{"history":[],"article":{"id":"bc8309e2-32ed-4b5d-85d1-5064b79010ae","title":"Thomas precession","slug":"thomas-precession","content":":::summary\n:::\n\n# \nIn [physics](https://bliptext.com/articles/physics), the **Thomas precession**, named after [Llewellyn Thomas](https://bliptext.com/articles/llewellyn-thomas), is a [relativistic](https://bliptext.com/articles/theory-of-relativity) correction that applies to the [spin](https://bliptext.com/articles/spin-physics) of an elementary particle or the rotation of a macroscopic [gyroscope](https://bliptext.com/articles/gyroscope) and relates the [angular velocity](https://bliptext.com/articles/angular-velocity) of the [spin](https://bliptext.com/articles/spin-physics) of a particle following a [curvilinear](https://bliptext.com/articles/curvilinear) orbit to the [angular velocity](https://bliptext.com/articles/angular-velocity) of the orbital motion.\nFor a given [inertial frame](https://bliptext.com/articles/inertial-frame), if a second frame is Lorentz-boosted relative to it, and a third boosted relative to the second, but non-collinear with the first boost, then the Lorentz transformation between the first and third frames involves a combined boost and rotation, known as the \"[Wigner rotation](https://bliptext.com/articles/wigner-rotation)\" or \"Thomas rotation\".\nFor accelerated motion, the accelerated frame has an inertial frame at every instant.\nTwo boosts a small time interval (as measured in the lab frame) apart leads to a Wigner rotation after the second boost.\nIn the limit the time interval tends to zero, the accelerated frame will rotate at every instant, so the accelerated frame rotates with an angular velocity.\nThe precession can be understood geometrically as a consequence of the fact that the [space](https://bliptext.com/articles/mathematical-space) of velocities in relativity is [hyperbolic](https://bliptext.com/articles/hyperbolic-geometry), and so [parallel transport](https://bliptext.com/articles/parallel-transport) of a vector (the gyroscope's angular velocity) around a circle (its linear velocity) leaves it pointing in a different direction, or understood algebraically as being a result of the [non-commutativity](https://bliptext.com/articles/non-commutativity) of [Lorentz transformations](https://bliptext.com/articles/lorentz-transformation).\nThomas precession gives a correction to the [spin–orbit interaction](https://bliptext.com/articles/spin-orbit-interaction) in [quantum mechanics](https://bliptext.com/articles/quantum-mechanics), which takes into account the [relativistic time dilation](https://bliptext.com/articles/relativistic-time-dilation) between the [electron](https://bliptext.com/articles/electron) and the [nucleus](https://bliptext.com/articles/[atom](https://bliptext.com/articles/atom)ic-nucleus) of an [atom](https://bliptext.com/articles/atom).\nThomas precession is a [kinematic effect](https://bliptext.com/articles/kinematic) in the [flat spacetime](https://bliptext.com/articles/flat-spacetime) of [special relativity](https://bliptext.com/articles/special-relativity).\nIn the curved spacetime of [general relativity](https://bliptext.com/articles/general-relativity), Thomas precession combines with a geometric effect to produce [de Sitter precession](https://bliptext.com/articles/de-sitter-precession).\nAlthough Thomas precession (net rotation after a trajectory that returns to its initial velocity) is a purely kinematic effect, it only occurs in curvilinear motion and therefore cannot be observed independently of some external force causing the curvilinear motion such as that caused by an [electromagnetic field](https://bliptext.com/articles/electromagnetic-field), a [gravitational field](https://bliptext.com/articles/gravitational-field) or a mechanical force, so Thomas precession is usually accompanied by [dynamical effects](https://bliptext.com/articles/dynamics-mechanics).\nIf the system experiences no external torque, e.g., in external scalar fields, its spin dynamics are determined only by the Thomas precession.\nA single discrete Thomas rotation (as opposed to the series of infinitesimal rotations that add up to the Thomas precession) is present in situations anytime there are three or more inertial frames in non-collinear motion, as can be seen using Lorentz transformations.\n# [Llewellyn Thomas (1903 – 1992) | upload.wikimedia.org/wikipedia/commons/a/a1/Thomas%2CLlewellyn///1926///Kopenhagen.jpg]\n\n# History\nThomas precession in relativity was already known to [Ludwik Silberstein](https://bliptext.com/articles/ludwik-silberstein), in 1914.\nBut the only knowledge Thomas had of relativistic precession came from [de Sitter](https://bliptext.com/articles/de-sitter)'s paper on the relativistic precession of the moon, first published in a book by [Eddington](https://bliptext.com/articles/arthur-stanley-eddington).\nIn 1925 Thomas relativistically recomputed the precessional frequency of the doublet separation in the fine structure of the atom.\nHe thus found the missing factor 1/2, which came to be known as the Thomas half.\nThis discovery of the relativistic precession of the electron spin led to the understanding of the significance of the relativistic effect.\nThe effect was consequently named \"Thomas precession\".\n\n# Introduction\n\n## Definition\nConsider a physical system moving through [Minkowski spacetime](https://bliptext.com/articles/minkowski-spacetime).\nAssume that there is at any moment an inertial system such that in it, the system is at rest.\nThis assumption is sometimes called the third postulate of relativity.\nThis means that at any instant, the coordinates and state of the system can be Lorentz transformed to the lab system through some Lorentz transformation.\nLet the system be subject to external forces that produce no [torque](https://bliptext.com/articles/torque) with respect to its center of mass in its (instantaneous) rest frame.\nThe condition of \"no torque\" is necessary to isolate the phenomenon of Thomas precession.\nAs a simplifying assumption one assumes that the external forces bring the system back to its initial velocity after some finite time.\nFix a Lorentz frame O such that the initial and final velocities are zero.\nThe **Pauli–Lubanski spin vector** Sμ is defined to be\n(0, Si)\nin the system's rest frame, with Si the angular-momentum three-vector about the center of mass.\nIn the motion from initial to final position, Sμ undergoes a rotation, as recorded in O, from its initial to its final value.\nThis continuous change is the Thomas precession.\n\n## Statement\n[[File:Thomas precession coefficient.svg|thumb|400px|Value of\nγ2/(γ + 1)\nas\nincreases, with v the instantaneous magnitude of the particle's velocity.\nThe Thomas rotation is negligible for\nβ < 0.5\n, increases steadily for\n0.5 < β < 0.8\n, then rapidly shoots to infinity as\nβ\ntends to 1.\nThe \"Thomas half\" is evident in the low-speed limit, and the rotation is only very clear for speeds approaching that of light.]]\nConsider the motion of a [particle](https://bliptext.com/articles/particle).\nIntroduce a [lab frame](https://bliptext.com/articles/lab-frame)\nΣ\nin which an observer can measure the relative motion of the particle.\nAt each instant of time the particle has an [inertial frame](https://bliptext.com/articles/inertial-frame) in which it is at rest.\nRelative to this lab frame, the instantaneous velocity of the particle is\nv(t)\nwith magnitude\nbounded by the [speed of light](https://bliptext.com/articles/speed-of-light) c, so that\n0 ≤ v < c\n. Here the time t is the [coordinate time](https://bliptext.com/articles/coordinate-time) as measured in the lab frame, not the [proper time](https://bliptext.com/articles/proper-time) of the particle.\nApart from the upper limit on magnitude, the velocity of the particle is arbitrary and not necessarily constant; its corresponding vector of [acceleration](https://bliptext.com/articles/acceleration) is\n. As a result of the Wigner rotation at every instant, the particle's frame precesses with an angular velocity given by the equation\nwhere × is the [cross product](https://bliptext.com/articles/cross-product) and\nis the instantaneous [Lorentz factor](https://bliptext.com/articles/lorentz-factor), a function of the particle's instantaneous velocity.\nLike any angular velocity,\nωT\nis a [pseudovector](https://bliptext.com/articles/pseudovector); its magnitude is the angular speed the particle's frame precesses (in [radians](https://bliptext.com/articles/radian) per second), and the direction points along the rotation axis.\nAs is usual, the right-hand convention of the cross product is used (see [right-hand rule](https://bliptext.com/articles/right-hand-rule)).\nThe precession depends on accelerated motion, and the non-[collinearity](https://bliptext.com/articles/collinear) of the particle's instantaneous velocity and acceleration.\nNo precession occurs if the particle moves with uniform velocity (constant\nv\nso\n), or accelerates in a straight line (in which case\nv\nand\na\nare parallel or antiparallel so their cross product is zero). The particle has to move in a curve, say an arc, [spiral](https://bliptext.com/articles/spiral), [helix](https://bliptext.com/articles/helix), or a [circular orbit](https://bliptext.com/articles/circular-orbit) or [elliptical orbit](https://bliptext.com/articles/elliptical-orbit), for its frame to precess. The angular velocity of the precession is a maximum if the velocity and acceleration vectors are perpendicular throughout the motion (a [circular orbit](https://bliptext.com/articles/circular-orbit)), and is large if their magnitudes are large (the magnitude of\nv\nis almost c).\nIn the non-relativistic limit,\nv → 0\nso\nγ → 1\n, and the angular velocity is approximately\nThe factor of 1/2 turns out to be the critical factor to agree with experimental results.\nIt is informally known as the \"Thomas half\".\n\n# Mathematical explanation\n\n## Lorentz transformations\nThe description of relative motion involves [Lorentz transformations](https://bliptext.com/articles/lorentz-transformation), and it is convenient to use them in [matrix](https://bliptext.com/articles/matrix-mathematics) form; symbolic [matrix](https://bliptext.com/articles/matrix-mathematics) expressions summarize the transformations and are easy to manipulate, and when required the full matrices can be written explicitly.\nAlso, to prevent extra factors of c cluttering the equations, it is convenient to use the definition\nwith magnitude\nsuch that\n0 ≤ β < 1\n.\nThe spacetime coordinates of the lab frame are collected into a 4×1 [column vector](https://bliptext.com/articles/column-vector), and the [boost](https://bliptext.com/articles/lorentz-transformation) is represented as a 4×4 [symmetric matrix](https://bliptext.com/articles/symmetric-matrix), respectively\nand turn\nis the Lorentz factor of\nβ\n. In other frames, the corresponding coordinates are also arranged into column vectors.\nThe [inverse matrix](https://bliptext.com/articles/inverse-matrix) of the boost corresponds to a boost in the opposite direction, and is given by\nB(β)−1 = B(−β)\n.\nAt an instant of lab-recorded time t measured in the lab frame, the transformation of spacetime coordinates from the lab frame\nΣ\nto the particle's frame Σ\n′\nis\nand at later lab-recorded time\nt + Δt\nwe can define a new frame\nΣ′′\nfor the particle, which moves with velocity\nβ + Δβ\nrelative to\nΣ\n, and the corresponding boost is\nThe vectors\nβ\nand\nΔβ\nare two separate vectors.\nThe latter is a small increment, and can be conveniently split into components parallel (‖) and perpendicular (⊥) to\nβ\nCombining and obtains the Lorentz transformation between\nΣ′\nand\nΣ′′\n,\nand this composition contains all the required information about the motion between these two lab times.\nNotice\nB(β + Δβ)B(−β)\nand\nB(β + Δβ)\nare infinitesimal transformations because they involve a small increment in the relative velocity, while\nB(−β)\nis not.\nThe composition of two boosts equates to a single boost combined with a [Wigner rotation](https://bliptext.com/articles/wigner-rotation) about an axis perpendicular to the relative velocities;\nThe rotation is given by is a 4×4 rotation matrix\nR\nin the [axis–angle representation](https://bliptext.com/articles/axis-angle-representation), and coordinate systems are taken to be [right-handed](https://bliptext.com/articles/orientation-vector-space).\nThis matrix rotates 3d vectors anticlockwise about an axis ([active transformation](https://bliptext.com/articles/active-transformation)), or equivalently rotates coordinate frames clockwise about the same axis (passive transformation).\nThe axis-angle vector\nΔθ\nparametrizes the rotation, its magnitude\nΔθ\nis the angle\nΣ′′\nhas rotated, and direction is parallel to the rotation axis, in this case the axis is parallel to the [cross product](https://bliptext.com/articles/cross-product)\n(−β)×(β + Δβ) = −β×Δβ\n. If the angles are negative, then the sense of rotation is reversed.\nThe inverse matrix is given by\nR(Δθ)−1 = R(−Δθ)\n.\nCorresponding to the boost is the (small change in the) boost vector\nΔb\n, with magnitude and direction of the relative velocity of the boost (divided by\nc\n). The boost\nB(Δb)\nand rotation\nR(Δθ)\nhere are infinitesimal transformations because\nΔb\nand rotation\nΔθ\nare small.\nThe rotation gives rise to the Thomas precession, but there is a subtlety.\nTo interpret the particle's frame as a co-moving inertial frame relative to the lab frame, and agree with the non-relativistic limit, we expect the transformation between the particle's instantaneous frames at times\nt\nand\nt + Δt\nto be related by a boost without rotation.\nCombining and and rearranging gives\nwhere another instantaneous frame\nΣ′′′\nis introduced with coordinates\nX′′′\n, to prevent conflation with\nΣ′′\n. To summarize the frames of reference: in the lab frame\nΣ\nan observer measures the motion of the particle, and three instantaneous inertial frames in which the particle is at rest are\nΣ′\n(at time t),\nΣ′′\n(at time\nt + Δt\n), and\nΣ′′′\n(at time\nt + Δt\n). The frames\nΣ′′\nand\nΣ′′′\nare at the same location and time, they differ only by a rotation.\nBy contrast\nΣ′\nand\nΣ′′′\ndiffer by a boost and lab time interval\nΔt\n.\nRelating the coordinates\nX′′′\nto the lab coordinates\nX\nvia and ;\nthe frame\nΣ′′′\nis rotated in the negative sense.\nThe rotation is between two instants of lab time.\nAs\nΔt → 0\n, the particle's frame rotates at every instant, and the continuous motion of the particle amounts to a continuous rotation with an [angular velocity](https://bliptext.com/articles/angular-velocity) at every instant.\nDividing\n−Δθ\nby\nΔt\n, and taking the [limit](https://bliptext.com/articles/limit-mathematics)\nΔt → 0\n, the angular velocity is by definition\nIt remains to find what\nΔθ\nprecisely is.\n\n## Extracting the formula\nThe composition can be obtained by explicitly calculating the matrix product.\nThe boost matrix of\nβ + Δβ\nwill require the magnitude and Lorentz factor of this vector.\nSince\nΔβ\nis small, terms of \"second order\"\n,\n(Δβx)2\n,\n(Δβy)2\n,\nΔβxΔβy\nand higher are negligible.\nTaking advantage of this fact, the magnitude squared of the vector is\nand expanding the Lorentz factor of\nβ + Δβ\nas a [power series](https://bliptext.com/articles/power-series) gives to first order in\nΔβ\n,\nusing the Lorentz factor\nγ\nof\nβ\nas above.\nIntroducing the boost generators\nand rotation generators\nalong with the [dot product](https://bliptext.com/articles/dot-product) · facilitates the coordinate independent expression\nwhich holds if\nβ\nand\nΔβ\nlie in any plane.\nThis is an infinitesimal Lorentz transformation in the form of a combined boost and rotation\nwhere\nAfter dividing\nΔθ\nby\nΔt\nand taking the limit as in, one obtains the instantaneous angular velocity\nwhere\na\nis the [acceleration](https://bliptext.com/articles/acceleration) of the particle as observed in the lab frame.\nNo forces were specified or used in the derivation so the precession is a kinematical effect - it arises from the geometric aspects of motion.\nHowever, forces cause accelerations, so the Thomas precession is observed if the particle is subject to forces.\nThomas precession can also be derived using the [Fermi-Walker transport equation](https://bliptext.com/articles/fermi-walker-transport).\nOne assumes uniform circular motion in flat Minkowski spacetime.\nThe spin 4-vector is orthogonal to the velocity 4-vector.\nFermi-Walker transport preserves this relation.\nOne finds that the dot product of the acceleration 4-vector with the spin 4-vector varies sinusoidally with time with an angular frequency γω, where ω is the angular frequency of the circular motion and γ=1/√(1-v^2/c^2), the [Lorentz factor](https://bliptext.com/articles/lorentz-factor).\nThis is easily shown by taking the second time derivative of that dot product.\nBecause this angular frequency exceeds ω, the spin precesses in the retrograde direction.\nThe difference (γ-1)ω is the Thomas precession angular frequency already given, as is simply shown by realizing that the magnitude of the 3-acceleration is ω v.\n\n# Applications\n\n## In electron orbitals\nIn quantum mechanics **Thomas precession** is a correction to the [spin-orbit interaction](https://bliptext.com/articles/spin-orbit-interaction), which takes into account the relativistic [time dilation](https://bliptext.com/articles/time-dilation) between the [electron](https://bliptext.com/articles/electron) and the [nucleus](https://bliptext.com/articles/atomic-nucleus) in [hydrogenic atoms](https://bliptext.com/articles/hydrogen-like-atom).\nBasically, it states that spinning objects [precess](https://bliptext.com/articles/precess) when they accelerate in [special relativity](https://bliptext.com/articles/special-relativity) because [Lorentz boosts](https://bliptext.com/articles/lorentz-boost) do not commute with each other.\nTo calculate the spin of a particle in a [magnetic field](https://bliptext.com/articles/magnetic-field), one must also take into account [Larmor precession](https://bliptext.com/articles/larmor-precession).\n\n## In a Foucault pendulum\nThe rotation of the swing plane of [Foucault pendulum](https://bliptext.com/articles/foucault-pendulum) can be treated as a result of [parallel transport](https://bliptext.com/articles/parallel-transport) of the pendulum in a 2-dimensional sphere of Euclidean space.\nThe [hyperbolic](https://bliptext.com/articles/hyperbolic-geometry) space of velocities in [Minkowski spacetime](https://bliptext.com/articles/minkowski-spacetime) represents a 3-dimensional (pseudo-) sphere with imaginary radius and imaginary timelike coordinate.\nParallel transport of a spinning particle in relativistic velocity space leads to Thomas precession, which is similar to the rotation of the swing plane of a Foucault pendulum.\nThe angle of rotation in both cases is determined by the area integral of curvature in agreement with the [Gauss–Bonnet theorem](https://bliptext.com/articles/gauss-bonnet-theorem).\nThomas precession gives a correction to the precession of a Foucault pendulum.\nFor a Foucault pendulum located in the city of Nijmegen in the Netherlands the correction is:\nNote that it is more than two orders of magnitude smaller than the precession due to the general-relativistic correction arising from [frame-dragging](https://bliptext.com/articles/frame-dragging), the [Lense–Thirring precession](https://bliptext.com/articles/lense-thirring-precession).\n\n# Remarks\n\n# Textbooks","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T21:39:43.831Z","updated_at":"2025-01-25T21:39:43.831Z","search_vector":"'precess':2 'thoma':1","search_content":" \n \n\n# \nIn [physics](https://bliptext.com/articles/physics), the **Thomas precession**, named after [Llewellyn Thomas](https://bliptext.com/articles/llewellyn-thomas), is a [relativistic](https://bliptext.com/articles/theory-of-relativity) correction that applies to the [spin](https://bliptext.com/articles/spin-physics) of an elementary particle or the rotation of a macroscopic [gyroscope](https://bliptext.com/articles/gyroscope) and relates the [angular velocity](https://bliptext.com/articles/angular-velocity) of the [spin](https://bliptext.com/articles/spin-physics) of a particle following a [curvilinear](https://bliptext.com/articles/curvilinear) orbit to the [angular velocity](https://bliptext.com/articles/angular-velocity) of the orbital motion.\nFor a given [inertial frame](https://bliptext.com/articles/inertial-frame), if a second frame is Lorentz-boosted relative to it, and a third boosted relative to the second, but non-collinear with the first boost, then the Lorentz transformation between the first and third frames involves a combined boost and rotation, known as the \"[Wigner rotation](https://bliptext.com/articles/wigner-rotation)\" or \"Thomas rotation\".\nFor accelerated motion, the accelerated frame has an inertial frame at every instant.\nTwo boosts a small time interval (as measured in the lab frame) apart leads to a Wigner rotation after the second boost.\nIn the limit the time interval tends to zero, the accelerated frame will rotate at every instant, so the accelerated frame rotates with an angular velocity.\nThe precession can be understood geometrically as a consequence of the fact that the [space](https://bliptext.com/articles/mathematical-space) of velocities in relativity is [hyperbolic](https://bliptext.com/articles/hyperbolic-geometry), and so [parallel transport](https://bliptext.com/articles/parallel-transport) of a vector (the gyroscope's angular velocity) around a circle (its linear velocity) leaves it pointing in a different direction, or understood algebraically as being a result of the [non-commutativity](https://bliptext.com/articles/non-commutativity) of [Lorentz transformations](https://bliptext.com/articles/lorentz-transformation).\nThomas precession gives a correction to the [spin–orbit interaction](https://bliptext.com/articles/spin-orbit-interaction) in [quantum mechanics](https://bliptext.com/articles/quantum-mechanics), which takes into account the [relativistic time dilation](https://bliptext.com/articles/relativistic-time-dilation) between the [electron](https://bliptext.com/articles/electron) and the [nucleus](https://bliptext.com/articles/[atom](https://bliptext.com/articles/atom)ic-nucleus) of an [atom](https://bliptext.com/articles/atom).\nThomas precession is a [kinematic effect](https://bliptext.com/articles/kinematic) in the [flat spacetime](https://bliptext.com/articles/flat-spacetime) of [special relativity](https://bliptext.com/articles/special-relativity).\nIn the curved spacetime of [general relativity](https://bliptext.com/articles/general-relativity), Thomas precession combines with a geometric effect to produce [de Sitter precession](https://bliptext.com/articles/de-sitter-precession).\nAlthough Thomas precession (net rotation after a trajectory that returns to its initial velocity) is a purely kinematic effect, it only occurs in curvilinear motion and therefore cannot be observed independently of some external force causing the curvilinear motion such as that caused by an [electromagnetic field](https://bliptext.com/articles/electromagnetic-field), a [gravitational field](https://bliptext.com/articles/gravitational-field) or a mechanical force, so Thomas precession is usually accompanied by [dynamical effects](https://bliptext.com/articles/dynamics-mechanics).\nIf the system experiences no external torque, e.g., in external scalar fields, its spin dynamics are determined only by the Thomas precession.\nA single discrete Thomas rotation (as opposed to the series of infinitesimal rotations that add up to the Thomas precession) is present in situations anytime there are three or more inertial frames in non-collinear motion, as can be seen using Lorentz transformations.\n# [Llewellyn Thomas (1903 – 1992) | upload.wikimedia.org/wikipedia/commons/a/a1/Thomas%2CLlewellyn///1926///Kopenhagen.jpg]\n\n# History\nThomas precession in relativity was already known to [Ludwik Silberstein](https://bliptext.com/articles/ludwik-silberstein), in 1914.\nBut the only knowledge Thomas had of relativistic precession came from [de Sitter](https://bliptext.com/articles/de-sitter)'s paper on the relativistic precession of the moon, first published in a book by [Eddington](https://bliptext.com/articles/arthur-stanley-eddington).\nIn 1925 Thomas relativistically recomputed the precessional frequency of the doublet separation in the fine structure of the atom.\nHe thus found the missing factor 1/2, which came to be known as the Thomas half.\nThis discovery of the relativistic precession of the electron spin led to the understanding of the significance of the relativistic effect.\nThe effect was consequently named \"Thomas precession\".\n\n# Introduction\n\n## Definition\nConsider a physical system moving through [Minkowski spacetime](https://bliptext.com/articles/minkowski-spacetime).\nAssume that there is at any moment an inertial system such that in it, the system is at rest.\nThis assumption is sometimes called the third postulate of relativity.\nThis means that at any instant, the coordinates and state of the system can be Lorentz transformed to the lab system through some Lorentz transformation.\nLet the system be subject to external forces that produce no [torque](https://bliptext.com/articles/torque) with respect to its center of mass in its (instantaneous) rest frame.\nThe condition of \"no torque\" is necessary to isolate the phenomenon of Thomas precession.\nAs a simplifying assumption one assumes that the external forces bring the system back to its initial velocity after some finite time.\nFix a Lorentz frame O such that the initial and final velocities are zero.\nThe **Pauli–Lubanski spin vector** Sμ is defined to be\n(0, Si)\nin the system's rest frame, with Si the angular-momentum three-vector about the center of mass.\nIn the motion from initial to final position, Sμ undergoes a rotation, as recorded in O, from its initial to its final value.\nThis continuous change is the Thomas precession.\n\n## Statement\n[[File:Thomas precession coefficient.svg|thumb|400px|Value of\nγ2/(γ + 1)\nas\nincreases, with v the instantaneous magnitude of the particle's velocity.\nThe Thomas rotation is negligible for\nβ < 0.5\n, increases steadily for\n0.5 < β < 0.8\n, then rapidly shoots to infinity as\nβ\ntends to 1.\nThe \"Thomas half\" is evident in the low-speed limit, and the rotation is only very clear for speeds approaching that of light.]]\nConsider the motion of a [particle](https://bliptext.com/articles/particle).\nIntroduce a [lab frame](https://bliptext.com/articles/lab-frame)\nΣ\nin which an observer can measure the relative motion of the particle.\nAt each instant of time the particle has an [inertial frame](https://bliptext.com/articles/inertial-frame) in which it is at rest.\nRelative to this lab frame, the instantaneous velocity of the particle is\nv(t)\nwith magnitude\nbounded by the [speed of light](https://bliptext.com/articles/speed-of-light) c, so that\n0 ≤ v < c\n. Here the time t is the [coordinate time](https://bliptext.com/articles/coordinate-time) as measured in the lab frame, not the [proper time](https://bliptext.com/articles/proper-time) of the particle.\nApart from the upper limit on magnitude, the velocity of the particle is arbitrary and not necessarily constant; its corresponding vector of [acceleration](https://bliptext.com/articles/acceleration) is\n. As a result of the Wigner rotation at every instant, the particle's frame precesses with an angular velocity given by the equation\nwhere × is the [cross product](https://bliptext.com/articles/cross-product) and\nis the instantaneous [Lorentz factor](https://bliptext.com/articles/lorentz-factor), a function of the particle's instantaneous velocity.\nLike any angular velocity,\nωT\nis a [pseudovector](https://bliptext.com/articles/pseudovector); its magnitude is the angular speed the particle's frame precesses (in [radians](https://bliptext.com/articles/radian) per second), and the direction points along the rotation axis.\nAs is usual, the right-hand convention of the cross product is used (see [right-hand rule](https://bliptext.com/articles/right-hand-rule)).\nThe precession depends on accelerated motion, and the non-[collinearity](https://bliptext.com/articles/collinear) of the particle's instantaneous velocity and acceleration.\nNo precession occurs if the particle moves with uniform velocity (constant\nv\nso\n), or accelerates in a straight line (in which case\nv\nand\na\nare parallel or antiparallel so their cross product is zero). The particle has to move in a curve, say an arc, [spiral](https://bliptext.com/articles/spiral), [helix](https://bliptext.com/articles/helix), or a [circular orbit](https://bliptext.com/articles/circular-orbit) or [elliptical orbit](https://bliptext.com/articles/elliptical-orbit), for its frame to precess. The angular velocity of the precession is a maximum if the velocity and acceleration vectors are perpendicular throughout the motion (a [circular orbit](https://bliptext.com/articles/circular-orbit)), and is large if their magnitudes are large (the magnitude of\nv\nis almost c).\nIn the non-relativistic limit,\nv → 0\nso\nγ → 1\n, and the angular velocity is approximately\nThe factor of 1/2 turns out to be the critical factor to agree with experimental results.\nIt is informally known as the \"Thomas half\".\n\n# Mathematical explanation\n\n## Lorentz transformations\nThe description of relative motion involves [Lorentz transformations](https://bliptext.com/articles/lorentz-transformation), and it is convenient to use them in [matrix](https://bliptext.com/articles/matrix-mathematics) form; symbolic [matrix](https://bliptext.com/articles/matrix-mathematics) expressions summarize the transformations and are easy to manipulate, and when required the full matrices can be written explicitly.\nAlso, to prevent extra factors of c cluttering the equations, it is convenient to use the definition\nwith magnitude\nsuch that\n0 ≤ β < 1\n.\nThe spacetime coordinates of the lab frame are collected into a 4×1 [column vector](https://bliptext.com/articles/column-vector), and the [boost](https://bliptext.com/articles/lorentz-transformation) is represented as a 4×4 [symmetric matrix](https://bliptext.com/articles/symmetric-matrix), respectively\nand turn\nis the Lorentz factor of\nβ\n. In other frames, the corresponding coordinates are also arranged into column vectors.\nThe [inverse matrix](https://bliptext.com/articles/inverse-matrix) of the boost corresponds to a boost in the opposite direction, and is given by\nB(β)−1 = B(−β)\n.\nAt an instant of lab-recorded time t measured in the lab frame, the transformation of spacetime coordinates from the lab frame\nΣ\nto the particle's frame Σ\n′\nis\nand at later lab-recorded time\nt + Δt\nwe can define a new frame\nΣ′′\nfor the particle, which moves with velocity\nβ + Δβ\nrelative to\nΣ\n, and the corresponding boost is\nThe vectors\nβ\nand\nΔβ\nare two separate vectors.\nThe latter is a small increment, and can be conveniently split into components parallel (‖) and perpendicular (⊥) to\nβ\nCombining and obtains the Lorentz transformation between\nΣ′\nand\nΣ′′\n,\nand this composition contains all the required information about the motion between these two lab times.\nNotice\nB(β + Δβ)B(−β)\nand\nB(β + Δβ)\nare infinitesimal transformations because they involve a small increment in the relative velocity, while\nB(−β)\nis not.\nThe composition of two boosts equates to a single boost combined with a [Wigner rotation](https://bliptext.com/articles/wigner-rotation) about an axis perpendicular to the relative velocities;\nThe rotation is given by is a 4×4 rotation matrix\nR\nin the [axis–angle representation](https://bliptext.com/articles/axis-angle-representation), and coordinate systems are taken to be [right-handed](https://bliptext.com/articles/orientation-vector-space).\nThis matrix rotates 3d vectors anticlockwise about an axis ([active transformation](https://bliptext.com/articles/active-transformation)), or equivalently rotates coordinate frames clockwise about the same axis (passive transformation).\nThe axis-angle vector\nΔθ\nparametrizes the rotation, its magnitude\nΔθ\nis the angle\nΣ′′\nhas rotated, and direction is parallel to the rotation axis, in this case the axis is parallel to the [cross product](https://bliptext.com/articles/cross-product)\n(−β)×(β + Δβ) = −β×Δβ\n. If the angles are negative, then the sense of rotation is reversed.\nThe inverse matrix is given by\nR(Δθ)−1 = R(−Δθ)\n.\nCorresponding to the boost is the (small change in the) boost vector\nΔb\n, with magnitude and direction of the relative velocity of the boost (divided by\nc\n). The boost\nB(Δb)\nand rotation\nR(Δθ)\nhere are infinitesimal transformations because\nΔb\nand rotation\nΔθ\nare small.\nThe rotation gives rise to the Thomas precession, but there is a subtlety.\nTo interpret the particle's frame as a co-moving inertial frame relative to the lab frame, and agree with the non-relativistic limit, we expect the transformation between the particle's instantaneous frames at times\nt\nand\nt + Δt\nto be related by a boost without rotation.\nCombining and and rearranging gives\nwhere another instantaneous frame\nΣ′′′\nis introduced with coordinates\nX′′′\n, to prevent conflation with\nΣ′′\n. To summarize the frames of reference: in the lab frame\nΣ\nan observer measures the motion of the particle, and three instantaneous inertial frames in which the particle is at rest are\nΣ′\n(at time t),\nΣ′′\n(at time\nt + Δt\n), and\nΣ′′′\n(at time\nt + Δt\n). The frames\nΣ′′\nand\nΣ′′′\nare at the same location and time, they differ only by a rotation.\nBy contrast\nΣ′\nand\nΣ′′′\ndiffer by a boost and lab time interval\nΔt\n.\nRelating the coordinates\nX′′′\nto the lab coordinates\nX\nvia and ;\nthe frame\nΣ′′′\nis rotated in the negative sense.\nThe rotation is between two instants of lab time.\nAs\nΔt → 0\n, the particle's frame rotates at every instant, and the continuous motion of the particle amounts to a continuous rotation with an [angular velocity](https://bliptext.com/articles/angular-velocity) at every instant.\nDividing\n−Δθ\nby\nΔt\n, and taking the [limit](https://bliptext.com/articles/limit-mathematics)\nΔt → 0\n, the angular velocity is by definition\nIt remains to find what\nΔθ\nprecisely is.\n\n## Extracting the formula\nThe composition can be obtained by explicitly calculating the matrix product.\nThe boost matrix of\nβ + Δβ\nwill require the magnitude and Lorentz factor of this vector.\nSince\nΔβ\nis small, terms of \"second order\"\n,\n(Δβx)2\n,\n(Δβy)2\n,\nΔβxΔβy\nand higher are negligible.\nTaking advantage of this fact, the magnitude squared of the vector is\nand expanding the Lorentz factor of\nβ + Δβ\nas a [power series](https://bliptext.com/articles/power-series) gives to first order in\nΔβ\n,\nusing the Lorentz factor\nγ\nof\nβ\nas above.\nIntroducing the boost generators\nand rotation generators\nalong with the [dot product](https://bliptext.com/articles/dot-product) · facilitates the coordinate independent expression\nwhich holds if\nβ\nand\nΔβ\nlie in any plane.\nThis is an infinitesimal Lorentz transformation in the form of a combined boost and rotation\nwhere\nAfter dividing\nΔθ\nby\nΔt\nand taking the limit as in, one obtains the instantaneous angular velocity\nwhere\na\nis the [acceleration](https://bliptext.com/articles/acceleration) of the particle as observed in the lab frame.\nNo forces were specified or used in the derivation so the precession is a kinematical effect - it arises from the geometric aspects of motion.\nHowever, forces cause accelerations, so the Thomas precession is observed if the particle is subject to forces.\nThomas precession can also be derived using the [Fermi-Walker transport equation](https://bliptext.com/articles/fermi-walker-transport).\nOne assumes uniform circular motion in flat Minkowski spacetime.\nThe spin 4-vector is orthogonal to the velocity 4-vector.\nFermi-Walker transport preserves this relation.\nOne finds that the dot product of the acceleration 4-vector with the spin 4-vector varies sinusoidally with time with an angular frequency γω, where ω is the angular frequency of the circular motion and γ=1/√(1-v^2/c^2), the [Lorentz factor](https://bliptext.com/articles/lorentz-factor).\nThis is easily shown by taking the second time derivative of that dot product.\nBecause this angular frequency exceeds ω, the spin precesses in the retrograde direction.\nThe difference (γ-1)ω is the Thomas precession angular frequency already given, as is simply shown by realizing that the magnitude of the 3-acceleration is ω v.\n\n# Applications\n\n## In electron orbitals\nIn quantum mechanics **Thomas precession** is a correction to the [spin-orbit interaction](https://bliptext.com/articles/spin-orbit-interaction), which takes into account the relativistic [time dilation](https://bliptext.com/articles/time-dilation) between the [electron](https://bliptext.com/articles/electron) and the [nucleus](https://bliptext.com/articles/atomic-nucleus) in [hydrogenic atoms](https://bliptext.com/articles/hydrogen-like-atom).\nBasically, it states that spinning objects [precess](https://bliptext.com/articles/precess) when they accelerate in [special relativity](https://bliptext.com/articles/special-relativity) because [Lorentz boosts](https://bliptext.com/articles/lorentz-boost) do not commute with each other.\nTo calculate the spin of a particle in a [magnetic field](https://bliptext.com/articles/magnetic-field), one must also take into account [Larmor precession](https://bliptext.com/articles/larmor-precession).\n\n## In a Foucault pendulum\nThe rotation of the swing plane of [Foucault pendulum](https://bliptext.com/articles/foucault-pendulum) can be treated as a result of [parallel transport](https://bliptext.com/articles/parallel-transport) of the pendulum in a 2-dimensional sphere of Euclidean space.\nThe [hyperbolic](https://bliptext.com/articles/hyperbolic-geometry) space of velocities in [Minkowski spacetime](https://bliptext.com/articles/minkowski-spacetime) represents a 3-dimensional (pseudo-) sphere with imaginary radius and imaginary timelike coordinate.\nParallel transport of a spinning particle in relativistic velocity space leads to Thomas precession, which is similar to the rotation of the swing plane of a Foucault pendulum.\nThe angle of rotation in both cases is determined by the area integral of curvature in agreement with the [Gauss–Bonnet theorem](https://bliptext.com/articles/gauss-bonnet-theorem).\nThomas precession gives a correction to the precession of a Foucault pendulum.\nFor a Foucault pendulum located in the city of Nijmegen in the Netherlands the correction is:\nNote that it is more than two orders of magnitude smaller than the precession due to the general-relativistic correction arising from [frame-dragging](https://bliptext.com/articles/frame-dragging), the [Lense–Thirring precession](https://bliptext.com/articles/lense-thirring-precession).\n\n# Remarks\n\n# Textbooks","revisionCount":0},"hasMore":false,"nextCursor":null}