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As far as I know, inertial frame of reference are the ones where the all the three Newton's laws of motion hold. Having this definition we can then identify all such frames of reference which are inertial, if we have an inertial frame of reference, to begin with, to observe them by applying Newton's first law of motion i.e.,

  • If S is an inertial frame of reference then we can conclude that S' is also an inertial frame of reference if velocity of S' is uniform/constant with respect to S.

Now from these, we can define a non-inertial frame of reference as a frame of reference where laws of motion are not valid in their current form and need to be modified so that they can be used (such as introduction of Fictitious force).

Now the question:

  • Given a non-inertial frame of reference what is(are) the condition(s) required to affirm whether another frame of reference (being observed from the current non-inertial frame) is inertial or non-inertial?

I think a brief background to the question is required. I thought of this situation while considering the following case: suppose we are observing an observer (in space) from Earth, how may I claim that the the reference frame attached to that observer is inertial or not? Clearly earth is a non-inertial frame of reference, hence the question.

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You don’t need a second frame to determine if a frame is inertial. Simply compare the coordinate acceleration in the frame to the proper acceleration measured by momentarily co-moving accelerometers. If they match then the frame is inertial. If they do not match then the frame is non-inertial and the difference between the coordinate acceleration and the proper acceleration is a fictitious force.

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  • $\begingroup$ I thought of this situation while considering the following case: suppose we are observing an observer (in space) from Earth, how may I claim that the the reference frame attached to that observer is inertial or not? Clearly earth is a non-inertial frame of reference, hence the question. $\endgroup$
    – Shyam Tripathi
    Commented Feb 2, 2022 at 12:29
  • $\begingroup$ @shyamtripathi As I said, use accelerometers in the observer's frame. Your own frame is irrelevant. $\endgroup$
    – Dale
    Commented Feb 2, 2022 at 13:23
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    $\begingroup$ I upvoted because you distinguished between proper motion and coordinate motion, and a lot of people don't see the difference. But I agree with @ShyamTripathi that you have not addressed the specific question asked. $\endgroup$
    – Arc
    Commented Feb 13, 2022 at 19:23
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Given a non-inertial frame of reference what is(are) the condition(s) required to affirm whether another frame of reference (being observed from the current non-inertial frame) is inertial or non-inertial?

I think a brief background to the question is required. I thought of this situation while considering the following case: suppose we are observing an observer (in space) from Earth, how may I claim that the the reference frame attached to that observer is inertial or not? Clearly earth is a non-inertial frame of reference, hence the question.

I assume you can't go to the space observer's frame of reference, so you have to do it at a distance.

Motion is more generally composed of inertial motion (field forces) and non-inertial motion (contact forces). In your example, you are on the surface of Earth observing someone in space, so you have to separate the components.

When you are in deep-space (gravity negligible) either "standing still" or travelling with constant velocity, then you are clearly in an inertial frame. The same hapens when you are in free fall in a gravitational field (either in a parabolic trajectory over a planet's surface, or in orbit around a large mass). In both scenarios, you don't feel motion, you are weightless, because gravity is a field force and thus your accelerometers measure zero, and you are always oriented to the same direction in space, just as a gyroscope.

The inertial motion due to gravity must be measured by an external referential, say the distant stars.

Then, you have to know how much of your motion is non-inertial. Since you are being held in the surface of Earth by the normal force, then you can feel weight, because the normal force is a contact force, and thus you can measure the proper acceleration it causes using an accelerometer.

When you have determined how is your motion composed, measuring both your proper acceleration due to the normal force using accelerometers, and the coordinate acceleration due to gravity using relative position to distant stars, then you have to measure the motion of the observer in space relative to you.

Now the trick part: the observer in space motion's may also be composed by inertial and non-inertial movement, so you have to estimate what the gravity field looks like in the observer in space's position.

Once you have a) your motion decomposed into inertial and non-inertial components, b) the relative motion between you and the observer in space mapped in detail - so you know what their movement relative to the distant stars is, and c) an estimate of the gravity field around the space observer, that is, what are the geodesics in their surroundings, then you can d) subtract their inertial motion, and whatever motion is left is their non-inertial motion.

Note that this is very difficult to do in practice, not only beacuse the form of the gravitational field may be very complicated (many celestial objects, near and far, dust clouds, small but heavy meteorites, etc.), but because so many factors can cause the space observer's proper motion (small amounts of gases being expelled from it, thermal radiation emission and absorption, and even anisotropic radiation pressure - see the Pioneer anomaly). See, for example, the controversy surrounding the non-gravitational trajectory of the ʻOumuamua object.

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  • $\begingroup$ "The same hapens when you are in free fall in a gravitational field" No. Thats not an inertial frame. $\endgroup$
    – MatterGauge
    Commented Feb 13, 2022 at 22:51
  • $\begingroup$ Yes! It is! That's exactly how you tell inertial frames from non-inertial ones! There can be an inertial frame of reference that's accelerating, if the acceleration is promoted by gravity. When you free fall, an accelerometer mesaures zero, you feel no weight as you follow a geodesic of the field, there's no proper acceleration and thus its an inertial frame of reference! See Proper acceleration. $\endgroup$
    – Arc
    Commented Feb 13, 2022 at 22:56
  • $\begingroup$ No, its not. If you fall freely to Earth you not in an inertial frame. $\endgroup$
    – MatterGauge
    Commented Feb 13, 2022 at 22:59
  • $\begingroup$ @Felicia, the main issue here is to distinguish proper acceleration from coordinate acceleration. Proper acceleration (in classical) mechanics is absolute, it does not depend on an external frame of reference, you can measure it from within, while coordinate acceleration by itself needs external sources of reference to be measured. $\endgroup$
    – Arc
    Commented Feb 13, 2022 at 23:00
  • $\begingroup$ Well, then can you explain why it is not an inertial frame? Please read the proper acceleration entry, you will understand it better. The correct way to tell if a frame of reference is inertial or not is by using an accelerometer, and when you free fall your accelerometer measures zero, thus the weightlessness. $\endgroup$
    – Arc
    Commented Feb 13, 2022 at 23:02
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As the OP has mentioned Newton's laws, I will not not use concepts of general relativity.

Besides the $3$ Newton's laws, there is also his force of gravity. So, if the planets including Earth are moving around the Sun, they are not inertial frames, they are accelerated by the gravity force. The Sun itself is not an inertial frame because it rotates around itself.

A good candidate at a first approximation (that considers the orbit of the Sun around the galaxy center as a second order effect) is a rocket that keeps the same position $(R,\theta,\phi)$ in a system of spherical polar coordinates with the Sun at center, and the fixed stars as angular coordinates reference. Or moving in a straight line in this frame.

The rocket should burn fuel to generate a force to balance the gravity attraction from the Sun (and other planets) at each location, so that the sum of forces are zero on it.

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  • $\begingroup$ This is incorrect. A planet, considering its orbit around the sun, is indeed in an inertial frame of reference. Gravity is a field force, and thus produces no internal compression, the planet simply slides through the field, You can verify this by comparing inertial rotation and non-inertial rotation: a person on a merry go round is subject to a contact force and thus is in a state of non-inertial rotation, the greater the angular velocity, and the greater the radius of rotation, the greater the centrifugal force. $\endgroup$
    – Arc
    Commented Feb 13, 2022 at 22:47
  • $\begingroup$ In the case of the orbit around the sun, we don't feel centrifugal forces because gravity is a field force - else we would be completely thorn apart a long ago. Another very distinctive feature of non-inertial rotation is the direction of the rotating body: the person on the merry go round keeps spinning, say facing the center, or facing outwards the rotation, while inertial rotation always face the same orientation in space. Rotation of the planet, or the sun, around its own axis is, of course, non-inertial rotation. $\endgroup$
    – Arc
    Commented Feb 13, 2022 at 22:50
  • $\begingroup$ @Arc I started my answer saying that I am talking about Newton's gravity, not GR. According to Newton, if the force of gravity is balanced by another force, the total force on the object is zero. The concept of field started in the $19^{th}$ century, probably with Faraday. $\endgroup$
    – Claudio Saspinski
    Commented Feb 13, 2022 at 23:16
  • $\begingroup$ Well, agree, but the 'modern' gravity field understanding of the 19th century still predates GR, right? Newton's strict version of the laws of motion and the law of gravity actually have inconsistencies which have been elucidated along the time, but too many people still take those literally. For instance, too many people still think that any acceleration results in non-inertial movement, and that's something that I think we should try to explain better to all (the distinction between proper acceleration and coordinate acceleration). $\endgroup$
    – Arc
    Commented Feb 13, 2022 at 23:29
  • $\begingroup$ @Arc notions like: orbits are inertial movements, or distinction between proper and coordinate acceleration don't have meaning out of GR, with its metrics, covariant derivatives and geodesics. $\endgroup$
    – Claudio Saspinski
    Commented Feb 14, 2022 at 1:26
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This is a variation of the answer by @Felicia.
(One can also attach accelerometers to the lab, as @Dale suggests.)

From my answer to How can one tell they are accelerating? ,
have them conduct an experiment as Ivey & Hume did:
If a ball that is dropped from the top of a stand
lands at the base of the stand, then the frame is inertial.

Here are a few frames (superimposed) from Ivey and Hume's Frames of Reference video https://archive.org/details/frames_of_reference

(You can probably find it on YouTube [with slightly different timestamps]. However, this archive.org URL should be more permanent than YouTube.)

  • At t=4m22s , this is a ball dropped from a cart at rest in the inertial-Lab frame. When released, there is no horizontal force on the ball, hence it has constant horizontal velocity in the Lab. It lands at the base of the stand.
    IveyHume-InertialAtRest-viewedInLab IveyHume-InertialInMotion-viewedInLab
    At t=5m25s , this is a ball dropped from a cart in uniform motion in the inertial-Lab frame. When released, there is no horizontal force on the ball, hence it has constant horizontal velocity. It lands at the base of the stand.... just like it was at rest-and-inertial.

From your non-inertial frame, you might find it difficult to write an expression for the trajectory of the falling ball... but what you want is the result... Does the ball end up at the base of the stand?

If you don't allow me to use gravity, do a variation where the projectile is sent across the room. Did the projectile end up at the end point of a segment tangent to its initial velocity?

For accelerated cart case,
continue to How can one tell they are accelerating? In that non-inertial case, the ball does not land at the base of the stand.

The full video treats the case of a rotating frame of reference and the Foucault pendulum.

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  • $\begingroup$ Note the question is: how to tell if a system is not inertial at a distance. If you could instruct the space observer to do some experiment, or go there yourself, then it would be far easier to answer. The premisse of the question is that the motion of the space observer is arbitrary, can be any, including the possibility of non-inertial motion. But you must tell that by just considering your motion on Earth, and the motion of the space observer relative to Earth. $\endgroup$
    – Arc
    Commented Feb 13, 2022 at 22:36
  • $\begingroup$ @Arc Observe all motions in that frame and look for the indications I suggested. Maybe there is a bouncing ball. The OP didn't set more restrictive "ground rules", so until then I think my answer and those of others are reasonable. (By the way, we make a similar assumption about the emission of light in atomic transitions in stars. We didn't setup an experiment there... we just observe some natural process and deduce what we can.) $\endgroup$
    – robphy
    Commented Feb 13, 2022 at 22:46
  • $\begingroup$ Ok, now you explained it better, this last comment of yours connects your answer with the question (which by the way you perhaps want to add to your reasoning). You are right that the OP didn't set that kind of rule: does the system being observed has any features that might indicate non-inertial motion? If it's a star, its emitting radiation, if its a planet with children playing ball you can see the balls' trajectories, say, but if it's an asteroid with no such indicators? $\endgroup$
    – Arc
    Commented Feb 13, 2022 at 23:21
  • $\begingroup$ @Arc ... for asteroid, it's harder to measure. Sure. But that's the art of experimental physics... find something to measure.... possibly after waiting for the technology and the analysis tools to be developed. In principle, there has to be at least one. (It's a different type of question if the OP asked: today, I need to measure this property of some system at a distance I have to no access to. But again, the OP made no such constraint thus far.) $\endgroup$
    – robphy
    Commented Feb 13, 2022 at 23:39
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Tell the people in the frame you look at to go to different positions without relative motions and fixed wrt to the axes of their frame. Tell them to hold a mass. Then tell them to unleash the masses they hold. If the masses stay stationary wrt one another and the frame, the frame is inertial. If not, the frame is non-inertial.

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  • $\begingroup$ Again, note the question is: how to tell if a system is not inertial at a distance (think of an asteroid, for instance). If you could instruct the space observer to do some experiment, or go there yourself, then it would be far easier to answer. The premisse of the question is that the motion of the space observer is arbitrary, can be any, including the possibility of non-inertial motion. But you must tell that by just considering your motion on Earth, and the motion of the space observer relative to Earth. $\endgroup$
    – Arc
    Commented Feb 13, 2022 at 22:40
  • $\begingroup$ @Arc You can see that the masses stay at the same place inside a rocket. I can look with a telescope inside your rocket. If the masses all stay at the same place wrt to the inside of the rocket then the inside of the rocket is an inertial frame. $\endgroup$
    – MatterGauge
    Commented Feb 13, 2022 at 22:50
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To confirm that another frame of reference $F$ is intertial while standing on Earth, take an accelerometer and point it in directions that shows one is accelerating, say with acceleration $\vec{a}$ (not explained by e.g. gravity). Measure the velocity of $F$ in that direction to confirm that it is accelerating with acceleration $-\vec{a}$. Turn the accelerometer in directions were it doesnt show acceleration. Check $Fs$ velocity in that direction to confirm that it is not accelerating.

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If you can move into the observer's frame, your own frame doesn't matter. You simply go to the other frame, assume the frame is inertial and calculate the acceleration of some body there with the laws of motion. Then you actually measure the acceleration in the observer's frame. If they match, then you can say that the reference frame is behaving inertially in that motion you investigated.

If you can't move into the observer's frame, you would need outside information. For instance, if there's someone there saying they are static in their own frame. Do this: Measure their acceleration as seen from your own frame, next you account for the effects of the acceleration of your own frame. If the static observer is still moving with some acceleration, then the outside frame is not inertial.

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  • $\begingroup$ Your first paragraph not only does not address the question, it's also wrong. You have to make a distinction between proper acceleration and coordinate acceleration in order to distinguish inertial reference frames from non-inertial ones. "You simply go to the other frame, assume the frame is inertial and calculate the acceleration ..." clearly is not scientific enough to tell. Also, everyone is static on their own frames, right? And how to you "account for the effects of the acceleration of your own frame"? $\endgroup$
    – Arc
    Commented Feb 13, 2022 at 19:27
  • $\begingroup$ So, unless i really slipped here i remember the title of this question being "How to tell if a system not inertial". And, per the tags, i used the Newtonian definition of an inertial system, which means a system where laws of Newton are valid. So, he knows he is in eath, an non-inertial system due to rotation, measures the relative motion on the observed space body, removes the effects of earth's own acceleration(the fictitious forces of rotation) and check if they match, if not, the space body has his own acceleration being in an non-inertial frame. $\endgroup$
    – Klaus3
    Commented Feb 13, 2022 at 22:01
  • $\begingroup$ I know that this would imply that a free falling body is accelerating, which is false in relativity, but again, the tags. I can delete the answer if so desired. And notice i didn't claim that correspondence with the laws of motion is enough to determine that the reference is inertial, i said that its behaving inertially in that case, just like our everyday earth frame is approximately inertial. $\endgroup$
    – Klaus3
    Commented Feb 13, 2022 at 22:04
  • $\begingroup$ I don't think you should remove the answer, perhaps improve it. You are right, no need to put relativity in the table to explain inertial frames of reference, my answer only resorts to classical mechanics. But my point is: your answer does not make a distiction between proper acceleration and coordinate acceleration, so its difficult to put it to practice. How exactly do you tell in practice if Newton's laws are valid in a given frame of reference? How do you know a priori the contribution of each fictious force? $\endgroup$
    – Arc
    Commented Feb 13, 2022 at 22:30
  • $\begingroup$ You can only do that by using accelerometers and gyroscopes, and the question states that you can only do that in Earth's frame, not in the space observer's frame. Any body can be accelerated by a field force like gravity (so you are just sliding in the spacetime curvature), or by contact forces (like a person on a rocket under the normal force due to the body of the rocket), but the effects are very different. You must be able to distinguish, and that is the nice point the question raises. $\endgroup$
    – Arc
    Commented Feb 13, 2022 at 22:31

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