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How does one determine whether one is in an inertial frame?

An inertial frame is one on which a particle with no force on it travels in a straight line.

But how does one determine that no forces are acting on it? After all, one might declare a force is acting on a particle because it is deviating from a straight line.

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    $\begingroup$ Doesn't your question simplify to how do you tell if a freely falling object is actually falling freely?. $\endgroup$ Commented May 7, 2015 at 16:32
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    $\begingroup$ @rennie: possibly; I wasn't thinking in terms of gravity; I was starting from Newton's first law. $\endgroup$ Commented May 7, 2015 at 16:52
  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/3193/2451 and links therein. $\endgroup$
    – Qmechanic
    Commented May 7, 2015 at 18:42
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    $\begingroup$ @MoziburUllah: the point is that in an inertial frame a freely falling object obeys Newton's first law - end of definition. What you're saying is aha, but how do we know the obect is really falling freely i.e. there isn't an external force acting on it?. And we don't. We just take sensible precautions to make sure there aren't any unaccounted for external forces. $\endgroup$ Commented May 7, 2015 at 21:37
  • $\begingroup$ If the particle is experiencing an acceleration by an external force, an observer riding on the particle can measure an acceleration. Look at Einstein's elevator experiment again and ask yourself what the meaning of the completely enclosed space is. $\endgroup$
    – CuriousOne
    Commented May 8, 2015 at 1:06

5 Answers 5

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Imagine a point mass attached to ends of six identical springs of relaxed length $L$. The springs are oriented in co-linear pairs, with these pairs mutually perpendicular to the other pairs, i.e., they form an $(x,y,z)$ coordinate system. Attach the other ends to the walls of a cube of dimension $2L$.

If the mass remains in the center of the cube, the cube is in an inertial reference frame.

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  • $\begingroup$ You are talking about an Accelerometer which can detect acceleration right? but an Accelerometer attached to a freely falling body which is in noninertial frame shows zero reading. $\endgroup$
    – Paul
    Commented May 7, 2015 at 18:01
  • $\begingroup$ There is no way one can distinguish between an inertial frame and an freely falling frame. (which is accelerating.) $\endgroup$
    – Paul
    Commented May 7, 2015 at 18:11
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    $\begingroup$ So, they mean the same thing. The freely falling frame has no local gravitational acceleration evident in the frame. Objects placed at rest will remain at rest in all coordinates. Objects set in motion will, after the impulse stops, travel at constant velocity. From a GR viewpoint, we live in a frame which is accelerating upward at 9.80 m/s$^2$. $\endgroup$
    – Bill N
    Commented May 7, 2015 at 21:40
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    $\begingroup$ What if the cube is spinning? $\endgroup$ Commented May 8, 2015 at 0:39
  • $\begingroup$ @user2357112 if the cube is spinning, the springs will be pulled into spinning as well; and as with all such confined spinning motions, they will experience a centrifugal force in the spinning frame. Also, any object placed at rest away from the center of the box will appear to make an orbit within the box, the center of which will be the axis of rotation of the box. $\endgroup$
    – Asher
    Commented May 8, 2015 at 3:36
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There are a certain number of known forces, so you could start by ruling each one out by experiment.

Otherwise the so-called fictitious forces - the centrifugal force and the Coriolis effect - have the characteristic that they provide the same acceleration to all objects, regardless of their mass. As you know, in classical mechanics $a = F/m$ so forces will in general accelerate objects with different masses at different rates. The only known real force that does that is gravity.

Bonus material: This last observation was in fact central in developing General Relativity. In GR, gravity is indeed seen as a fictitious as the centrifugal force, and due to the way the observer lays the spacetime coordinated.

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  • $\begingroup$ Have I misunderstood you here 'forces will in general accelerate objects with different masses at different rates'; but here on the moon a cannon-ball and an acorn fall exactly at the same rate. $\endgroup$ Commented May 7, 2015 at 23:14
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    $\begingroup$ @MoziburUllah see the last paragraph. Under GR, Gravity is not a real force either. $\endgroup$
    – k_g
    Commented May 8, 2015 at 1:47
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An inertial frame of reference is one that is not accelerating. A frame that is accelerating with respect to another inertial frame is a non-inertial frame of reference.

An inertial frame moves with steady velocity in respect to other inertial frames, such that an observer within the frame can not detect any movement of the frame, unless she looks outside the frame. There is no difference in the attributes of any direction within the frame.

A clock placed in any part of the inertial frame can be synchronized with a clock in any other part of the frame, and all clocks so synchronized tell the same time.

If a particle deviates from a straight line, start looking for forces within the inertial frame that would cause the deviation. If you can't find any, you quite possibly are NOT in an inertial frame of reference. However, even though an object deviates from a straight line in a gravitational field, you may consider the surface area of the gravitating body to be an inertial frame of reference, as the normal force between the surface and objects on the surface cancels out acceleration.

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    $\begingroup$ But you haven't defined any specific physical method of measuring "velocity" or "movement" or "straight line" to check if it's constant or changing. According to the rulers and clocks of a set of Rindler observers (see the paragraph starting 'We can imagine a flotilla...' here for details), for example, an inertial particle will have a changing velocity, whereas each of the Rindler observers is at rest according to the (non-inertial) rulers and clocks defining the Rindler coordinate system. $\endgroup$
    – Hypnosifl
    Commented May 7, 2015 at 19:15
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Short answer:

One cannot determine whether a reference frame is inertial. It is simply not possible. This situation is avoided in General Relativity (or any of its extensions), since in GR the physics does not depend on the reference frame (any reference frame), and the concept of inertial frame is not necessary. Of course one can determine an inertial frame which is good enough for a given system, but one cannot determine, for example, an inertial frame to describe the motion of galaxies in the whole universe.

Long answer:

As pointed out by Andrea Di Biagio, one can reason by exclusion. If a particle moves on a straight line and no force is applied, the reference frame is inertial. Since there is a finite number of known forces, one can in principle rule out each of them and reduce to a situation where no force is applied to the particle. There are at least two weak points in this reasoning:

1) The list of known forces in modern physics is: gravitational, electromagnetic, weak and strong interactions. No one knows if this list is complete of course, therefore in principle reasoning by exclusion cannot work.

2) In classical mechanics, two distant bodies can interact over a long distance through the electromagnetic or gravitational field. Therefore to exclude, say, electromagnetic and gravitational forces, one should in principle know the charge and mass distribution of the entire universe. Consider the following situation. In a hypothetical laboratory in deep space, far away from galaxies and other visible mass densities, a neutron is observed to travel in a certain reference frame. Is this reference inertial? What if a very massive structure is placed just beyond the portion of universe observable by the laboratory? Will this mass distribution exert a force on the neutron in the laboratory? As one can see, the concept of inertial frame is very problematic, and the paradoxes which arise from this concept are not curable in the framework of classical (Newton) mechanics.

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  • $\begingroup$ +1, the "really, on can't" was not yet stated like this in the answer section. (But I'm not sure what "one needs General Relativity, since..." really means, though - Newton didn't seem to need it either. When it comes to pure understanding of things, I think it's always unfinished business: The question what the right energy tensor for a physical situation should be is open enough to claim, I think, that the problem is just moved.) $\endgroup$
    – Nikolaj-K
    Commented Jul 8, 2015 at 13:57
  • $\begingroup$ In Newtonian mechanics the inertial frame is a kind of circular reasoning. One writes the laws of dynamics which are valid in an inertial frame, but then one defines the inertial frame as one where the laws are valid... $\endgroup$
    – sintetico
    Commented Jul 8, 2015 at 14:04
  • $\begingroup$ Yes, that's right. But it doesn't follow that one needs general relativity, just because the latter is a theory that doesn't exhibit this unnice circularity. One needs that particular alternative IF what? Maybe you mean if you want to describe gravity without inertial frames? You could just drop the obligation and state that the research relevant theory of general relativity doesn't make use of those frames. $\endgroup$
    – Nikolaj-K
    Commented Jul 8, 2015 at 14:09
  • $\begingroup$ Ok, I just mean that in GR there are no such paradoxes. Physics are the same whatever one choses the reference frame. I agree that GR does not solve all problems though. $\endgroup$
    – sintetico
    Commented Jul 8, 2015 at 14:14
  • $\begingroup$ I like this related list. $\endgroup$
    – Nikolaj-K
    Commented Jul 8, 2015 at 14:15
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I asked someone who studied physics this question and was told that the following is a definition:

A reference frame in which a mass point thrown from the same point in three different (non co-planar) directions follows rectilinear paths each time it is thrown is called an inertial frame.

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