1 The Relativistic Vlasov Equation with current and charge density given by by Alec Johnson, January 2011 X X J = qpvpδxp σ = qpδxp I. Recitation of basic electrodynamics p p X δxp X δxp = q p ; = q γ ; A. Classical electrodynamics p p γ p p γ p p p p Classical electrodynamics is governed by Newton's sec- here p (i.e. γ v ) is the proper velocity, where γ is the ond law (for particle motion), p p p p rate at which time t elapses with respect to the proper time τ of a clock moving with particle p. Dropping the particle mpdtvp = Fp; vp := dtxp; p index, (where p is particle index, t is time, mp is particle mass, γ := d t; and xp(t) is particle position, and Fp(t) is the force on the τ particle), the Lorentz force law, p := dτ x = (dτ t)dtx = γv: Fp = qp (E + v × B) A concrete expression relating γ to (proper) velocity is (where E(t; x) = electric field, B(t; x) = magnetic field, and qp is particle charge), and Maxwell's equations, γ2 = 1 + (p=c)2; i.e. (dividing by γ2), −2 2 @tB + r × E = 0; r · B = 0; 1 = γ + (v=c) ; i.e. (solving for γ), 2 −1=2 @tE − c r × B = −J/, r · E = σ/, γ = 1 − (v=c)2 : with current and charge density given by II. Vlasov equation X X J := qpvpδxp ; σ := qpδxp ; The Vlasov equation asserts that particles are conserved p p in phase space and that the only force acting on particles is the electromagnetic force. where c is light speed, 0 is electric permittivity, J is net current density, σ is net charge density, δ (t; x) := δ(x − xp The Vlasov equation writes xp(t)) is the particle density function, and δ is the Dirac delta function (unit spike). @tfs + rx · (vfs) + rp · (asfs) = 0; (1) B. Relativistic electrodynamics where fs(t; x; p) is the particle density function of species s and as = dtp = (qs=ms)(E+v ×B) is particle acceleration Newton's second law is invariant under Galilean transfor- and we recall that v = dtx is velocity and p = γv is proper mations, but Maxwell's equations are not (because they velocity. imply the existence of a light speed c). The Maxwell source terms are provided by the relations Einstein (and previously Lorentz) characterized the set of affine coordinate transformations, called Lorentz transfor- mations, which leave light speed constant and which satisfy X Z X Z J = q f vdp σ := q f dp the property that if system A has velocity v in system B s s s s then system B has velocity −v in system A. s s X Z dp X Z dp = q f p ; = q f γ : s s γ s s γ Einstein then modified the laws of electrodynamics to make s s them Lorentz-invariant. To do so we replace velocity with proper velocity in Newton's second law. So the laws of relativistic electrodynamics are: If we regard fs as a linear superposition of spike functions then the Vlasov system (1) with Maxwell's equations is Newton's second law, equivalent to the fundamental equations of electrodynam- ics written in terms of individual particles. mpdtpp = Fp; vp := dtxp; A. Lorentz-invariant form the Lorentz force law, 1. fs is a scalar. Fp = qp (E + v × B) ; (In this section without loss of generality we take c = 1.) and Maxwell's equations, The particle density function fs(t; x; p) is a scalar. This means that it remains invariant under the transforma- @tB + r × E = 0; r · B = 0; 2 tion of phase space that is implied by a Lorentz trans- @tE − c r × B = −J/, r · E = σ/, formation (i.e. a relativistic inertial transformation). It 2 takes a little work to see this. By definition, the quantity This confirms that the Vlasov equation (1) is indeed phys- 3 3 dfs := fs(t; x; p)d x ^ d p counts the number of particles ical and Lorentz-invariant. 3 3 in a region (x0 +d x; p0 +d p) in phase space. (Note that d3x ^ d3p denotes an infinitesimal surface of simultaneous B. Moments points in (t; x; p)-space.) We argue that dfs is a scalar, then argue that d3x ^ d3p is a scalar, and then conclude Let χ(p) be a generic moment to evaluate: that fs is a scalar. The essence of the argument is to con- Z 3 µ k d p sider a Lorentz boost which transforms into the reference χ @xµ (p fs) + γrpk (as fs) = Cs : 3 3 3 3 0 γ frame of d x^d p (i.e. which transforms p0 +d p to d p , p so that the transformed range of velocities is centered on Integrate by parts to get zero). Z 3 Z 3 Z 3 µ d p k d p d p @ µ χp f = f a r k χ + C χ : x s γ ses p γ s γ To see that such a boost leaves dfs invariant, consider p p p the particle event paths (\world lines") that intersect 3 3 d x ^ d p. The transformed region of phase space will Use p⊗n := ⊗np to denote the nth tensor power. In be intersected by exactly the same particles. Although case χ = p⊗n = Sym (p⊗n), we have (at least for spa- the transformed region of phase space is not simultane- tial indices, and then by tensoriality of the expression for ous, orthogonal projection to make it simultaneous does k ⊗n−1 temporal indices as well): aes rpk χ = n Sym (p aes) = not change the number of particles intersecting it because qs n Sym (F•p⊗n); where I adopt the notation (F•p)µ = ms their velocity is approximately zero. µν qs F pν and use that as = F• p, where F is the electro- e ms magnetic four-tensor with components To see that the same boost leaves d3x ^ d3p invariant, 3 3 0 −ET =c note that orthogonality of d x and d p is invariant under F µν := : the boost, that the simultaneous region d3x is mapped E=c B × I 3 to a (nonsimultaneous) region of size γ0d x (where γ0 is 3 computed from the boost velocity), and that d p is shrunk So for χ = p⊗n (n ≥ 1) we have by a factor of γ because it is transformed to a region where 2 Z 3 Z 3 dγ = 0 (because γdγ = p · dp (because γ = 1 + p · p) ⊗n µ d p qs ⊗n d p 0 @xµ fsp p = n Sym F• fsp and p0 = 0 post-transformation). Again, projecting onto p γ ms p γ 3 simultaneous points does not change d x. In summary, Z 3 3 3 ⊗n d p we have argued that γ0d x and d p/γ0 are scalars, and + Csp : therefore their product d3x ^ d3p is a scalar. p γ 2. Lorentz-invariant form For χ = 1 we get balance of particle four-flow: Z 3 Z 3 µ µ d p d p Observe that @tfs + rx · (vfs) = @xµ (v fs), which would @xµ p fs = Cs : be Lorentz-invariant if we replaced vµ with γvµ = pµ. p γ p γ This indicates that to put the Vlasov equation (1) in a Separating time and space derivatives, manifestly Lorentz-invariant form we need to multiply it Z Z Z d3p by γ. Then we have @ f + r · f v = C : ct s s s γ µ k p p p @xµ (p fs) + γrpk (as fs) = 0; the first term is manifestly Lorentz-invariant; for the sec- More generally, for χ = p⊗n, separating time and space ond term we rewrite rpk in terms of derivatives with re- derivatives reveals a moment hierarchy: µ 0 k spect to the four-vector p by regarding γ = p and p as Z Z independent quantities (that is, we extend the definition of @ct fsχ + r· fsvχ all quantities (arbitrarily) beyond the quadratic manifold p p 2 2 Z Z γ = 1 + p ). Then the chain rule for partial derivatives qs χ χ = n Sym F• fs + Cs : says ms p γ p γ @γ pk @ k 7! @ + @ k = @ + @ k ; p @pk γ p γ γ p Explicitly, omitting the collision term, the moment hierar- so chy is: Z Z Z k qs χ k p k k χ :@ct fsχ + r· fsvχ = n Sym F• fs γr k (a f ) 7! γ @ (a f ) + r k (a f ) : ms γ p s s γ γ s s p s s p p p Z Z Let a := d p = γa be the proper acceleration. Then 1 : @ct fs + r· fsv = 0; es τ s p p p · a = p · d p = γd γ = γa0; that is, p · a = a0; so the Z Z Z e τ τ e s e qs acceleration term can indeed be cast into the form appear- p : @ct fsp + r· fsvp = F• fsv ; p p ms p ing in the following manifestly Lorentz-invariant version of Z Z Z qs the Vlasov equation, pp : @ct fspp + r· fsvpp = 2 Sym F• fsvp ; p p ms p µ µ @xµ (p fs) + @pµ (aes fs) = 0; ::: 3 sary for all the flux moments and not just the highest flux moment). Note that (1) the evolved moments are not tensorial (be- dp To separate space and time components of vectors, use that cause γ , not dp, is tensorial) and (2) the fluxes are not identical to the evolved moments (so closures are neces- v = (1; v) and that p = (γ; p); then the moment hierarchy is: ⊗n R ⊗n R ⊗n qs −1 R ⊗n−1 R ⊗n−1 p : @ct p fs + r · vp fs = n Sym c E p fs + vp fs × B p p ms p p ⊗n−1 R ⊗n−1 R ⊗n qs −1 R ⊗n−1 γp : @ct γp fs + r · p fs = n Sym c E · vp fs p p ms p R R 1 : @ct pfs + r· pvfs = 0; R R qs −1 R R p : @ct pfs + r· vpfs = c E fs + vfs × B ; p p ms p p R R qs −1 R γ : @ct γfs + r· pfs = c E · vfs; p p ms p R R qs −1 R R pp : @ct fspp + r· fsvpp = 2 Sym c E · p + vpfs × B ; p p ms p p R R qs −1 R γp : @ct γpfs + r · ppfs = 2 Sym c E · vpfs p p ms p ::: C.