Editing Inertial frame of reference (section)

===Inertial frames and rotation===

In an inertial frame, [[Newton's first law]], the ''law of inertia'', is satisfied: Any free motion has a constant magnitude and direction.<ref name=LandauMechanics>{{cite book|last1=Landau|first1=L. D.|last2=Lifshitz|first2=E. M.|title=Mechanics|url=https://archive.org/download/landau-and-lifshitz-physics-textbooks-series/Vol%201%20-%20Landau%2C%20Lifshitz%20-%20Mechanics%20%283rd%20ed%2C%201976%29.pdf#page=31|date=1960|publisher=Pergamon Press|pages=4–6}}</ref> [[Newton's second law]] for a [[Point particle|particle]] takes the form:

:<math>\mathbf{F} = m \mathbf{a} \ ,</math>

with '''F''' the net force (a [[Euclidean vector|vector]]), ''m'' the mass of a particle and '''a''' the [[acceleration]] of the particle (also a vector) which would be measured by an observer at rest in the frame. The force '''F''' is the [[vector sum]] of all "real" forces on the particle, such as [[contact force]]s, electromagnetic, gravitational, and nuclear forces.

In contrast, Newton's second law in a [[rotating frame of reference]] (a '''non-inertial frame of reference'''), rotating at angular rate ''Ω'' about an axis, takes the form:

:<math>\mathbf{F}' = m \mathbf{a} \ ,</math>

which looks the same as in an inertial frame, but now the force '''F'''′ is the resultant of not only '''F''', but also additional terms (the paragraph following this equation presents the main points without detailed mathematics):

:<math>\mathbf{F}' = \mathbf{F} - 2m \mathbf{\Omega} \times \mathbf{v}_{B} - m \mathbf{\Omega} \times (\mathbf{\Omega} \times \mathbf{x}_B ) - m \frac{d \mathbf{\Omega}}{dt} \times \mathbf{x}_B \ , </math>

where the angular rotation of the frame is expressed by the vector '''Ω''' pointing in the direction of the axis of rotation, and with magnitude equal to the angular rate of rotation ''Ω'', symbol × denotes the [[vector cross product]], vector '''x'''<sub>''B''</sub> locates the body and vector '''v'''<sub>''B''</sub> is the [[velocity]] of the body according to a rotating observer (different from the velocity seen by the inertial observer).

The extra terms in the force '''F'''′ are the "fictitious" forces for this frame, whose causes are external to the system in the frame. The first extra term is the [[Coriolis force]], the second the [[centrifugal force (rotating reference frame)|centrifugal force]], and the third the [[Euler force]]. These terms all have these properties: they vanish when ''Ω'' = 0; that is, they are zero for an inertial frame (which, of course, does not rotate); they take on a different magnitude and direction in every rotating frame, depending upon its particular value of '''Ω'''; they are ubiquitous in the rotating frame (affect every particle, regardless of circumstance); and they have no apparent source in identifiable physical sources, in particular, [[matter]]. Also, fictitious forces do not drop off with distance (unlike, for example, [[nuclear force]]s or [[electrical force]]s). For example, the centrifugal force that appears to emanate from the axis of rotation in a rotating frame increases with distance from the axis.

All observers agree on the real forces, '''F'''; only non-inertial observers need fictitious forces. The laws of physics in the inertial frame are simpler because unnecessary forces are not present.

In Newton's time the [[fixed stars]] were invoked as a reference frame, supposedly at rest relative to [[absolute space]]. In reference frames that were either at rest with respect to the fixed stars or in uniform translation relative to these stars, [[Newton's laws of motion]] were supposed to hold. In contrast, in frames accelerating with respect to the fixed stars, an important case being frames rotating relative to the fixed stars, the laws of motion did not hold in their simplest form, but had to be supplemented by the addition of [[fictitious forces]], for example, the [[Coriolis force]] and the [[centrifugal force]]. Two experiments were devised by Newton to demonstrate how these forces could be discovered, thereby revealing to an observer that they were not in an inertial frame: the example of the tension in the cord linking [[rotating spheres|two spheres rotating]] about their center of gravity, and the example of the curvature of the surface of water in a [[bucket argument|rotating bucket]]. In both cases, application of [[Newton's second law]] would not work for the rotating observer without invoking centrifugal and Coriolis forces to account for their observations (tension in the case of the spheres; parabolic water surface in the case of the rotating bucket).

As now known, the fixed stars are not fixed. Those that reside in the [[Milky Way]] turn with the galaxy, exhibiting [[proper motion]]s. Those that are outside our galaxy (such as nebulae once mistaken to be stars) participate in their own motion as well, partly due to [[expansion of the universe]], and partly due to [[peculiar velocity|peculiar velocities]].<ref name=Balbi>{{Cite book|title=The Music of the Big Bang |author=Amedeo Balbi |isbn=978-3-540-78726-6 |publisher=Springer |date=2008 |page= 59 |url=https://books.google.com/books?id=vEJM7s909CYC&q=CMB+%22rotation+of+the+universe%22&pg=PA58 }}</ref> For instance, the [[Andromeda Galaxy]] is on [[Andromeda–Milky Way collision|collision course with the Milky Way]] at a speed of 117&nbsp;km/s.<ref>{{Cite journal |title=Constraints on the proper motion of the Andromeda Galaxy based on the survival of its satellite M33 |pages=894–898 |author1=Abraham Loeb |author2=Mark J. Reid |author3=Andreas Brunthaler |author4=Heino Falcke |journal=The Astrophysical Journal |volume=633 |date=2005 |url=http://www.mpifr-bonn.mpg.de/staff/abrunthaler/pub/loeb.pdf |doi=10.1086/491644 |bibcode=2005ApJ...633..894L |arxiv=astro-ph/0506609 |issue=2 |s2cid=17099715 |access-date=15 December 2008 |archive-date=11 August 2017 |archive-url=https://web.archive.org/web/20170811143825/http://www3.mpifr-bonn.mpg.de/staff/abrunthaler/pub/loeb.pdf |url-status=live }}</ref> The concept of inertial frames of reference is no longer tied to either the fixed stars or to absolute space. Rather, the identification of an inertial frame is based on the simplicity of the laws of physics in the frame. 
The laws of nature take a simpler form in inertial frames of reference because in these frames one did not have to introduce inertial forces when writing down Newton's law of motion.<ref name=Stachel>{{Cite book|pages= 235–236 |url=https://books.google.com/books?id=OAsQ_hFjhrAC&q=%22laws+of+nature+took+a+simpler+form%22&pg=PA235 |title=Einstein from "B" to "Z" |author=John J. Stachel |isbn=0-8176-4143-2 |publisher=Springer |date=2002}}</ref>

In practice, using a frame of reference based upon the fixed stars as though it were an inertial frame of reference introduces little discrepancy. For example, the centrifugal acceleration of the Earth because of its rotation about the Sun is about thirty million times greater than that of the Sun about the galactic center.<ref name=Graneau>{{Cite book|title=In the Grip of the Distant Universe |author1=Peter Graneau |author2=Neal Graneau |page= 147 |url=https://books.google.com/books?id=xpIJZxDkWAUC&q=universe+%22fixed+stars%22+date:2004-2010&pg=PA144 |isbn=981-256-754-2 |publisher=World Scientific |date=2006}}</ref>

To illustrate further, consider the question: "Does the Universe rotate?" An answer might explain the shape of the [[Milky Way]] galaxy using the laws of physics,<ref name=Genz>{{Cite book |title=Nothingness |author=Henning Genz |page=275 |url=https://books.google.com/books?id=Cn_Q9wbDOM0C&q=%22rotation+of+the+universe%22&pg=PA274 |isbn=0-7382-0610-5 |date=2001 |publisher=Da Capo Press }}{{Dead link|date=January 2023 |bot=InternetArchiveBot |fix-attempted=yes }}</ref> although other observations might be more definitive; that is, provide larger [[Observational error|discrepancies]] or less [[measurement uncertainty]], like the anisotropy of the [[microwave background radiation]] or [[Big Bang nucleosynthesis]].<ref name=Thompson>{{Cite book|title=Advances in Astronomy |chapter-url= https://books.google.com/books?id=3TrsMTmbr-sC&q=CMB+%22rotation+of+the+universe%22&pg=PA32 |author=J Garcio-Bellido|editor=J. M. T. Thompson |publisher=Imperial College Press |date=2005 |page= 32, §9 |chapter=The Paradigm of Inflation |isbn=1-86094-577-5}}</ref><ref name=Szydlowski>{{Cite journal|title=Dark energy and global rotation of the Universe |author1=Wlodzimierz Godlowski |author2=Marek Szydlowski |arxiv=astro-ph/0303248 |date=2003 |doi=10.1023/A:1027301723533 |journal=General Relativity and Gravitation |volume=35 |pages=2171–2187|issue=12|bibcode = 2003GReGr..35.2171G |s2cid=118988129 }}</ref> The flatness of the Milky Way depends on its rate of rotation in an inertial frame of reference. If its apparent rate of rotation is attributed entirely to rotation in an inertial frame, a different "flatness" is predicted than if it is supposed that part of this rotation is actually due to rotation of the universe and should not be included in the rotation of the galaxy itself. Based upon the laws of physics, a model is set up in which one parameter is the rate of rotation of the Universe. If the laws of physics agree more accurately with observations in a model with rotation than without it, we are inclined to select the best-fit value for rotation, subject to all other pertinent experimental observations. If no value of the rotation parameter is successful and theory is not within observational error, a modification of physical law is considered, for example, [[dark matter]] is invoked to explain the [[galactic rotation curve]]. So far, observations show any rotation of the universe is very slow, no faster than once every {{val|6|e=13}} years (10<sup>−13</sup>&nbsp;rad/yr),<ref name=Birch>{{cite journal |url=http://www.nature.com/nature/journal/v298/n5873/abs/298451a0.html |first=P. |last=Birch |title=Is the Universe rotating? |journal=[[Nature (journal)|Nature]] |volume=298 |pages=451–454 |date=29 July 1982 |issue=5873 |doi=10.1038/298451a0 |bibcode=1982Natur.298..451B |s2cid=4343095 |access-date=14 December 2008 |archive-date=5 March 2016 |archive-url=https://web.archive.org/web/20160305064307/http://www.nature.com/nature/journal/v298/n5873/abs/298451a0.html |url-status=live |url-access=subscription }}</ref> and debate persists over whether there is ''any'' rotation. However, if rotation were found, interpretation of observations in a frame tied to the universe would have to be corrected for the fictitious forces inherent in such rotation in classical physics and special relativity, or interpreted as the curvature of spacetime and the motion of matter along the geodesics in general relativity.<ref>{{citation|title=Mach's Principle II|first1=James G.|last1=Gilson|date=1 September 2004|arxiv=physics/0409010|bibcode = 2004physics...9010G }}</ref>

When [[quantum mechanics|quantum]] effects are important, there are additional conceptual complications that arise in [[quantum reference frame]]s.