Why isn't the idea of an inertial frame relative? [closed]

Question

I truly apologise if this has been asked to death somewhere, I imagine it has, but I'm yet to find an answer that completely satisfies me.

In short, I don't see why our chosen inertial frames are "correct". They're defined as those that do not require "fictitious" forces to explain motion, but that's also relative, no? Indeed, the forces seem pretty real in the "accelerating" frames, after all. You could say there's no identifiable force causing motion in those frames, but just because we can't identify a cause doesn't quite convince me.

Setting aside the fact that even "stationary" frames on Earth are accelerating and that we already make this sort of approximation, I just don't see how we can claim there to be an absolute 0 acceleration not defined in some relative way. In our case, relative to our (seemingly) arbitrary laws.

All of our inertial frames have the same laws of physics, but so do all frames with any given acceleration relative to our inertial frames, do they not? An acceleration doesn't preserve our laws, but it does change them, and from there, Galilean transformations (or some tweaked equivalent?) should preserve those other laws, yes?

Is there any real reason other than for convenience to assume our definitions of inertia? I feel like any argument I've heard to the contrary can be refuted by simply tweaking some law in some way, or some definition. I suppose it's ideal to assume the simplest laws work ok, but, returning to the example of the approximate frames we use, our understanding of forces and causes is always evolving, and frames that feel or seem inertial are regularly shown to not be thanks to discoveries, like the Earth's acceleration around the Sun, our solar system's "acceleration" through space, and indeed the curvature of spacetime nullifying the notion of gravity as a causing "force". Would it be fair to say we're just defining things as we go along, and in terms of the identifiable causes we can find??

I also apologise if this question isn't very practical - I'm from a mathematical background, after all...

Answer 1

They're defined as those that do not require "fictitious" forces to explain motion, but that's also relative, no?

Whether a force is fictitious or real is not relative. Fictitious forces cannot be detected with accelerometers. The categorization is operationally very clear and unambiguous. If a force produces a local acceleration that is measurable with an accelerometer then it is a real force. If not then it is a fictitious force locally. All frames will agree with this distinction, whether inertial or not, and all frames can thus use this distinction to characterize themselves as inertial or not.

Answer 2

Yes, the semi philosophical question "does an inertial frame even exist" is a valid one, and, in fact, if one is in a closed box, it is impossible to distinguish between the supposed "deep space inertial frame" and free fall in gravity (just like one cannot distinguish between the box being on the ground or in deep space being thrusted by a rocket).

That said, I would say that we all know what we mean by inertial frame, and as such it provides a relevant thought experiment from which we can deduce what the physical laws must look like.

It is worth noting that in the examples like the one above, the observer needs to "be in a box" because otherwise the view of the universe will make no sense. We can tell very easily if we are actually in orbit around the earth if the space craft has a window, because then it would make no sense for all of the observable universe to be going around in circles (in phase!) of radius that just happens to equal the distance to the center of the big round rock below us (including that rock itself, by the way!). So, in that case, we obviously say "I am not stationary, I am in an accelerating frame in free fall of just the right conditions to be in a Kepler orbit" because the alternative makes no sense. We would need to add forces to every other object we can see to explain why they are traveling in circles.

A keen reader would then note that it is not space and time itself that defines the inertial frame, but rather all the mass around us. To that, there actually is no good response to refute it. From an experimental perspective, we define the inertial frames to be those frames that put all the astronomical objects into a frame with no external forces.

Answer 3

An inertial frame is one for which accelerometers comoving with the frame read zero. It doesn't matter what your velocity with respect to the accelerometer is.

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To the implied subordinate question, "are there transformation rules for converting from inertial to non-inertial frames?" - yes, the transformation rule is to add one or more fictitious forces which account for the acceleration relative to an initially comoving inertial frame in the same place. For a laboratory on Earth, these are gravity, centrifugal force, Coriolis force, and Euler force. Subtract these forces to transform back to an inertial frame.

Answer 4

You seem to not realise that there is a need for coherence in a physical theory.

The reason why we pay attention to a concept called momentum is that, in a wide class of frames that we call inertial, momentum is a strictly conserved quantity. If things can just randomly change, we do not spend so much effort keeping track of them. In fact, this happened in our own history; Energy as a concept took very long to gestate precisely because it seemed to be conserved in some circumstances, and not conserved in some other circumstances, until it was realised that we had to include enough different forms of energy and also heat as a form of energy before it can be always conserved. At least, until cosmological scales come in.

All of our inertial frames have the same laws of physics, but so do all frames with any given acceleration relative to our inertial frames, do they not? An acceleration doesn't preserve our laws, but it does change them

Let us entertain this. In an accelerating frame, there will be some crazy but specific law asserting what changes that momentum will suffer. In such a frame, we can study what changes in frames behave like. Then, by considering a change into yet another accelerating frame (relative to this one), we can undo all those changes, leaving us with a frame in which momentum is strictly conserved. We can then call those frames inertial, and be back to square one.

Inertial frames are simply privileged. Zero is privileged, even in mathematics. It is the natural setting from which to base a physical theory upon. We derive what actually happens in an accelerating frame from considerations within inertial frames.

You could say there's no identifiable force causing motion in those frames, but just because we can't identify a cause doesn't quite convince me.

That is you just misunderstanding the gravity of this situation. It is not that we silly humans cannot identify a cause. It is that it is physically impossible for a suitable physical interaction to be correct.

Consider a simple rotating frame. As you simply increase the distance from the radial centre of the rotation, an object's momentum would suffer ever increasing fictitious forces. There is no possible physical interaction that can have this behaviour. If an object is "staying stationary" in such a rotating frame, then there is a maximum radial distance that it can have, because otherwise it would be moving faster than the speed of light in vacuum in the inertial frame. Yet, we clearly can observe distant objects even in a rotating frame; such unphysical artifacts come about just because we are entertaining non-inertial frames.

Other commentators have also repeatedly told you that real forces are experimentally measurable. Fictitious forces cannot be. With fictitious stuff, you can only play a game of deduction and curve-fitting. With real forces, we can discover the laws of electromagnetism, we can discover the laws of gravitation, we can use Hooke's Law to measure the forces involved. This is a strict difference and you are repeatedly not showing any understanding that this is important.

our understanding of forces and causes is always evolving, and frames that feel or seem inertial are regularly shown to not be thanks to discoveries, like the Earth's acceleration around the Sun, our solar system's "acceleration" through space, and indeed the curvature of spacetime nullifying the notion of gravity as a causing "force". Would it be fair to say we're just defining things as we go along, and in terms of the identifiable causes we can find??

This is nothing to do with the topic at hand and really just a fact: Physics is a science, not mathematics. Science deals with observations and might have to change when new observations show us that older theories are incomplete.

There is no amount of maths-ing or pure philosophising that can get us to the truth in Physics. On the contrary, it is a sign of philosophical growth that one stops seeking the impossible and learn to accept that the nature of epistemology is the one that has to comply with physics, rather than the other way around.

Answer 5

What makes fictitious forces different from real forces is that ficticious forces are actually acceleration terms in the equations of motion that one pretends are forces for convenience.

It may be helpful to start from the definition of an inertial frame. Here's a well accepted phrasing for a classical inertial frame:

An inertial frame of reference is one in which the motion of a particle not subject to forces is in a straight line at constant speed.

Note that for this to be true, the acceleration of said particle with no forces applied must be zero. That stems from the definition of motion in a straight line at constant speed.

Now we have two ways to deviate from this definition, creating a non-inertial frame. The first is to say that a particle does not move in a straight line. In a rotating reference frame, this is true. Odd effects like the Coriolis effect abound, causing particles to travel in anything but straight lines. So clearly in that sense the inertial frames are special and distinguishable from non-inertial frames.

But what if we say there's forces applied to it instead of saying there's an acceleration. As alluded earlier, we can always swap out an acceleration for a "fictitious force" simply by multiplying by the mass of an object. Now we have a force on said particle, so its allowed to move in a non-straight path, right? Well, almost. Remember Netwon's third law: for every action there is an equal and opposite reaction. If there is a force on the particle, the particle must be exerting an equal and opposite reaction on another object. But what is that object? There really isn't one. The fictitious force isn't a real force, but rather a mathematical artifact of treating an acceleration as-if it were a force.

Consider a free body diagram of an object in a centrifuge. Obviously the outer wall of the centrifuge is applying an inward force on the object, and the object is applying its equal and opposite reactionary force outward on the wall. Those are real forces. So for the free body diagram of the object, you have a force pointing inward (from the wall), and you have a fictitious centripetal force pointing outward. Those two are balanced, as the object is pinned to the wall of the centrifuge and thus not moving in the rotating frame. And obviously we have one force pair (the wall on the object and the object on the wall), but try as we might, there is no equal and opposite reaction to the centripetal force.

So its not that we can't identify the force causing these effects. It's that the effects cannot be a force, because there is no equal and opposite reaction, which is part of the definition of a force. It comes up just shy of being a fully-fledged force.

And yes, the rules for inertial frames are a little arbitrary. The inertial frames of reference are privileged in the sense that the equations of motion are simpler in them. They're the frames with the least non-force related accelerations in them. Its just like how you can develop a geocentric model of the solar system, but the math is cluttered and obnoxious compared with the raw elegance of a solarcentric model. In a very informal sense, there's something preferable about the solarcentric model's explanatory power.