Editing Inertial frame of reference (section)

===Newtonian mechanics===
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{{Main|Newton's laws of motion}}<-->
Classical theories that use the [[Galilean transformation]] postulate the equivalence of all inertial reference frames. The Galilean transformation transforms coordinates from one inertial reference frame, <math>\mathbf{s}</math>, to another, <math>\mathbf{s}^{\prime}</math>, by simple addition or subtraction of coordinates:

:<math>
\mathbf{r}^{\prime} = \mathbf{r} - \mathbf{r}_{0} - \mathbf{v} t
</math>

:<math>
t^{\prime} = t - t_{0}
</math>

where '''r'''<sub>0</sub> and ''t''<sub>0</sub> represent shifts in the origin of space and time, and '''v''' is the relative velocity of the two inertial reference frames. Under Galilean transformations, the time ''t''<sub>2</sub> − ''t''<sub>1</sub> between two events is the same for all reference frames and the [[distance]] between two simultaneous events (or, equivalently, the length of any object, |'''r'''<sub>2</sub> − '''r'''<sub>1</sub>|) is also the same.
[[Image:Inertial frames.svg|250px|thumbnail|'''Figure 1''': Two frames of reference moving with relative velocity <math>\stackrel{\vec v}{}</math>. Frame ''S' '' has an arbitrary but fixed rotation with respect to frame ''S''. They are both ''inertial frames'' provided a body not subject to forces appears to move in a straight line. If that motion is seen in one frame, it will also appear that way in the other.]]
Within the realm of Newtonian mechanics, an [[inertia]]l frame of reference, or inertial reference frame, is one in which [[Newton's laws of motion#Newton's first law|Newton's first law of motion]] is valid.<ref name=Moeller>{{Cite book|author=C Møller |title=The Theory of Relativity |publisher=Oxford University Press |location=Oxford UK |isbn=0-19-560539-X |date=1976 |page=1 |oclc=220221617 |edition=Second}}</ref> However, the [[#principle|principle of special relativity]] generalizes the notion of an inertial frame to include all physical laws, not simply Newton's first law.

Newton viewed the first law as valid in any reference frame that is in uniform motion (neither rotating nor accelerating) relative to [[absolute space]]; as a practical matter, "absolute space" was considered to be the [[fixed stars]]<ref>For a discussion of the role of fixed stars, see {{Cite book |title=Nothingness: The Science of Empty Space |author=Henning Genz |page=150 |isbn=0-7382-0610-5 |publisher=Da Capo Press |date=2001 |url=https://books.google.com/books?id=Cn_Q9wbDOM0C&q=frame+Newton+%22fixed+stars%22&pg=PA150 }}{{Dead link|date=January 2023 |bot=InternetArchiveBot |fix-attempted=yes }}</ref><ref name=Resnick>{{Cite book|title=Physics |page=Volume 1, Chapter 3 |isbn=0-471-32057-9 |url=https://archive.org/details/fundamentalsofph02hall
|url-access=registration |quote=physics resnick. |publisher=Wiley |date=2001 |edition=5th |author1=Robert Resnick |author2=David Halliday |author3=Kenneth S. Krane |no-pp=true }}</ref> In the theory of relativity the notion of [[absolute space]] or a [[privileged frame]] is abandoned, and an inertial frame in the field of [[classical mechanics]] is defined as:<ref name=Takwale>{{Cite book|url=https://books.google.com/books?id=r5P29cN6s6QC&q=fixed+stars+%22inertial+frame%22&pg=PA70 |title=Introduction to classical mechanics |page=70 |author=RG Takwale |publisher=Tata McGraw-Hill|date=1980 |isbn=0-07-096617-6 |location=New Delhi}}</ref><ref name=Woodhouse>{{Cite book|url=https://books.google.com/books?id=ggPXQAeeRLgC |title=Special relativity |page=6 |author=NMJ Woodhouse |publisher=Springer |date=2003 |isbn=1-85233-426-6 |location=London/Berlin}}</ref>

{{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>}}

Hence, with respect to an inertial frame, an object or body [[acceleration|accelerates]] only when a physical [[force]] is applied, and (following [[Newton's laws of motion|Newton's first law of motion]]), in the absence of a net force, a body at [[rest (physics)|rest]] will remain at rest and a body in motion will continue to move uniformly—that is, in a straight line and at constant [[speed]]. Newtonian inertial frames transform among each other according to the [[Galilean transformation|Galilean group of symmetries]].

If this rule is interpreted as saying that [[straight-line motion]] is an indication of zero net force, the rule does not identify inertial reference frames because straight-line motion can be observed in a variety of frames. If the rule is interpreted as defining an inertial frame, then being able to determine when zero net force is applied is crucial. The problem was summarized by Einstein:<ref name=Einstein5>{{Cite book|title=The Meaning of Relativity |author=A Einstein |page=58 |date=1950 |url=https://books.google.com/books?num=10&btnG=Google+Search|publisher=Princeton University Press}}</ref>

{{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}}

There are several approaches to this issue. One approach is to argue that all real forces drop off with distance from their sources in a known manner, so it is only needed that a body is far enough away from all sources to ensure that no force is present.<ref name=Rosser>{{Cite book|title=Introductory Special Relativity |author=William Geraint Vaughan Rosser |page=3 |url=https://books.google.com/books?id=zpjBEBbIjAIC&q=reference+%22laws+of+physics%22&pg=PA94
|isbn=0-85066-838-7 |date=1991 |publisher=CRC Press }}</ref> A possible issue with this approach is the historically long-lived view that the distant universe might affect matters ([[Mach's principle]]). Another approach is to identify all real sources for real forces and account for them. A possible issue with this approach is the possibility of missing something, or accounting inappropriately for their influence, perhaps, again, due to Mach's principle and an incomplete understanding of the universe. A third approach is to look at the way the forces transform when shifting reference frames. Fictitious forces, those that arise due to the acceleration of a frame, disappear in inertial frames and have complicated rules of transformation in general cases. Based on the universality of physical law and the request for frames where the laws are most simply expressed, inertial frames are distinguished by the absence of such fictitious forces.

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}}

This principle differs from the [[#principle|special principle]] in two ways: first, it is restricted to mechanics, and second, it makes no mention of simplicity. It shares the special principle of the invariance of the form of the description among mutually translating reference frames.<ref name=note1>However, in the Newtonian system the Galilean transformation connects these frames and in the special theory of relativity the [[Lorentz transformation]] connects them. The two transformations agree for speeds of translation much less than the [[speed of light]].</ref> The role of fictitious forces in classifying reference frames is pursued further below.