Is the definition of inertial reference frame circular?

Question

In elementary physics classes, inertial reference frames are defined as a coordinate system which is in constant rectilinear motion (or at least that is how it was defined by my professor). How then are we to know that the said reference frame undergoes that constant rectilinear motion, wouldn't we need another reference frame to determine if the current reference frame is truly inertial? Just seems very ad hoc to define it that way.

Answer 1

An inertial frame is one in which the laws of physics take their simplest form. In a non-inertial frame, you also see fictitious forces that don't have any apparent source. For example, if a system contains only a single particle then that particle should travel at constant velocity ($\ddot{x} = 0$) if observed from an inertial frame, and this is the simplest law that is consistent with how the universe actually behaves. A non-inertial observer may see the particle apparently accelerating ($\ddot{x} \ne 0$) and its acceleration may depend on its current position.

It may be observed that fictitious forces can be recharacterized as non-fictitious if we introduce a new field into our theory, which is responsible for exerting those forces the same way that the electromagnetic field exerts electromagnetic forces. This field is the connection in general relativity. If you can find a frame in which the connection vanishes everywhere, that's an inertial frame because the laws of physics are simpler when the additional field is not present at all. If you can find one such frame, then any other frame that moves with constant velocity relative to the former is also inertial.

However, in the presence of gravitational sources, no inertial frames exist at all, except locally, and the metric and/or connection always appear in the laws of physics. When the metric and connection are put in, the laws of physics are the same in all reference frames.

Answer 2

As far as Newtonian physics goes, and in particular not considering General Relativity*, defining an inertial frame as one undergoing "constant rectilinear motion"** is correct and not at all circular. The reason is that it is experimentally testable whether or not you are in such a frame. In this context, it is worth mentioning that this definition of an inertial frame is what I would call a physically grounded definition, as opposed to a mathematical one. It relies on an empirically testable condition. How can it be tested? An accelerometer is actually a fancier example, for all such devices/experiments rely on very simple principles. For example, if you throw a ball (into empty space) and it doesn't keep traveling in a straight line with constant velocity, you're not in an inertial frame. If you have an approximately ideal spring and it doesn't obey Hooke's law, you're not in an inertial frame, etc.

If we attempt to formulate this physical idea purely mathematically, we are at a disadvantage from the get-go, because the requirement that all objects obey $\ddot{\vec{r}}=0$ in the absence of real forces, while correct, is not sufficient to account for the possible presence of compressive forces that can be experimentally detected in a non-inertial frame of reference. The reason is that in the framework of Newtonian mechanics, we focus mainly and indeed, often exclusively on the movements of either idealized point particles or center of masses of extended rigid objects. Such objects obeying $\ddot{\vec{r}}=0$ does not conclusively demonstrate that the system is inertial because a point particle or a perfectly rigid body, by definition, does not respond to a compression force in any manner. The simplest Newtonian system that does, is the ideal spring, and indeed a mechanical spring is often used in simple accelerometers and in weight scales. Such devices can and do reveal when a reference frame is not inertial, and the direction in which such compressive forces act.

^{* In the context of General Relativity it is subtler to define what an inertial frame means. Since gravity is no longer considered a force in this context, you either 1. Define inertial frames as existing only locally in (generally speaking: curved) spacetime or 2. Define it as any frame following a geodesic worldline, which is nice but you then have to extend the definition of inertial to accommodate for phenomena such as tidal forces. There's a lot more to say about this topic which is beyond the scope of this question.}

^{** It should be added that the phrase "constant rectilinear motion" in itself is of course meaningless to an observer moving with the frame itself. What makes it valid is that having verified via the mentioned experimental methods that he is indeed in an inertial frame, the observer then can verify that any other frame that is moving with respect to him with uniform velocity is also demonstrably inertial. Hence the association between "constant rectilinear motion" and inertiality.}

Answer 3

I think this question can be generalized: is all of science circular reasoning?

An essential factor in why the endeavour of science isn't circular reasoning is that there is applied science.

With all of our technology we are surrounded by applied science. Let me take an example that I think is particularly vivid: spacecrafts that travel the Solar System by way of a series of gravity assists.

If the reasoning that underlies the planning and executing of gravity assists would be in any way circular then the plans would not work as intended. As we know: gravity assists are executed according to plan all the time.

I think it's essential to always think of science in terms of the validation that comes from applied science.

Conversely, to not take facts from applied science into consideration is to deprive yourself of necessary input.

In the book Gravitation by Misner, Thorne and Wheeler there is a discussion that, I believe, identifies what it is that gives the appearance of circular reasoning.

Misner, Thorne and Wheeler do not discuss the case of newtonian mechanics specifically - here is how I paraphrase their point for newtonian mechanics:
(Further down I will present the more general discussion by MTW)

For newtonian mechanics:

In order to formulate a theory of motion it is necessary to recognize the concept of equivalence class of inertial coordinate systems. The members of this class are related by Galilean transformation.

At this point I need to insert discussion of using a non-inertial coordinate system.

As example the case of using a rotating coordinate system: then the process is as follows: with a rotating coordinate system the equation of motion contains additional terms, a centrifugal term and a coriolis term. Those extra terms contain a factor $\Omega$ for the angular velocity of the rotating coordinate system with respect to the equivalence class of inertial coordinate systems. That is: using a rotating coordinate system is possible by virtue of still using the inertial coordinate system as underlying reference.

So:
While it is possible to use any non-inertial coordinate system, you can do so if and only if your process relies on the equivalence class of inertial coordinate systems for comprehensive underlying reference.

Conversely: in newtonian mechanics, if you were to avoid using the equivalence class of inertial coordinate systems as comprehensive underlying reference then you cannot formulate a theory of motion at all.

Validation

The validation of the concept of (equivalence class of) inertial coordinate systems comes from the very laws of motion. The equivalence class of inertial coordinate systems is that system which makes it possible to formulate laws of motion.

Specific example:
For the motions of the celestial bodies comprising our Solar System: what is the appropriate inertial coordinate system for the Solar System? For that the very laws of motion are used as criterion. The planet are seen to move along Kepler orbits if and only if an inertial coordinate system is used to represent the motions of the planets.

The book 'Gravitation' by Misner, Thorne and Wheeler is available on archive.org

section 3.1. The Lorentz force and the electromagnetic field tensor

Here and elsewhere in science, as stressed not least by Henri Poincare, that view is out of date which used to say, “Define your terms before you proceed.” All the laws and theories of physics, including the Lorentz force law, have this deep and subtle character, that they both define the concepts they use (here B and E) and make statements about these concepts. Contrariwise, the absence of some body of theory, law, and principle deprives one of the means properly to define or even to use concepts.

MTW return to this philosophy of science in section 12.3

All the laws and theories of physics, including Newton’s laws of gravity, have this deep and subtle character, that they both define the concepts they use (here Galilean coordinates) and make statements about these concepts.

I find the case that MTW are making a very compelling case.

In any science the theory that is in usage performs two functions, of equal importance:

• providing operational definition of the concepts used
• making statements about the concepts used

It is simply not possible to have separation of definitions and statements.

That is why I put emphasis on application of the theory in technology. It's the validation from applied science that makes the reasoning not circular.

For more discussion of that quote from MTW:
Stackexchange question:
Help understanding quote on theory and knowledge in Gravitation (Misner, Thorne and Wheeler)