Misplaced Pages

#372627

133-501: A gimbal mounted mechanical gyroscope remains pointing in the same direction after spinning up, and thus can be used as a rotational reference for an inertial navigation system . With the development of so-called laser gyroscopes and fiber optic gyroscopes based on the Sagnac effect, bulky mechanical gyroscopes can be replaced by those with no moving parts in many modern inertial navigation systems. A conventional gyroscope relies on

266-406: A Cardan suspension after Italian mathematician and physicist Gerolamo Cardano (1501–1576) who described it in detail. However, Cardano did not invent the gimbal, nor did he claim to. The device has been known since antiquity, first described in the 3rd c. BC by Philo of Byzantium , although some modern authors support the view that it may not have a single identifiable inventor. The gimbal

399-445: 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 can be observed without the need for acceleration correction. 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

532-400: 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. Here the relation between inertial and non-inertial observational frames of reference

665-573: A "valid" reference frame), and in a footnote he wrote "a system which rotates in respect to a valid system K 0 {\displaystyle K^{0}} is not valid". Assuming constant light speed c {\displaystyle c} , and setting the rotational velocity as ω {\displaystyle \omega } , he computed the propagation time τ + {\displaystyle \tau _{+}} of one ray and τ − {\displaystyle \tau _{-}} of

798-399: A century long debate on its meaning and interpretation, much of this debate being surprising since the effect is perfectly well understood in the context of special relativity. The shift in interference fringes in a ring interferometer can be viewed intuitively as a consequence of the different distances that light travels due to the rotation of the ring.(Fig. 3) The simplest derivation

931-441: 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. 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

1064-722: A closed optical path. They differ considerably in various cost, reliability, size, weight, power, and other performance characteristics that need to be considered when evaluating these distinct technologies for a particular application. RLGs require accurate machining, use of precision mirrors, and assembly under clean room conditions. Their mechanical dithering assemblies add somewhat to their weight but not appreciably. RLGs are capable of logging in excess of 100,000 hours of operation in near-room temperature conditions. Their lasers have relatively high power requirements. Interferometric FOGs are purely solid-state, require no mechanical dithering components, do not require precision machining, have

1197-415: A counterpart to Sagnac effect physics. In the actual Hafele–Keating experiment the mode of transport (long-distance flights) gave rise to time dilation effects of its own, and calculations were needed to separate the various contributions. For the (theoretical) case of clocks that are transported so slowly that time dilation effects arising from the transport are negligible the amount of time difference between

1330-437: A flexible geometry, and can be made very small. They use many standard components from the telecom industry. In addition, the major optical components of FOGs have proven performance in the telecom industry, with lifespans measured in decades. However, the assembly of multiple optical components into a precision gyro instrument is costly. Analog FOGs offer the lowest possible cost but are limited in performance; digital FOGs offer

1463-482: A footnote regarding discussions with German physicist, Wilhelm Wien . The reason for looking at General Relativity is because Einstein's Theory of General Relativity predicted that light would slow down in a gravitational field which is why it could predict the curvature of light around a massive body. Under General Relativity, there is the equivalence principle which states that gravity and acceleration are equivalent. Spinning or accelerating an interferometer creates

SECTION 10

#1732772630373

1596-407: A frame 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 v 2 − v 1 = 8 m/s . To catch up to the first car, it will take a time of ⁠ d / v 2 − v 1 ⁠ = ⁠ 200 / 8 ⁠ s , that is, 25 seconds, as before. Note how much easier the problem becomes by choosing

1729-466: 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 "relativists", even at the time of Newton, felt that absolute space was a defect of the formulation, and should be replaced. The expression inertial frame of reference ( German : Inertialsystem ) was coined by Ludwig Lange in 1885, to replace Newton's definitions of "absolute space and time" with

1862-456: A frame, 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 frames are known as inertial. Some physicists, like Isaac Newton , originally thought that one of these 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

1995-452: A giant ring interferometer be constructed to measure the rotation of the Earth; a similar suggestion was made by Albert Abraham Michelson in 1904. They hoped that with such an interferometer, it would be possible to decide between a stationary aether, versus aethers which are partially or completely dragged by the Earth. That is, if the hypothetical aether were carried along by the Earth (or by

2128-516: A gravitational effect. "There are, however, two different types of such [non-inertial] motion; it may for instance be acceleration in a straight line, or circular motion with constant speed." Also, Irwin Shapiro in 1964 explained General Relativity saying, "the speed of a light wave depends on the strength of the gravitational potential along its path". This is called the Shapiro delay . However, since

2261-719: A lens around its center of gravity , thus allowing for easy and smooth manipulation while tracking moving subjects. Very large gimbal mounts in the form 2 or 3 axis altitude-altitude mounts are used in satellite photography for tracking purposes. Gyrostabilized gimbals which house multiple sensors are also used for airborne surveillance applications including airborne law enforcement, pipe and power line inspection, mapping , and ISR ( intelligence, surveillance, and reconnaissance ). Sensors include thermal imaging , daylight, low light cameras as well as laser range finder , and illuminators . Gimbal systems are also used in scientific optics equipment. For example, they are used to rotate

2394-401: A loop of an optical fiber, see Figure 4. The loop may have an arbitrary shape, and can move arbitrarily in space. The only restriction is that it is not allowed to stretch. (The case of a circular ring interferometer rotating about its center in free space is recovered by taking the index of refraction of the fiber to be 1.) Consider a small segment of the fiber, whose length in its rest frame

2527-414: A material sample along an axis to study their angular dependence of optical properties. Handheld 3-axis gimbals are used in stabilization systems designed to give the camera operator the independence of handheld shooting without camera vibration or shake. There are two versions of such stabilization systems: mechanical and motorized. Mechanical gimbals have the sled, which includes the top stage where

2660-414: A measure of difference in arrival time is obtained by producing interference fringes, and observing the fringe shift. In the case of a relay of pulses around the world the difference in arrival time is obtained directly from the actual arrival time of the pulses. In both cases the mechanism of the difference in arrival time is the same: the Sagnac effect. The Hafele–Keating experiment is also recognized as

2793-468: A more operational definition : 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. The inadequacy of the notion of "absolute space" in Newtonian mechanics is spelled out by Blagojevich: The utility of operational definitions was carried much further in

SECTION 20

#1732772630373

2926-490: 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. Consequently, modern special relativity

3059-476: 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

3192-470: A portable warming stove to Empress Wu Zetian (r. 690–705) which employed gimbals. Extant specimens of Chinese gimbals used for incense burners date to the early Tang dynasty (618–907), and were part of the silver -smithing tradition in China. The authenticity of Philo's description of a cardan suspension has been doubted by some authors on the ground that the part of Philo's Pneumatica which describes

3325-428: A rapid rate (hundreds of hertz ), lock-in will only occur during the brief instances where the rotational velocity is close to zero; the errors thereby induced approximately cancel each other between alternating dead periods. Fibre optic gyros (FOGs) and ring laser gyros (RLGs) both operate by monitoring the difference in propagation time between beams of light traveling in clockwise and counterclockwise directions about

3458-504: A ring laser cavity, it is helpful to discuss the physics of the laser process in a laser setup with continuous generation of light. As the laser excitation is started, the molecules inside the cavity emit photons, but since the molecules have a thermal velocity, the light inside the laser cavity is at first a range of frequencies, corresponding to the statistical distribution of velocities. The process of stimulated emission makes one frequency quickly outcompete other frequencies, and after that

3591-403: A suitable frame of reference. The third possible frame 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 frame of reference, moving in a complicated manner, but this would have served to complicate

3724-412: A system depend therefore on the observer's frame of reference (you might say that the bus arrived at 5 past three, when in fact it arrived at three). 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

3857-628: 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. This simplicity manifests itself in that inertial frames have self-contained physics without the need for external causes, while physics in non-inertial frames 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

3990-418: A view recently also shared by the classicist Andrew Wilson (2002). The ancient Roman author Athenaeus Mechanicus , writing during the reign of Augustus (30 BC–14 AD), described the military use of a gimbal-like mechanism, calling it "little ape" ( pithêkion ). When preparing to attack coastal towns from the sea-side, military engineers used to yoke merchant-ships together to take the siege machines up to

4123-601: Is d ℓ ′ {\displaystyle d\ell '} . The time intervals, d t ± ′ {\displaystyle dt'_{\pm }} , it takes the left and right moving light rays to traverse the segment in the rest frame coincide and are given by d t ± ′ = n c d ℓ ′ {\displaystyle dt'_{\pm }={n \over c}d\ell '} Let d ℓ = | d x | {\textstyle d\ell =|d\mathbf {x} |} be

Sagnac effect - Misplaced Pages Continue

4256-503: Is invariant , the transformation between inertial frames 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

4389-403: Is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body". Einstein specifically stated that light speed is only constant in the vacuum of empty space, using equations that only held in linear and parallel inertial frames. However, when Einstein started to investigate accelerated reference frames, he noticed that "the principle of

4522-418: 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: The weakness of the principle of inertia lies in this, that it involves an argument in

4655-587: Is considered. The basic difference between these frames is the need in non-inertial frames for fictitious forces, as described below. 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 frame is accelerating. This idea was introduced in Einstein's 1907 article "Principle of Relativity and Gravitation" and later developed in 1911. Support for this principle

4788-418: Is constant, what acceleration does Betsy measure? If Betsy's velocity v is constant, she is in an inertial frame of reference, and she will find the acceleration to be the same as Alfred in her frame 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′ =

4921-425: Is for a circular ring of radius R, with a refractive index of one, rotating at an angular velocity of ω {\displaystyle \omega } , but the result is general for loop geometries with other shapes. If a light source emits in both directions from one point on the rotating ring, light traveling in the same direction as the rotation direction needs to travel more than one circumference around

5054-570: 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 10 . 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

5187-434: Is invalid, however, if the light source's path in space does not follow that of the light signals, for example in the standard rotating platform case (FOG) but with a non-circular light path. In this case the phase difference formula necessarily involves the area enclosed by the light path due to Stokes' theorem . Consider a ring interferometer where two counter-propagating light beams share a common optical path determined by

5320-428: Is misleading: no gimbal is restrained. All three gimbals can still rotate freely about their respective axes of suspension. Nevertheless, because of the parallel orientation of two of the gimbals' axes there is no gimbal available to accommodate rotation about one axis. Inertial reference frame In classical physics and special relativity , an inertial frame of reference (also called an inertial space or

5453-511: 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

Sagnac effect - Misplaced Pages Continue

5586-403: Is one in which Newton's first law of motion is valid. However, the 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"

5719-415: 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 frames, called inertial frames. This fact represents

5852-443: 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

5985-587: Is safe to assume the Arabic version is a faithful copying of Philo's original, and credits Philon explicitly with the invention. So does his colleague Michael Lewis (2001). In fact, research by the latter scholar (1997) demonstrates that the Arab copy contains sequences of Greek letters which fell out of use after the 1st century, thereby strengthening the case that it is a faithful copy of the Hellenistic original,

6118-449: Is sensitive to its orientation. Because of this, chronometers were normally mounted on gimbals, in order to isolate them from the rocking motions of a ship at sea. Gimbal lock is the loss of one degree of freedom in a three-dimensional, three-gimbal mechanism that occurs when the axes of two of the three gimbals are driven into a parallel configuration, "locking" the system into rotation in a degenerate two-dimensional space. The word lock

6251-407: Is the area of the ring. Although this simple derivation is for a circular ring with an index of refraction of one, the result holds true for any shape of rotating loop with area  A .(Fig. 4) For more complicated shapes, or other refractive index values, the same result can be derived by calculating the optical phase shift in each direction using Fermat's principle and taking into account

6384-438: Is the oriented area of the loop and λ {\displaystyle \lambda } the wavelength of light. The effect is a consequence of the different times it takes right and left moving light beams to complete a full round trip in the interferometer ring. The difference in travel times, when multiplied by the optical frequency c / λ {\displaystyle c/\lambda } , determines

6517-451: Is the principle of the ring laser gyroscope , widely used in modern inertial navigation systems . In passive ring interferometers, the fringe displacement is proportional to the first derivative of angular position; careful calibration is required to determine the fringe displacement that corresponds to zero angular velocity of the ring interferometer setup. On the other hand, ring laser interferometers do not require calibration to determine

6650-418: Is well understood in the context of special relativity where from the rotating light source's point of view the phase difference is due to the line of simultaneity along the light path not forming a closed loop in spacetime. Modified versions of the experiment have been proposed with the light source allowed to move along a (not necessarily circular) light path. This configuration introduces another reason for

6783-457: The Fizeau interference formula. A relay of pulses that circumnavigates the Earth, verifying precise synchronization, is also recognized as a case requiring correction for the Sagnac effect. In 1984 a verification was set up that involved three ground stations and several GPS satellites, with relays of signals both going eastward and westward around the world. In the case of a Sagnac interferometer

SECTION 50

#1732772630373

6916-442: The Galilean transformation postulate the equivalence of all inertial reference frames. The Galilean transformation transforms coordinates from one inertial reference frame, s {\displaystyle \mathbf {s} } , to another, s ′ {\displaystyle \mathbf {s} ^{\prime }} , by simple addition or subtraction of coordinates: where r 0 and t 0 represent shifts in

7049-502: The Little Key of Painting' ( mappae clavicula ). The French inventor Villard de Honnecourt depicts a set of gimbals in his sketchbook (see right). In the early modern period, dry compasses were suspended in gimbals. In inertial navigation, as applied to ships and submarines, a minimum of three gimbals are needed to allow an inertial navigation system (stable table) to remain fixed in inertial space, compensating for changes in

7182-601: 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 frame. 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 frames of reference . 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 : Special principle of relativity: If

7315-504: The principle of special relativity , all physical laws look the same in all inertial reference frames, and no inertial frame is privileged over another. Measurements of objects in one inertial frame 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

7448-403: The sidereal day , which can also be termed "mode 1". Global navigation satellite systems (GNSSs), such as GPS , GLONASS , COMPASS or Galileo , need to take the rotation of the Earth into account in the procedures of using radio signals to synchronize clocks. Fibre optic gyroscopes are sometimes referred to as 'passive ring interferometers'. A passive ring interferometer uses light entering

7581-531: The Galilean transformation as the speed of light approaches infinity or as the relative velocity between frames approaches zero. 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

7714-404: The Sagnac effect is a simple consequence of the source independence of the speed of light. In other words, the Sagnac experiment does not distinguish between pre-relativistic physics and relativistic physics. When light propagates in fibre optic cable, the setup is effectively a combination of a Sagnac experiment and the Fizeau experiment . In glass the speed of light is slower than in vacuum, and

7847-669: The Sagnac effect, Harress had realized the presence of an "unexpected bias" in his measurements, but was unable to explain its cause. Harress' analysis of the results contained an error, and they were reanalyzed in 1914 by Paul Harzer , who claimed the results were at odds with special relativity. This was rebutted by Einstein. Harress himself died during the First World War, and his results were not publicly available until von Laue persuaded Otto Knopf, whose assistant Harress had been, to publish them in 1920. Harress' results were published together with an analysis by von Laue, who showed

7980-465: 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. 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

8113-487: The camera is attached, the post which in most models can be extended, with the monitor and batteries at the bottom to counterbalance the camera weight. This is how the Steadicam stays upright, by simply making the bottom slightly heavier than the top, pivoting at the gimbal. This leaves the center of gravity of the whole rig, however heavy it may be, exactly at the operator's fingertip, allowing deft and finite control of

SECTION 60

#1732772630373

8246-476: The camera to seem as if it is floating through the air, an effect achieved by a Steadicam in the past. Gimbals can be mounted to cars and other vehicles such as drones , where vibrations or other unexpected movements would make tripods or other camera mounts unacceptable. An example which is popular in the live TV broadcast industry, is the Newton 3-axis camera gimbal . The rate of a mechanical marine chronometer

8379-413: The cars is accelerating, we can determine their positions by the following formulas, where x 1 ( t ) {\displaystyle x_{1}(t)} is the position in meters of car one after time t in seconds and x 2 ( t ) {\displaystyle x_{2}(t)} is the position of car two after time t . Notice that these formulas predict at t = 0 s

8512-419: The case of a ring laser the same applies: the number of cycles of the laser light's frequency is the same in both directions. This quality of the same number of cycles in both directions is preserved when the ring laser setup is rotating. The image illustrates that there is wavelength shift (hence a frequency shift) in such a way that the number of cycles is the same in both directions of propagation. By bringing

8645-459: The clocks when they arrive back at the starting point will be equal to the time difference that is found for a relay of pulses that travels around the world: 207 nanoseconds. The Sagnac effect is employed in current technology. One use is in inertial guidance systems . Ring laser gyroscopes are extremely sensitive to rotations, which need to be accounted for if an inertial guidance system is to return accurate results. The ring laser also can detect

8778-404: The constancy of light must be modified" for accelerating frames of reference. Max von Laue in his 1920 paper gave serious consideration to the effect of General Relativity on the Sagnac effect stating, "General relativity would of course be capable of giving some statements about it, and we want to show at first that no noticeable influences of acceleration are expected according to it." He makes

8911-428: The counter-propagating ray, and consequently obtained the time difference Δ τ = τ + − τ − {\displaystyle \Delta \tau =\tau _{+}-\tau _{-}} . He concluded that this interferometer experiment would indeed produce (when restricted to terms of first order in v / c {\displaystyle v/c} )

9044-422: The counter-rotating beam and away from the co-rotating beam. Consequently the beams will reach the detector at slightly different times, and slightly out of phase, producing optical interference 'fringes' that can be observed and measured." In 1926, an ambitious ring interferometry experiment was set up by Albert Michelson and Henry Gale . The aim was to find out whether the rotation of the Earth has an effect on

9177-553: 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, inertial frames of reference are related by the Galilean group of symmetries. Newton posited an absolute space considered well-approximated by

9310-435: The different phase velocities for the different propagation directions in an inertial laboratory frame, which can be calculated using relativistic addition of velocities. We imagine a screen for viewing fringes placed at the light source (or we use a beamsplitter to send light from the source point to the screen). Given a steady light source, interference fringes will form on the screen with a fringe displacement proportional to

9443-457: 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

9576-472: The essence of the Galilean principle of relativity:    The laws of mechanics have the same form in all inertial frames. However, this definition of inertial frames is understood to apply in the Newtonian realm and ignores relativistic effects. In practical terms, the equivalence of inertial reference frames means that scientists within a box moving with a constant absolute velocity cannot determine this velocity by any experiment. Otherwise,

9709-427: 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 {\displaystyle x_{1}=x_{2}} . Therefore, we set x 1 = x 2 {\displaystyle x_{1}=x_{2}} and solve for t {\displaystyle t} , that is: Alternatively, we could choose

9842-510: The first factor is the frequency of light. This gives the generalized Sagnac formula Δ ϕ ≈ 4 π λ c ∮ v ⋅ d x {\displaystyle \Delta \phi \approx {\frac {4\pi }{\lambda c}}\oint \mathbf {v} \cdot d\mathbf {x} } In the special case that the fiber moves like a rigid body with angular frequency ω {\displaystyle {\boldsymbol {\omega }}} ,

9975-415: The frequencies of the counter-propagating laser modes become almost identical. In this case, crosstalk between the counter-propagating beams can result in injection locking , so that the standing wave "gets stuck" in a preferred phase, locking the frequency of each beam to each other rather than responding to gradual rotation. By rotationally dithering the laser cavity back and forth through a small angle at

10108-433: The gravitational field would have to be significant, Laue (1920) concluded it is more likely that the effect is a result of changing the distance of the path by its movement through space. "The beam traveling around the loop in the direction of rotation will have farther to go than the beam traveling counter to the direction of rotation, because during the period of travel the mirrors and detector will all move (slightly) toward

10241-412: The innermost gimbal to remain independent of the rotation of its support (e.g. vertical in the first animation). For example, on a ship, the gyroscopes , shipboard compasses , stoves , and even drink holders typically use gimbals to keep them upright with respect to the horizon despite the ship's pitching and rolling . The gimbal suspension used for mounting compasses and the like is sometimes called

10374-409: 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 frame of reference. The Coriolis effect can deflect certain forms of motion as seen from Earth , and

10507-528: The interference fringes is proportional to the platform's angular frequency ω {\displaystyle {\boldsymbol {\omega }}} and is given by a formula originally derived by Sagnac: Δ ϕ ≈ 8 π λ c ω ⋅ A {\displaystyle \Delta \phi \approx {\frac {8\pi }{\lambda c}}{\boldsymbol {\omega }}\cdot \mathbf {A} } where A {\displaystyle \mathbf {A} }

10640-410: The interferometer) the result would be negative, while a stationary aether would give a positive result. The first description of the Sagnac effect in the framework of special relativity was done by Max von Laue in 1911, two years before Sagnac conducted his experiment. By continuing the theoretical work of Michelson (1904), von Laue confined himself to an inertial frame of reference (which he called

10773-417: The left, and a modified fibre optic conveyor, shown on the right, conform to the equation Δ t  = 2 vL / c , whose derivation is based on the constant speed of light. It is evident from this formula that the total time delay is equal to the cumulative time delays along the entire length of fibre, regardless whether the fibre is in a rotating section of the conveyor, or a straight section. This equation

10906-504: The length of this small segment in the lab frame. By the relativistic length contraction formula, d ℓ ′ = γ d ℓ ≈ d ℓ {\textstyle d\ell '=\gamma d\ell \approx d\ell } correct to first order in the velocity v {\displaystyle \mathbf {v} } of the segment. The time intervals d t ± {\displaystyle dt_{\pm }} for traversing

11039-435: The light is very close to monochromatic. For the sake of simplicity, assume that all emitted photons are emitted in a direction parallel to the ring. Fig. 7 illustrates the effect of the ring laser's rotation. In a linear laser, an integer multiple of the wavelength fits the length of the laser cavity. This means that in traveling back and forth the laser light goes through an integer number of cycles of its frequency. In

11172-401: The light through a closed path (Fig. 2). If the platform on which the ring interferometer is mounted is rotating, the interference fringes are displaced compared to their position when the platform is not rotating. The amount of displacement is proportional to the angular velocity of the rotating platform. The axis of rotation does not have to be inside the enclosed area. The phase shift of

11305-399: 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

11438-464: The optical cable is the moving medium. In that case the relativistic velocity addition rule applies. Pre-relativistic theories of light propagation cannot account for the Fizeau effect. (By 1900 Lorentz could account for the Fizeau effect, but by that time his theory had evolved to a form where in effect it was mathematically equivalent to special relativity.) Since emitter and detector are traveling at

11571-431: The orientation, an effect which was shown by Langevin in another paper (1937). This does not contradict special relativity and the above explanation by von Laue that the speed of light is not affected by accelerations. Because this apparent variable light speed in rotating frames only arises if rotating coordinates are used, whereas if the Sagnac effect is described from the viewpoint of an external inertial coordinate frame

11704-460: The origin of space and time, and v is the relative velocity of the two inertial reference frames. Under Galilean transformations, the time t 2 − t 1 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 2 − r 1 |) is also the same. Within the realm of Newtonian mechanics, an inertial frame of reference, or inertial reference frame,

11837-426: The output that corresponds to zero angular velocity. Ring laser interferometers are self-calibrating. The beat frequency will be zero if and only if the ring laser setup is non-rotating with respect to inertial space. Fig. 8 illustrates the physical property that makes the ring laser interferometer self-calibrating. The grey dots represent molecules in the laser cavity that act as resonators. Along every section of

11970-411: The person driving the car. Betsy, in choosing her frame of reference, defines her location as the origin, the direction to her right as the positive x -axis, and the direction in front of her as the positive y -axis. In this frame 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

12103-454: The phase difference Δ ϕ {\displaystyle \Delta \phi } . The rotation thus measured is an absolute rotation , that is, the platform's rotation with respect to an inertial reference frame . The Michelson–Morley experiment of 1887 had suggested that the hypothetical luminiferous aether , if it existed, was completely dragged by the Earth . To test this hypothesis, Oliver Lodge in 1897 proposed that

12236-464: The phase difference: according to the light source the two signals now follow different paths in space. Some authors refer to this effect as Sagnac effect although in this case the discrepancy need not be due to the lines of simultaneity not forming closed loops. An example of the modified configuration is shown in Fig. 5, the measured phase difference in both a standard fibre optic gyroscope, shown on

12369-550: The pot is turned. In Ancient China , the Han dynasty (202 BC – 220 AD) inventor and mechanical engineer Ding Huan created a gimbal incense burner around 180 AD. There is a hint in the writing of the earlier Sima Xiangru (179–117 BC) that the gimbal existed in China since the 2nd century BC. There is mention during the Liang dynasty (502–557) that gimbals were used for hinges of doors and windows, while an artisan once presented

12502-409: The principle of conservation of angular momentum whereas the sensitivity of the ring interferometer to rotation arises from the invariance of the speed of light for all inertial frames of reference . Typically three or more mirrors are used, so that counter-propagating light beams follow a closed path such as a triangle or square (Fig. 1). Alternatively fiber optics can be employed to guide

12635-449: 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

12768-478: The propagation of light in the vicinity of the Earth. The Michelson–Gale–Pearson experiment was a very large ring interferometer, (a perimeter of 1.9 kilometer), large enough to detect the angular velocity of the Earth. The outcome of the experiment was that the angular velocity of the Earth as measured by astronomy was confirmed to within measuring accuracy. The ring interferometer of the Michelson–Gale experiment

12901-429: 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 objects 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 physical forces caused by

13034-416: 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 frames 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 frame of reference, Alfred defines

13167-445: The ring before it catches up with the light source from behind. The time t 1 {\displaystyle t_{1}} that it takes to catch up with the light source is given by: Δ L {\displaystyle \Delta L} is the distance (black bold arrow in Fig. 3) that the mirror has moved in that same time: Eliminating Δ L {\displaystyle \Delta L} from

13300-431: The ring cavity, the speed of light is the same in both directions. When the ring laser device is rotating, then it rotates with respect to that background. In other words: invariance of the speed of light provides the reference for the self-calibrating property of the ring laser interferometer. Ring laser gyroscopes suffer from an effect known as "lock-in" at low rotation rates (less than 100°/h). At very low rotation rates,

13433-516: The role of the Sagnac effect in the experiment. Laue said that in the Harress experiment there was a calculable difference in time due to both the dragging of light (which follows from the relativistic velocity addition in moving media , i.e. in moving glass) and "the fact that every part of the rotating apparatus runs away from one ray, while it approaches the other one", i.e. the Sagnac effect. He acknowledged that this latter effect alone could cause

13566-404: The same positive result for both special relativity and the stationary aether (the latter he called "absolute theory" in reference to the 1895-theory of Lorentz ). He also concluded that only complete-aether-drag models (such as the ones of Stokes or Hertz ) would give a negative result. The first interferometry experiment aimed at observing the correlation of angular velocity and phase-shift

13699-557: The same speeds, Doppler effects cancel out, so the Sagnac effect does not involve the Doppler effect. In the case of ring laser interferometry, it is important to be aware of this. When the ring laser setup is rotating, the counterpropagating beams undergo frequency shifts in opposite directions. This frequency shift is not a Doppler shift, but is rather an optical cavity resonance effect, as explained below in Ring lasers . The Sagnac effect

13832-418: The second car to catch up with the first, there are three obvious "frames of reference" that we could choose. First, we could observe the two cars from the side of the road. We define our "frame 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

13965-612: The segment in the lab frame are given by Lorentz transformation as: d t ± = γ ( d t ′ ± v ⋅ d x ′ c 2 ) ≈ n c d ℓ ± v c 2 ⋅ d x {\displaystyle dt_{\pm }=\gamma \left(dt'\pm {\frac {\mathbf {v} \cdot d\mathbf {x} '}{c^{2}}}\right)\approx {\frac {n}{c}}d\ell \pm {\frac {\mathbf {v} }{c^{2}}}\cdot d\mathbf {x} } correct to first order in

14098-437: The setup from outside. The interference pattern that is obtained is a fringe pattern, and what is measured is a phase shift. It is also possible to construct a ring interferometer that is self-contained, based on a completely different arrangement. This is called a ring laser or ring laser gyroscope . The light is generated and sustained by incorporating laser excitation in the path of the light. To understand what happens in

14231-458: The ship's yaw, pitch, and roll. In this application, the inertial measurement unit (IMU) is equipped with three orthogonally mounted gyros to sense rotation about all axes in three-dimensional space. The gyro outputs are kept to a null through drive motors on each gimbal axis, to maintain the orientation of the IMU. To accomplish this, the gyro error signals are passed through " resolvers " mounted on

14364-426: The special principle of the invariance of the form of the description among mutually translating reference frames. The role of fictitious forces in classifying reference frames is pursued further below. Einstein's theory of special relativity , like Newtonian mechanics, postulates the equivalence of all inertial reference frames. However, because special relativity postulates that the speed of light in free space

14497-415: The special theory of relativity. Some historical background including Lange's definition is provided by DiSalle, who says in summary: 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. Classical theories that use

14630-459: The speed of light of course remains constant – so the Sagnac effect arises no matter whether one uses inertial coordinates (see the formulas in section § Theories below) or rotating coordinates (see the formulas in section § Reference frames below). That is, special relativity in its original formulation was adapted to inertial coordinate frames, not rotating frames. Albert Einstein in his paper introducing special relativity stated, "light

14763-406: The spot where he is standing as the origin, 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 frame of reference is considered an inertial frame because he is not accelerating, ignoring effects such as Earth's rotation and gravity. Now consider Betsy,

14896-498: The three gimbal rings to align with their pivot axes in a single plane. When this occurs, it is no longer possible to maintain the sensing platform's orientation. In spacecraft propulsion , rocket engines are generally mounted on a pair of gimbals to allow a single engine to vector thrust about both the pitch and yaw axes; or sometimes just one axis is provided per engine. To control roll, twin engines with differential pitch or yaw control signals are used to provide torque about

15029-451: The three gimbals, roll, pitch and yaw. These resolvers perform an automatic matrix transformation according to each gimbal angle, so that the required torques are delivered to the appropriate gimbal axis. The yaw torques must be resolved by roll and pitch transformations. The gimbal angle is never measured. Similar sensing platforms are used on aircraft. In inertial navigation systems, gimbal lock may occur when vehicle rotation causes two of

15162-611: The time difference is independent of the refraction index n {\displaystyle n} and the velocity of light in the fiber. Imagine a screen for viewing fringes placed at the light source (alternatively, use a beamsplitter to send light from the source point to the screen). Given a steady light source, interference fringes will form on the screen with a fringe displacement given by Δ ϕ ≈ 2 π c λ Δ T {\textstyle \Delta \phi \approx {\frac {2\pi c}{\lambda }}\Delta T} where

15295-486: The time differences required for the two counter-rotating beams to traverse the circuit. The phase shift is Δ ϕ = 2 π c Δ t λ {\displaystyle \Delta \phi ={\frac {2\pi c\,\Delta t}{\lambda }}} , which causes fringes to shift in proportion to A {\displaystyle A} and ω {\displaystyle \omega } . At non-relativistic speeds,

15428-403: The time variance and, therefore, "the accelerations connected with the rotation in no way influence the speed of light". While Laue's explanation is based on inertial frames, Paul Langevin (1921, 1937) and others described the same effect when viewed from rotating reference frames (in both special and general relativity, see Born coordinates ). So when the Sagnac effect should be described from

15561-434: The two equations above we get: Likewise, the light traveling in the opposite direction of the rotation will travel less than one circumference before hitting the light source on the front side. So the time for this direction of light to reach the moving source again is: The time difference is For R ω = v ≪ c {\displaystyle R\omega =v\ll c} , this reduces to where A

15694-446: The two frequencies of laser light to interference a beat frequency can be obtained; the beat frequency is the difference between the two frequencies. This beat frequency can be thought of as an interference pattern in time. (The more familiar interference fringes of interferometry are a spatial pattern). The period of this beat frequency is linearly proportional to the angular velocity of the ring laser with respect to inertial space. This

15827-417: 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

15960-520: The use of the gimbal survived only in an Arabic translation of the early 9th century. Thus, as late as 1965, the sinologist Joseph Needham suspected Arab interpolation . However, Carra de Vaux, author of the French translation which still provides the basis for modern scholars, regards the Pneumatics as essentially genuine. The historian of technology George Sarton (1959) also asserts that it

16093-418: The vehicle's roll axis. Gimbals are also used to mount everything from small camera lenses to large photographic telescopes. In portable photography equipment, single-axis gimbal heads are used in order to allow a balanced movement for camera and lenses. This proves useful in wildlife photography as well as in any other case where very long and heavy telephoto lenses are adopted: a gimbal head rotates

16226-740: The velocity v {\displaystyle \mathbf {v} } . In general, the two beams will visit a given segment at slightly different times, but, in the absence of stretching, the length d ℓ {\textstyle d\ell } is the same for both beams. It follows that the time difference for completing a cycle for the two beams is Δ T = ∫ ( d t + − d t − ) ≈ 2 c 2 ∮ v ⋅ d x {\displaystyle \Delta T=\int \left(dt_{+}-dt_{-}\right)\approx {\frac {2}{c^{2}}}\oint \mathbf {v} \cdot d\mathbf {x} } Remarkably,

16359-1254: The velocity is v = ω × x {\textstyle \mathbf {v} ={\boldsymbol {\omega }}\times \mathbf {x} } and the line integral can be computed in terms of the area of the loop: ∮ v ⋅ d x = ∮ ω × x ⋅ d x = ∮ ω ⋅ x × d x = 2 ∮ ω ⋅ d A = 2 ω ⋅ A {\displaystyle \oint \mathbf {v} \cdot d\mathbf {x} =\oint {\boldsymbol {\omega }}\times \mathbf {x} \cdot d\mathbf {x} =\oint {\boldsymbol {\omega }}\cdot \mathbf {x} \times d\mathbf {x} =2\oint {\boldsymbol {\omega }}\cdot d\mathbf {A} =2{\boldsymbol {\omega }}\cdot \mathbf {A} } This gives Sagnac formula for ring interferometers of arbitrary shape and geometry Δ ϕ ≈ 8 π λ c ω ⋅ A {\displaystyle \Delta \phi \approx {\frac {8\pi }{\lambda c}}{\boldsymbol {\omega }}\cdot \mathbf {A} } If one also allows for stretching one recovers

16492-551: The viewpoint of a corotating frame, one can use ordinary rotating cylindrical coordinates and apply them to the Minkowski metric , which results into the so-called Born metric or Langevin metric. From these coordinates, one can derive the different arrival times of counter-propagating rays, an effect which was shown by Paul Langevin (1921). Or when these coordinates are used to compute the global speed of light in rotating frames, different apparent light speeds are derived depending on

16625-540: The walls. But to prevent the shipborne machinery from rolling around the deck in heavy seas, Athenaeus advises that "you must fix the pithêkion on the platform attached to the merchant-ships in the middle, so that the machine stays upright in any angle". After antiquity , gimbals remained widely known in the Near East . In the Latin West, reference to the device appeared again in the 9th century recipe book called

16758-506: The whole system with the lightest of touches on the gimbal. Powered by three brushless motors , motorized gimbals have the ability to keep the camera level on all axes as the camera operator moves the camera. An inertial measurement unit (IMU) responds to movement and utilizes its three separate motors to stabilize the camera. With the guidance of algorithms, the stabilizer is able to notice the difference between deliberate movement such as pans and tracking shots from unwanted shake. This allows

16891-450: The wide dynamic ranges and accurate scale factor corrections required for stringent applications. Use of longer and larger coils increases sensitivity at the cost of greater sensitivity to temperature variations and vibrations. Gimbal A gimbal is a pivoted support that permits rotation of an object about an axis. A set of three gimbals, one mounted on the other with orthogonal pivot axes, may be used to allow an object mounted on

17024-438: Was considered to be the fixed stars 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: 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. Hence, with respect to an inertial frame, an object or body accelerates only when

17157-522: Was first described by the Greek inventor Philo of Byzantium (280–220 BC). Philo described an eight-sided ink pot with an opening on each side, which can be turned so that while any face is on top, a pen can be dipped and inked — yet the ink never runs out through the holes of the other sides. This was done by the suspension of the inkwell at the center, which was mounted on a series of concentric metal rings so that it remained stationary no matter which way

17290-553: 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. In an inertial frame, Newton's first law ,

17423-451: Was not calibrated by comparison with an outside reference (which was not possible, because the setup was fixed to the Earth). From its design it could be deduced where the central interference fringe ought to be if there would be zero shift. The measured shift was 230 parts in 1000, with an accuracy of 5 parts in 1000. The predicted shift was 237 parts in 1000. The Sagnac effect has stimulated

17556-414: Was performed by the French scientist Georges Sagnac in 1913. Its purpose was to detect "the effect of the relative motion of the ether". Sagnac believed that his results constituted proof of the existence of a stationary aether. However, as explained above, von Laue already showed in 1911 that this effect is consistent with special relativity. Unlike the carefully prepared Michelson–Morley experiment which

17689-509: Was set up to prove an aether wind caused by earth drag, the Sagnac experiment could not prove this type of aether wind because a universal aether would affect all parts of the rotating light equally. Einstein was aware of the phenomenon of the Sagnac effect through the earlier experiments of Franz Harress in 1911. Harress' experiment had been aimed at making measurements of the Fresnel drag of light propagating through moving glass. Not aware of

#372627