Closing a window for massive photons

0.26 MB·English

by  Sergio A. Hojman

Closing the window for massive photons
Sergio A. Hojman and Benjamin Koch
(Dated: September 25, 2012)
Working with the assumption of non-zero photon mass and a trajectory that is described by the
relativistic world-line of a spinning top we find, by deriving new astrophysical bounds, that this
assumptionisincontradictionwithtodaysexperimentalresults. Thisyieldstheconclusionthatthe
photon has to be exactly massless.
PACSnumbers: 04.62.+v,03.65.Ta
I. INTRODUCTION
2
1 Although there is good theoretical reason to believe that the photon mass should be exactly zero, there is no
0 experimental proof of this believe. A long series of very different experiments lead to the current experimental upper
2 bound on the photon mass m <10−18eV. However, even with further improvement of the experimental technology
γ
p and precision a complete exclusion of a non-zero photon mass by those techniques will never be possible. In order to
e improveonthissituationwewillworkwiththeassumptionthatthephotonmassisdifferentfromzerom (cid:54)=0. Since
γ
S
the photon is a particle with spin one, its motion in a gravitational field has to be described by spin-gravity coupled

1 equations of motion. A very generic approach to this kind of problem is to consider a spinning top as a point on a
2 world-line to which a rotating frame has been attached [1–7]. Exact solutions to those equations show that spinning
tops without mass follow geodesics, while massive spinning tops have different trayectories. Further implications and
]
h discussions of the spinning top approach can be found in [8–13].
p
-
p
II. ASTROPHYSICAL BOUND
e
h
[ As master equation for the trajectory of massive particles with spin we use [4–7]

1 (cid:18) (cid:19)(cid:18) (cid:19)
dφ 2η+1 P
v = φ . (1)
7 dr η−1 r2Pr
0
9 where one has to insert the following definitions
4
. J2r
9 η = 0 , (2)
0 2m2c2r3
γ
2
1 where J =(cid:126) is the photon spin. For a given angular momentum j, the momenta are
:
v
−j+(± )EJ/(m c2)
i P = φ γ , (3)
X φ 1−η
r
a
E−(± )jJr /(2m r3)
P = t 0 γ , (4)
t 1−η
and
(cid:34) (cid:32) (cid:33) (cid:35)1/2
P2 P2 (cid:16) r (cid:17)
Pr =± t − φ +m2c2 1− 0 . (5)
c2 r2 r
Theorientationofthisproblemischosensuchthattheangleθ =π/2isconstantalongthetrajectories,whichimplies
P =0 . (6)
θ
Those definitions are such that P Pµ =m2c2 is fulfilled.
µ γ
2
For small spin corrections and large radii one can approximate
(cid:18)dr(cid:19)2 (cid:18) 1 2(cid:126) (cid:19) (cid:18) (± )(cid:126)(cid:19)
=r4 +(± ) −r2+rr 1− t . (7)
dφ j2 φ cj3m 0 cjm
γ γ
In this equation one can easily verify that for spin-less case ((cid:126) → 0), the geodesic trajectory is recovered. One can
express the angular momentum j in terms of the minimal radial distance r
m
2cE(cid:126)m(± r +2± (r −r ))r4 (cid:112)−(r −r )r
j = t 0 φ 0 m m + 0 m m ·
c(cid:126)2+r2r +4c3m2(r −r )r4 c(cid:126)2+r2r +4c3m2(r −r )r4
0 m 0 m m 0 m 0 m m
(cid:113)
4E2(cid:126)4+r2r4 +16c8m6(r −r )r9 +16c6(cid:126)2m4r r6 (r −r )+c2(cid:126)2 (8)
0 m 0 m m 0 m m 0
(cid:113)
· ± r ((cid:126)4± r3−16E2m2± r7 )+4c4m2r3 (4E2m2r7 −(cid:126)4r2((−1+)r +r )) .
t 0 t 0 φ m m m 0 0 m
Due to the steep behavior of dφ/dr at the minimal radius r one has to use (8), since one can not rely a priory on
√ m
the approximate value j ≈r3/2/ r −r .
m m 0
Due to the involved form of (8), the complete angular deviation has to be computed essentially numerically, as it
was done here. In order to gain a better intuition of the numerical results one can however do approximations and
expansions for small (cid:126), mc2/E, and r /r
0 m
(cid:90) ∞(cid:18)dφ(cid:19) r (cid:18) (cid:126) (cid:19)
∆φ=2( dr)−π ≈2 0 1+ , (9)
dr r r cm
rm m m γ
where ± =−1 and ± =1 was used. One observes that the standard result 2r /r is modified by a spin-dependent
t φ 0 m
correction. In order to be in agreement with the experimentally well confirmed gravitational lensing effect, this
dimensionless modification has to be much smaller than one
(cid:126)
1(cid:29)∆ ≈ . (10)
J r cm
m γ
Thismodificationisinverselyproportionaltothephotonmassm . Thisinverseproportionalityimpliesthatpractically
γ
no deviations from the usual trajectories are observable for massive standard model particles. Please note that this
non perturbative feature in m is also known from theories of massive gravity where the limit m →0 does not give
γ g
standard gravity where the graviton was massless right from the start m = 0. It is further interesting to note that
g
∆ actually increases the usual angular deflection. However, a deviation from the geodesic bending of light has been
J
excluded to high precision [16]. Thus, the relation (10) can be interpreted as an estimate for the numerical lower
photon mass limit
(cid:126)
m (cid:29) ≈3×10−16eV . (11)
γ r c
m
In figure 1 we compare this estimate to the observed value [14, 15] and to the precise numerical results by using the
solar radius r =6.96×108 m, the solar Schwarzschild radius r =2964 m, and a photon energy of E =1 eV. One
m 0
finds that the estimated deviation is in very good agreement with the lower numerical result. By using conservative
exclusion ranges for ∆φ one can read from figure a more precise limit
m >1.1×10−15eV . (12)
γ
The lower limit (12) can be combined with the upper limit for a photon mass from the Particle Data Group [17]
m < 10−18eV. Some of the limits in the PDG tables [17] are however derived by assuming a spin-less coupling
γ
of the photon to gravity and to matter. But this might not be true. When deriving the limit (12), we found the
gravitational coupling of the massive particles with spin, is different from a spin-less or mass-less coupling. Thus, out
of the limits in [17] only those can be applied straight forwardly, where the gravitational coupling is not relevant. For
example, results from laboratory experiments, that do not involve astrophysical or gravitational components [18–22]
can be used directly. Also results from other experiments, that do involve astrophysical components but where the
actual trajectory of the photon does not play any role [23–31] should be applicable, but a sound revision is at place.
Nevertheless, the bounds on m from some experiments are not directly applicable because they are not sufficiently
γ
general [32, 33], or because they make explicit use of the photon trajectory in a gravitational field [22, 34–37] (Note
that those experimental references are ordered by the strength of their respective bounds on m ). The experimental
γ
3
FIG. 1: Numerical results for the angular deviation ∆φ as a function of m . The horizontal lines represent a
γ
conservative range for the measured gravitational lensing by the sun [14]. Please note that this range can be largely
improved by measurements using radio astronomy [15, 16]. The red lines are the numerical values for the allowed
values of ± and ± . The black dotted line is the analytic estimate (9). The vertical line is the deduced minimal
φ t
value for m given in (12).
γ
bounds, that are directly applicable are
m <{1.2×10−18eV/c2 [18],5.6×10−17eV/c2 [19],×10−14eV/c2 [20],4.5×10−10eV/c2 [21],1.6×10−4eV/c2 [22]}.
γ
Thus, threeoftheapplicableexperimentallimits[20–22], combinedwiththelimit(12),leaveonlyawindowforthe
photon mass. However, if one refers only to the more stringent upper limits on m [18, 19] one finds
γ
m <{1.2×10−18eV/c2, 5.6×10−17eV/c2} , (13)
γ
which is in clear contradiction to (12). Therefore, one can conclude that confronting the assumption of a non-zero
photonmasswithanastrophysicallowerboundandtheestablishedupperboundsexcludestheexistenceofanyphoton
mass. Thus, the photon has to be massless right from the start
m =0 . (14)
γ
III. DISCUSSION AND CONCLUSION
Under the assumption that the trajectory of a photon can be described by the world-line of a top with spin one, it
wasshownthatsuchaphotonhastohaveeitherzeromassm =0oraminimalnon-zeromassm >1.1×10−15 eV,
γ γ
inordertobeinagreementwiththeobservedgravitationallightbendingbythesun[14–16]. Combiningthoseresults
withtheexperimentalboundsfromlaboratoryexperimentssuchasm <5.6×10−17 eV[18,19], onefindsthatthere
γ
is no room for a photon mass different from zero.
[1] M. Mathisson, Acta Physica Polonica 6, 163 (1937); 6, 218 (1937).
[2] A. Papapetrou, Proc. Roy. Soc. Lond. A 209, 248 (1951).
[3] E. Corinaldesi and A. Papapetrou, Proc. Roy. Soc. Lond. A 209, 259 (1951).
[4] A. J. Hanson and T. Regge, Annals of Physics 87, 498 (1974).
[5] S. Hojman, Phys. Rev. D 18, 2741 (1978).
[6] S. Hojman, M. Rosenbaum, M. P. Ryan and L. C. Shepley, Phys. Rev. D 17, 3141 (1978).
[7] S. A. Hojman and F. A. Asenjo, arXiv:1203.5008 [physics.gen-ph].
[8] R. M. Wald, Phys. Rev. D 6, 406 (1972).
4
[9] B. M. Barker and R. F. O’Connell, Phys. Rev. D 12, 329 (1975).
[10] A. Barducci, R. Casalbuoni and L. Lusanna, Nucl. Phys. B 124, 93 (1977).
[11] F. W. Hehl, P. Von Der Heyde, G. D. Kerlick and J. M. Nester, Rev. Mod. Phys. 48, 393 (1976).
[12] D. Piriz, M. Roy and J. Wudka, Phys. Rev. D 54, 1587 (1996) [hep-ph/9604403].
[13] Q. G. Bailey, Phys. Rev. D 82, 065012 (2010) [arXiv:1005.1435 [gr-qc]].
[14] S. Weinberg, “Gravitation and Cosmology”, ed. John Wiley and Sons, (1972).
[15] C. M. Will, “Theory and experiment in gravitational physics”, ed. Cambridge University Press, (1993).
[16] E. Fomalont et al., Astrophys. J. 699, 1395 (2009).
[17] K. Nakamura et al. (Particle Data Group), JP G 37, 075021 (2010) and 2011 partial update for the 2012 edition.
[18] G. Spavieri, J. Quintero, G. T. Gillies and M. Rodriguez, Eur. Phys. J. D 61, 531 (2011); G. Spavieri and M. Rodriguez,
Phys. Rev. A 75, 052113 (2007) [arxiv.org/abs/0705.1101v1].
[19] P. A. Franken and G. W. Ampulski, Phys. Rev. Lett. 26, 115 (1971).
[20] E. R. Williams, J. E. Faller and H. A. Hill, Phys. Rev. Lett. 26, 721 (1971).
[21] M. A. Chernikov, C. J. Gerber, H. R. Ott and H. J. Gerber, Phys. Rev. Lett. 68, 3383 (1992) [Erratum-ibid. 69, 2999
(1992)].
[22] A. Accioly, J. Helayel-Neto and E. Scatena, Phys. Rev. D 82, 065026 (2010) [arXiv:1012.2717 [hep-th]].
[23] J. Luo, L.J. Tu, Z.K. Hu, and E.J. Luan PRL 90 081801; PRL 91 149102.
[24] D. D. Ryutov, Phys. Rev. Lett. 103, 201803 (2009).
[25] R. Lakes, Phys. Rev. Lett. 80, 1826 (1998).
[26] L. Davis, Jr., A. S. Goldhaber and M. M. Nieto, Phys. Rev. Lett. 35, 1402 (1975).
[27] J. V. Hollweg, Phys. Rev. Lett. 32, 961 (1974).
[28] E. Fischbach, H. Kloor, R. A. Langel, A. T. Y. Liu and M. Peredo, Phys. Rev. Lett. 73, 514 (1994).
[29] M. A. Gintsburg, Sov. Astron. 7, 536 (1964) [Astron. Zh. 40, 703 (1963)].
[30] V. L. Patel, Phys. Lett. 14, 105 (1965).
[31] A. S. Goldhaber and M. M. Nieto, Rev. Mod. Phys. 82, 939 (2010) [arXiv:0809.1003 [hep-ph]].
[32] G. V. Chibisov, Sov. Phys. Usp. 19, 624 (1976) [Usp. Fiz. Nauk 119, 551 (1976)].
[33] E. Adelberger, G. Dvali and A. Gruzinov, Phys. Rev. Lett. 98, 010402 (2007) [hep-ph/0306245].
[34] M. Fullekrug, Phys. Rev. Lett. 93, 043901 (2004).
[35] D. Robertson, W. Carter, and W. Dillenger, Nature 349, 768 (1991).
[36] S. S. Shapiro, J. L. Davis, D. E. Lebach and J. S. Gregory, Phys. Rev. Lett. 92, 121101 (2004).
[37] S. B. Lambert and C. Le Poncin-Lafitte, arXiv:0903.1615 [gr-qc].