keywords: elementary inertial frames, center-of-mass inertial frames, comoving frames.

Yours truly has tried in the following to explicate correctly, comprehensively, and concisely inertial frames in 3 classes: elementary inertial frames, center-of-mass inertial frames, and comoving frames. It is a hard task for yours truly.

Explication:

1. Inertial frames are reference frames relative to which all physical laws are referenced, except general relativity which tells us what they are according to our modern understanding which is NOT the original understanding of their first proposer Isaac Newton (1643--1727). They are NOT just arbitrary coordinate systems.

   Note for example, Newtonian physics is defined with respect to inertial frames---which is something NOT usually explicated in high-school physics---but it really should be for Newtonian physics to make any sense.

   The upshot is that inertial frames are the reference frames used for the analysis of the motions of systems of objects (e.g., systems of astro-bodies).

2. From general relativity, elementary inertial frames are free-fall frames in a UNIFORM EXTERNAL gravitational field NOT rotating relative to the observable universe (i.e., to the bulk mass-energy of observable universe).

   A spacecraft in orbit (which means it is in free fall) is a simple example of an elementary inertial frame.

   Any reference frame NOT accelerated relative to an elementary inertial frame for some system in the same UNIFORM EXTERNAL gravitational field is also an elementary inertial frame for the system.

   Note a rotating frame is an accelerated reference frame though in a somewhat complex way and is therefore NOT an elementary inertial frame though it can be treated as an non-elementary inertial frame as we discuss below.

   Note also all elementary inertial frames/a> in the observable universe do NOT rotate with respect to each other, except in very strong gravitational fields like near black holes???? (Wikipedia: Inertial frame of reference: General relativity). This, of course, is a theory, but it has NEVER been falsified and may be absolutely true in the limit of weak enough gravitational fields???.

3. How do you do physics if you do NOT have an elementary inertial frame for a system of objects (e.g., a system of astro-bodies)? You do NOT have an elementary inertial frame because the system is being accelerated in some more or less complex way and/or EXTERNAL gravitational field is NOT UNIFORM. Such reference frames are non-inertial frames in an elementary sense, but they can always be converted to non-elementary inertial frames in some way or you can avoid using them, and so the term non-inertial frame is NOT very necessary.

   In the classical limit (i.e., the limit in which Newtonian physics applies), you can use a center-of-mass inertial frames.

   Note we often use the term center-of-mass inertial frame as a synonym for the system of objects it is used to analyze. This usage just seems natural and context tells you the specific meaning of center-of-mass inertial frame.

   First, you identify the center of mass of your system and it becomes the most conventional origin for the center-of-mass inertial frame.

   You then divide forces on the system into EXTERNAL (arising from outside the system) and INTERNAL (arising from inside the system). The center of mass motion is determined only by the NET EXTERNAL force.

   The forces can be ORDINARY forces or inertial forces which arise from non-elementary inertial frame effects.

   The INTERNAL motions of the system are determined by the EXTERNAL and INTERNAL forces.

   The above procedure is complex in general, but in many useful special cases it is simple.

4. For a system of astro-bodies (which are our main interest here), the usually dominant and often overwhelmingly dominant force is the gravitational force.

   In the simplest case, the EXTERNAL gravitational field is UNIFORM and there are NO other EXTERNAL forces to accelerate the system and the center-of-mass inertial frame reduces to being an elementary inertial frame.

   In fact, planetary systems are usually so small relative to variations in the EXTERNAL gravitational field that they can to be treated as in an elementary inertial frame.

   But for most other systems of astro-bodies (e.g., planet-moon systems, star clusters, galaxies, and galaxy clusters), the EXTERNAL gravitational field is NOT sufficiently UNIFORM.

   In these cases, the EXTERNAL gravitational field gives rise to the tidal force which is essentially a stretching force due the variations in the gravitational field over the system. The tidal force is also partially an inertial force though this is often NOT mentioned since its complication to do so. For example, the tidal force causes the Earth's tides as we will discuss a bit further below.

5. For single compact astro-body (i.e., one NOT supported against self-gravity by kinetic energy), it is usually convenient to use a rotating frame that rotates with the astro-body's rotation (relative to the observable universe). In this case, one must use special rotating frame inertial forces. These are detailed in file Mechanics file: frame_rotating.html. We will NOT go into them here.

   But how does the idea of center-of-mass inertial frame apply to the Earth's surface which is a sufficiently good inertial frame for most purposes without using rotating frame inertial forces or the tidal force?

   The Earth's center is the center of mass of the Earth's center-of-mass inertial frame. If you imagine the Earth as NOT rotating and in an absolutely UNIFORM EXTERNAL gravitational field, then the Earth's center-of-mass inertial frame would be an elementary inertial frame and then any point on the Earth's surface could also be used as the origin of an elementary inertial frame. This is just what we do all the time doing physics on the small scale: i.e., the scale much smaller than the whole Earth's surface. We can do this since the effect of inertial forces and the tidal force is very small on that scale.

   But if you have to deal with the exact shape of the Earth and the weather, then you have to use the rotating frame inertial forces. If you have to deal with the tides, then you have to use the tidal forces of the Moon and secondarily the Sun.

6. Our main interest is in systems of astro-bodies. To be more specific gravitationally bound systems in which the astro-bodies orbit the center of mass of the systems.
   A gravitationally bound system is one where the astro-bodies CANNOT escape to infinity, except via ejection processes like gravitational assists (AKA gravitational slingshot maneuvers).

   Of course, we can analyze such systems using center-of-mass inertial frames.

   In fact, there is whole hierarchy of center-of-mass inertial frames. Each kind of center-of-mass inertial frame is nested in a larger one. For an important example of systems in different levels of the hierarchy, there are center-of-mass inertial frames for:

   1. The Earth.
   2. The Earth-Moon system.
   3. The Solar System.
   4. The Milky Way.
   5. The Local Group.
   6. A local comoving frame which is actually NOT a gravitationally bound system, but is a system that participates in the expansion of the universe, but that only approximates more or less closely the mean the expansion of the universe.

7. The hierarchy of center-of-mass inertial frames top outs with comoving frames.

   In fact, you have infinite choice for a comoving frame.

   Take any point in space and surround it by large enough sphere of radius certainly ⪆ 100 Mpc and, depending on the level of accuracy needed, maybe ⪆ 370 Mpc (Wikipedia: Cosmological principle: Violations of homogeneity). The sphere is big enough to have approximately UNIFORM density on a scale much bigger than that of the largest galaxy clusters and to be expanding with the expansion of the universe. The chosen point is then the center of mass of the center-of-mass inertial frame for the sphere which is the comoving frame for the sphere. Note the sphere is a gravitationally unbound system of astro-bodies. You can use the comoving frame to analyze all the INTERNAL motions of the sphere. Ideally (i.e., in the asymptotic limit as the comoving frame is made large enough), the EXTERNAL forces have NO effect on the INTERNAL motions and the center of mass has NO recession velocity NOR recession velocity acceleration relative to the observable universe (i.e., to the bulk mass-energy of observable universe). Note recession velocity is NOT ordinary velocity, but a rate of growth of space, but on the scale of the comoving frame can be treated as an ordinary velocity in the classical limit.

   For instance, you could choose the Milky Way center (which at Milky Way center distance = 7.4--8.7 kpc = 24--28.4 kly) as your chosen point. But there is a fine point to make. The velocity and acceleration of the Milky Way center of mass relative to the comoving frame center of mass can measured, but they are NOT the recession velocity and recession velocity acceleration of the comoving frame center of mass itself even though the two centers of mass are coincident in space for at our moment in cosmic time. As aforementoned, ideally, there is NO recession velocity NOR recession velocity acceleration relative to the observable universe. However, any actual sphere used for the comoving frame is probably too small to make those quantities exactly zero. But currently they are too small to measure. However, in the 2030s, they may become measurable (see Roos el al. 2024).

8. Can all the foregoing explication be vastly compacted? Yes, in an equation of motion form. But the equation of motion form has to be decompacted into a word explication to one degree or another to order to understand it.