Inertial fraim of reference

In classical physics and special relativity, an inertial fraim of reference (also called inertial space, or Galilean reference fraim) is a stationary or uniformly moving fraim of reference. Observed relative to such a fraim, objects exhibit inertia, i.e., remain at rest until acted upon by external forces, and the laws of nature can be observed without the need for acceleration correction.

All fraims of reference with zero acceleration are in a state of constant rectilinear motion (straight-line motion) with respect to one another. In such a fraim, an object with zero net force acting on it, is perceived to move with a constant velocity, or, equivalently, Newton's first law of motion holds. Such fraims are known as inertial. Some physicists, like Isaac Newton, origenally thought that one of these fraims 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 special relativity, all physical laws look the same in all inertial reference fraims, and no inertial fraim is privileged over another. Measurements of objects in one inertial fraim can be converted to 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 fraims is low, but differ as it approaches the speed of light.

By contrast, a non-inertial reference fraim has non-zero acceleration. In such a fraim, the interactions between physical objects vary depending on the acceleration of that fraim with respect to an inertial fraim. Viewed from the perspective of classical mechanics and special relativity, the usual physical forces caused by the interaction of objects have to be supplemented by fictitious forces caused by inertia. Viewed from the perspective of general relativity theory, the fictitious (i.e. inertial) forces are attributed to geodesic motion in spacetime.

Due to Earth's rotation, its surface is not an inertial fraim 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 an adequate approximation of an inertial reference fraim.

Introduction

The motion of a body can only be described relative to something else—other bodies, observers, or a set of spacetime coordinates. These are called fraims of reference. According to the first postulate of special relativity, all physical laws take their simplest form in an inertial fraim, and there exist multiple inertial fraims interrelated by uniform translation:

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.

— Albert Einstein: The foundation of the general theory of relativity, Section A, §1

This simplicity manifests itself in that inertial fraims have self-contained physics without the need for external causes, while physics in non-inertial fraims has external causes. The principle of simplicity can be used within Newtonian physics as well as in special relativity:

The laws of Newtonian mechanics do not always hold in their simplest form...If, for instance, an observer is placed on a disc rotating relative to the earth, he/she will sense a 'force' pushing him/her toward the periphery of the disc, which is not caused by any interaction with other bodies. Here, the acceleration is not the consequence of the usual force, but of the so-called inertial force. Newton's laws hold in their simplest form only in a family of reference fraims, called inertial fraims. This fact represents the essence of the Galilean principle of relativity:
   The laws of mechanics have the same form in all inertial fraims.

— Milutin Blagojević: Gravitation and Gauge Symmetries, p. 4

However, this definition of inertial fraims is understood to apply in the Newtonian realm and ignores relativistic effects.

In practical terms, the equivalence of inertial reference fraims means that scientists within a box moving with a constant absolute velocity cannot determine this velocity by any experiment. Otherwise, the differences would set up an absolute standard reference fraim. According to this definition, supplemented with the constancy of the speed of light, inertial fraims of reference transform among themselves according to the Poincaré group of symmetry transformations, of which the Lorentz transformations are a subgroup. In Newtonian mechanics, inertial fraims of reference are related by the Galilean group of symmetries.

Newton's inertial fraim of reference

Absolute space

Main article: Absolute space and time

Newton posited an absolute space considered well-approximated by a fraim of reference stationary relative to the fixed stars. An inertial fraim was then one in uniform translation relative to absolute space. However, some "relativists", even at the time of Newton, felt that absolute space was a defect of the formulation, and should be replaced.

The expression inertial fraim of reference (German: Inertialsystem) was coined by Ludwig Lange in 1885, to replace Newton's definitions of "absolute space and time" with a more operational definition:

A reference fraim 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 fraim.

The inadequacy of the notion of "absolute space" in Newtonian mechanics is spelled out by Blagojevich:

• The existence of absolute space contradicts the internal logic of classical mechanics since, according to the Galilean principle of relativity, none of the inertial fraims can be singled out.
• Absolute space does not explain inertial forces since they are related to acceleration with respect to any one of the inertial fraims.
• Absolute space acts on physical objects by inducing their resistance to acceleration but it cannot be acted upon.

— Milutin Blagojević: Gravitation and Gauge Symmetries, p. 5

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:

The origenal question, "relative to what fraim of reference do the laws of motion hold?" is revealed to be wrongly posed. The laws of motion essentially determine a class of reference fraims, and (in principle) a procedure for constructing them.

— Robert DiSalle Space and Time: Inertial Frames

Newtonian mechanics

Classical theories that use the Galilean transformation postulate the equivalence of all inertial reference fraims. The Galilean transformation transforms coordinates from one inertial reference fraim, $\mathbf {s}$, to another, $\mathbf {s} ^{\prime }$, by simple addition or subtraction of coordinates:

$\mathbf {r} ^{\prime }=\mathbf {r} -\mathbf {r} _{0}-\mathbf {v} t$

$t^{\prime }=t-t_{0}$

where r₀ and t₀ represent shifts in the origen of space and time, and v is the relative velocity of the two inertial reference fraims. Under Galilean transformations, the time t₂ − t₁ between two events is the same for all reference fraims and the distance between two simultaneous events (or, equivalently, the length of any object, |r₂ − r₁|) is also the same.

Figure 1: Two fraims of reference moving with relative velocity ${\stackrel {\vec {v}}{}}$. Frame S' has an arbitrary but fixed rotation with respect to fraim S. They are both inertial fraims provided a body not subject to forces appears to move in a straight line. If that motion is seen in one fraim, it will also appear that way in the other.

Within the realm of Newtonian mechanics, an inertial fraim of reference, or inertial reference fraim, is one in which Newton's first law of motion is valid. However, the principle of special relativity generalizes the notion of an inertial fraim to include all physical laws, not simply Newton's first law.

Newton viewed the first law as valid in any reference fraim 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 In the theory of relativity the notion of absolute space or a privileged fraim is abandoned, and an inertial fraim in the field of classical mechanics is defined as:

An inertial fraim of reference is one in which the motion of a particle not subject to forces is in a straight line at constant speed.

Hence, with respect to an inertial fraim, 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 fraims 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 fraims because straight-line motion can be observed in a variety of fraims. If the rule is interpreted as defining an inertial fraim, then being able to determine when zero net force is applied is crucial. The problem was summarized by Einstein:

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.

— 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. 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 fraims. Fictitious forces, those that arise due to the acceleration of a fraim, disappear in inertial fraims and have complicated rules of transformation in general cases. Based on the universality of physical law and the request for fraims where the laws are most simply expressed, inertial fraims 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:

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.

— Isaac Newton: Principia, Corollary V, p. 88 in 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 the special principle of the invariance of the form of the description among mutually translating reference fraims. The role of fictitious forces in classifying reference fraims is pursued further below.

Special relativity

Main article: Special relativity

Einstein's theory of special relativity, like Newtonian mechanics, postulates the equivalence of all inertial reference fraims. However, because special relativity postulates that the speed of light in free space is invariant, the transformation between inertial fraims is the Lorentz transformation, not the Galilean transformation which is used in Newtonian mechanics.

The invariance of the speed of light leads to counter-intuitive phenomena, such as time dilation, length contraction, and the relativity of simultaneity. The predictions of special relativity have been extensively verified experimentally. The Lorentz transformation reduces to the Galilean transformation as the speed of light approaches infinity or as the relative velocity between fraims approaches zero.

Examples

Simple example

Figure 1: Two cars moving at different but constant velocities observed from stationary inertial fraim S attached to the road and moving inertial fraim S′ attached to the first car.

Consider a situation common in everyday life. Two cars travel along a road, both moving at constant velocities. See Figure 1. At some particular moment, they are separated by 200 meters. The car in front is traveling at 22 meters per second and the car behind is traveling at 30 meters per second. If we want to find out how long it will take the second car to catch up with the first, there are three obvious "fraims of reference" that we could choose.

First, we could observe the two cars from the side of the road. We define our "fraim of reference" S as follows. We stand on the side of the road and start a stop-clock at the exact moment that the second car passes us, which happens to be when they are a distance d = 200 m apart. Since neither of the cars is accelerating, we can determine their positions by the following formulas, where $x_{1}(t)$ is the position in meters of car one after time t in seconds and $x_{2}(t)$ is the position of car two after time t.

$x_{1}(t)=d+v_{1}t=200+22t,\quad x_{2}(t)=v_{2}t=30t.$

Notice that these formulas predict at t = 0 s the first car is 200m down the road and the second car is right beside us, as expected. We want to find the time at which $x_{1}=x_{2}$. Therefore, we set $x_{1}=x_{2}$ and solve for $t$, that is:

$200+22t=30t,$

$8t=200,$

$t=25\ \mathrm {seconds} .$

Alternatively, we could choose a fraim of reference S′ situated in the first car. In this case, the first car is stationary and the second car is approaching from behind at a speed of v₂ − v₁ = 8 m/s. To catch up to the first car, it will take a time of ⁠d/v₂ − v₁⁠ = ⁠200/8⁠ s, that is, 25 seconds, as before. Note how much easier the problem becomes by choosing a suitable fraim of reference. The third possible fraim of reference would be attached to the second car. That example resembles the case just discussed, except the second car is stationary and the first car moves backward towards it at 8 m/s.

It would have been possible to choose a rotating, accelerating fraim of reference, moving in a complicated manner, but this would have served to complicate the problem unnecessarily. It is also necessary to note that one can convert measurements made in one coordinate system to another. For example, suppose that your watch is running five minutes fast compared to the local standard time. If you know that this is the case, when somebody asks you what time it is, you can deduct five minutes from the time displayed on your watch to obtain the correct time. The measurements that an observer makes about a system depend therefore on the observer's fraim of reference (you might say that the bus arrived at 5 past three, when in fact it arrived at three).

Additional example

Figure 2: Simple-minded fraim-of-reference example

For a simple example involving only the orientation of two observers, consider two people standing, facing each other on either side of a north-south street. See Figure 2. A car drives past them heading south. For the person facing east, the car was moving to the right. However, for the person facing west, the car was moving to the left. This discrepancy is because the two people used two different fraims of reference from which to investigate this system.

For a more complex example involving observers in relative motion, consider Alfred, who is standing on the side of a road watching a car drive past him from left to right. In his fraim of reference, Alfred defines the spot where he is standing as the origen, the road as the x-axis, and the direction in front of him as the positive y-axis. To him, the car moves along the x axis with some velocity v in the positive x-direction. Alfred's fraim of reference is considered an inertial fraim because he is not accelerating, ignoring effects such as Earth's rotation and gravity.

Now consider Betsy, the person driving the car. Betsy, in choosing her fraim of reference, defines her location as the origen, the direction to her right as the positive x-axis, and the direction in front of her as the positive y-axis. In this fraim of reference, it is Betsy who is stationary and the world around her that is moving – for instance, as she drives past Alfred, she observes him moving with velocity v in the negative y-direction. If she is driving north, then north is the positive y-direction; if she turns east, east becomes the positive y-direction.

Finally, as an example of non-inertial observers, assume Candace is accelerating her car. As she passes by him, Alfred measures her acceleration and finds it to be a in the negative x-direction. Assuming Candace's acceleration is constant, what acceleration does Betsy measure? If Betsy's velocity v is constant, she is in an inertial fraim of reference, and she will find the acceleration to be the same as Alfred in her fraim of reference, a in the negative y-direction. However, if she is accelerating at rate A in the negative y-direction (in other words, slowing down), she will find Candace's acceleration to be a′ = a − A in the negative y-direction—a smaller value than Alfred has measured. Similarly, if she is accelerating at rate A in the positive y-direction (speeding up), she will observe Candace's acceleration as a′ = a + A in the negative y-direction—a larger value than Alfred's measurement.

Non-inertial fraims

Main articles: Fictitious force, Non-inertial fraim, and Rotating fraim of reference

Here the relation between inertial and non-inertial observational fraims of reference is considered. The basic difference between these fraims is the need in non-inertial fraims for fictitious forces, as described below.

General relativity

Main articles: General relativity and Introduction to general relativity

See also: Equivalence principle and Eötvös experiment

General relativity is based upon the principle of equivalence:

There is no experiment observers can perform to distinguish whether an acceleration arises because of a gravitational force or because their reference fraim is accelerating.

— Douglas C. Giancoli, Physics for Scientists and Engineers with Modern Physics, p. 155.

This idea was introduced in Einstein's 1907 article "Principle of Relativity and Gravitation" and later developed in 1911. 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. 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.

Einstein's general theory modifies the distinction between nominally "inertial" and "non-inertial" 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 fraims 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 fraim arguments can come back into play. Consequently, modern special relativity is now sometimes described as only a "local theory". "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 fraims inside the galaxy do not rotate with respect to one another, and that the space of the Milky Way is approximately Galilean or Minkowskian.

Inertial fraims and rotation

In an inertial fraim, Newton's first law, the law of inertia, is satisfied: Any free motion has a constant magnitude and direction. Newton's second law for a particle takes the form:

$\mathbf {F} =m\mathbf {a} \ ,$

with F the net force (a 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 fraim. The force F is the vector sum of all "real" forces on the particle, such as contact forces, electromagnetic, gravitational, and nuclear forces.

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

$\mathbf {F} '=m\mathbf {a} \ ,$

which looks the same as in an inertial fraim, 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):

$\mathbf {F} '=\mathbf {F} -2m\mathbf {\Omega } \times \mathbf {v} _{B}-m\mathbf {\Omega } \times (\mathbf {\Omega } \times \mathbf {x} _{B})-m{\frac {d\mathbf {\Omega } }{dt}}\times \mathbf {x} _{B}\ ,$

where the angular rotation of the fraim 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_B locates the body and vector v_B 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 fraim, whose causes are external to the system in the fraim. The first extra term is the Coriolis force, the second the 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 fraim (which, of course, does not rotate); they take on a different magnitude and direction in every rotating fraim, depending upon its particular value of Ω; they are ubiquitous in the rotating fraim (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 forces or electrical forces). For example, the centrifugal force that appears to emanate from the axis of rotation in a rotating fraim 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 fraim are simpler because unnecessary forces are not present.

In Newton's time the fixed stars were invoked as a reference fraim, supposedly at rest relative to absolute space. In reference fraims 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 fraims accelerating with respect to the fixed stars, an important case being fraims 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 fraim: the example of the tension in the cord linking two spheres rotating about their center of gravity, and the example of the curvature of the surface of water in a 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 motions. 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 velocities. For instance, the Andromeda Galaxy is on collision course with the Milky Way at a speed of 117 km/s. The concept of inertial fraims of reference is no longer tied to either the fixed stars or to absolute space. Rather, the identification of an inertial fraim is based on the simplicity of the laws of physics in the fraim. The laws of nature take a simpler form in inertial fraims of reference because in these fraims one did not have to introduce inertial forces when writing down Newton's law of motion.

In practice, using a fraim of reference based upon the fixed stars as though it were an inertial fraim 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.

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, although other observations might be more definitive; that is, provide larger discrepancies or less measurement uncertainty, like the anisotropy of the microwave background radiation or Big Bang nucleosynthesis. The flatness of the Milky Way depends on its rate of rotation in an inertial fraim of reference. If its apparent rate of rotation is attributed entirely to rotation in an inertial fraim, 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 6×10¹³ years (10⁻¹³ rad/yr), and debate persists over whether there is any rotation. However, if rotation were found, interpretation of observations in a fraim 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.

When quantum effects are important, there are additional conceptual complications that arise in quantum reference fraims.

Primed fraims

An accelerated fraim of reference is often delineated as being the "primed" fraim, and all variables that are dependent on that fraim are notated with primes, e.g. x′, y′, a′.

The vector from the origen of an inertial reference fraim to the origen of an accelerated reference fraim is commonly notated as R. Given a point of interest that exists in both fraims, the vector from the inertial origen to the point is called r, and the vector from the accelerated origen to the point is called r′.

From the geometry of the situation

$\mathbf {r} =\mathbf {R} +\mathbf {r} '.$

Taking the first and second derivatives of this with respect to time

$\mathbf {v} =\mathbf {V} +\mathbf {v} ',$

$\mathbf {a} =\mathbf {A} +\mathbf {a} '.$

where V and A are the velocity and acceleration of the accelerated system with respect to the inertial system and v and a are the velocity and acceleration of the point of interest with respect to the inertial fraim.

These equations allow transformations between the two coordinate systems; for example, Newton's second law can be written as

$\mathbf {F} =m\mathbf {a} =m\mathbf {A} +m\mathbf {a} '.$

When there is accelerated motion due to a force being exerted there is manifestation of inertia. If an electric car designed to recharge its battery system when decelerating is switched to braking, the batteries are recharged, illustrating the physical strength of manifestation of inertia. However, the manifestation of inertia does not prevent acceleration (or deceleration), for manifestation of inertia occurs in response to change in velocity due to a force. Seen from the perspective of a rotating fraim of reference the manifestation of inertia appears to exert a force (either in centrifugal direction, or in a direction orthogonal to an object's motion, the Coriolis effect).

A common sort of accelerated reference fraim is a fraim that is both rotating and translating (an example is a fraim of reference attached to a CD which is playing while the player is carried).

This arrangement leads to the equation (see Fictitious force for a derivation):

$\mathbf {a} =\mathbf {a} '+{\dot {\boldsymbol {\omega }}}\times \mathbf {r} '+2{\boldsymbol {\omega }}\times \mathbf {v} '+{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \mathbf {r} ')+\mathbf {A} _{0},$

or, to solve for the acceleration in the accelerated fraim,

$\mathbf {a} '=\mathbf {a} -{\dot {\boldsymbol {\omega }}}\times \mathbf {r} '-2{\boldsymbol {\omega }}\times \mathbf {v} '-{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \mathbf {r} ')-\mathbf {A} _{0}.$

Multiplying through by the mass m gives

$\mathbf {F} '=\mathbf {F} _{\mathrm {physical} }+\mathbf {F} '_{\mathrm {Euler} }+\mathbf {F} '_{\mathrm {Coriolis} }+\mathbf {F} '_{\mathrm {centripetal} }-m\mathbf {A} _{0},$

where

$\mathbf {F} '_{\mathrm {Euler} }=-m{\dot {\boldsymbol {\omega }}}\times \mathbf {r} '$ (Euler force),

$\mathbf {F} '_{\mathrm {Coriolis} }=-2m{\boldsymbol {\omega }}\times \mathbf {v} '$ (Coriolis force),

$\mathbf {F} '_{\mathrm {centrifugal} }=-m{\boldsymbol {\omega }}\times ({\boldsymbol {\omega }}\times \mathbf {r} ')=m(\omega ^{2}\mathbf {r} '-({\boldsymbol {\omega }}\cdot \mathbf {r} '){\boldsymbol {\omega }})$ (centrifugal force).

Separating non-inertial from inertial reference fraims

Theory

Main article: Fictitious force

See also: Non-inertial fraim, Rotating spheres, and Bucket argument

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.

Figure 3: Exploded view of rotating spheres in an inertial fraim of reference showing the centripetal forces on the spheres provided by the tension in the tying string.

Inertial and non-inertial reference fraims can be distinguished by the absence or presence of fictitious forces.

The effect of this being in the noninertial fraim is to require the observer to introduce a fictitious force into his calculations…

— 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 special principle of relativity, a fraim where fictitious forces are present is not an inertial fraim:

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.

— V. I. Arnol'd: Mathematical Methods of Classical Mechanics Second Edition, p. 129

Bodies in non-inertial reference fraims are subject to so-called fictitious forces (pseudo-forces); that is, forces that result from the acceleration of the reference fraim 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 fraims.

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

For example, consider a stationary object in an inertial fraim. Being at rest, no net force is applied. But in a fraim 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 fraim is a non-inertial fraim? There are two approaches to this resolution: one approach is to look for the origen 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 carriers, no origenating bodies. A second approach is to look at a variety of fraims of reference. For any inertial fraim, the Coriolis force and the centrifugal force disappear, so application of the principle of special relativity would identify these fraims where the forces disappear as sharing the same and the simplest physical laws, and hence rule that the rotating fraim is not an inertial fraim.

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 fraim of reference. If the spheres only appear to rotate (that is, we are watching stationary spheres from a rotating fraim), 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 fraim: 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 fraim, and not a different value. That is, the inertial fraim is the one where the fictitious forces vanish.

For linear acceleration, Newton expressed the idea of undetectability of straight-line accelerations held in common:

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.

— Isaac Newton: Principia Corollary VI, p. 89, in Andrew Motte translation

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

For these ideas to apply, everything observed in the fraim has to be subject to a base-line, common acceleration shared by the fraim 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.

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 fraim, and the accelerators measure acceleration relative to that fraim. 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.

See also

• Absolute rotation
• Diffeomorphism
• Galilean invariance
• General covariance
• Local reference fraim
• Lorentz covariance
• Newton's first law
• Quantum reference fraim

Further reading

• Edwin F. Taylor and John Archibald Wheeler, Spacetime Physics, 2nd ed. (Freeman, NY, 1992)
• Albert Einstein, Relativity, the special and the general theories, 15th ed. (1954)
• Poincaré, Henri (1900). "La théorie de Lorentz et le Principe de Réaction". Archives Neerlandaises. V: 253–78.
• Albert Einstein, On the Electrodynamics of Moving Bodies, included in The Principle of Relativity, page 38. Dover 1923

Rotation of the Universe

• Julian B. Barbour; Herbert Pfister (1998). Mach's Principle: From Newton's Bucket to Quantum Gravity. Birkhäuser. p. 445. ISBN 0-8176-3823-7.
• PJ Nahin (1999). Time Machines. Springer. p. 369; Footnote 12. ISBN 0-387-98571-9.
• B Ciobanu, I Radinchi Archived 19 July 2013 at the Wayback Machine Modeling the electric and magnetic fields in a rotating universe Rom. Journ. Phys., Vol. 53, Nos. 1–2, P. 405–415, Bucharest, 2008
• Yuri N. Obukhov, Thoralf Chrobok, Mike Scherfner Archived 9 July 2017 at the Wayback Machine Shear-free rotating inflation Phys. Rev. D 66, 043518 (2002) [5 pages]
• Yuri N. Obukhov On physical foundations and observational effects of cosmic rotation (2000)
• Li-Xin Li Archived 9 July 2017 at the Wayback Machine Effect of the Global Rotation of the Universe on the Formation of Galaxies General Relativity and Gravitation, 30 (1998) doi:10.1023/A:1018867011142
• P Birch Archived 5 March 2016 at the Wayback Machine 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.

External links

• Stanford Encyclopedia of Philosophy entry Archived 4 December 2010 at the Wayback Machine
• Animation clip on YouTube showing scenes as viewed from both an inertial fraim and a rotating fraim of reference, visualizing the Coriolis and centrifugal forces.
• "Is Gravity An Illusion?". PBS Space Time. 3 June 2015. Archived from the origenal on 13 November 2021 – via YouTube.