Explosive astrophysical events at high red shift can be used to place severe
limits on the fractional variation in the speed of light ($\Delta c/c$), the
photon mass ($m_{\gamma}$), and the energy scale of quantum gravity ($E_{QG}$).
I find $\Delta c/c < 6.3 \times 10^{-21}$ based on the simultaneous arrival of
a flare in GRB 930229 with a rise time of $220 \pm 30 \mu s$ for photons of 30
keV and 200 keV. The limit on $m_{\gamma}$ is $4.2 \times 10^{-44} g$ for GRB
980703 from radio to gamma ray observations. The limit on $E_{QG}$ is $8.3
\times 10^{16}$ GeV for GRB 930131 from 30 keV to 80 MeV photons.Comment: 7 pages, Submitted to PR

Schaefer, Bradley E.

English

arXiv

Bradley E.  Schaefer

Crossref

Severe Limits on Variations of the Speed of Light with Frequency

Explosive astrophysical events at high redshift can be used to place severe limits on the fractional variation in the speed of light with frequency (Δc/c), the photon mass (mγ), and the energy scale of quantum gravity (EQG). I find Δc/c\u3c6.3×10-21 based on the simultaneous arrival of a flare in GRB 930229 with a rise time of 220±30μs for photons of 30 and 200 keV. The limit on mγ is 4.2×10-44g for GRB 980703 from radio to gamma ray observations. The limit on EQG is 8.3×1016GeV for GRB 930131 from 30 keV to 80 MeV photons. © 1999 The American Physical Society

LSU Scholarly Repository (Louisiana State Univ.)

Louisiana State University LSU Scholarly Repository Faculty Publications Department of Physics & Astronomy 1-1-1999 Severe limits on variations of the speed of light with frequency Bradley E. Schaefer Yale University Follow this and additional works at: https://repository.lsu.edu/physics_astronomy_pubs Recommended Citation Schaefer, B. (1999). Severe limits on variations of the speed of light with frequency. Physical Review Letters, 82 (25), 4964-4966. https://doi.org/10.1103/PhysRevLett.82.4964 This Article is brought to you for free and open access by the Department of Physics & Astronomy at LSU Scholarly Repository. It has been accepted for inclusion in Faculty Publications by an authorized administrator of LSU Scholarly Repository. For more information, please contact ir@lsu.edu. arXiv:astro-ph/9810479v1  29 Oct 1998Severe Limits on Variations of the Speed of Light with FrequencyBradley E. SchaeferPhysics Department, Yale University, PO Box 208121, New Haven CT 06520-8121schaefer@grb2.physics.yale.edu(July 9, 2018)Explosive astrophysical events at high red shift can be used to place severelimits on the fractional variation in the speed of light (∆c/c), the photonmass (mγ), and the energy scale of quantum gravity (EQG). I find ∆c/c <6.3× 10−21 based on the simultaneous arrival of a flare in GRB 930229 witha rise time of 220 ± 30µs for photons of 30 keV and 200 keV. The limit onmγ is 4.2 × 10−44g for GRB 980703 from radio to gamma ray observations.The limit on EQG is 8.3× 1016 GeV for GRB 930131 from 30 keV to 80 MeVphotons.The question of whether the speed of light varies with frequency is of fundamental andcurrent interest: 1) Einstein’s postulate of the invariance of light is the cornerstone of muchof modern physics, so tests of the correctness of this postulate should be pushed as far aspossible. 2) With the recent evidence from Super-Kamiokande that the neutrino likely hasa mass [1], the question of the mass of the photon should be re-examined. 3) Quantumgravity models suggest [2] [3] that the speed of light has an effective energy dependence.Classical textbooks [4] and review articles [5] report the status as of the middle 1970’s.Laboratory and accelerator experiments were not able test for fractional variations in thespeed of light (∆c/c) to better than roughly 10−8. The first limit that took advantage ofastronomical distances compared the arrival time of radio and optical emission from flarestars to constrain ∆c/c < 10−6 [6]. Then B. Warner and R. Nather measured the phasedifference for pulses of the Crab Pulsar between optical wavelengths of 0.35 and 0.55 microns1to be less than 10 microseconds [7]. At a distance of 2 kpc, this limits ∆c/c to be less than5× 10−17.Surprisingly, no further improvements on ∆c/c have been made on the Crab Pulsar limit,and the topic has received little subsequent discussion in the literature. Indeed, the speedof light has been defined to be a constant for purposes of metrology. In the meantime,a key assumption for the Crab Pulsar limit has been severely undermined since five outof six pulsars detected at high energy have pulse structures that vary strongly both inshape and phase as a function of frequency [8]. Within the last year, several groups haveindependently realized that Gamma Ray Bursts (GRBs) provide a means to look for delaysin light traversing extremely large distances. Amelino-Camelia et al. [2] set approximatelimits on the dispersion scale for quantum gravity, while Biller et al. [9] set stricter limitsby analysis of a short TeV flare seen in a nearby active galaxy (Mkn 421).In this paper, I present new limits based on a variety of explosive events at high red shifts.My original motivation was the discovery last year of a flare in a Gamma Ray Burst (GRB930229) with a 220±30µs rise time that occurs simultaneously from 30 to 200 keV [10], andthe realization that this constrains the dispersion of light to be less than a millisecond outof a Hubble Time. However, the relevant limit depends on the assumed functional form forthe frequency dependence of ‘c’, so different events are the most restrictive in the variouscases.In Table 1, I have gathered the data for the most restrictive events of various classes.These include short duration GRBs, GRBs with GeV photons, GRBs with associated x-ray/optical/radio transients, high red shift Type Ia supernovae, the active galaxy Mkn 421,and the Crab Pulsar. The first seven columns give the source name, the observed bands, theobserved maximum delay ∆t between the rise of the light curve at two different frequencies,the reference for the observation, the low frequency, the high frequency, and the sourcedistance D. Cosmological distances were calculated from the look-back time as a functionof red shift for the most conservative reasonable case of H0 = 80km · s−1Mpc−1 and Ω = 1.Observed GRBs are known to have a relatively narrow luminosity function (as required2by the fast turnover in the LogN-LogP relation [11]), limits on host galaxies forces thepeak luminosity to be much greater than 1058photons · s−1 [12], and the three bursts withmeasured red shifts all have luminosities > 4 × 1057photons · s−1 [12]. So I have adopted apeak luminosity of 1057photons · s−1 as a conservative minimum for use in Fenimore’s peakflux versus red shift relation [13].All the events at cosmological distances are explosive, so presumably the light curvesof different wavelengths will start to rise simultaneously in the absence of dispersion. Thelack of flux or the lack of observations might produce an apparent delay not associated withdispersion. Also, the explosive system might have an intrinsic delay in emission which couldcounter any effects of dispersion. Nevertheless, I will assume that no conspiracies of delayshides the effect of dispersion. The number of strict limits from widely disparate classes ofevents argues that any such conspiracy is unlikely.The frequency dependence of ‘c’ is not known, so no single number can represent thelimit on variations in the speed of light. The most model independent parameter is one withno reference to the observed frequencies, ∆c/c. This limit on ∆c/c will be c · ∆t/D. Fora general dispersion relation like V = c(1 + Aν−2)0.5 with V as the velocity of light withfrequency ν and both c and A as constants [4] [14], the limit on A is (2·c·∆t/D)·(ν−22 −ν−21 )−1for observations at frequencies ν1 and ν2. This dispersion can be related to the photon massas mγ = A0.5h/c2. For quantum gravity models [2], the characteristic energy EQG is greaterthan h · (ν2− ν1) ·D/(c ·∆t). The limits on ∆c/c, mγ , and EQG for each event are presentedin columns 8-10 of Table 1.The limits in Table 1 are many orders of magnitude past those from the Crab Pulsar.The strictest limit on ∆c/c is 6.3× 10−21 for GRB 930229, with second place at 1.4× 10−20for GRB 910711. The two lowest limits on the photon mass are 4.2×10−44g and 1.5×10−43gfor GRB 980703 and GRB 970508, both from radio to gamma ray constraints for bursts withmeasured red shifts. The tightest limit on EQG is 8.3× 1016 GeV for GRB 930131 followedclosely by 6.0× 1016 GeV for Mkn 421.The new limit on ∆c/c is close to 104 times better than the Crab Pulsar limit, and it3is comforting to know that Einstein’s postulate is vindicated to this level. The limit on thephoton mass is 104 times worse than that obtained by considering the Earth’s magnetic field[15], however this result uses virtual photons with ν = 0 so an independent method is stillof interest [14]. The limits on EQG are not yet close to the expected ∼ 1019 GeV [2] [9].Significant improvements in limits on ∆c/c are unlikely since it we are already dealingwith delays of under a millisecond out of a Hubble time. Limits on the photon mass canbe improved by several orders of magnitude with radio detection of prompt emission fromGRBs (as with the FLIRT telescope [16] ) or with radio studies of millisecond pulsars. Thedetection of ∼ 100 GeV photons near the start of a GRB of moderate brightness is possiblewith the GLAST satellite mission [17], and this would test the expected quantum gravitythreshold.[1] Y. Fukuda et al., Phys. Rev. Lett. 81, 1562 (1998).[2] G. Amelino-Camelia et al., Nature 393, 763 (1998).[3] R. Gambini and J. Pullin, gr-qc/9809038 (1998).[4] A. P. French, Special Relativity (W. W. Norton, New York, 1968); J. D. Jackson, ClassicalElectrodynamics (Wiley, New York, 1975).[5] J. L. Hall, in Atomic Masses and Fundamental Constants, edited by J. H. Sanders and A. H.Wapstra (Plenum, New York, 1976) p. 322; R. H. Garstang, A. J. 79, 1260 (1974).[6] B. Lovell, F. Whipple, and L. Solomon, Nature 202, 377 (1964).[7] B. Warner and R. Nather, Nature 222, 157 (1969).[8] D. J. Thompson, in Pulsars, Problems and Progress, IAU Coll. 160, edited by S. Johnston, M.Walker, and M. Bailes (San Francisco, ASP, 1996), p. 307.[9] S. D. Biller et al., Phys. Rev. Lett. (submitted, gr-qc/9810044).4[10] B. E. Schaefer and K. C. Walker, Ap. J. Lett. (submitted, astro-ph/9810270).[11] J. M. Horack, A. G. Emslie, and C. A. Meegan, Ap. J. 426, L5 (1994).[12] B. E. Schaefer, Ap. J. Lett. (submitted, astro-ph/9810424).[13] E. E. Fenimore et al., Nature 357, 140 (1992).[14] G. Feinberg, Science 166, 879 (1969); Z. Bay, and J. A. White, Phys. Rev. D 5, 796 (1972).[15] A. S. Goldhaber and M. M. Nieto, Rev. Mod. Phys. 43, 277 (1971).[16] R. J. Balsano et al., in Gamma Ray Bursts, AIP Conf. Proc. 428, edited by C. A. Meegan, R.D. Preece, and T. M. Koshut (AIP, New York, 1998), p. 585.[17] J. T. Bonnell et al., in Gamma Ray Bursts, AIP Conf. Proc. 428, edited by C. A. Meegan, R.D. Preece, and T. M. Koshut (AIP, New York, 1998), p. 884.[18] P. N. Bhat et al., Nature 359, 217 (1992).[19] M. Sommer et al., Ap. J. 422, L63 (1994).[20] K. Hurley et al., Nature 372, 652 (1994).[21] L. Piro et al., IAU Circ. 6656 (1997).[22] A. J. Castro-Tirado et al., IAU Circ. 6657 (1997).[23] D. A. Frail et al., Nature 389, 261 (1997).[24] T. J. Galama et al., GCN#127 (1998).[25] P. M. Vreeswijk et al., GCN#132 (1998).[26] D. A. Frail et al., GCN #128 (1998).[27] S. Perlmutter et al., Nature 391, 51 (1998).[28] S. Perlmutter et al., Ap. J. 483, 565 (1997).5TABLE I. Limits on ∆c/c, mγ , and EQG from explosive events at high redshift.EVENT Bands ∆t Ref. ν1(Hz) ν2(Hz) D (Mpc) ∆c/c mγ(g) EQG(GeV)GRB 930229 γ − γ 0.5 ms [10] 7.2 × 1018 4.8× 1019 791 6.3× 10−21 6.1× 10−39 2.7× 1016GRB 910711 γ − γ 2 ms [18] 7.2 × 1018 1.2× 1020 1413 1.4× 10−20 9.0× 10−39 3.3× 1016GRB 910625 γ − γ 4 ms · · · 7.2 × 1018 1.2× 1020 1344 3.0× 10−20 1.3× 10−38 1.6× 1016GRB 910607 γ − γ 8 ms · · · 7.2 × 1018 1.2× 1020 1677 4.8× 10−20 1.7× 10−38 9.9× 1015GRB 930131 γ −GeV 25 ms [19] 7.2 × 1018 1.9× 1022 260 9.6× 10−19 7.4× 10−38 8.3× 1016GRB 930131 γ −GeV 0.5 s [19] 7.2 × 1018 1.1× 1023 260 1.9× 10−17 3.3× 10−37 2.4× 1016GRB 940217 γ −GeV 4800 s [20] 7.2 × 1018 4.3× 1024 385 1.2× 10−13 2.7× 10−35 1.4× 1014GRB 970508 X − γ 5.6 hrs [21] 2.4 × 1017 1.2× 1020 1493 1.4× 10−13 9.2× 10−37 3.7× 109GRB 970508 U − γ 4.4 hrs [22] 8.2 × 1014 1.2× 1020 1493 1.1× 10−13 2.8× 10−39 4.7× 109GRB 970508 Radio− γ 121 hrs [23] 8.6× 109 1.2× 1020 1493 2.9× 10−12 1.5× 10−43 1.7× 108GRB 980703 X − γ 22 hrs [24] 2.4 × 1017 1.2× 1020 1592 5.0× 10−13 1.8× 10−36 1.0× 109GRB 980703 I − γ 21 hrs [25] 3.3 × 1014 1.2× 1020 1592 4.8× 10−13 2.4× 10−39 1.0× 109GRB 980703 Radio− γ 29 hrs [26] 5.0× 109 1.2× 1020 1592 6.6× 10−13 4.2× 10−44 7.6× 108SN 1997ap I −R 240 hrs [27] 3.3 × 1014 4.3× 1014 1489 5.8× 10−12 1.3× 10−38 6.8× 101SN 1994G I −R 72 hrs [28] 3.3 × 1014 4.3× 1014 1029 2.5× 10−12 8.8× 10−28 1.6× 102Mkn 421 TeV − TeV 280s [9] 1.2 × 1026 4.8× 1026 112 2.5× 10−14 2.1× 10−28 6.0× 1016Crab V − U 0.01ms [7] 5.5 × 1014 8.2× 1014 0.002 5.0× 10−17 5.4× 10−41 2.3× 1076

Severe Limits on Variations of the Speed of Light with Frequency

Abstract

Similar works

Full text

Available Versions

Crossref

LSU Scholarly Repository (Louisiana State Univ.)