Force is defined by acceleration , and acceleration requires the determination of" inertial frames".
But an inertial frames also requires the knowledge of forces which requires measuring acceleration, but with respect to what ?
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2$\begingroup$ Does this answer your question? What is the fundamental definition of force? $\endgroup$– Cathartic EncephalopathyCommented May 14, 2023 at 11:35
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$\begingroup$ More on the definition of force in Newtonian mechanics. $\endgroup$– Qmechanic ♦Commented May 14, 2023 at 11:54
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$\begingroup$ I don't think this is as trivial as it seems $\endgroup$– GaugeCommented May 14, 2023 at 12:49
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1$\begingroup$ IIRC the most rigorous treatment of this issue I've seen is (unsurprisingly) in Arnold's Mathematical Methods of Classical Mechanics. Taylor's Classical Mechanics also has a fairly good and somewhat more accessible discussion. I don't have access to either one of these just now but I'll try to remember to come back to this and write an answer later. $\endgroup$– Michael SeifertCommented May 14, 2023 at 13:02
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$\begingroup$ Both statements in the first sentence are not generally true. $\endgroup$– Ján LalinskýCommented May 14, 2023 at 13:04
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2 Answers
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acceleration requires the determination of" inertial frames" .
That is not correct. In modern physics there are two distinct concepts of acceleration. Neither requires the determination of an inertial frame.
Proper acceleration is the acceleration measured by an accelerometer. It is an invariant quantity, so it is the same in any frame whether inertial or not. Proper acceleration is one of the particularly important concepts because it is experimentally measurable.
Coordinate acceleration is the second time derivative of position in some specified reference frame. It obviously is frame dependent, but it is not required to identify whether the given frame is inertial or not.
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1$\begingroup$ The common accelerometer is a force gauge coupled to a mass. $F=ma$ is thus assumed. Coordinate acceleration is what Galileo, Newton, et al. studied, and in the environments where they studied it (inertial frames in flat spacetime), it is the same as proper acceleration. $F=ma$ emerged from these experiments and observations. This allowed acceleration to be determined through force measurements, but it was only in the 20th century that this was divorced from coordinate acceleration. Although it's important in my own work, the distinction is rarely useful in real life. $\endgroup$ Commented May 14, 2023 at 11:58
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$\begingroup$ @JohnDoty there are many types of accelerometers, not all function that way. Physics is an experimental science, and experimental measurements are foundational. The devices that produce such experimental measurements are categorized according to the meaning of the measurand, not how they operate. The resulting experimental measurements are primitive concepts, more foundational than any theoretical concept like Newton’s laws. The experimental measurements come first, theories (like Newtons laws) come second $\endgroup$– DaleCommented May 14, 2023 at 12:40
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$\begingroup$ The way instruments operate determine what is measured, so that is more foundational than any abstract concept. $\endgroup$ Commented May 14, 2023 at 12:44
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1$\begingroup$ @LeoKovacic covariant derivatives are not owned by nor restricted to general relativity. They are indeed part of special relativity. And (although it is rarely done) they can be made part of Newtonian mechanics also through Newton-Cartan gravity. But yes, I believe that this concept is necessary for a sound foundation for classical mechanics $\endgroup$– DaleCommented May 14, 2023 at 17:21
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1$\begingroup$ I remember reading about such geometric and covariant axioms for classical mechanics in Misners ls gravitation.. $\endgroup$– GaugeCommented May 14, 2023 at 18:28
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The direct way to measure force is a force gauge. A force gauge is a spring whose extension you measure. No acceleration involved.
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$\begingroup$ Can you discern the difference between a neutral tensionless spring from a tense spring acted on by force or acceleration , with only local observations both spatially local and temporally ? What if you are on an "upward accelerating ship , the spring will be compressed but who knows what would it's natural state be , or , in a completely empty space , how would you even tell the difference then, between inertial movement and accelerated one $\endgroup$– GaugeCommented May 14, 2023 at 12:48
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$\begingroup$ You make that measurement by comparing an unloaded spring to the same spring with a mass attached. If its length doesn't change, your local frame is inertial. $\endgroup$ Commented May 14, 2023 at 12:52
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1$\begingroup$ So exactly what does this mean , you need to include springs in definition of intertial frames ? $\endgroup$– GaugeCommented May 14, 2023 at 13:11