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Inertial frame of reference

Framing}} An inertial frame of reference, in classical physics, is a frame of reference in which bodies, whose net force acting upon them is zero, are not accelerated, that is they are at rest or they move at a constant velocity in a straight line. In analytical terms, it is a frame of reference that describes time and space homogeneously, isotropically, and in a time-independent manner. Conceptually, in classical physics and special relativity, the physics of a system in an inertial frame have no causes external to the system. An inertial frame of reference may also be called an inertial reference frame, inertial frame, Galilean reference frame, or inertial space. All inertial frames are in a state of constant, motion with respect to one another; an accelerometer moving with any of them would detect zero acceleration. Measurements in one inertial frame can be converted to measurements in another by a simple transformation (the Galilean transformation in Newtonian physics and the Lorentz transformation in special relativity). In general relativity, in any region small enough for the curvature of spacetime and tidal forces Extract of page 219 to be negligible, one can find a set of inertial frames that approximately describe that region. In a non-inertial reference frame in classical physics and special relativity, the physics of a system vary depending on the acceleration of that frame with respect to an inertial frame, and the usual physical forces must be supplemented by fictitious forces.{{Cite book|title=Discovering the Natural Laws: The Experimental Basis of Physics |author= Milton A. Rothman |page=23 |url=https://books.google.com/?id=Wdp-DFK3b5YC&pg=PA23&vq=inertial&dq=reference+%22laws+of+physics%22 |isbn=0-486-26178-6 |publisher=Courier Dover Publications |date=1989}} In contrast, systems in non-inertial frames in general relativity don't have external causes, because of the principle of geodesic motion. In classical physics, for example, a ball dropped towards the ground does not go exactly straight down because the Earth is rotating, which means the frame of reference of an observer on Earth is not inertial. The physics must account for the Coriolis effect—in this case thought of as a force—to predict the horizontal motion. Another example of such a fictitious force associated with rotating reference frames is the centrifugal effect, or centrifugal force.

Background

A brief comparison of inertial frames in special relativity and in Newtonian mechanics, and the role of absolute space is next.

A set of frames where the laws of physics are simple

According to the first postulate of special relativity, all physical laws take their simplest form in an inertial frame, and there exist multiple inertial frames interrelated by uniform translation: REF -->{{Cite book|title=The Principle of Relativity: a collection of original memoirs on the special and general theory of relativity |author=Einstein, A., Lorentz, H. A., Minkowski, H., & Weyl, H. |page=111 |url=https://books.google.com/?id=yECokhzsJYIC&pg=PA111&dq=postulate+%22Principle+of+Relativity%22 |isbn=0-486-60081-5 |publisher=Courier Dover Publications |date=1952 }} This simplicity manifests in that inertial frames have self-contained physics without the need for external causes, while physics in non-inertial frames have external causes. The principle of simplicity can be used within Newtonian physics as well as in special relativity; see Nagel and also Blagojević. In practical terms, the equivalence of inertial reference frames means that scientists within a box moving uniformly cannot determine their absolute velocity by any experiment (otherwise the differences would set up an absolute standard reference frame). According to this definition, supplemented with the constancy of the speed of light, inertial frames of reference transform among themselves according to the Poincaré group of symmetry transformations, of which the Lorentz transformations are a subgroup. In Newtonian mechanics, which can be viewed as a limiting case of special relativity in which the speed of light is infinite, inertial frames of reference are related by the Galilean group of symmetries.

Absolute space

Newton posited an absolute space considered well approximated by a frame of reference stationary relative to the fixed stars. An inertial frame was then one in uniform translation relative to absolute space. However, some scientists (called "relativists" by Mach), even at the time of Newton, felt that absolute space was a defect of the formulation, and should be replaced. Indeed, the expression inertial frame of reference () was coined by Ludwig Lange in 1885, to replace Newton's definitions of "absolute space and time" by a more operational definition.{{Cite journal |author=Lange, Ludwig |date=1885 |title=Über die wissenschaftliche Fassung des Galileischen Beharrungsgesetzes |journal=Philosophische Studien |volume=2}}{{Cite book|author=Julian B. Barbour |title=The Discovery of Dynamics |edition=Reprint of 1989 Absolute or Relative Motion? |pages=645–646 |url=https://books.google.com/?id=WQidkYkleXcC&pg=PA645&dq=Ludwig+Lange+%22operational+definition%22 |isbn=0-19-513202-5 |publisher=Oxford University Press |date=2001 }} As translated by Iro, Lange proposed the following definition:L. Lange (1885) as quoted by Max von Laue in his book (1921) Die Relativitätstheorie, p. 34, and translated by A discussion of Lange's proposal can be found in Mach. The inadequacy of the notion of "absolute space" in Newtonian mechanics is spelled out by Blagojević: The utility of operational definitions was carried much further in the special theory of relativity. Some historical background including Lange's definition is provided by DiSalle, who says in summary:{{Cite book |author =Robert DiSalle |chapter =Space and Time: Inertial Frames |title =The Stanford Encyclopedia of Philosophy |editor=Edward N. Zalta |url=http://plato.stanford.edu/archives/sum2002/entries/spacetime-iframes/#Oth |date=Summer 2002}}

Newton's inertial frame of reference

Within the realm of Newtonian mechanics, an inertial frame of reference, or inertial reference frame, is one in which Newton's first law of motion is valid. However, the principle of special relativity generalizes the notion of 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 relative to the fixed stars;The question of "moving uniformly relative to what?" was answered by Newton as "relative to absolute space". As a practical matter, "absolute space" was considered to be the fixed stars. 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/?id=Cn_Q9wbDOM0C&pg=PA150&dq=frame+Newton+%22fixed+stars%22 }} that is, neither rotating nor accelerating relative to the stars.{{Cite book|title=Physics |page=Volume 1, Chapter 3 |isbn=0-471-32057-9 |url=https://books.google.com/?id=CucFAAAACAAJ&dq=intitle:physics+inauthor:resnick |publisher=Wiley |date=2001 |edition=5th |author1=Robert Resnick |author2=David Halliday |author3=Kenneth S. Krane |nopp=true }} Today the notion of " absolute space" is abandoned, and an inertial frame in the field of classical mechanics is defined as: Hence, with respect to an inertial frame, an object or body accelerates only when a physical force is applied, and (following Newton's first law of motion), in the absence of a net force, a body at 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 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 we have to be able to determine when zero net force is applied. The problem was summarized by Einstein: 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 we have only to be sure that a body is far enough away from all sources to ensure that no force is present.{{Cite book|title=Introductory Special Relativity |author=William Geraint Vaughan Rosser |page=3 |url=https://books.google.com/?id=zpjBEBbIjAIC&pg=PA94&dq=reference+%22laws+of+physics%22 |isbn=0-85066-838-7 |date=1991 |publisher=CRC Press }} 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 that we might miss something, or account 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 we shift 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. On the basis of 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:See the Principia on line at Andrew Motte Translation This principle differs from the special principle in two ways: first, it is restricted to mechanics, and second, it makes no mention of simplicity. It shares with the special principle the invariance of the form of the description among mutually translating reference frames.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. The role of fictitious forces in classifying reference frames is pursued further below.

Separating non-inertial from inertial reference frames

Theory

Inertial and non-inertial reference frames can be distinguished by the absence or presence of fictitious forces, as explained shortly. The presence of fictitious forces indicates the physical laws are not the simplest laws available so, in terms of the special principle of relativity, a frame where fictitious forces are present is not an inertial frame: Bodies in non-inertial reference frames are subject to so-called fictitious forces (pseudo-forces); that is, forces that result from the acceleration of the reference frame itself and not from any physical force acting on the body. Examples of fictitious forces are the centrifugal force and the Coriolis force in rotating reference frames. How then, are "fictitious" forces to be separated from "real" forces? It is hard to apply the Newtonian definition of an inertial frame without this separation. 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 (which is made up of the Coriolis force and the centrifugal force). How can we decide 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). We will find there are no sources for these forces, no associated force carriers, no originating bodies.For example, there is no body providing a gravitational or electrical attraction. 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.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. 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. So much for fictitious forces due to rotation. However, for linear acceleration, Newton expressed the idea of undetectability of straight-line accelerations held in common: 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, we can define inertial frames collectively 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. In 1899 the astronomer Karl Schwarzschild pointed out an observation about double stars. The motion of two stars orbiting each other is planar, the two orbits of the stars of the system lie in a plane. In the case of sufficiently near double star systems, it can be seen from Earth whether 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. The logical inference is that just like gyroscopes, the angular momentum of all celestial bodies is angular momentum with respect to a universal inertial space. In the Shadow of the Relativity Revolution Section 3: The Work of Karl Schwarzschild (2.2 MB PDF-file)

Applications

Inertial navigation systems used a cluster of gyroscopes 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. 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. A gyrocompass, employed for navigation of seagoing vessels, finds the geometric north. It does so, not by sensing the Earth's magnetic field, but by using inertial space as its reference. The outer casing of the gyrocompass device is held in such a way that it remains aligned with the local plumb line. When the gyroscope wheel inside the gyrocompass device is spun up, the way the gyroscope wheel is suspended causes the gyroscope wheel to gradually align its spinning axis with the Earth's axis. Alignment with the Earth's axis is the only direction for which the gyroscope's spinning axis can be stationary with respect to the Earth and not be required to change direction with respect to inertial space. After being spun up, a gyrocompass can reach the direction of alignment with the Earth's axis in as little as a quarter of an hour.

Newtonian mechanics

Classical mechanics, which includes relativity, assumes the equivalence of all inertial reference frames. Newtonian mechanics makes the additional assumptions of absolute space and absolute time. Given these two assumptions, the coordinates of the same event (a point in space and time) described in two inertial reference frames are related by a Galilean transformation. \mathbf{r}^{\prime} = \mathbf{r} - \mathbf{r}_{0} - \mathbf{v} t t^{\prime} = t - t_{0} where r0 and t0 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 t2 − t1 between two events is the same for all inertial reference frames and the distance between two simultaneous events (or, equivalently, the length of any object, |r2 − r1|) is also the same.

Special relativity

Einstein's theory of special relativity, like Newtonian mechanics, assumes the equivalence of all inertial reference frames, but makes an additional assumption, foreign to Newtonian mechanics, namely, that in free space light always is propagated with the speed of light c0, a defined value independent of its direction of propagation and its frequency, and also independent of the state of motion of the emitting body. This second assumption has been verified experimentally Extract of page 27 and leads to counter-intuitive deductions including:
These deductions are logical consequences of the stated assumptions, and are general properties of space-time, typically without regard to a consideration of properties pertaining to the structure of individual objects like atoms or stars, nor to the mechanisms of clocks. These effects are expressed mathematically by the Lorentz transformation x^{\prime} = \gamma \left(x - v t \right) y^{\prime} = y z^{\prime} = z t^{\prime} = \gamma \left(t - \frac{v x}{c_0^{2}}\right) where shifts in origin have been ignored, the relative velocity is assumed to be in the x-direction and the Lorentz factor γ is defined by: \gamma \ \stackrel{\mathrm{def}}{=}\ \frac{1}{\sqrt{1 - (v/c_0)^2}} \ \ge 1. The Lorentz transformation is equivalent to the Galilean transformation in the limit c0 → ∞ (a hypothetical case) or v → 0 (low speeds). Under Lorentz transformations, the time and distance between events may differ among inertial reference frames; however, the Lorentz scalar distance s between two events is the same in all inertial reference frames s^{2} = \left( x_{2} - x_{1} \right)^{2} + \left( y_{2} - y_{1} \right)^{2} + \left( z_{2} - z_{1} \right)^{2} - c_0^{2} \left(t_{2} - t_{1}\right)^{2} From this perspective, the speed of light is only accidentally a property of light, and is rather a property of spacetime, a conversion factor between conventional time units (such as seconds) and length units (such as meters). Incidentally, because of the limitations on speeds faster than the speed of light, notice that in a rotating frame of reference (which is a non-inertial frame, of course) stationarity is not possible at arbitrary distances because at large radius the object would move faster than the speed of light.

General relativity

General relativity is based upon the principle of equivalence:{{Cite book|title=Physics for Scientists and Engineers with Modern Physics |author=Douglas C. Giancoli |url=https://books.google.com/?id=xz-UEdtRmzkC&pg=PA155&dq=%22principle+of+equivalence%22 |page=155 |date=2007 |publisher=Pearson Prentice Hall |isbn=0-13-149508-9 }} This idea was introduced in Einstein's 1907 article "Principle of Relativity and Gravitation" and later developed in 1911.A. Einstein, "On the influence of gravitation on the propagation of light", Annalen der Physik, vol. 35, (1911) : 898-908 Support for this principle is found in the Eötvös experiment, which determines whether the ratio of inertial to gravitational mass is the same for all bodies, regardless of size or composition. To date no difference has been found to a few parts in 1011.{{Cite book|title=Physics Through the Nineteen Nineties: Overview |page=15 |url=https://books.google.com/?id=Hk1wj61PlocC&pg=PA15&dq=equivalence+gravitation |isbn=0-309-03579-1 |date=1986 |author=National Research Council (US) |publisher=National Academies Press }} For some discussion of the subtleties of the Eötvös experiment, such as the local mass distribution around the experimental site (including a quip about the mass of Eötvös himself), see Franklin.{{Cite book|title=No Easy Answers: Science and the Pursuit of Knowledge |author=Allan Franklin |page=66 |url=https://books.google.com/?id=_RN-v31rXuIC&pg=PA66&dq=%22Eotvos+experiment%22 |isbn=0-8229-5968-2 |date=2007 |publisher=University of Pittsburgh Press }} Einstein’s general theory modifies the distinction between nominally "inertial" and "noninertial" effects by replacing special relativity's "flat" Minkowski Space with a metric that produces non-zero curvature. In general relativity, the principle of inertia is replaced with the principle of geodesic motion, whereby objects move in a way dictated by the curvature of spacetime. As a consequence of this curvature, it is not a given in general relativity that inertial objects moving at a particular rate with respect to each other will continue to do so. This phenomenon of geodesic deviation means that inertial frames of reference do not exist globally as they do in Newtonian mechanics and special relativity. 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. Extract of page 154 Extract of page 116 Consequently, modern special relativity is now sometimes described as only a "local theory". Extract of page 329

References

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• Yuri N. Obukhov, Thoralf Chrobok, Mike Scherfner Shear-free rotating inflation Phys. Rev. D 66, 043518 (2002) pages
• Yuri N. Obukhov On physical foundations and observational effects of cosmic rotation (2000)
• Li-Xin Li Effect of the Global Rotation of the Universe on the Formation of Galaxies General Relativity and Gravitation, 30 (1998)
• P Birch Is the Universe rotating? Nature 298, 451 - 454 (29 July 1982)
• Kurt Gödel An example of a new type of cosmological solutions of Einstein’s field equations of gravitation Rev. Mod. Phys., Vol. 21, p. 447, 1949.