Editing Inertial frame of reference (section)

===Theory===
{{Main|Fictitious force}}
{{See also|Non-inertial frame|Rotating spheres|Bucket argument}}
[[Image:Rotating spheres.svg|thumb|180px|'''Figure 2''': Two spheres tied with a string and rotating at an angular rate ω. Because of the rotation, the string tying the spheres together is under tension.]]
[[Image:Rotating-sphere forces.svg|thumb|'''Figure 3''': Exploded view of rotating spheres in an inertial frame of reference showing the centripetal forces on the spheres provided by the tension in the tying string.]]
Inertial and non-inertial reference frames can be distinguished by the absence or presence of [[fictitious force]]s.<ref name="Rothman"/><ref name="Borowitz"/> 
{{blockquote|<i>The effect of this being in the noninertial frame is to require the observer to introduce a fictitious force into his calculations…</i>|Sidney Borowitz and Lawrence A Bornstein in ''A Contemporary View of Elementary Physics'', p. 138}}

The presence of fictitious forces indicates the physical laws are not the simplest laws available, in terms of the [[#principle|special principle of relativity]], a frame where fictitious forces are present is not an inertial frame:<ref name=Arnold2>{{Cite book|title=Mathematical Methods of Classical Mechanics |page=129 |author=V. I. Arnol'd |authorlink=Vladimir Arnold|isbn=978-0-387-96890-2 |date=1989 |url=https://books.google.com/books?num=10&btnG=Google+Search|publisher=Springer}}</ref>

{{blockquote|<i>The equations of motion in a non-inertial system differ from the equations in an inertial system by additional terms called inertial forces. This allows us to detect experimentally the non-inertial nature of a system.</i>|[[Vladimir Arnold|V. I. Arnol'd]]: ''[[Mathematical Methods of Classical Mechanics]]'' Second Edition, p. 129}}

Bodies in [[non-inertial reference frame]]s are subject to so-called ''fictitious'' forces (pseudo-forces); that is, [[force]]s that result from the acceleration of the [[Frame of reference|reference frame]] itself and not from any physical force acting on the body. Examples of fictitious forces are the [[centrifugal force (fictitious)|centrifugal force]] and the [[Coriolis force]] in [[rotating reference frame]]s.

To apply the Newtonian definition of an inertial frame, the understanding of separation between "fictitious" forces and "real" forces must be made clear.

For example, consider a stationary object in an inertial frame. Being at rest, no net force is applied. But in a frame rotating about a fixed axis, the object appears to move in a circle, and is subject to centripetal force. How can it be decided that the rotating frame is a non-inertial frame? There are two approaches to this resolution: one approach is to look for the origin of the fictitious forces (the Coriolis force and the centrifugal force). It will be found there are no sources for these forces, no associated [[force carrier]]s, no originating bodies.<ref name="note2">For example, there is no body providing a gravitational or electrical attraction.</ref> A second approach is to look at a variety of frames of reference. For any inertial frame, the Coriolis force and the centrifugal force disappear, so application of the principle of special relativity would identify these frames where the forces disappear as sharing the same and the simplest physical laws, and hence rule that the rotating frame is not an inertial frame.

Newton examined this problem himself using rotating spheres, as shown in Figure 2 and Figure 3. He pointed out that if the spheres are not rotating, the tension in the tying string is measured as zero in every frame of reference.<ref name=tension>That is, the universality of the laws of physics requires the same tension to be seen by everybody. For example, it cannot happen that the string breaks under extreme tension in one frame of reference and remains intact in another frame of reference, just because we choose to look at the string from a different frame.</ref> If the spheres only appear to rotate (that is, we are watching stationary spheres from a rotating frame), the zero tension in the string is accounted for by observing that the centripetal force is supplied by the centrifugal and Coriolis forces in combination, so no tension is needed. If the spheres really are rotating, the tension observed is exactly the centripetal force required by the circular motion. Thus, measurement of the tension in the string identifies the inertial frame: it is the one where the tension in the string provides exactly the centripetal force demanded by the motion as it is observed in that frame, and not a different value. That is, the inertial frame is the one where the fictitious forces vanish.

For [[linear acceleration]], Newton expressed the idea of undetectability of straight-line accelerations held in common:<ref name=Principia/>

{{blockquote|<i>If bodies, any how moved among themselves, are urged in the direction of parallel lines by equal accelerative forces, they will continue to move among themselves, after the same manner as if they had been urged by no such forces.</i>|Isaac Newton: ''Principia'' Corollary VI, p. 89, in Andrew Motte translation }}

This principle generalizes the notion of an inertial frame. For example, an observer confined in a free-falling lift will assert that he himself is a valid inertial frame, even if he is accelerating under gravity, so long as he has no knowledge about anything outside the lift. So, strictly speaking, inertial frame is a relative concept. With this in mind, inertial frames can collectively be defined as a set of frames which are stationary or moving at constant velocity with respect to each other, so that a single inertial frame is defined as an element of this set.

For these ideas to apply, everything observed in the frame has to be subject to a base-line, common acceleration shared by the frame itself. That situation would apply, for example, to the elevator example, where all objects are subject to the same gravitational acceleration, and the elevator itself accelerates at the same rate.