I Inertial reference frames

Summary
What is inertial
I have a really stupid question.

Suppose I am in a car and the car is going around a loop.

Yes, the earth is not inertial -- it rotates -- but let me ignore that and, momentarily, designate the earth as inertial.

Now, we know the frame of the car is not inertial--the car is turning. We approximate the frame of the earth as inertial.

But what prevents me from assuming the world is INSIDE the car, and the EARTH (compared to the car) is rotating around the car?

I sort of "feel" the centrifugal (fictitious force) inside the car. But that is a feeling.

Why is it wrong to assert that in comparison to the car frame, the earth frame is not inertial, but the car frame is?
 

Doc Al

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But what prevents me from assuming the world is INSIDE the car, and the EARTH (compared to the car) is rotating around the car?
Nothing at all. (But the analysis is a bit more complicated from such a non-inertial frame.)
Why is it wrong to assert that in comparison to the car frame, the earth frame is not inertial, but the car frame is?
An inertial frame is one in which the law of inertia applies. For example: Drop a ball. If there are no horizontal forces acting on it, it should drop straight down. It does so when you drop the ball while standing on the earth, but not when you drop it in that accelerating car.
 

Ibix

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Why is it wrong to assert that in comparison to the car frame, the earth frame is not inertial, but the car frame is?
An inertial frame is one in which a body with no net forces acting on it moves in a straight line. You can put an accelerometer on the road outside your car and it will show no acceleration and hence no net force, but it moves in circles around you in your rest frame. Hence your rest frame is non-inertial.
 
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Summary: What is inertial

I sort of "feel" the centrifugal (fictitious force) inside the car. But that is a feeling.
This is the key. You don’t have to rely on feelings, you can use an accelerometer. For an inertial frame you can place an accelerometer at rest anywhere and it will read 0. That is not true in the car frame, therefore it is non inertial.
 
Thank you to all three of you.

I was searching for a systematic test on what is inertial.

It now seems that I cannot make the test through a theoretical analysis (a proof -- of the mind).

The test must involve a physical observation (an experiment).
 
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It now seems that I cannot make the test through a theoretical analysis (a proof -- of the mind).
Hmm, I am not sure what you mean about a proof of the mind. Could you explain what you mean? Perhaps it is possible.
 

HallsofIvy

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"Physics" is not "mathematics"! In mathematics we have assumed starting points (axioms, postulates, definitions) from which we prove theorems. That can be done "in the mind. Physics has to be based on experimental results.
 

vanhees71

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And also let me stress that the question is everything else than stupid. It has been discussed since the centuries following Newton, and it has come to a kind of conclusion only quite recently with the discovery of general relativity and its application to cosmology.

Indeed, as was stressed, according to Newton, an inertial frame is defined as one where the special principle of relativity (Newton's 1st postulate) is valid. It's just an assumption, you can formulate as follows:

There exists one (and as one can show then a whole class of) inertial reference frames, where you can define a clock measuring (absolute) time and three rigid rods fixed at some arbitrary point in (absolute Euclidean) space that a body is moving with constant velocity, if no force acts on it.

This is an assumption. The big question is, how to really establish such a inertial reference frame as an experimental physicist! As you rightly said, the Earth is approximately an inertial reference frame. Bodies "fall down", i.e., are acclerated, because there's a force acting, Newton's universal force of gravity. Taking this "true force" into account, the laws of motion within inertial reference frames hold true quite well by just choosing a reference frame at rest relative to the Earth.

Now, you know very well that Earth is rotating once a day around its axis and running on some orbit around the Sun (this is also a problematic statement, but let's make it for the moment to make the argument). Rotation is always an accelerated motion, i.e., it's somewhat strange that a restframe relative to the Earth should be an inertial frame. Indeed, the Foucault-pendulum experiment shows that its motion is not only due to the influence of the gravitation interaction with the Earth, but there's also a Coriolis force, predicted to be exist in a rotating (i.e., non-inertial) reference frame, and with this the motion is well described again.

So we can safely state "the Earth provides a reference frame, which is rotating relative to the class of inertial frames, aka "the absolute space". But this again leads to a dilemma: Which physical observation/object defines "the absolute space". It's obviously not the restframe of a point on Earth (thanks to Foucaul's pendulum we know). But what is it?

Mach, and now we are already in the 20th century, had the idea that the inertial forces on a mass in a non-inertial frame are somehow due to the presence of all other masses in the universe, when the mass under investigation is accelerated relative to all these other masses. So an inertial frame should rather be defined by some rest frame of all the mass in the universe. This somehow however also makes the hidden assumption that the center of all masses in the universe is at rest relative to the absolute space, just assumed to exist according to Newton. So again, we haven't solved our problem of physically "finding" the "absolute space" through some objective observation.

Then came Einstein with his solution of the problem with Maxwell's electrodynamics being not invariant under Galilei transformations. First the physicists had the idea that this is not a problem but rather provides the solution of the problem of absolute space: Since Maxwell's electrodynamics is not Galilei invariant, it provides an observable consequence of the absolute space, i.e., the reference frame, where the Maxwell equations as usually stated are right when a restframe relative to absolute space is chosen. This makes the absolute space observable and determinable by checking whether in the chosen reference frame the Maxwell laws hold true in the usual form.

Now at this time, many physicists also thought that there must be a medium, the electromagnetic waves predicted by Maxwell and discovered by H. Hertz must propagate analogous to how sound waves propagate through air oder water waves are propagating through water etc. This matter they called aether, and the absolute space was thus thought to be the restframe of the aether (assuming of course again that there's a global restframe of this ominous aether!). This brought Michelson and Morley to check whether they could measure the motion of the Earth around the Sun by checking, whether there's the aether wind (assuming that it's rather the restframe of the Sun that defines the reference frame at rest to the aether than the restframe of the Earth!), and famously they couldn't find anything like that.

Then Einstein came to the rescue in formulating his special theory of relativity. He could make Maxwell's theory consistent with the special principle of relativity by just redefining how to describe space and time. Still there's of course a special class of intertial reference frames. Since this also made the aether obsolete since now Maxwell's theory fulfilled the special principle of relativity and thus cannot define the restframe of absolute space.

10 more years later, Einstein also had solved the puzzle, how to describe gravitational interactions by his General Theory of Relativity, according to which there are no global inertial frames anymore but only local ones. It's given by a small box freely falling (i.e., being not subject to forces other than gravity, which was no longer a usual force but just due to the curvature of space-time itself). An example is the international space station, ISS, where the astronautes don't feel any gravity and a body put to rest relative to the ISS stays at rest relative to the ISS (in very good approximation at least). The scale of the box, providing a good approximation for an inertial frame is given by the scale over which the gravitational field can be considered homogeneous, i.e., as far as in the free-falling ISS tidal forces can be neglected.

Now there's also cosmology, according to which there was a "big bang" (unavoidable spacetime singularity within GR, as famously proven by Hawking and Penrose), of which the cosmic microwave background radiation (CMBF) is a remnant, and indeed averaged over large enough scales of spacetime, the universe is well-described by a Robertson-Walker-Metric, defining a reference frame where the CMBR is homogeneous and isotropic, and derived under the assumption of the cosmological principle, i.e., that (at a large scale average) the universe is homogeneous and isotropic for a class of "fundamental observers", and this frame is simply defined as being the (local) rest frame of the CMBR (indeed we know since satellite measurements of the CMBR like COBE, WMAP, and PLANCK how fast our solar system is moving relative to the CMBR), and the CMBR indeed is up to small fluctuations of its temperature of the relative order of ##10^{-5}## homogeneous and isotropic, and the small fluctuations are rather a feature than a bug, giving us some information what's in the universe (assuming that the cosmological principle and GR are good assumptions).

In this sense, we can say that the best physical definition of a (necessarilly only local!) inertial reference frame is the rest frame of the CMBR or moving with constant velocity relative to it.
 
Thank you again to everyone.

To Dale: ignore my statement as manifested confusion. My confusion is gone. Thank you, vanhees71
 

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