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Absolute rotation
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In physics, the concept of absolute rotation—rotation independent of any external reference—is a topic of debate about relativity, cosmology, and the nature of physical laws.
For the concept of absolute rotation to be scientifically meaningful, it must be measurable. In other words, can an observer distinguish between the rotation of an observed object and their own rotation? Newton suggested two experiments to resolve this problem. One is the bucket argument, regarding the effects of centrifugal force upon the shape of the surface of water rotating in a bucket; it is equivalent to the phenomenon of rotational gravity used in proposals for human spaceflight. The second is the effect of centrifugal force upon the tension in a string joining two spheres rotating about their center of mass.
Classical mechanics
[edit]Newton's bucket argument
[edit]
Newton suggested the shape of the surface of the water indicates the presence or absence of absolute rotation relative to absolute space: rotating water has a curved surface, still water has a flat surface. Because rotating water has a concave surface, if the surface you see is concave, and the water does not seem to you to be rotating, then you are rotating with the water.
Centrifugal force is needed to explain the concavity of the water in a co-rotating frame of reference (one that rotates with the water) because the water appears stationary in this frame, and so should have a flat surface. Thus, observers looking at the stationary water need the centrifugal force to explain why the water surface is concave and not flat. The centrifugal force pushes the water toward the sides of the bucket, where it piles up deeper and deeper, Pile-up is arrested when any further climb costs as much work against gravity as is the energy gained from the centrifugal force, which is greater at larger radius.
If you need a centrifugal force to explain what you see, then you are rotating. Newton's conclusion was that rotation is absolute.[1]
Other thinkers suggest that pure logic implies only relative rotation makes sense. For example, Bishop Berkeley and Ernst Mach (among others) suggested that it is relative rotation with respect to the fixed stars that matters, and rotation of the fixed stars relative to an object has the same effect as rotation of the object with respect to the fixed stars.[2] Newton's arguments do not settle this issue; his arguments may be viewed, however, as establishing centrifugal force as a basis for an operational definition of what we actually mean by absolute rotation.[3]
Rotating spheres
[edit]
Newton also proposed another experiment to measure one's rate of rotation: using the tension in a cord joining two spheres rotating about their center of mass. Non-zero tension in the string indicates rotation of the spheres, whether or not the observer thinks they are rotating. This experiment is simpler than the bucket experiment in principle, because it need not involve gravity.
Beyond a simple "yes or no" answer to rotation, one may actually calculate one's rotation. To do that, one takes one's measured rate of rotation of the spheres and computes the tension appropriate to this observed rate. This calculated tension then is compared to the measured tension. If the two agree, one is in a stationary (non-rotating) frame. If the two do not agree, to obtain agreement, one must include a centrifugal force in the tension calculation; for example, if the spheres appear to be stationary, but the tension is non-zero, the entire tension is due to centrifugal force. From the necessary centrifugal force, one can determine one's speed of rotation; for example, if the calculated tension is greater than measured, one is rotating in the sense opposite to the spheres, and the larger the discrepancy the faster this rotation.
The tension in the wire is the required centripetal force to sustain the rotation. What is experienced by the physically rotating observer is the centripetal force and the physical effect arising from his own inertia. The effect arising from inertia is referred to as reactive centrifugal force.
Whether or not the effects from inertia are attributed to a fictitious centrifugal force is a matter of choice.
Relativity
[edit]Special relativity
[edit]French physicist Georges Sagnac in 1913 conducted an experiment that was similar to the Michelson–Morley experiment, which was intended to observe the effects of rotation. Sagnac set up this experiment to prove the existence of the luminiferous aether that Einstein's 1905 theory of special relativity had discarded.
The Sagnac experiment and later similar experiments showed that a stationary object on the surface of the Earth will rotate once every rotation of the Earth when using stars as a stationary reference point. Rotation was thus concluded to be absolute rather than relative.[citation needed]
General relativity
[edit]Mach's principle is the name given by Einstein to a hypothesis often credited to the physicist and philosopher Ernst Mach.
The idea is that the local motion of a rotating reference frame is determined by the large-scale distribution of matter in the universe. Mach's principle says that there is a physical law that relates the motion of the distant stars to the local inertial frame. If you see all the stars whirling around you, Mach suggests that there is some physical law which would make it so you would feel a centrifugal force. The principle is often stated in vague ways, like "mass out there influences inertia here".
The example considered by Einstein was the rotating elastic sphere. Like a rotating planet bulging at the equator, a rotating sphere deforms into an oblate (squashed) spheroid depending on its rotation.
In classical mechanics, an explanation of this deformation requires external causes in a frame of reference in which the spheroid is not rotating, and these external causes may be taken as "absolute rotation" in classical physics and special relativity.[4] In general relativity, no external causes are invoked. The rotation is relative to the local geodesics, and since the local geodesics eventually channel information from the distant stars, there appears to be absolute rotation relative to these stars.[5]
See also
[edit]References
[edit]- ^ Max Born and Günther Leibfried (January 1962). Einstein's Theory of Relativity. Courier Dover Publications. pp. 78–79. ISBN 0-486-60769-0.
{{cite book}}: ISBN / Date incompatibility (help) - ^ BK Ridley (1995). Time, Space, and Things (3 ed.). Cambridge University Press. p. 146. ISBN 0-521-48486-3.
- ^ Rather than justifying a causal link between rotation and centrifugal effects, Newton's arguments may be viewed as defining "absolute rotation" by stating a procedure for its detection and measurement involving centrifugal force. See Robert Disalle (2002). I. Bernard Cohen & George E. Smith (ed.). The Cambridge Companion to Newton. Cambridge University Press. pp. 44–45. ISBN 0-521-65696-6.
- ^ Ferraro, Rafael (2007), "Chapter 8: Inertia and Gravity", Einstein's Space-Time: An Introduction to Special and General Relativity, Springer Science & Business Media, ISBN 9780387699462
- ^ Gilson, James G. (September 1, 2004), Mach's Principle II, arXiv:physics/0409010, Bibcode:2004physics...9010G
Absolute rotation
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Historical and Classical Foundations
Newton's Concept of Absolute Space
In his Philosophiæ Naturalis Principia Mathematica (1687), Isaac Newton introduced the concept of absolute space in the Scholium following the definitions, describing it as a fixed and immutable entity that exists independently of any external relations.[3] Absolute space, in its own nature, without relation to anything external, remains always similar and immovable, serving as the unchanging backdrop against which all physical phenomena occur.[4] This contrasts with relative space, which Newton defined as a movable dimension or measure of the absolute spaces, determined by our senses through its position relative to surrounding bodies.[3] Newton extended this framework to motion, positing absolute motion as the translation of a body from one absolute place into another, detectable through its intrinsic effects rather than mere observations of relative positions.[4] Specifically, absolute rotation manifests through forces such as centrifugal effects, which arise independently of the surrounding environment and indicate true motion relative to absolute space, as opposed to relative rotation that depends on observable changes in position among bodies.[3] Thus, while relative motion describes apparent shifts perceptible to observers, absolute motion represents the genuine change in a body's location within the immutable framework of absolute space, essential for explaining dynamical phenomena like inertial forces in rotating systems.[4] This conception arose in the 17th-century context as Newton's direct response to René Descartes' relational view of space and motion, which treated space as merely the arrangement of material bodies without an independent existence.[5] Descartes argued that motion is inherently relative, defined solely by changes in the positions of bodies with respect to one another, denying any absolute reference frame.[5] Newton rejected this relationalism, insisting that absolute space provides the necessary fixed arena to account for the uniformity of natural laws and the observability of true motions, thereby laying the philosophical groundwork for classical mechanics.[4]The Bucket Argument
In Newton's thought experiment, known as the bucket argument, a cylindrical bucket suspended by a long twisted rope is filled with water and then released to rotate about its vertical axis. Initially, as the bucket begins to spin, the water remains at rest relative to the surrounding environment, maintaining a flat surface. Gradually, friction causes the water to acquire the same angular velocity as the bucket, at which point the surface becomes concave, with the water level rising at the edges and dipping at the center. This concavity arises due to the centrifugal force acting on the water particles, pushing them outward from the axis of rotation.[6] Newton interpreted this phenomenon as evidence of absolute rotation, arguing that the observed effects occur even when the water and bucket are rotating together with no relative motion between them. If rotation were merely relative to surrounding bodies, the flat surface should persist once the water co-rotates with the bucket, as there would be no differential motion. Instead, the concavity demonstrates that the water is rotating relative to an absolute space, which remains fixed and undetectable directly but manifests through these inertial effects. This distinguishes true circular motion, characterized by forces tending to recede from the axis, from mere relative translation. In Newtonian mechanics, the centrifugal force is a fictitious force appearing in the rotating reference frame of the bucket, but Newton viewed the resulting concavity as proof of genuine absolute rotation against an inertial frame defined by absolute space. To find the equilibrium shape of the water surface, consider the balance between the gravitational potential and the centrifugal potential in the rotating frame. The effective potential energy per unit mass is , where is the height, is gravitational acceleration, is the angular velocity, and is the radial distance from the axis. At equilibrium, the surface is an equipotential, so , yielding a parabolic profile relative to the lowest point at the center. This derivation confirms the concavity's dependence on absolute rotation rate , independent of relative motion to the bucket walls. The bucket argument was first published in the Scholium following the Definitions in Book I of Newton's Philosophiæ Naturalis Principia Mathematica in 1687, where it served as a key defense of absolute space against relational theories of motion, such as those proposed by Descartes. By linking observable physical effects to rotation relative to an absolute frame, it established a foundational criterion for identifying true motion in classical mechanics, influencing subsequent debates on space and inertia.[7]The Rotating Spheres Experiment
In Newton's second thought experiment on absolute rotation, described in the scholium following the definitions in Philosophiæ Naturalis Principia Mathematica, two identical globes of equal mass are connected by a taut cord and caused to revolve around their common center of gravity in an otherwise empty space devoid of external bodies or influences. If the system is at rest or in uniform rectilinear motion relative to absolute space, the cord remains slack with no tension, as the globes exhibit no tendency to separate. However, upon imparting rotation to the system, the cord becomes tense, indicating an internal force that stretches it, even though the globes maintain a fixed distance from one another and rotate together without relative motion between them.[8] Newton argued that this tension arises solely from the absolute rotation of the globes with respect to absolute space, rather than any relative motion between the globes themselves, since they co-rotate as a rigid body without approaching or receding from each other. He posited that the tension reveals the "endeavor" of each globe to recede from the axis of rotation, a manifestation of inertial forces detectable independently of external references, thereby distinguishing true (absolute) circular motion from apparent motion.[8] By measuring the tension and applying forces to the faces of the globes, one could further determine the quantity and direction (clockwise or counterclockwise) of this absolute rotation. Mechanically, the tension in the cord counteracts the radial acceleration required for circular motion. Consider two spheres of mass $ m $ connected by a massless cord of total length $ 2L $, rotating with angular velocity $ \omega $ about the midpoint (their common center of gravity). In an inertial frame, each sphere undergoes uniform circular motion at radius $ L $ from the axis, requiring a centripetal acceleration $ a = \omega^2 L $ directed inward toward the center.[9] The cord provides this centripetal force via tension $ T $, so for each sphere, Newton's second law gives $ T = m a = m \omega^2 L $. To derive this, start from the tangential velocity $ v = \omega L $; the centripetal acceleration is then $ a = v^2 / L = (\omega L)^2 / L = \omega^2 L $. Thus, the net inward force on each sphere is $ T \omega = 0 $), $ T = 0 $, and the cord slackens.[9] Newton emphasized that even in a void, the tension persists, underscoring its origin in absolute space. This setup parallels the bucket argument by similarly revealing centrifugal effects through internal stresses, but focuses on rigid-body tension rather than fluid deformation. In the 18th century, contemporaries like Colin Maclaurin refined these ideas in discussions of orbital mechanics, exploring how the tension in such rotating systems implies absolute motion in celestial bodies, such as the mutual revolutions of planets and satellites, and integrating it with gravitational theories to explain stable orbits without relying solely on relative positions.[10]Philosophical Criticisms and Alternatives
Leibniz and Berkeley's Objections
In the correspondence between Gottfried Wilhelm Leibniz and Samuel Clarke from 1715 to 1716, Leibniz articulated a relational view of space as an ideal order among coexisting bodies, rejecting Newton's absolute space as an unnecessary and unobservable entity.[11] He contended that absolute rotation or motion lacks meaning without reference to other bodies, as differences in absolute states would be indiscernible and violate the principle of sufficient reason, making such motion a mere fiction.[11] For instance, Leibniz argued that if the entire universe rotated uniformly in absolute space, no observable effects would distinguish it from rest, rendering absolute rotation empirically empty.[11] George Berkeley extended these relationalist critiques in his 1721 treatise De Motu, where he dismissed absolute space and motion as imperceptible and thus irrelevant to natural philosophy.[12] Berkeley specifically addressed rotational phenomena, asserting that effects like the concavity in Newton's rotating bucket arise from the water's relative motion against the surrounding air or container, not any absolute inertial frame.[12] He emphasized that true motion involves changes in situation relative to sensible objects and impressed forces from interactions, eliminating the need for an invisible absolute space.[12] At the core of both philosophers' objections lies relationalism, which posits that space and motion derive solely from relations among bodies, with forces emerging from their interactions rather than an absolute frame.[13] This framework reinterprets Newton's experiments: the bucket's water surface tension results from relative motion to the Earth or laboratory, while the tension in rotating spheres stems from adjustments in their mutual attractions due to relative velocities, without invoking absolute rotation.[13] These 18th-century critiques shifted philosophical and scientific debates toward relational explanations, prompting empirical investigations into motion's dependence on distributed matter throughout the 18th and 19th centuries.[13]Mach's Principle
Ernst Mach, in his 1883 work The Science of Mechanics, proposed that local inertial frames are defined relative to the distant stars and the overall distribution of matter in the universe, rendering the concept of absolute rotation meaningless without such cosmic references. He argued that inertia originates not from an abstract absolute space but from the interactions of a body with all other matter in the cosmos, emphasizing that "motion is completely determined by the entire universe." This formulation challenged Newtonian mechanics by tying the determination of rotation and inertial effects to the relative configuration of massive bodies, particularly the fixed stars, rather than an independent spatial framework.[14] Mach applied this idea directly to Newton's rotating bucket experiment, suggesting that the concavity of the water's surface arises from its rotation relative to the mass of the Earth and other celestial bodies, not against an absolute space. He famously questioned the experiment's outcome by positing an alternative: "Try to fix Newton's bucket and rotate the heaven of fixed stars and then it is easy to see that the water will take the form of a concave surface," implying that the centrifugal effects would reverse if the universe rotated instead. In this view, the bucket's result demonstrates relational motion against the universe's mass distribution, dismissing Newton's invocation of empty absolute space as an unphysical and arbitrary fiction.[15][16] Mach's ideas profoundly influenced Albert Einstein, who credited them with inspiring key aspects of general relativity, including the equivalence principle and the notion that matter curves spacetime to define inertial paths. Einstein explicitly coined the term "Mach's principle" in a 1918 address, viewing it as a guiding heuristic for relativizing inertia through the global distribution of mass. This connection briefly manifests in general relativity's description of geodesic motion, where distant matter influences local frames.[15][17] Despite its impact, Mach's principle has faced criticisms for lacking a precise mathematical quantification of how distant matter exactly determines local inertia. Modern tests, such as those confirming frame-dragging effects in general relativity via satellites like Gravity Probe B, indicate partial validity by showing that rotating masses influence nearby inertial frames, though the full cosmic dependence of inertia remains unresolved.[15]Relativistic Perspectives
Special Relativity
Special relativity, formulated by Albert Einstein in 1905, revolutionized the classical notion of absolute space by establishing that there is no privileged inertial frame of reference and eliminating the idea of an absolute rest frame. In his foundational paper "On the Electrodynamics of Moving Bodies," Einstein addressed the inconsistencies between Newtonian mechanics and Maxwell's electromagnetism by introducing two postulates: the laws of physics are identical in all inertial frames, and the speed of light in vacuum is constant regardless of the source's motion. This framework discarded the luminiferous aether, previously invoked as an absolute medium for light propagation, thereby rendering absolute space unnecessary and undetectable.[18] At its core, special relativity treats rotation not as an absolute property but as a form of acceleration relative to inertial frames, with no preferred frame to define absolute rotational motion. All inertial observers are equivalent, and uniform linear motion is undetectable without external references, but rotation introduces non-inertial effects that can be locally measured. In a rotating frame, fictitious forces such as centrifugal and Coriolis forces emerge due to the frame's acceleration, yet these are relational, depending on the choice of inertial frame, and do not indicate an absolute rotation against the universe. Special relativity thus affirms the relativity of inertial motion while preserving the detectability of acceleration, including rotation, through local experiments.[19][20] Lorentz transformations, which underpin special relativity, mix spatial and temporal coordinates between inertial frames, ensuring that concepts like simultaneity and length are frame-dependent. For rotational scenarios, these transformations—particularly boosts—affect the measurement of angular velocity, as the relativity of simultaneity prevents a unique, observer-independent definition of angular displacement over time. An observer undergoing a Lorentz boost relative to a rotating system will perceive a different angular velocity due to the non-invariance of rotation rates under such transformations, underscoring that absolute angular velocity cannot exist without specifying the reference frame. In rotating frames, this manifests as the impossibility of globally synchronizing clocks, since Einstein synchronization is non-transitive around a closed loop, further highlighting the local, relative nature of rotation.[20][21] The implications of these principles extend to thought experiments like the twin paradox, where path-dependent proper time arises from non-inertial trajectories. In a rotational variant, one twin remains in an inertial frame while the other follows a circular path at relativistic speeds; upon reunion, the rotating twin has aged less due to the integrated effects of velocity changes along the accelerated worldline, demonstrating that rotational motion leads to differential aging relative to inertial paths. This effect, analyzable within special relativity for the acceleration phases, reinforces the absence of absolute rotation by tying aging differences to the geometry of spacetime paths rather than an intrinsic rotational absolute. The Sagnac effect offers a key experimental confirmation, manifesting as a phase shift in light propagating in opposite directions within a rotating apparatus, detectable locally but relative to the inertial frame.[22][20]General Relativity
In general relativity, the equivalence principle asserts that locally, the effects of a uniform gravitational field are indistinguishable from those experienced in an accelerated reference frame, including rotational acceleration.[23] This principle extends to rotation, where a rotating observer in flat spacetime cannot locally differentiate their motion from the influence of a gravitational field, thereby relativizing acceleration and eliminating absolute notions of rotation within small regions of spacetime.[23] Frame-dragging, also known as the Lense-Thirring effect, describes how a rotating mass generates a gravitomagnetic field that drags nearby spacetime, rendering rotation relative to the distribution of cosmic matter rather than an absolute frame.[24] In the weak-field approximation, the precession rate of a gyroscope or orbit due to this effect is given by
where is the gravitational constant, is the moment of inertia of the rotating body, is its angular velocity, is the speed of light, and is the distance from the body.[24] This effect demonstrates that inertial frames are influenced by the rotation of distant masses, aligning rotation with the overall geometry of spacetime.
Geodesic motion in general relativity further underscores the absence of absolute rotation, as freely falling particles follow geodesics defined by spacetime curvature, with locally non-rotating inertial frames determined relative to the distant stars through this curvature rather than a fixed absolute space.[1] Einstein reinterpreted Newton's rotating bucket experiment in this framework, attributing the concave water surface not to motion against absolute space but to the geometry of curved spacetime induced by the bucket's rotation and the universe's mass distribution.[25]
General relativity incorporates Machian elements by suggesting that inertia arises from interactions with the universe's mass distribution, though it does not fully implement Mach's principle, as spacetime curvature can exist independently of matter and absolute rotational solutions persist in certain limits.[26] Einstein viewed this partial alignment as a step toward relativizing inertia, influenced by Mach's ideas during the theory's development.[26]
Modern Experimental Tests
Interferometry and Sagnac Effect
In 1913, Georges Sagnac conducted a pivotal experiment using an interferometer mounted on a rotating turntable to investigate the propagation of light in a rotating frame. A coherent light beam from a source was split by a beam splitter and directed along opposite paths around a closed polygonal loop formed by mirrors, before recombining to produce interference fringes. When the apparatus rotated, the counter-propagating beams experienced different travel times due to the motion of the mirrors, resulting in a observable shift in the interference pattern proportional to the rotation rate. This setup demonstrated a phase difference that Sagnac initially attributed to an "optical whirlwind" in the luminiferous aether dragged by the rotation.[27] The phase shift in the Sagnac interferometer is described by the formula
where is the enclosed area of the loop, is the angular rotation rate, is the speed of light in vacuum, and is the wavelength of the light. This shift arises from the path length difference between the two beams in the rotating frame, leading to a time delay and thus . The derivation follows from considering the velocity addition in the non-inertial frame, where one beam travels with the rotation while the other opposes it. Sagnac's original measurements on a turntable with an area of approximately 0.08 m² and rotation rates on the order of 2 Hz (approximately 12.6 rad/s) yielded fringe shifts on the order of 0.07 fringes, confirming the predicted dependence on and .[27]
Subsequent replications solidified the effect's reliability. In 1925, Albert Michelson and Henry Gale constructed a massive rectangular interferometer (dimensions 640 m by 320 m) to measure Earth's rotation, observing a fringe shift of about 0.23 fringes at a latitude of 41.4°N, consistent with the Sagnac formula using Earth's angular velocity of rad/s relative to distant stars as the inertial reference. This demonstrated the effect's sensitivity to rotation against the cosmic background, interpreted as an inertial frame. Later turntable-based tests, such as those by Dufour and Prunier in 1937, replicated the setup on controlled platforms and confirmed the phase shift's independence from linear translation, as no shift occurred under pure uniform motion but appeared solely with angular rotation. These experiments underscored that the Sagnac effect detects rotation relative to a non-rotating inertial frame, not absolute motion in a Newtonian sense.[27]
Modern applications exploit the Sagnac effect in fiber-optic gyroscopes (FOGs), which use coiled optical fibers as the interferometer loop to measure rotational velocity for navigation in aircraft, ships, and spacecraft. In a FOG, counter-propagating laser beams in the fiber experience the phase shift, converted to an electrical signal proportional to , enabling drift-free operation without moving parts and accuracies down to 0.001°/h. These devices confirm the effect's robustness in practical settings, such as inertial navigation systems that account for Earth's rotation relative to inertial space. However, the Sagnac effect measures only relative rotation with respect to distant inertial frames, offering no evidence for an absolute space and aligning with special relativity's treatment of non-inertial frames, where the shift reflects synchronization differences rather than a preferred absolute rest frame.[28][29]