As a computational matter, when we are in a rotating frame such as the Earth, we have a choice. We could work with inertial frames where Newton’s laws and Maxwell’s equation are valid, use real forces only, and make sure to include all the accelerations introduced due to the rotational motions. Or we could use a frame that is embedded in the Earth and thus rotating but treat it as an inertial frame by including via the fictitious Coriolis and centrifugal forces the non-inertial effects caused by its rotation. The two methods are mathematically equivalent but conceptually different. It is sometimes easier to treat the Earth as an inertial frame that is not spinning and incorporate fictitious forces and that is often done in the field of meteorology.

It is the search for genuine inertial frames that is of interest in this series of posts because it is important in the resolution of the radiation paradoxes.

So what would constitute a genuine inertial frame? To identify one, we would need to observe the behavior of an object that we could be sure was not subject to any forces at all. If we see such an object as being at rest or moving with constant velocity, then we could say that we (the observers) are in an inertial frame. The problem is that while it is possible to ensure that an object is not subject to almost all known forces by making sure that it is not in contact with anything and shielding it from non-contact forces like electricity and magnetism, the one exception is gravity. It is not possible to shield an object from the force of gravity, which is one of the features that makes gravity such a special force, distinct from all others.

To get around this, at least theoretically, we can construct a thought experiment in which we imagine that we can observe an object Q that is floating in space, far away from any sources of force, gravitational and otherwise. We can assume that the net force on such an object is zero since the stars and planets are too far away to exert any measurable gravitational forces on it. If we now observe this object to be at rest or moving with a constant velocity, we can say that we are in an inertial frame. We will call this frame I to signify that it is an inertial frame. All frames that move with a uniform velocity (i.e., with a constant speed that does not change direction) with respect to I are then also inertial frames, since in those frames too the object will be at rest or moving with constant velocity. All these inertial frames are equivalent as far as the laws of physics are concerned. Incidentally, this is why we say that it is impossible to identify absolute velocity through space, since whether an object is at rest or moving with constant velocity is purely an artifact of the inertial frame in which one observes it.

In the Newtonian system, space is something fixed and an inertial frame is one which is at rest or moving with constant velocity with respect to that space. But we cannot tell what the constant velocity of the inertial frame through that space is because the laws of physics hold in inertial frames and all inertial frames are equivalent as far as the laws of physics are concerned. If you saw Q accelerating, i.e., violating the law of inertia, that must be because you are in a non-inertial frame in which the laws of physics need not hold.

Note that while the laws of physics are the same in all inertial frames, the values of specific measured quantities need not be. The velocity of an ant crawling along the floor of a train will be different depending on whether the velocity is measured by a person on the train or a person on the ground. But the relationships between quantities as required by Newton’s laws of motion and Maxwell’s equations will be the same in both frames.

Can we go further and distinguish if Q is at at rest or is moving with constant velocity? In other words, determine absolute velocity rather than merely relative velocity? The fact that it is at rest with respect to us is insufficient since it may be that we are moving along with Q. To try to solve that problem, let us assume that we can see distant stars that are far enough away to not exert any gravitational forces on Q but visible enough to enable us to know that Q is either at rest with respect to those stars or is moving with constant velocity. The stars serve as a passive backdrop, like wallpaper, providing us with a measurement guide without affecting the dynamics of Q.

But the background of distant stars, while enabling us to get a sense that we are moving (like people in a train get by looking out the window at something on the ground) cannot serve as indicator of absolute motion of Q unless we assume that the stars are fixed in space, so that being at rest or moving with constant velocity with respect to the stars is equivalent to being at rest or moving with constant velocity with respect to space. But there is no a priori reason to make such an assumption. Furthermore, the assumption that the distant stars are just purely passive bystanders not playing any dynamical role in the motion of Q is also questionable. As we will see later, it can be argued that they play a role in arriving at a deeper understanding of space and inertial frames.

(As a digression, high quality science fiction films like 2001: A Space Odyssey have to deal with the background stars problem. The spaceship they used was a large model that was kept stationary while the camera was moved slowly, giving the impression of the spaceship moving, thus taking advantage of Galilean relativity that says that when it comes to two objects moving with constant velocity with respect to each other, we cannot tell which is moving with respect to which. The problem is that moving the camera meant that the painted background of stars would also appear to move and this anomaly would be quickly pointed out by annoying nitpickers like me. So they used matte techniques to black out the background when the camera was moving and then reinserting the stars later.)

(To be continued.)

1. GerrardOfTitanServer says

I feel so confused. Like, isn’t free-fall also an inertia frame according to Einstein? I thought the whole point of inertia frames is that you could tell if you were in an inertia frame or not via experiments that could be done inside of a closed box, e.g. an accelerometer. Unless you meant to restrict yourself to Newtonian physics only for this discussion.

2. Rob Grigjanis says

Gerrard @1:

isn’t free-fall also an inertia frame according to Einstein?

Yes. Strictly speaking, ‘free-fall’ frames are the closest thing to proper inertial frames. Other frames can approximate inertial frames, depending on the type of experiment, accuracy desired, etc, etc.

But in a curved spacetime, real inertial frames are strictly local. A lab in orbit around the Earth is a good approximation to an inertial frame. So is a lab falling towards the Earth. But those two frames are also undergoing nowhere near uniform motion with respect to each other. But, the physical laws they infer from experiments done within their labs (i.e. locally) will agree with each other very well.

3. birgerjohansson says

As I process visual input, my brain plays tricks on me. As I read the title of the post , I associated to Ian McCollums Youtube blog Forgotten weapons- “is this about Walther P7 ?” (a system relying on the inertia of the frame). 🙂

4. birgerjohansson says

In ordinary articles aimed at non-specialists, you might use terms like “the centrifugal pseudoforce” to make the text scientifically accurate but not lose the lay readers.
Another annoying detail -- it is non-trivial to translate physics texts from one language to another unless you have encyclopedic knowledge of the terms in both languages. A lexicon may give an ambiguous explanation..