[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/relativistic-angular-momentum-wikipedia\/#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/en\/wiki24\/relativistic-angular-momentum-wikipedia\/","headline":"Relativistic angular momentum – Wikipedia","name":"Relativistic angular momentum – Wikipedia","description":"before-content-x4 Angular momentum in special and general relativity after-content-x4 In physics, relativistic angular momentum refers to the mathematical formalisms and","datePublished":"2018-04-20","dateModified":"2018-04-20","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/en\/wiki24\/author\/lordneo\/","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/7\/7c\/Angular_momentum_bivector_and_pseudovector.svg\/275px-Angular_momentum_bivector_and_pseudovector.svg.png","url":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/7\/7c\/Angular_momentum_bivector_and_pseudovector.svg\/275px-Angular_momentum_bivector_and_pseudovector.svg.png","height":"188","width":"275"},"url":"https:\/\/wiki.edu.vn\/en\/wiki24\/relativistic-angular-momentum-wikipedia\/","wordCount":37097,"articleBody":" (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4Angular momentum in special and general relativity (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4In physics, relativistic angular momentum refers to the mathematical formalisms and physical concepts that define angular momentum in special relativity (SR) and general relativity (GR). The relativistic quantity is subtly different from the three-dimensional quantity in classical mechanics.Angular momentum is an important dynamical quantity derived from position and momentum. It is a measure of an object’s rotational motion and resistance to changes in its rotation. Also, in the same way momentum conservation corresponds to translational symmetry, angular momentum conservation corresponds to rotational symmetry \u2013 the connection between symmetries and conservation laws is made by Noether’s theorem. While these concepts were originally discovered in classical mechanics, they are also true and significant in special and general relativity. In terms of abstract algebra, the invariance of angular momentum, four-momentum, and other symmetries in spacetime, are described by the Lorentz group, or more generally the Poincar\u00e9 group.Physical quantities that remain separate in classical physics are naturally combined in SR and GR by enforcing the postulates of relativity. Most notably, the space and time coordinates combine into the four-position, and energy and momentum combine into the four-momentum. The components of these four-vectors depend on the frame of reference used, and change under Lorentz transformations to other inertial frames or accelerated frames. (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Relativistic angular momentum is less obvious. The classical definition of angular momentum is the cross product of position x with momentum p to obtain a pseudovector x \u00d7 p, or alternatively as the exterior product to obtain a second order antisymmetric tensor x \u2227 p. What does this combine with, if anything? There is another vector quantity not often discussed \u2013 it is the time-varying moment of mass polar-vector (not the moment of inertia) related to the boost of the centre of mass of the system, and this combines with the classical angular momentum pseudovector to form an antisymmetric tensor of second order, in exactly the same way as the electric field polar-vector combines with the magnetic field pseudovector to form the electromagnetic field antisymmetric tensor. For rotating mass\u2013energy distributions (such as gyroscopes, planets, stars, and black holes) instead of point-like particles, the angular momentum tensor is expressed in terms of the stress\u2013energy tensor of the rotating object.In special relativity alone, in the rest frame of a spinning object, there is an intrinsic angular momentum analogous to the “spin” in quantum mechanics and relativistic quantum mechanics, although for an extended body rather than a point particle. In relativistic quantum mechanics, elementary particles have spin and this is an additional contribution to the orbital angular momentum operator, yielding the total angular momentum tensor operator. In any case, the intrinsic “spin” addition to the orbital angular momentum of an object can be expressed in terms of the Pauli\u2013Lubanski pseudovector.[1]Table of Contents (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4Definitions[edit]Orbital 3d angular momentum[edit]Dynamic mass moment[edit]Special relativity[edit]Coordinate transformations for a boost in the x direction[edit]Vector transformations for a boost in any direction[edit]4d angular momentum as a bivector[edit]Rigid body rotation[edit]Spin in special relativity[edit]Four-spin[edit]The Pauli\u2013Lubanski pseudovector[edit]Spin\u2013orbital decomposition[edit]Angular momentum of a mass\u2013energy\u2013momentum distribution[edit]Angular momentum from the mass\u2013energy\u2013momentum tensor[edit]Angular momentum about the centre of mass[edit]Angular momentum conservation[edit]Torque in special relativity[edit]Angular momentum as the generator of spacetime boosts and rotations[edit]Angular momentum in general relativity[edit]See also[edit]References[edit]Further reading[edit]Special relativity[edit]General relativity[edit]External links[edit]Definitions[edit] The 3-angular momentum as a bivector (plane element) and axial vector, of a particle of mass m with instantaneous 3-position x and 3-momentum p.Orbital 3d angular momentum[edit]For reference and background, two closely related forms of angular momentum are given.In classical mechanics, the orbital angular momentum of a particle with instantaneous three-dimensional position vector x = (x, y, z) and momentum vector p = (px, py, pz), is defined as the axial vectorL=x\u00d7p{displaystyle mathbf {L} =mathbf {x} times mathbf {p} }which has three components, that are systematically given by cyclic permutations of Cartesian directions (e.g. change x to y, y to z, z to x, repeat)Lx=ypz\u2212zpy,Ly=zpx\u2212xpz,Lz=xpy\u2212ypx.{displaystyle {begin{aligned}L_{x}&=yp_{z}-zp_{y},,\\L_{y}&=zp_{x}-xp_{z},,\\L_{z}&=xp_{y}-yp_{x},.end{aligned}}}A related definition is to conceive orbital angular momentum as a plane element. This can be achieved by replacing the cross product by the exterior product in the language of exterior algebra, and angular momentum becomes a contravariant second order antisymmetric tensor[2]L=x\u2227p{displaystyle mathbf {L} =mathbf {x} wedge mathbf {p} }or writing x = (x1, x2, x3) = (x, y, z) and momentum vector p = (p1, p2, p3) = (px, py, pz), the components can be compactly abbreviated in tensor index notationLij=xipj\u2212xjpi{displaystyle L^{ij}=x^{i}p^{j}-x^{j}p^{i}}where the indices i and j take the values 1, 2, 3. On the other hand, the components can be systematically displayed fully in a 3 \u00d7 3 antisymmetric matrixL=(L11L12L13L21L22L23L31L32L33)=(0LxyLxzLyx0LyzLzxLzy0)=(0Lxy\u2212Lzx\u2212Lxy0LyzLzx\u2212Lyz0)=(0xpy\u2212ypx\u2212(zpx\u2212xpz)\u2212(xpy\u2212ypx)0ypz\u2212zpyzpx\u2212xpz\u2212(ypz\u2212zpy)0){displaystyle {begin{aligned}mathbf {L} &={begin{pmatrix}L^{11}&L^{12}&L^{13}\\L^{21}&L^{22}&L^{23}\\L^{31}&L^{32}&L^{33}\\end{pmatrix}}={begin{pmatrix}0&L_{xy}&L_{xz}\\L_{yx}&0&L_{yz}\\L_{zx}&L_{zy}&0end{pmatrix}}={begin{pmatrix}0&L_{xy}&-L_{zx}\\-L_{xy}&0&L_{yz}\\L_{zx}&-L_{yz}&0end{pmatrix}}\\&={begin{pmatrix}0&xp_{y}-yp_{x}&-(zp_{x}-xp_{z})\\-(xp_{y}-yp_{x})&0&yp_{z}-zp_{y}\\zp_{x}-xp_{z}&-(yp_{z}-zp_{y})&0end{pmatrix}}end{aligned}}}This quantity is additive, and for an isolated system, the total angular momentum of a system is conserved.Dynamic mass moment[edit]In classical mechanics, the three-dimensional quantity for a particle of mass m moving with velocity u[2][3]N=m(x\u2212tu)=mx\u2212tp{displaystyle mathbf {N} =mleft(mathbf {x} -tmathbf {u} right)=mmathbf {x} -tmathbf {p} }has the dimensions of mass moment \u2013 length multiplied by mass. It is equal to the mass of the particle or system of particles multiplied by the distance from the space origin to the centre of mass (COM) at the time origin (t=0), as measured in the lab frame. There is no universal symbol, nor even a universal name, for this quantity. Different authors may denote it by other symbols if any (for example \u03bc), may designate other names, and may define N to be the negative of what is used here. The above form has the advantage that it resembles the familiar Galilean transformation for position, which in turn is the non-relativistic boost transformation between inertial frames.This vector is also additive: for a system of particles, the vector sum is the resultant\u2211nNn=\u2211nmn(xn\u2212tun)=(xCOM\u2211nmn\u2212t\u2211nmnun)=Mtot(xCOM\u2212uCOMt){displaystyle sum _{n}mathbf {N} _{n}=sum _{n}m_{n}left(mathbf {x} _{n}-tmathbf {u} _{n}right)=left(mathbf {x} _{mathrm {COM} }sum _{n}m_{n}-tsum _{n}m_{n}mathbf {u} _{n}right)=M_{text{tot}}(mathbf {x} _{mathrm {COM} }-mathbf {u} _{mathrm {COM} }t)}where the system’s centre of mass position and velocity and total mass are respectivelyxCOM=\u2211nmnxn\u2211nmn,uCOM=\u2211nmnun\u2211nmn,Mtot=\u2211nmn.{displaystyle {begin{aligned}mathbf {x} _{mathrm {COM} }&={frac {sum _{n}m_{n}mathbf {x} _{n}}{sum _{n}m_{n}}},\\[3pt]mathbf {u} _{mathrm {COM} }&={frac {sum _{n}m_{n}mathbf {u} _{n}}{sum _{n}m_{n}}},\\[3pt]M_{text{tot}}&=sum _{n}m_{n}.end{aligned}}}For an isolated system, N is conserved in time, which can be seen by differentiating with respect to time. The angular momentum L is a pseudovector, but N is an “ordinary” (polar) vector, and is therefore invariant under inversion.The resultant Ntot for a multiparticle system has the physical visualization that, whatever the complicated motion of all the particles are, they move in such a way that the system’s COM moves in a straight line. This does not necessarily mean all particles “follow” the COM, nor that all particles all move in almost the same direction simultaneously, only that the motion of all the particles are constrained in relation to the centre of mass.In special relativity, if the particle moves with velocity u relative to the lab frame, thenE=\u03b3(u)m0c2,p=\u03b3(u)m0u{displaystyle {begin{aligned}E&=gamma (mathbf {u} )m_{0}c^{2},&mathbf {p} &=gamma (mathbf {u} )m_{0}mathbf {u} end{aligned}}}where\u03b3(u)=11\u2212u\u22c5uc2{displaystyle gamma (mathbf {u} )={frac {1}{sqrt {1-{frac {mathbf {u} cdot mathbf {u} }{c^{2}}}}}}}is the Lorentz factor and m is the mass (i.e. the rest mass) of the particle. The corresponding relativistic mass moment in terms of m, u, p, E, in the same lab frame isN=Ec2x\u2212pt=m\u03b3(u)(x\u2212ut).{displaystyle mathbf {N} ={frac {E}{c^{2}}}mathbf {x} -mathbf {p} t=mgamma (mathbf {u} )(mathbf {x} -mathbf {u} t).}The Cartesian components areNx=mx\u2212pxt=Ec2x\u2212pxt=m\u03b3(u)(x\u2212uxt)Ny=my\u2212pyt=Ec2y\u2212pyt=m\u03b3(u)(y\u2212uyt)Nz=mz\u2212pzt=Ec2z\u2212pzt=m\u03b3(u)(z\u2212uzt){displaystyle {begin{aligned}N_{x}=mx-p_{x}t&={frac {E}{c^{2}}}x-p_{x}t=mgamma (u)(x-u_{x}t)\\N_{y}=my-p_{y}t&={frac {E}{c^{2}}}y-p_{y}t=mgamma (u)(y-u_{y}t)\\N_{z}=mz-p_{z}t&={frac {E}{c^{2}}}z-p_{z}t=mgamma (u)(z-u_{z}t)end{aligned}}}Special relativity[edit]Coordinate transformations for a boost in the x direction[edit]Consider a coordinate frame F\u2032 which moves with velocity v = (v, 0, 0) relative to another frame F, along the direction of the coincident xx\u2032 axes. The origins of the two coordinate frames coincide at times t = t\u2032 = 0. The mass\u2013energy E = mc2 and momentum components p = (px, py, pz) of an object, as well as position coordinates x = (x, y, z) and time t in frame F are transformed to E\u2032 = m\u2032c2, p\u2032 = (px\u2032, py\u2032, pz\u2032), x\u2032 = (x\u2032, y\u2032, z\u2032), and t\u2032 in F\u2032 according to the Lorentz transformationst\u2032=\u03b3(v)(t\u2212vxc2),E\u2032=\u03b3(v)(E\u2212vpx)x\u2032=\u03b3(v)(x\u2212vt),px\u2032=\u03b3(v)(px\u2212vEc2)y\u2032=y,py\u2032=pyz\u2032=z,pz\u2032=pz{displaystyle {begin{aligned}t’&=gamma (v)left(t-{frac {vx}{c^{2}}}right),,quad &E’&=gamma (v)left(E-vp_{x}right)\\x’&=gamma (v)(x-vt),,quad &p_{x}’&=gamma (v)left(p_{x}-{frac {vE}{c^{2}}}right)\\y’&=y,,quad &p_{y}’&=p_{y}\\z’&=z,,quad &p_{z}’&=p_{z}\\end{aligned}}}The Lorentz factor here applies to the velocity v, the relative velocity between the frames. This is not necessarily the same as the velocity u of an object.For the orbital 3-angular momentum L as a pseudovector, we haveLx\u2032=y\u2032pz\u2032\u2212z\u2032py\u2032=LxLy\u2032=z\u2032px\u2032\u2212x\u2032pz\u2032=\u03b3(v)(Ly\u2212vNz)Lz\u2032=x\u2032py\u2032\u2212y\u2032px\u2032=\u03b3(v)(Lz+vNy){displaystyle {begin{aligned}L_{x}’&=y’p_{z}’-z’p_{y}’=L_{x}\\L_{y}’&=z’p_{x}’-x’p_{z}’=gamma (v)(L_{y}-vN_{z})\\L_{z}’&=x’p_{y}’-y’p_{x}’=gamma (v)(L_{z}+vN_{y})\\end{aligned}}}DerivationFor the x-componentLx\u2032=y\u2032pz\u2032\u2212z\u2032py\u2032=ypz\u2212zpy=Lx{displaystyle L_{x}’=y’p_{z}’-z’p_{y}’=yp_{z}-zp_{y}=L_{x}}the y-componentLy\u2032=z\u2032px\u2032\u2212x\u2032pz\u2032=z\u03b3(px\u2212vEc2)\u2212\u03b3(x\u2212vt)pz=\u03b3[zpx\u2212zvEc2\u2212xpz+vtpz]=\u03b3[(zpx\u2212xpz)+v(pzt\u2212zEc2)]=\u03b3(Ly\u2212vNz){displaystyle {begin{aligned}L_{y}’&=z’p_{x}’-x’p_{z}’\\&=zgamma left(p_{x}-{frac {vE}{c^{2}}}right)-gamma left(x-vtright)p_{z}\\&=gamma left[zp_{x}-z{frac {vE}{c^{2}}}-xp_{z}+vtp_{z}right]\\&=gamma left[left(zp_{x}-xp_{z}right)+vleft(p_{z}t-z{frac {E}{c^{2}}}right)right]\\&=gamma left(L_{y}-vN_{z}right)end{aligned}}}and z-componentLz\u2032=x\u2032py\u2032\u2212y\u2032px\u2032=\u03b3(x\u2212vt)py\u2212y\u03b3(px\u2212vEc2)=\u03b3[xpy\u2212vtpy\u2212ypx+yvEc2]=\u03b3[(xpy\u2212ypx)+v(yEc2\u2212tpy)]=\u03b3(Lz+vNy){displaystyle {begin{aligned}L_{z}’&=x’p_{y}’-y’p_{x}’\\&=gamma left(x-vtright)p_{y}-ygamma left(p_{x}-{frac {vE}{c^{2}}}right)\\&=gamma left[xp_{y}-vtp_{y}-yp_{x}+y{frac {vE}{c^{2}}}right]\\&=gamma left[left(xp_{y}-yp_{x}right)+vleft(y{frac {E}{c^{2}}}-tp_{y}right)right]\\&=gamma left(L_{z}+vN_{y}right)end{aligned}}}In the second terms of Ly\u2032 and Lz\u2032, the y and z components of the cross product v \u00d7 N can be inferred by recognizing cyclic permutations of vx = v and vy = vz = 0 with the components of N,\u2212vNz=vzNx\u2212vxNz=(v\u00d7N)yvNy=vxNy\u2212vyNx=(v\u00d7N)z{displaystyle {begin{aligned}-vN_{z}&=v_{z}N_{x}-v_{x}N_{z}=left(mathbf {v} times mathbf {N} right)_{y}\\vN_{y}&=v_{x}N_{y}-v_{y}N_{x}=left(mathbf {v} times mathbf {N} right)_{z}\\end{aligned}}}Now, Lx is parallel to the relative velocity v, and the other components Ly and Lz are perpendicular to v. The parallel\u2013perpendicular correspondence can be facilitated by splitting the entire 3-angular momentum pseudovector into components parallel (\u2225) and perpendicular (\u22a5) to v, in each frame,L=L\u2225+L\u22a5,L\u2032=L\u2225\u2032+L\u22a5\u2032.{displaystyle mathbf {L} =mathbf {L} _{parallel }+mathbf {L} _{perp },,quad mathbf {L} ‘=mathbf {L} _{parallel }’+mathbf {L} _{perp }’,.}Then the component equations can be collected into the pseudovector equationsL\u2225\u2032=L\u2225L\u22a5\u2032=\u03b3(v)(L\u22a5+v\u00d7N){displaystyle {begin{aligned}mathbf {L} _{parallel }’&=mathbf {L} _{parallel }\\mathbf {L} _{perp }’&=gamma (mathbf {v} )left(mathbf {L} _{perp }+mathbf {v} times mathbf {N} right)\\end{aligned}}}Therefore, the components of angular momentum along the direction of motion do not change, while the components perpendicular do change. By contrast to the transformations of space and time, time and the spatial coordinates change along the direction of motion, while those perpendicular do not.These transformations are true for all v, not just for motion along the xx\u2032 axes.Considering L as a tensor, we get a similar resultL\u22a5\u2032=\u03b3(v)(L\u22a5+v\u2227N){displaystyle mathbf {L} _{perp }’=gamma (mathbf {v} )left(mathbf {L} _{perp }+mathbf {v} wedge mathbf {N} right)}wherevzNx\u2212vxNz=(v\u2227N)zxvxNy\u2212vyNx=(v\u2227N)xy{displaystyle {begin{aligned}v_{z}N_{x}-v_{x}N_{z}&=left(mathbf {v} wedge mathbf {N} right)_{zx}\\v_{x}N_{y}-v_{y}N_{x}&=left(mathbf {v} wedge mathbf {N} right)_{xy}\\end{aligned}}}The boost of the dynamic mass moment along the x direction isNx\u2032=m\u2032x\u2032\u2212px\u2032t\u2032=NxNy\u2032=m\u2032y\u2032\u2212py\u2032t\u2032=\u03b3(v)(Ny+vLzc2)Nz\u2032=m\u2032z\u2032\u2212pz\u2032t\u2032=\u03b3(v)(Nz\u2212vLyc2){displaystyle {begin{aligned}N_{x}’&=m’x’-p_{x}’t’=N_{x}\\N_{y}’&=m’y’-p_{y}’t’=gamma (v)left(N_{y}+{frac {vL_{z}}{c^{2}}}right)\\N_{z}’&=m’z’-p_{z}’t’=gamma (v)left(N_{z}-{frac {vL_{y}}{c^{2}}}right)\\end{aligned}}}DerivationFor the x-componentNx\u2032=E\u2032c2x\u2032\u2212t\u2032px\u2032=\u03b3c2(E\u2212vpx)\u03b3(x\u2212vt)\u2212\u03b3(t\u2212xvc2)\u03b3(px\u2212vEc2)=\u03b32[1c2(E\u2212vpx)(x\u2212vt)\u2212(t\u2212xvc2)(px\u2212vEc2)]=\u03b32[Exc2\u2212Evtc2\u2212vpxxc2+vpxvtc2\u2212tpx+xvc2px+tvEc2\u2212xvc2vEc2]=\u03b32[Exc2\u2212Evtc2\u2212vpxxc2+v2c2pxt\u2212tpx+xvc2px+tvEc2\u2212v2c2Exc2]=\u03b32[(Exc2\u2212tpx)+v2c2(pxt\u2212Exc2)]=\u03b32[1\u2212v2c2]Nx=\u03b321\u03b32Nx{displaystyle {begin{aligned}N_{x}’&={frac {E’}{c^{2}}}x’-t’p_{x}’\\&={frac {gamma }{c^{2}}}(E-vp_{x})gamma (x-vt)-gamma left(t-{frac {xv}{c^{2}}}right)gamma left(p_{x}-{frac {vE}{c^{2}}}right)\\&=gamma ^{2}left[{frac {1}{c^{2}}}left(E-vp_{x}right)(x-vt)-left(t-{frac {xv}{c^{2}}}right)left(p_{x}-{frac {vE}{c^{2}}}right)right]\\&=gamma ^{2}left[{frac {Ex}{c^{2}}}-{frac {Evt}{c^{2}}}-{frac {vp_{x}x}{c^{2}}}+{frac {vp_{x}vt}{c^{2}}}-tp_{x}+{frac {xv}{c^{2}}}p_{x}+t{frac {vE}{c^{2}}}-{frac {xv}{c^{2}}}{frac {vE}{c^{2}}}right]\\&=gamma ^{2}left[{frac {Ex}{c^{2}}}{cancel {-{frac {Evt}{c^{2}}}}}{cancel {-{frac {vp_{x}x}{c^{2}}}}}+{frac {v^{2}}{c^{2}}}p_{x}t-tp_{x}{cancel {+{frac {xv}{c^{2}}}p_{x}}}{cancel {+t{frac {vE}{c^{2}}}}}-{frac {v^{2}}{c^{2}}}{frac {Ex}{c^{2}}}right]\\&=gamma ^{2}left[left({frac {Ex}{c^{2}}}-tp_{x}right)+{frac {v^{2}}{c^{2}}}left(p_{x}t-{frac {Ex}{c^{2}}}right)right]\\&=gamma ^{2}left[1-{frac {v^{2}}{c^{2}}}right]N_{x}\\&=gamma ^{2}{frac {1}{gamma ^{2}}}N_{x}end{aligned}}}the y-componentNy\u2032=E\u2032c2y\u2032\u2212t\u2032py\u2032=1c2\u03b3(E\u2212vpx)y\u2212\u03b3(t\u2212xvc2)py=\u03b3[1c2(E\u2212vpx)y\u2212(t\u2212xvc2)py]=\u03b3[1c2Ey\u22121c2vpxy\u2212tpy+xvc2py]=\u03b3[(1c2Ey\u2212tpy)+vc2(xpy\u2212ypx)]=\u03b3(Ny+vc2Lz){displaystyle {begin{aligned}N_{y}’&={frac {E’}{c^{2}}}y’-t’p_{y}’\\&={frac {1}{c^{2}}}gamma (E-vp_{x})y-gamma left(t-{frac {xv}{c^{2}}}right)p_{y}\\&=gamma left[{frac {1}{c^{2}}}(E-vp_{x})y-left(t-{frac {xv}{c^{2}}}right)p_{y}right]\\&=gamma left[{frac {1}{c^{2}}}Ey-{frac {1}{c^{2}}}vp_{x}y-tp_{y}+{frac {xv}{c^{2}}}p_{y}right]\\&=gamma left[left({frac {1}{c^{2}}}Ey-tp_{y}right)+{frac {v}{c^{2}}}(xp_{y}-yp_{x})right]\\&=gamma left(N_{y}+{frac {v}{c^{2}}}L_{z}right)end{aligned}}}and z-componentNz\u2032=E\u2032c2z\u2032\u2212t\u2032pz\u2032=1c2\u03b3(E\u2212vpx)z\u2212\u03b3(t\u2212xvc2)pz=\u03b3[1c2(E\u2212vpx)z\u2212(t\u2212xvc2)pz]=\u03b3[1c2Ez\u22121c2vpzz\u2212tpz+xvc2pz]=\u03b3[(1c2Ez\u2212tpz)+vc2(xpz\u2212zpx)]=\u03b3(Nz\u2212vc2Ly){displaystyle {begin{aligned}N_{z}’&={frac {E’}{c^{2}}}z’-t’p_{z}’\\&={frac {1}{c^{2}}}gamma (E-vp_{x})z-gamma left(t-{frac {xv}{c^{2}}}right)p_{z}\\&=gamma left[{frac {1}{c^{2}}}(E-vp_{x})z-left(t-{frac {xv}{c^{2}}}right)p_{z}right]\\&=gamma left[{frac {1}{c^{2}}}Ez-{frac {1}{c^{2}}}vp_{z}z-tp_{z}+{frac {xv}{c^{2}}}p_{z}right]\\&=gamma left[left({frac {1}{c^{2}}}Ez-tp_{z}right)+{frac {v}{c^{2}}}(xp_{z}-zp_{x})right]\\&=gamma left(N_{z}-{frac {v}{c^{2}}}L_{y}right)end{aligned}}}Collecting parallel and perpendicular components as beforeN\u2225\u2032=N\u2225N\u22a5\u2032=\u03b3(v)(N\u22a5\u22121c2v\u00d7L){displaystyle {begin{aligned}mathbf {N} _{parallel }’&=mathbf {N} _{parallel }\\mathbf {N} _{perp }’&=gamma (mathbf {v} )left(mathbf {N} _{perp }-{frac {1}{c^{2}}}mathbf {v} times mathbf {L} right)\\end{aligned}}}Again, the components parallel to the direction of relative motion do not change, those perpendicular do change.Vector transformations for a boost in any direction[edit]So far these are only the parallel and perpendicular decompositions of the vectors. The transformations on the full vectors can be constructed from them as follows (throughout here L is a pseudovector for concreteness and compatibility with vector algebra).Introduce a unit vector in the direction of v, given by n = v\/v. The parallel components are given by the vector projection of L or N into nL\u2225=(L\u22c5n)n,N\u2225=(N\u22c5n)n{displaystyle mathbf {L} _{parallel }=(mathbf {L} cdot mathbf {n} )mathbf {n} ,,quad mathbf {N} _{parallel }=(mathbf {N} cdot mathbf {n} )mathbf {n} }while the perpendicular component by vector rejection of L or N from nL\u22a5=L\u2212(L\u22c5n)n,N\u22a5=N\u2212(N\u22c5n)n{displaystyle mathbf {L} _{perp }=mathbf {L} -(mathbf {L} cdot mathbf {n} )mathbf {n} ,,quad mathbf {N} _{perp }=mathbf {N} -(mathbf {N} cdot mathbf {n} )mathbf {n} }and the transformations areL\u2032=\u03b3(v)(L+vn\u00d7N)\u2212(\u03b3(v)\u22121)(L\u22c5n)nN\u2032=\u03b3(v)(N\u2212vc2n\u00d7L)\u2212(\u03b3(v)\u22121)(N\u22c5n)n{displaystyle {begin{aligned}mathbf {L} ‘&=gamma (mathbf {v} )(mathbf {L} +vmathbf {n} times mathbf {N} )-(gamma (mathbf {v} )-1)(mathbf {L} cdot mathbf {n} )mathbf {n} \\mathbf {N} ‘&=gamma (mathbf {v} )left(mathbf {N} -{frac {v}{c^{2}}}mathbf {n} times mathbf {L} right)-(gamma (mathbf {v} )-1)(mathbf {N} cdot mathbf {n} )mathbf {n} \\end{aligned}}}or reinstating v = vn,L\u2032=\u03b3(v)(L+v\u00d7N)\u2212(\u03b3(v)\u22121)(L\u22c5v)vv2N\u2032=\u03b3(v)(N\u22121c2v\u00d7L)\u2212(\u03b3(v)\u22121)(N\u22c5v)vv2{displaystyle {begin{aligned}mathbf {L} ‘&=gamma (mathbf {v} )(mathbf {L} +mathbf {v} times mathbf {N} )-(gamma (mathbf {v} )-1){frac {(mathbf {L} cdot mathbf {v} )mathbf {v} }{v^{2}}}\\mathbf {N} ‘&=gamma (mathbf {v} )left(mathbf {N} -{frac {1}{c^{2}}}mathbf {v} times mathbf {L} right)-(gamma (mathbf {v} )-1){frac {(mathbf {N} cdot mathbf {v} )mathbf {v} }{v^{2}}}\\end{aligned}}}These are very similar to the Lorentz transformations of the electric field E and magnetic field B, see Classical electromagnetism and special relativity.Alternatively, starting from the vector Lorentz transformations of time, space, energy, and momentum, for a boost with velocity v,t\u2032=\u03b3(v)(t\u2212v\u22c5rc2),r\u2032=r+\u03b3(v)\u22121v2(r\u22c5v)v\u2212\u03b3(v)tv,p\u2032=p+\u03b3(v)\u22121v2(p\u22c5v)v\u2212\u03b3(v)Ec2v,E\u2032=\u03b3(v)(E\u2212v\u22c5p),{displaystyle {begin{aligned}t’&=gamma (mathbf {v} )left(t-{frac {mathbf {v} cdot mathbf {r} }{c^{2}}}right),,\\mathbf {r} ‘&=mathbf {r} +{frac {gamma (mathbf {v} )-1}{v^{2}}}(mathbf {r} cdot mathbf {v} )mathbf {v} -gamma (mathbf {v} )tmathbf {v} ,,\\mathbf {p} ‘&=mathbf {p} +{frac {gamma (mathbf {v} )-1}{v^{2}}}(mathbf {p} cdot mathbf {v} )mathbf {v} -gamma (mathbf {v} ){frac {E}{c^{2}}}mathbf {v} ,,\\E’&=gamma (mathbf {v} )left(E-mathbf {v} cdot mathbf {p} right),,\\end{aligned}}}inserting these into the definitionsL\u2032=r\u2032\u00d7p\u2032,N\u2032=E\u2032c2r\u2032\u2212t\u2032p\u2032{displaystyle {begin{aligned}mathbf {L} ‘&=mathbf {r} ‘times mathbf {p} ‘,,&mathbf {N} ‘&={frac {E’}{c^{2}}}mathbf {r} ‘-t’mathbf {p} ‘end{aligned}}}gives the transformations.4d angular momentum as a bivector[edit]In relativistic mechanics, the COM boost and orbital 3-space angular momentum of a rotating object are combined into a four-dimensional bivector in terms of the four-position X and the four-momentum P of the object[4][5]M=X\u2227P{displaystyle mathbf {M} =mathbf {X} wedge mathbf {P} }In componentsM\u03b1\u03b2=X\u03b1P\u03b2\u2212X\u03b2P\u03b1{displaystyle M^{alpha beta }=X^{alpha }P^{beta }-X^{beta }P^{alpha }}which are six independent quantities altogether. Since the components of X and P are frame-dependent, so is M. Three componentsMij=xipj\u2212xjpi=Lij{displaystyle M^{ij}=x^{i}p^{j}-x^{j}p^{i}=L^{ij}}are those of the familiar classical 3-space orbital angular momentum, and the other threeM0i=x0pi\u2212xip0=c(tpi\u2212xiEc2)=\u2212cNi{displaystyle M^{0i}=x^{0}p^{i}-x^{i}p^{0}=c,left(tp^{i}-x^{i}{frac {E}{c^{2}}}right)=-cN^{i}}are the relativistic mass moment, multiplied by \u2212c. The tensor is antisymmetric;M\u03b1\u03b2=\u2212M\u03b2\u03b1{displaystyle M^{alpha beta }=-M^{beta alpha }}The components of the tensor can be systematically displayed as a matrixM=(M00M01M02M03M10M11M12M13M20M21M22M23M30M31M32M33)=(0\u2212N1c\u2212N2c\u2212N3cN1c0L12\u2212L31N2c\u2212L120L23N3cL31\u2212L230)=(0\u2212NcNTcx\u2227p){displaystyle {begin{aligned}mathbf {M} &={begin{pmatrix}M^{00}&M^{01}&M^{02}&M^{03}\\M^{10}&M^{11}&M^{12}&M^{13}\\M^{20}&M^{21}&M^{22}&M^{23}\\M^{30}&M^{31}&M^{32}&M^{33}end{pmatrix}}\\[3pt]&=left({begin{array}{c|ccc}0&-N^{1}c&-N^{2}c&-N^{3}c\\hline N^{1}c&0&L^{12}&-L^{31}\\N^{2}c&-L^{12}&0&L^{23}\\N^{3}c&L^{31}&-L^{23}&0end{array}}right)\\[3pt]&=left({begin{array}{c|c}0&-mathbf {N} c\\hline mathbf {N} ^{mathrm {T} }c&mathbf {x} wedge mathbf {p} \\end{array}}right)end{aligned}}}in which the last array is a block matrix formed by treating N as a row vector which matrix transposes to the column vector NT, and x \u2227 p as a 3 \u00d7 3 antisymmetric matrix. The lines are merely inserted to show where the blocks are.Again, this tensor is additive: the total angular momentum of a system is the sum of the angular momentum tensors for each constituent of the system:Mtot=\u2211nMn=\u2211nXn\u2227Pn.{displaystyle mathbf {M} _{text{tot}}=sum _{n}mathbf {M} _{n}=sum _{n}mathbf {X} _{n}wedge mathbf {P} _{n},.}Each of the six components forms a conserved quantity when aggregated with the corresponding components for other objects and fields.The angular momentum tensor M is indeed a tensor, the components change according to a Lorentz transformation matrix \u039b, as illustrated in the usual way by tensor index notationM\u2032\u03b1\u03b2=X\u2032\u03b1P\u2032\u03b2\u2212X\u2032\u03b2P\u2032\u03b1=\u039b\u03b1\u03b3X\u03b3\u039b\u03b2\u03b4P\u03b4\u2212\u039b\u03b2\u03b4X\u03b4\u039b\u03b1\u03b3P\u03b3=\u039b\u03b1\u03b3\u039b\u03b2\u03b4(X\u03b3P\u03b4\u2212X\u03b4P\u03b3)=\u039b\u03b1\u03b3\u039b\u03b2\u03b4M\u03b3\u03b4,{displaystyle {begin{aligned}{M’}^{alpha beta }&={X’}^{alpha }{P’}^{beta }-{X’}^{beta }{P’}^{alpha }\\&={Lambda ^{alpha }}_{gamma }X^{gamma }{Lambda ^{beta }}_{delta }P^{delta }-{Lambda ^{beta }}_{delta }X^{delta }{Lambda ^{alpha }}_{gamma }P^{gamma }\\&={Lambda ^{alpha }}_{gamma }{Lambda ^{beta }}_{delta }left(X^{gamma }P^{delta }-X^{delta }P^{gamma }right)\\&={Lambda ^{alpha }}_{gamma }{Lambda ^{beta }}_{delta }M^{gamma delta }\\end{aligned}},}where, for a boost (without rotations) with normalized velocity \u03b2 = v\/c, the Lorentz transformation matrix elements are\u039b00=\u03b3\u039bi0=\u039b0i=\u2212\u03b3\u03b2i\u039bij=\u03b4ij+\u03b3\u22121\u03b22\u03b2i\u03b2j{displaystyle {begin{aligned}{Lambda ^{0}}_{0}&=gamma \\{Lambda ^{i}}_{0}&={Lambda ^{0}}_{i}=-gamma beta ^{i}\\{Lambda ^{i}}_{j}&={delta ^{i}}_{j}+{frac {gamma -1}{beta ^{2}}}beta ^{i}beta _{j}end{aligned}}}and the covariant \u03b2i and contravariant \u03b2i components of \u03b2 are the same since these are just parameters.In other words, one can Lorentz-transform the four position and four momentum separately, and then antisymmetrize those newly found components to obtain the angular momentum tensor in the new frame.Rigid body rotation[edit]For a particle moving in a curve, the cross product of its angular velocity \u03c9 (a pseudovector) and position x give its tangential velocityu=\u03c9\u00d7x{displaystyle mathbf {u} ={boldsymbol {omega }}times mathbf {x} }which cannot exceed a magnitude of c, since in SR the translational velocity of any massive object cannot exceed the speed of light c. Mathematically this constraint is 0 \u2264 |u| < c, the vertical bars denote the magnitude of the vector. If the angle between \u03c9 and x is \u03b8 (assumed to be nonzero, otherwise u would be zero corresponding to no motion at all), then |u| = |\u03c9| |x| sin \u03b8 and the angular velocity is restricted by0\u2264|\u03c9|{displaystyle 0leq |{boldsymbol {omega }}|\u21ccLi=Iij\u03c9j{displaystyle mathbf {L} =mathbf {I} cdot {boldsymbol {omega }}quad rightleftharpoons quad L_{i}=I_{ij}omega _{j}}(the dot \u00b7 denotes tensor contraction on one index). The relativistic angular momentum is also limited by the size of the object.Spin in special relativity[edit]Four-spin[edit]A particle may have a “built-in” angular momentum independent of its motion, called spin and denoted s. It is a 3d pseudovector like orbital angular momentum L.The spin has a corresponding spin magnetic moment, so if the particle is subject to interactions (like electromagnetic fields or spin-orbit coupling), the direction of the particle’s spin vector will change, but its magnitude will be constant.The extension to special relativity is straightforward.[6] For some lab frame F, let F\u2032 be the rest frame of the particle and suppose the particle moves with constant 3-velocity u. Then F\u2032 is boosted with the same velocity and the Lorentz transformations apply as usual; it is more convenient to use \u03b2 = u\/c. As a four-vector in special relativity, the four-spin S generally takes the usual form of a four-vector with a timelike component st and spatial components s, in the lab frameS\u2261(S0,S1,S2,S3)=(st,sx,sy,sz){displaystyle mathbf {S} equiv left(S^{0},S^{1},S^{2},S^{3}right)=(s_{t},s_{x},s_{y},s_{z})}although in the rest frame of the particle, it is defined so the timelike component is zero and the spatial components are those of particle’s actual spin vector, in the notation here s\u2032, so in the particle’s frameS\u2032\u2261(S\u20320,S\u20321,S\u20322,S\u20323)=(0,sx\u2032,sy\u2032,sz\u2032){displaystyle mathbf {S} ‘equiv left({S’}^{0},{S’}^{1},{S’}^{2},{S’}^{3}right)=left(0,s_{x}’,s_{y}’,s_{z}’right)}Equating norms leads to the invariant relationst2\u2212s\u22c5s=\u2212s\u2032\u22c5s\u2032{displaystyle s_{t}^{2}-mathbf {s} cdot mathbf {s} =-mathbf {s} ‘cdot mathbf {s} ‘}so if the magnitude of spin is given in the rest frame of the particle and lab frame of an observer, the magnitude of the timelike component st is given in the lab frame also.Vector transformations derived from the tensor transformationsThe boosted components of the four spin relative to the lab frame areS\u20320=\u039b0\u03b1S\u03b1=\u039b00S0+\u039b0iSi=\u03b3(S0\u2212\u03b2iSi)=\u03b3(ccS0\u2212uicSi)=1cU0S0\u22121cUiSiS\u2032i=\u039bi\u03b1S\u03b1=\u039bi0S0+\u039bijSj=\u2212\u03b3\u03b2iS0+[\u03b4ij+\u03b3\u22121\u03b22\u03b2i\u03b2j]Sj=Si+\u03b32\u03b3+1\u03b2i\u03b2jSj\u2212\u03b3\u03b2iS0{displaystyle {begin{aligned}{S’}^{0}&={Lambda ^{0}}_{alpha }S^{alpha }={Lambda ^{0}}_{0}S^{0}+{Lambda ^{0}}_{i}S^{i}=gamma left(S^{0}-beta _{i}S^{i}right)\\&=gamma left({frac {c}{c}}S^{0}-{frac {u_{i}}{c}}S^{i}right)={frac {1}{c}}U_{0}S^{0}-{frac {1}{c}}U_{i}S^{i}\\[3pt]{S’}^{i}&={Lambda ^{i}}_{alpha }S^{alpha }={Lambda ^{i}}_{0}S^{0}+{Lambda ^{i}}_{j}S^{j}\\&=-gamma beta ^{i}S^{0}+left[delta _{ij}+{frac {gamma -1}{beta ^{2}}}beta _{i}beta _{j}right]S^{j}\\&=S^{i}+{frac {gamma ^{2}}{gamma +1}}beta _{i}beta _{j}S^{j}-gamma beta ^{i}S^{0}end{aligned}}}Here \u03b3 = \u03b3(u). S\u2032 is in the rest frame of the particle, so its timelike component is zero, S\u20320 = 0, not S0. Also, the first is equivalent to the inner product of the four-velocity (divided by c) and the four-spin. Combining these facts leads toS\u20320=1cU\u03b1S\u03b1=0{displaystyle {S’}^{0}={frac {1}{c}}U_{alpha }S^{alpha }=0}which is an invariant. Then this combined with the transformation on the timelike component leads to the perceived component in the lab frame;S0=\u03b2iSi{displaystyle S^{0}=beta _{i}S^{i}}The inverse relations areS0=\u03b3(S\u20320+\u03b2iS\u2032i)Si=S\u2032i+\u03b32\u03b3+1\u03b2i\u03b2jS\u2032j+\u03b3\u03b2iS\u20320{displaystyle {begin{aligned}S^{0}&=gamma left({S’}^{0}+beta _{i}{S’}^{i}right)\\S^{i}&={S’}^{i}+{frac {gamma ^{2}}{gamma +1}}beta _{i}beta _{j}{S’}^{j}+gamma beta ^{i}{S’}^{0}end{aligned}}}The covariant constraint on the spin is orthogonality to the velocity vector,U\u03b1S\u03b1=0{displaystyle U_{alpha }S^{alpha }=0}In 3-vector notation for explicitness, the transformations arest=\u03b2\u22c5ss\u2032=s+\u03b32\u03b3+1\u03b2(\u03b2\u22c5s)\u2212\u03b3\u03b2st{displaystyle {begin{aligned}s_{t}&={boldsymbol {beta }}cdot mathbf {s} \\mathbf {s} ‘&=mathbf {s} +{frac {gamma ^{2}}{gamma +1}}{boldsymbol {beta }}left({boldsymbol {beta }}cdot mathbf {s} right)-gamma {boldsymbol {beta }}s_{t}end{aligned}}}The inverse relationsst=\u03b3\u03b2\u22c5s\u2032s=s\u2032+\u03b32\u03b3+1\u03b2(\u03b2\u22c5s\u2032){displaystyle {begin{aligned}s_{t}&=gamma {boldsymbol {beta }}cdot mathbf {s} ‘\\mathbf {s} &=mathbf {s} ‘+{frac {gamma ^{2}}{gamma +1}}{boldsymbol {beta }}left({boldsymbol {beta }}cdot mathbf {s} ‘right)end{aligned}}}are the components of spin the lab frame, calculated from those in the particle’s rest frame. Although the spin of the particle is constant for a given particle, it appears to be different in the lab frame.The Pauli\u2013Lubanski pseudovector[edit]The Pauli\u2013Lubanski pseudovectorS\u03c1=12\u03b5\u03bb\u03bc\u03bd\u03c1P\u03bbJ\u03bc\u03bd,{displaystyle S_{rho }={frac {1}{2}}varepsilon _{lambda mu nu rho }P^{lambda }J^{mu nu },}applies to both massive and massless particles.Spin\u2013orbital decomposition[edit]In general, the total angular momentum tensor splits into an orbital component and a spin component,J\u03bc\u03bd=M\u03bc\u03bd+S\u03bc\u03bd\u00a0.{displaystyle J^{mu nu }=M^{mu nu }+S^{mu nu }~.}This applies to a particle, a mass\u2013energy\u2013momentum distribution, or field.Angular momentum of a mass\u2013energy\u2013momentum distribution[edit]Angular momentum from the mass\u2013energy\u2013momentum tensor[edit]The following is a summary from MTW.[7] Throughout for simplicity, Cartesian coordinates are assumed.In special and general relativity, a distribution of mass\u2013energy\u2013momentum, e.g. a fluid, or a star, is described by the stress\u2013energy tensor T\u03b2\u03b3 (a second order tensor field depending on space and time). Since T00 is the energy density, Tj0 for j = 1, 2, 3 is the jth component of the object’s 3d momentum per unit volume, and Tij form components of the stress tensor including shear and normal stresses, the orbital angular momentum density about the position 4-vector X\u03b2 is given by a 3rd order tensorM\u03b1\u03b2\u03b3=(X\u03b1\u2212X\u00af\u03b1)T\u03b2\u03b3\u2212(X\u03b2\u2212X\u00af\u03b2)T\u03b1\u03b3{displaystyle {mathcal {M}}^{alpha beta gamma }=left(X^{alpha }-{bar {X}}^{alpha }right)T^{beta gamma }-left(X^{beta }-{bar {X}}^{beta }right)T^{alpha gamma }}This is antisymmetric in \u03b1 and \u03b2. In special and general relativity, T is a symmetric tensor, but in other contexts (e.g., quantum field theory), it may not be.Let \u03a9 be a region of 4d spacetime. The boundary is a 3d spacetime hypersurface (“spacetime surface volume” as opposed to “spatial surface area”), denoted \u2202\u03a9 where “\u2202” means “boundary”. Integrating the angular momentum density over a 3d spacetime hypersurface yields the angular momentum tensor about X,M\u03b1\u03b2(X\u00af)=\u222e\u2202\u03a9M\u03b1\u03b2\u03b3d\u03a3\u03b3{displaystyle M^{alpha beta }left({bar {X}}right)=oint _{partial Omega }{mathcal {M}}^{alpha beta gamma }dSigma _{gamma }}where d\u03a3\u03b3 is the volume 1-form playing the role of a unit vector normal to a 2d surface in ordinary 3d Euclidean space. The integral is taken over the coordinates X, not X. The integral within a spacelike surface of constant time isMij=\u222e\u2202\u03a9Mij0d\u03a30=\u222e\u2202\u03a9[(Xi\u2212Yi)Tj0\u2212(Xj\u2212Yj)Ti0]dxdydz{displaystyle M^{ij}=oint _{partial Omega }{mathcal {M}}^{ij0}dSigma _{0}=oint _{partial Omega }left[left(X^{i}-Y^{i}right)T^{j0}-left(X^{j}-Y^{j}right)T^{i0}right]dx,dy,dz}which collectively form the angular momentum tensor.Angular momentum about the centre of mass[edit]There is an intrinsic angular momentum in the centre-of-mass frame, in other words, the angular momentum about any eventXCOM=(XCOM0,XCOM1,XCOM2,XCOM3){displaystyle mathbf {X} _{text{COM}}=left(X_{text{COM}}^{0},X_{text{COM}}^{1},X_{text{COM}}^{2},X_{text{COM}}^{3}right)}on the wordline of the object’s center of mass. Since T00 is the energy density of the object, the spatial coordinates of the center of mass are given byXCOMi=1m0\u222b\u2202\u03a9XiT00dxdydz{displaystyle X_{text{COM}}^{i}={frac {1}{m_{0}}}int _{partial Omega }X^{i}T^{00}dxdydz}Setting Y = XCOM obtains the orbital angular momentum density about the centre-of-mass of the object.Angular momentum conservation[edit]The conservation of energy\u2013momentum is given in differential form by the continuity equation\u2202\u03b3T\u03b2\u03b3=0{displaystyle partial _{gamma }T^{beta gamma }=0}where \u2202\u03b3 is the four-gradient. (In non-Cartesian coordinates and general relativity this would be replaced by the covariant derivative). The total angular momentum conservation is given by another continuity equation\u2202\u03b3J\u03b1\u03b2\u03b3=0{displaystyle partial _{gamma }{mathcal {J}}^{alpha beta gamma }=0}The integral equations use Gauss’ theorem in spacetime\u222bV\u2202\u03b3T\u03b2\u03b3cdtdxdydz=\u222e\u2202VT\u03b2\u03b3d3\u03a3\u03b3=0\u222bV\u2202\u03b3J\u03b1\u03b2\u03b3cdtdxdydz=\u222e\u2202VJ\u03b1\u03b2\u03b3d3\u03a3\u03b3=0{displaystyle {begin{aligned}int _{mathcal {V}}partial _{gamma }T^{beta gamma },cdt,dx,dy,dz&=oint _{partial {mathcal {V}}}T^{beta gamma }d^{3}Sigma _{gamma }=0\\int _{mathcal {V}}partial _{gamma }{mathcal {J}}^{alpha beta gamma },cdt,dx,dy,dz&=oint _{partial {mathcal {V}}}{mathcal {J}}^{alpha beta gamma }d^{3}Sigma _{gamma }=0end{aligned}}}Torque in special relativity[edit]The torque acting on a point-like particle is defined as the derivative of the angular momentum tensor given above with respect to proper time:[8][9]\u0393=dMd\u03c4=X\u2227F{displaystyle {boldsymbol {Gamma }}={frac {dmathbf {M} }{dtau }}=mathbf {X} wedge mathbf {F} }or in tensor components:\u0393\u03b1\u03b2=X\u03b1F\u03b2\u2212X\u03b2F\u03b1{displaystyle Gamma _{alpha beta }=X_{alpha }F_{beta }-X_{beta }F_{alpha }}where F is the 4d force acting on the particle at the event X. As with angular momentum, torque is additive, so for an extended object one sums or integrates over the distribution of mass.Angular momentum as the generator of spacetime boosts and rotations[edit]The angular momentum tensor is the generator of boosts and rotations for the Lorentz group.[10][11]Lorentz boosts can be parametrized by rapidity, and a 3d unit vector n pointing in the direction of the boost, which combine into the “rapidity vector”\u03b6=\u03b6n=ntanh\u22121\u2061\u03b2{displaystyle {boldsymbol {zeta }}=zeta mathbf {n} =mathbf {n} tanh ^{-1}beta }where \u03b2 = v\/c is the speed of the relative motion divided by the speed of light. Spatial rotations can be parametrized by the axis\u2013angle representation, the angle \u03b8 and a unit vector a pointing in the direction of the axis, which combine into an “axis-angle vector”\u03b8=\u03b8a{displaystyle {boldsymbol {theta }}=theta mathbf {a} }Each unit vector only has two independent components, the third is determined from the unit magnitude. Altogether there are six parameters of the Lorentz group; three for rotations and three for boosts. The (homogeneous) Lorentz group is 6-dimensional.The boost generators K and rotation generators J can be combined into one generator for Lorentz transformations; M the antisymmetric angular momentum tensor, with componentsM0i=\u2212Mi0=Ki,Mij=\u03b5ijkJk.{displaystyle M^{0i}=-M^{i0}=K_{i},,quad M^{ij}=varepsilon _{ijk}J_{k},.}and correspondingly, the boost and rotation parameters are collected into another antisymmetric four-dimensional matrix \u03c9, with entries:\u03c90i=\u2212\u03c9i0=\u03b6i,\u03c9ij=\u03b5ijk\u03b8k,{displaystyle omega _{0i}=-omega _{i0}=zeta _{i},,quad omega _{ij}=varepsilon _{ijk}theta _{k},,}where the summation convention over the repeated indices i, j, k has been used to prevent clumsy summation signs. The general Lorentz transformation is then given by the matrix exponential\u039b(\u03b6,\u03b8)=exp\u2061(12\u03c9\u03b1\u03b2M\u03b1\u03b2)=exp\u2061(\u03b6\u22c5K+\u03b8\u22c5J){displaystyle Lambda ({boldsymbol {zeta }},{boldsymbol {theta }})=exp left({frac {1}{2}}omega _{alpha beta }M^{alpha beta }right)=exp left({boldsymbol {zeta }}cdot mathbf {K} +{boldsymbol {theta }}cdot mathbf {J} right)}and the summation convention has been applied to the repeated matrix indices \u03b1 and \u03b2.The general Lorentz transformation \u039b is the transformation law for any four vector A = (A0, A1, A2, A3), giving the components of this same 4-vector in another inertial frame of referenceA\u2032=\u039b(\u03b6,\u03b8)A{displaystyle mathbf {A} ‘=Lambda ({boldsymbol {zeta }},{boldsymbol {theta }})mathbf {A} }The angular momentum tensor forms 6 of the 10 generators of the Poincar\u00e9 group, the other four are the components of the four-momentum for spacetime translations.Angular momentum in general relativity[edit]The angular momentum of test particles in a gently curved background is more complicated in GR but can be generalized in a straightforward manner. If the Lagrangian is expressed with respect to angular variables as the generalized coordinates, then the angular momenta are the functional derivatives of the Lagrangian with respect to the angular velocities. Referred to Cartesian coordinates, these are typically given by the off-diagonal shear terms of the spacelike part of the stress\u2013energy tensor. If the spacetime supports a Killing vector field tangent to a circle, then the angular momentum about the axis is conserved.One also wishes to study the effect of a compact, rotating mass on its surrounding spacetime. The prototype solution is of the Kerr metric, which describes the spacetime around an axially symmetric black hole. It is obviously impossible to draw a point on the event horizon of a Kerr black hole and watch it circle around. However, the solution does support a constant of the system that acts mathematically similarly to an angular momentum.See also[edit]References[edit]^ D.S.A. Freed; K.K.A. Uhlenbeck (1995). Geometry and quantum field theory (2nd\u00a0ed.). Institute For Advanced Study (Princeton, N.J.): American Mathematical Society. ISBN\u00a00-8218-8683-5.^ a b R. Penrose (2005). The Road to Reality. vintage books. p.\u00a0433. ISBN\u00a0978-0-09-944068-0. Penrose includes a factor of 2 in the wedge product, other authors may also.^ M. Fayngold (2008). Special Relativity and How it Works. John Wiley & Sons. p.\u00a0138. ISBN\u00a0978-3-527-40607-4.^ R. Penrose (2005). The Road to Reality. vintage books. pp.\u00a0437\u2013438, 566\u2013569. ISBN\u00a0978-0-09-944068-0. Note: Some authors, including Penrose, use Latin letters in this definition, even though it is conventional to use Greek indices for vectors and tensors in spacetime.^ M. Fayngold (2008). Special Relativity and How it Works. John Wiley & Sons. pp.\u00a0137\u2013139. ISBN\u00a0978-3-527-40607-4.^ Jackson, J. D. (1975) [1962]. “Chapter 11”. Classical Electrodynamics (2nd\u00a0ed.). John Wiley & Sons. pp.\u00a0556\u2013557. ISBN\u00a00-471-43132-X. Jackson’s notation: S (spin in F, lab frame), s (spin in F\u2032, rest frame of particle), S0 (timelike component in lab frame), S\u20320 = 0 (timelike component in rest frame of particle), no symbol for 4-spin as a 4-vector^ J.A. Wheeler; C. Misner; K.S. Thorne (1973). Gravitation. W.H. Freeman & Co. pp.\u00a0156\u2013159, \u00a75.11. ISBN\u00a00-7167-0344-0.^ S. Aranoff (1969). “Torque and angular momentum on a system at equilibrium in special relativity”. American Journal of Physics. 37 (4): 453\u2013454. Bibcode:1969AmJPh..37..453A. doi:10.1119\/1.1975612. This author uses T for torque, here we use capital Gamma \u0393 since T is most often reserved for the stress\u2013energy tensor.^ S. Aranoff (1972). “Equilibrium in special relativity” (PDF). Nuovo Cimento. 10 (1): 159. Bibcode:1972NCimB..10..155A. doi:10.1007\/BF02911417. S2CID\u00a0117291369. Archived from the original (PDF) on 2012-03-28. Retrieved 2013-10-27.^ E. Abers (2004). Quantum Mechanics. Addison Wesley. pp.\u00a011, 104, 105, 410\u2013411. ISBN\u00a0978-0-13-146100-0.^ H.L. Berk; K. Chaicherdsakul; T. Udagawa (2001). “The Proper Homogeneous Lorentz Transformation Operator eL = e\u2212 \u03c9\u00b7S \u2212 \u03be\u00b7K, Where’s It Going, What’s the Twist” (PDF). American Journal of Physics. 69 (996). doi:10.1119\/1.1371919.Further reading[edit]Special relativity[edit]General relativity[edit]External links[edit] (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/en\/wiki24\/relativistic-angular-momentum-wikipedia\/#breadcrumbitem","name":"Relativistic angular momentum – Wikipedia"}}]}]