Inertial Frames

We are now finally prepared to tackle a very difficult concept. All of our dynamics so far is based on the notion that we can formulate it in an inertial frame. It’s right there in Newton’s Laws – valid only in inertial frames, and we can now clearly see that if we are not in such a frame we have to account for pseudoforces before we can solve Newtonian problems in that frame.

This is not a trivial question. The Universe doesn’t come with a frame attached – frames are something we imagine, a part of the conceptual map we are trying to build in our minds in an accurate correspondence with our experience of that Universe. When we look out of our window, the world appears flat so we invent a Cartesian flat Earth. Later, further experience on longer length scales reveals that the world is really a curved, approximately spherical object that is only locally flat -a manifold⁶⁷ in fact.

Similarly we do simple experiments suspending masses from strings, observing blocks sliding down inclined planes, firing simple projectiles and observing their trajec tories – under the assumption that our experiential coordinates associated with the Earth’s surface form an inertial frame, and Newton’s Laws appear to work pretty well in them at first. But then comes the day when we fire a naval cannon at a target twenty kilometers to the north of us and all of our shots consistently miss to the east of it because of that pesky coriolis force – the pseudoforce in the rotating frame of the earth and we learn of our mistake.

We learn to be cautious in our system of beliefs. We are always doing our experiments and making our observations in a sort of “box”, a box of limited range and resolution. We have to accept the fact that any set of coordinates we might choose might or might not be inertial, might or might not be “flat”, that at best they might be locally flat and inertial within the box we can reach so far.

In this latter, highly conservative point of view, how do we determine that the coordi nates we are using are truly inertial? To put it another way, is there a rest frame for the Universe, a Universal inertial frame S from which we can transform to all other frames S’, inertial or not?

The results of the last section provide us with one possible way. If we systematically determine the force laws of nature, Newton tells us that all of those laws involve two ob jects (at least). I cannot be pushed unless I push back on something. In more appropriate language (although not so conceptually profound) all of the force laws are laws of interac tion.

I interact with the Earth by means of gravity, and with a knowledge of the force law I can compute the force I exort on the Earth and the force the Earth exerts on me given only a knowledge of our mutual relative coordinates in any coordinate system.

Later we will learn that the same is more or less true for the electromagnetic interaction – it is a lot more complicated, in the end, than gravity, but it is still true that a knowledge of the trajectories of two charged objects suffices to determine their electromagnetic interaction. and there is a famous paper by Wheeler and Feynman that suggests that even “radiation reaction” (something that locally appears as a one-body self-force) is really a consequence of interaction with a Universe of charge pairs.

This, then, allows us to cleanly differentiate real forces and pseudoforces. Real forces always involve two objects, where pseudolorces have no Newton’s Third Law partner! In the Elevator and Boxcar examples below, real gravity is identified by the fact that while the Earth pulls down on the mass in question, the mass pulls up on the Earth! Where the normal force acts on it, it pushes back on the object exerting the normal force. When tension in a string pulis it, it pulls back on the string.

There is no Newton’s third law partner to the F -ma pseudoforce arising from the acceleration of the frame! Furthermore, this “pseudogravity” behaves differently from actual gravity- it is (for example) perfectly uniform within the trame no matter how large the frame might be where actual gravity drops off (however slightly) as one moves verti cally, with the field lines slowly diverging (because the gravitational field diverges from its massive sources).

One has to work a bit harder to make this operational distinction clear when one builds a (relativistic, quantum) field theory, but throughout physics it remains the case that forces (or their quantum equivalents, “interactions”) never accur in isolation, while

pseudoforces “just happen” with nothing else making them happen.

The point is that in the end, the operational definition of an inertial frame is that it is a frame where Newton’s Laws are true for a closed, finite set of (force) Law of Nature that all involve well-defined interaction in the coordinates of the inertial frame. In that case we can add up all of the actual forces acting on any mass.

If the observed movement of that mass is in correspondence with Newton’s Laws given that total force, the frame must be inertial! Otherwise, there must be at least one “force” that we cannot identify in terms of any interaction pair, and examined closely, it will have a structure that suggests some sort of acceleration of the frame rather than interaction per se with a (perhaps still undiscovered) interaction law.

There is little more that we can do, and even this will not generally suffice to prove that any given frame is truly inertial. For example, consider the “rest frame” of the visible Universe, which can be thought of as a sphere some 13.7 billion Light Years⁶⁸ in radius surrounding the Earth.

To the best of our ability to tell, there is no compelling asymmetry of velocity or relative acceleration within that sphere-all motion is reasonably well accounted for by means of the known forces plus an as yet unknown force, the force associated with “dark matter” and “dark energy”, that still appears to be a local interaction but one we do not yet understand.

How could we tell if the entire sphere were uniformly accelerating in some direction, however? Note well that we can only observe near-Earth gravity by its opposition – in a freely falling box all motion in box coordinates is precisely what one would expect if the box were not falling! The pseudoforce associated with the motion only appears when relating the box coordinates back to the actual unknown inertial frame.

If all of this gives you a headache, well, it gives me a bit of a headache too. The point is once again that an inertial frame is practically speaking a frame where Newton’s Laws hold, and that while the coordinate frame of the visible Universe is probably the best that we can do for a Universal rest frame, we cannot be certain that it is truly inertial on a much larger length scale – we might be able to detect it if it wasn’t, but then, we might not.

Einstein extended these general meditations upon the invariance of frames to invent first special relativity (frame transformations that leave Maxwell’s Equations form invariant and hence preserve the speed of light in all inertial frames) and then general relativity, which is discussed a bit further below.