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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 ?

Qmechanic
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Gauge
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2 Answers2

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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.

Dale
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    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. – John Doty May 14 '23 at 11:58
  • @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 – Dale May 14 '23 at 12:40
  • The way instruments operate determine what is measured, so that is more foundational than any abstract concept. – John Doty May 14 '23 at 12:44
  • And yet accelerometers operate in different ways. So how can it be “foundational” to say accelerometers operate in one way when demonstrably that doesn’t hold for all accelerometers? – Dale May 14 '23 at 12:50
  • Proper acceleration is literally acceleration relative to a freely falling or inertial observer.. – Gauge May 14 '23 at 12:51
  • @LeoKovacic but that is the definition of an inertial observer, not proper acceleration. An inertial observer is defined as one whose proper acceleration is zero. – Dale May 14 '23 at 12:53
  • @Dale Well, that's the usual problem of metrology. How do we know what a clock is, except by comparing to other clocks? Still, we have no other way to get into contact with the "primitive concept" of time. Without a clock, it's nothing but a mathematical abstraction, not physical. – John Doty May 14 '23 at 12:56
  • Ok so what exactly is proper acceleration mathematicallly? – Gauge May 14 '23 at 13:13
  • @JohnDoty exactly, although I wouldn’t characterize it as a problem. That is simply what experimental sciences like physics are based on. Far from being a problem, it is what ties science to measurable reality. – Dale May 14 '23 at 15:04
  • @LeoKovacic proper acceleration is the acceleration measured by an accelerometer. In our theories we represent it mathematically as the second covariant derivative of the accelerometer’s worldline. Being a covariant derivative it is independent of the reference frame. So both the physical and mathematical proper acceleration are independent of the reference frame – Dale May 14 '23 at 15:16
  • Ok , but what about special relativity , there are no coderivatives there – Gauge May 14 '23 at 15:54
  • Or Newtonian mechanics for that matter . Is general relativity needed to have a sound foundation for dynamics ? – Gauge May 14 '23 at 15:56
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    @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 – Dale May 14 '23 at 17:21
  • Well that's interesting .. – Gauge May 14 '23 at 18:26
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    I remember reading about such geometric and covariant axioms for classical mechanics in Misners ls gravitation.. – Gauge May 14 '23 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.

John Doty
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  • 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 – Gauge May 14 '23 at 12:48
  • 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. – John Doty May 14 '23 at 12:52
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    So exactly what does this mean , you need to include springs in definition of intertial frames ? – Gauge May 14 '23 at 13:11