrifugal force (or so small as to be undetectable).

But this is a global procedure, it requires us to see the photons from extremely far off stars and by using those photons determine where those stars appear to be relative to us, and then construct our coordinate system. Is that essential to the procedure, that the apparent position of far off masses is involved?

Suppose instead we want to construct a coordinate system that is local to us, that doesn't involve masses that are enormous distances away. Can we do it? How?

Imagine we are conducting an experiment to observe some masses and determine whether they experience centrifugal (and/or coriolis) forces relative to our coordinate system. Let's first define a time-scale for our experiment $t^*$ is the time it takes by our lab clock to conduct the whole experiment. To make this concrete, suppose we're interested in human scale objects and processes, so $t^*$ is somewhere between say 1 second and 1 day. Or in any case, it's isn't something like a billion years over which we're supposed to observe galactic events and in which the far off stars actually clearly move.

Next we will define a length scale. So, there is one obvious length scale, which is $r^*=ct^*$. This defines a communication horizon, motions we carry out after the start of the experiment can not be felt by objects farther away than this until after the experiment is done. But we may be able to define another length scale which could be of use. Define the force applied by any object $i$ on our objects of interest $j$ as $F_{ij}$. Now define a length scale by $\min r^+ : \max_i F_{ij} \cong 0 \land \forall \{k\} \sum_k F_{kj} \cong 0$ where $i$ ranges over objects farther away than $r^+$ and $k$ is any subset of objects beyond distance $r^+$. Ok, in reality physical measurements are always representable as standard numbers, the use of infinitesimal nonstandard numbers must be interpreted in terms of quantities too small to measure, the threshold is different for different measuring instruments, but we take it for granted that for any given laboratory there is some unspecified level below which forces will be so small as to be unmeasurable. For example good luck measuring the gravitational attraction of two electrons in a lab here on earth.

So we have a length scale that defines a horizon beyond which no individual object produces any appreciable force, and the sum of any group of objects produces no appreciable net force. This is I think a stronger horizon. Why? Well, for example I might have some delicate apparatus here on earth, and it measures the motion of some tiny slightly charged droplets of water in still air for example. The whole experiment might take say 1 minute, but I could imagine being able to just barely detect the tidal effect of the sun in this experiment even though the sun is 8 light minutes away. That is, the retarded field of the sun is still relevant even if the sun doesn't feel the wiggle of my droplet's motion until another 7 minutes go by.

Nevertheless, this radius is distinctly local, in that it's much smaller than the distance to the nearest galaxy for example. In fact, if we can define some cosmic radius $R$ as encompassing essentially all the visible mass in the universe, then $r^+/R \cong 0$ for our purposes.

Now, we can observe our objects doing their dynamic dance, whatever they are, perhaps we're in a spaceship and we are spinning a bucket of water, or watching charged particles interact, or playing catch the stick with a quadrocopter (see link for YouTube example). Overall we'd like to determine from what point of view do the dynamics occur without any superfluous centrifugal or coriolis accelerations? To figure this out, we can make reference to the positions of any masses within our horizon (but not to the "distant fixed stars"). The important thing here is that we need a notion of "superfluous". Clearly, if we're here on earth and want to study a spinning bucket of water we will say that there are centrifugal forces involved. But if we're doing it while seated on a merry go round there will be additional rotational effects involved. It would be simpler to view the whole thing from the ground rather than the merry go round itself. If we're watching the weather, it might be simpler to describe the forces from the point of view of a coordinate system that doesn't rotate with the earth around it's axis of spin. For example a coordinate system centered at the center of mass of the earth and with one axis pointed between the center of mass of the earth and the center of mass of the sun, the other being the axis of the earth's rotation, and the third being perpendicular to the other two. (yes, I know the earth also orbits the sun, but for the sake of argument perhaps that effect is infinitesimal on our time scale)

So, clearly the notion of a "preferred" or "inertial" reference frame is related to the complexity of the forces involved. So we can fix a coordinate system in our lab, and then determine a rotating coordinate system relative to the lab coordinate system by specifying a vector $\Omega$ around which our inertial coordinate system is rotating and the length of this vector is the rate of rotation. If we orient our measurement instrument along that axis and rotate it at the appropriate angular velocity all the dynamics will be "simplest" in terms of rotational forces. Perhaps we can define the net rotational force as something like $\sum_i |C_i|$ where $C_i$ are the individual centrifugal forces.

The next question is, in this coordinate system, what is the distribution of angular velocities of all masses within our local shell? If this procedure essentially constructs a zero mean angular velocity coordinate system for the objects that have appreciable affects on our observed experiment, then when you expand your shell to the universe it's not surprising that the distant stars (which make up the vast majority of the universe) are "fixed".

For the moment, that's as much time as I have to devote to this topic, but perhaps I'll come back or get some interesting comments here.

2 Responses leave one →
1. December 2, 2013

Note one thing, since in our coordinate system if there is "superfluous" centrifugal force it will be related to the angular velocity, mass of the object, and distance to the object via $m \Omega \times (\Omega \times r)$. "superfluous" angular acceleration in that context will be strongly controlled by large far away masses. If there are a lot of large far away masses within your horizon, they are almost guaranteed to have negligible angular velocity in constructing a coordinate system that minimizes the sum of the size of the centrifugal forces (or a similar metric).

2. December 2, 2013

"Newton more or less says to consider the distant stars as fixed, create a coordinate system relative to them, and determine if we are rotating relative to that coordinate system."

Just to be clear: Newton believed in absolute space and inertial frames were frames of reference moving at constant velocity relative to it. He also thought you could trivially tell locally whether you were accelerating with respect to absolute space. When you spin a bucket full of water the presence of centrifugal forces was evidence the bucket was spinning relative to absolute space. You could be in a locked room and have never seen the stars, but still know when something was spinning relative to absolute space.

For Newton, the fact that the frame of the fixed stars was an inertial frame was just a happy coincidence.