Inertial frame of reference: Difference between revisions

In [[classical physics]] and [[special relativity]], an '''[[inertia]]linertial frame of reference''' (also called an '''inertial space''', or a '''Galilean reference frame''') is a [[frame of reference]] in which objects exhibit [[inertia]]: they remain at rest or in uniform motion relative to the frame until acted upon by external forces. In such a frame, the laws of nature takecan onbe aobserved without the need to particularlycorrect simplefor formacceleration.

All frames of reference with zero acceleration are in a state of constant, [[rectilinear motion]] (straight -line motion) with respect to one another. In such a frame, an object with zero [[net force]] acting on it, is perceived to move with a constant [[velocity]], or, equivalently, [[Newton's laws of motion#Newton's first law|Newton's first law of motion]] holds. ThoseSuch frames are theknown frames calledas inertial. Originally, someSome physicists, like [[Isaac Newton]], originally thought that one of thosethese frames was absolute — the one approximated by the [[fixed stars]]. However, this is not required for the definition, and it is now known that those stars are in fact moving.

According to the [[Principle of relativity#Special principle of relativity|special principle of special relativity]], all [[physical laws]] look the same in all inertial reference frames, and no inertial frame is privileged over the otheranother. [[Measurement|Measurements]] of objects in one inertial frame can be converted to [[Measurement|measurements]] in another by a simple transformation — the [[Galilean transformation]] in [[Newtonian physics]] or the [[Lorentz transformation]] (combined with a translation) in [[special relativity]]; these approximately match when the relative speed of the frames is low, but differ as it approaches the [[speed of light]].

By contrast, a ''[[non-inertial reference frame]]'' has non-zero acceleration. In such a frame, the interactions between [[physical object]]s vary depending on the acceleration of that frame with respect to an inertial frame. Viewed from the perspective of [[classical mechanics]] and [[special relativity]], the 'usual' [[Fundamental forces|physical forces]] caused by the interaction of objects have to be supplemented by [[fictitious force]]s caused by [[inertia]].<ref name="Rothman">{{Cite book|title=Discovering the Natural Laws: The Experimental Basis of Physics |author= Milton A. Rothman |page=[https://archive.org/details/discoveringnatur0000roth/page/n37/mode/2up 23-24] |url=https://archive.org/details/discoveringnatur0000roth

Viewed from the perspective of [[General relativity|general relativity theory]], the inertialfictitious (i.e. fictitiousinertial) forces are attributed to [[geodesic (general relativity)|geodesic motion in spacetime]].

Due to [[Earth's rotation]], its surface is not an inertial frame of reference. The [[Coriolis effect]] can deflect certain forms of motion as seen from [[Earth]], and the [[centrifugal force]] will reduce the effective [[gravity]] at the [[equator]]. Nevertheless, for many applications the Earth is aan goodadequate [[approximation]] of an inertial reference frame, adequate in many applications.

|isbn=0-486-60081-5 |publisher=Courier Dover Publications |date=1952 }}</ref>{{blockquote|<i>Special principle of relativity: If a system of coordinates K is chosen so that, in relation to it, physical laws hold good in their simplest form, the same laws hold good in relation to any other system of coordinates K' moving in uniform translation relatively to K.</i>|Albert Einstein: ''The foundation of the general theory of relativity'', Section A, §1}}

The expression ''inertial frame of reference'' ({{lang-langx|de|Inertialsystem}}) was coined by [[Ludwig Lange (physicist)|Ludwig Lange]] in 1885, to replace Newton's definitions of "absolute space and time" with a more [[Operational definition#Science|operational definition]].:<ref>{{Cite journal

{{blockquote|<i>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.</i><ref name=Iro>L. Lange (1885) as quoted by Max von Laue in his book (1921) ''Die Relativitätstheorie'', p. 34, and translated by {{Cite book|page=169 |title=A Modern Approach to Classical Mechanics |author=Harald Iro |url=https://books.google.com/books?id=-L5ckgdxA5YC&q=inertial+noninertial&pg=PA179 |isbn=981-238-213-5 |date=2002 |publisher=World Scientific}}</ref>}}

The inadequacy of the notion of "absolute space" in Newtonian mechanics is spelled out by BlagojevićBlagojevich:<ref name="Blagojević2">{{Cite book|title=Gravitation and Gauge Symmetries |author=Milutin Blagojević |page=5 |url=https://books.google.com/books?id=N8JDSi_eNbwC&q=inertial+frame+%22absolute+space%22&pg=PA5 |isbn=0-7503-0767-6 |publisher=CRC Press |date=2002}}</ref>

{{blockquote|<i>

</i>| Milutin Blagojević: ''Gravitation and Gauge Symmetries'', p. 5}}

{{blockquote|<i>The original question, "relative to what frame of reference do the laws of motion hold?" is revealed to be wrongly posed. The laws of motion essentially determine a class of reference frames, and (in principle) a procedure for constructing them.</i>|[http://plato.stanford.edu/archives/sum2002/entries/spacetime-iframes/#Oth Robert DiSalle ''Space and Time: Inertial Frames'']}}

{{blockquote|<i>An inertial frame of reference is one in which the motion of a particle not subject to forces is in a straight line at constant speed.</i>}}

{{blockquote|<i>The weakness of the principle of inertia lies in this, that it involves an argument in a circle: a mass moves without acceleration if it is sufficiently far from other bodies; we know that it is sufficiently far from other bodies only by the fact that it moves without acceleration.</i>|Albert Einstein: ''[[The Meaning of Relativity]]'', p. 58}}

Newton enunciated a principle of relativity himself in one of his corollaries to the laws of motion:<ref name=Feynman2>{{Cite book |title=Six not-so-easy pieces: Einstein's relativity, symmetry, and space-time |author=Richard Phillips Feynman |page=50 |isbn=0-201-32842-9 |date=1998 |publisher=Basic Books |url=https://books.google.com/books?id=ipY8onVQWhcC&q=%22The+Principle+of+Relativity%22&pg=PA49 }}{{Dead link|date=January 2023 |bot=InternetArchiveBot |fix-attempted=yes }}</ref><ref name=Principia>See the ''Principia'' on line at [https://archive.org/stream/newtonspmathema00newtrich#page/n7/mode/2up Andrew Motte Translation]</ref>

{{blockquote|<i>The motions of bodies included in a given space are the same among themselves, whether that space is at rest or moves uniformly forward in a straight line.</i>|Isaac Newton: ''Principia'', Corollary V, p. 88 in Andrew Motte translation}}

Notice that these formulas predict at ''t'' = 0 s the first car is 200 m200m down the road and the second car is right beside us, as expected. We want to find the time at which <math>x_1=x_2</math>. Therefore, we set <math>x_1=x_2</math> and solve for <math>t</math>, that is:

|page=155 |date=2007 |publisher=Pearson Prentice Hall |isbn=978-0-13-149508-1 }}</ref>
{{blockquote|<i>There is no experiment observers can perform to distinguish whether an acceleration arises because of a gravitational force or because their reference frame is accelerating.</i>|Douglas C. Giancoli, ''Physics for Scientists and Engineers with Modern Physics'', p. 155.}}

However, the general theory reduces to the special theory over sufficiently small regions of [[spacetime]], where curvature effects become less important and the earlier inertial frame arguments can come back into play.<ref>{{cite book |title=Information Theory and Quantum Physics: Physical Foundations for Understanding the Conscious Process |first1=Herbert S. |last1=Green |publisher=Springer |date=2000 |isbn=354066517X |page=154 |url=https://books.google.com/books?id=CUJiQjSVCu8C}} [https://books.google.com/books?id=CUJiQjSVCu8C&pg=PA154 Extract of page 154]</ref><ref>{{cite book |title=Theory of Special Relativity |first1=Nikhilendu |last1=Bandyopadhyay |publisher=Academic Publishers |date=2000 |isbn=8186358528 |page=116 |url=https://books.google.com/books?id=qMOyfi_i0j8C}} [https://books.google.com/books?id=qMOyfi_i0j8C&pg=PA116 Extract of page 116]</ref> Consequently, modern special relativity is now sometimes described as only a "local theory".<ref>{{cite book |title=Cosmological Inflation and Large-Scale Structure |first1=Andrew R. |last1=Liddle |first2=David H. |last2=Lyth |publisher=Cambridge University Press |date=2000 |isbn=0-521-57598-2 |page=329 |url=https://books.google.com/books?id=XmWauPZSovMC}} [https://books.google.com/books?id=XmWauPZSovMC&pg=PA329 Extract of page 329]</ref> "Local" can encompass, for example, the entire [[Milky Way galaxy]]: The astronomer [[Karl Schwarzschild]] observed the motion of pairs of stars orbiting each other. He found that the two orbits of the stars of such a system lie in a plane, and the perihelion of the orbits of the two stars remains pointing in the same direction with respect to the [[Solar System]]. Schwarzschild pointed out that that was invariably seen: the direction of the [[angular momentum]] of all observed double star systems remains fixed with respect to the direction of the angular momentum of the Solar System. These observations allowed him to conclude that inertial frames inside the galaxy do not rotate with respect to one another, and that the space of the Milky Way is approximately Galilean or Minkowskian.<ref>[http://www.mpiwg-berlin.mpg.de/Preprints/P271.PDF In the Shadow of the Relativity Revolution] {{Webarchive|url=https://web.archive.org/web/20170520084821/http://www.mpiwg-berlin.mpg.de/Preprints/P271.PDF |date=20 May 2017 }} Section 3: The Work of Karl Schwarzschild (2.2 MB PDF-file)</ref>

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}}

{{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}}

{{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 }}

[[Inertial navigation system]]s used a cluster of [[gyroscope]]s and accelerometers to determine accelerations relative to inertial space. After a gyroscope is spun up in a particular orientation in inertial space, the law of conservation of angular momentum requires that it retain that orientation as long as no external forces are applied to it.<ref>{{cite book|last=Chatfield|first=Averil B.|title=Fundamentals of High Accuracy Inertial Navigation, Volume 174|date=1997|publisher=AIAA|isbn=9781600864278}}</ref>{{rp|59}} Three orthogonal gyroscopes establish an inertial reference frame, and the accelerators measure acceleration relative to that frame. The accelerations, along with a clock, can then be used to calculate the change in position. Thus, inertial navigation is a form of [[dead reckoning]] that requires no external input, and therefore cannot be jammed by any external or internal signal source.<ref>{{cite book|editor-last1=Kennie|editor-first1=T.J.M.|title=Engineering Surveying Technology|date=1993|publisher=Taylor & Francis|location=Hoboken|isbn=9780203860748|page=95|edition=pbk.|editor-last2=Petrie|editor-first2=G.}}</ref>