GOLDHABER AND NIETO Limits on the Photon Mass 285 Equations (4.7)and (4.8)show that the dispersion An amusing particular case of Eq.(4.11)arises for due to a plasma of one electron per cm2 would equal the =-2.This case is intriguing because,as discussed dispersion of a photon with the Plimpton-Lawton in Sec.II.B,such a form of D would produce exact mass.Despite this limitation,as Feinberg (1969)has cancellation of the lowest-order effects (proportional to pointed out,the observed dispersion in arrival time of 2)in electrostatics or magnetostatics.For this special radio signals from pulsars provides the most stringent form,the limit Eq.(4.12)on Au2 becomes a limit on "dynamic"'test of the photon mass to date.These data 2.If combined with Eq.(4.10),this result implies (assuming u=0)may be used to deduce an average h1≤10-2cm-/42,41<<5X109cm-1. interstellar plasma density of <0.028 electron/cm3 for We conclude that pulsar data do more than give the radiation from the Crab pulsar NP0532.If the disper- best dynamic (velocity-dispersion)limit on u.They sion is partly a photon mass effect,then we have also provide stringent limits on violation of energy conservation associated with a "multicomponent" 4≤10-4g=3X10-7cm-1=6X10-12eV.(4.9) photon having two or more different small masses. Feinberg also makes the interesting point that pulse B.Magnetostatic Effects arrival times show no sign of any dispersion,except that implied by the simple quadratic formula Eq.(4.6), 1.Schrodinger's External Field Method over the whole range of frequency from radio to optical. For the Crab pulsar the departure from Eq.(4.6)is Schrodinger (1943b),following an observation of △/c<10-l4(△arrival<10-8sec). McConnell,proposed a method using the earth's static We may ask for limits on the kind of low mass magnetic field that has yielded the best photon mass "structure"of the photon associated with violation of limit to date.Let us begin with a discussion of the principles on which the method is based.As mentioned energy conservation,for example,two poles in D(k2) separated by Au2.As discussed at the end of Sec.II.B in Sec.II.A,the qualitative effect of a photon of mass such structure could show itself in two ways.The first u on static fields is to cause an extra "Yukawa"de- crease in field strength as e-",where r is distance from is by a spreading in duration of low frequency pulses the source.However,we also have seen that even in At/T=Au2[2(/c)2]1.Here T is the flight time from massive electrodynamics,the divergence of the mag- source to receiver of the radiation.The most accurate netic field H must vanish.This is simply a consequence test on this phenomenon is again supplied by the Crab pulsar,with its very narrow (<1 msec)pulse peaks. of reflection symmetry in electrodynamics,and the absence of magnetic sources:Consider the magnetic In fact,there is an observed pulse broadening of about 10 msec at 74 MHz frequency (Rankin et al.,1970; field at a point r produced by electric currents J at points r'.Any divergence of H would be a pseudoscalar Rankin,private communication).This broadening is function of I and r-r',but there is no such function believed due to"scintillations''or fluctuating irregulari- ties in the interstellar medium.If one assumes that part The dependence only on r-r'is a consequence of assuming that the equations of physics are independent of it is due to a photon mass spread Au2,then we learn that of the choice of origin of coordinates.10 Applying these thoughts to the magnetic dipole field of the Earth,we (Au2)1/255X10-9 cm-1(pulse broadening).(4.10) note V.H=0 means that the flux in each field line is conserved.Now a field line is farther out at the mag- The second effect of structure in D is variation of netic equator than it is near the pole.Hence,the intensity of the signal at a given frequency,as a function Yukawa exponential decrease affects the field line most of time after emission.If the power in the signal is at the equator.To keep constant flux,the field pattern distributed smoothly over a broad range of frequencies, must change shape,allowing flux lines to move in some- then this effect can also be detected by an oscillation of what at the equator.This compression of the equatorial intensity as a function of frequency at a fixed receiver. field lines has the effect,on a sphere of fixed radius,of For the case of two poles in D,one at u2 with residue increasing the field at the equator relative to the field (1+e)-1,and one at ua2 with residue (1+e)-le,one may at the pole.The effect is the same as that of a constant derive the modulation of intensity as a function of external field parallel to Hequatorial.Of course,the field frequency: of the Earth is not pure magnetic dipole.However,for the massless Maxwell theory it is a theorem that only a I(@)=1-{1-[(1-e)/(1+e)]}sin2(△kr/2), true external current can produce a uniform field over △k=(412-2)[2(w/c)]于1+0(4). (4.11) the surface of a sphere.11 In the absence of such currents, For the Crab pulsar,the lack of conspicuous oscillations 10 These comments simply express in familiar three-dimensional terms the arguments of Sec.II.B,which were given in relativistic, down to v=w/2=74 MHz implies eAkr<4m at v=100 four-dimensional notation. MHz or,if e is of order unity, This is most easily proven by noting that if there are no external currents,H outside the sphere is the gradient of a solution of the Laplace equation,and then expanding that (Au2)12<10-cm-(intensity oscillations).(4.12) solution in spherical harmonics