arXiv:1203.0102v1 [gr-qc] 1 Mar 2012 l aitoso udmna osat,priual the constant particularly constants, fine-structure fundamental of variations ble bevbeuies 2.If [2]. universe observable ennl,suiso usrasrto pcr ugs a suggest spectra in absorption change quasar small of studies null, been ytoersls htwudipyacag na least at in comprise change that a parameters imply the would of that one results, those by arsoi hoyo rvt,gnrlrltvt el us a tells as relativity about Nevertheless, general little gravity, space. of free theory the in macroscopic to bodies is Einstein’s fields of of it gravitational accelerations variation universe relating a the is principle in principle equivalence falling when This and freely [5]. where performed experiment local, of nongravitational independent in any is of that outcome states the frames which (LPI), ance’ constant on Planck’s bounds of establish terrestrial invariance to the prior [4] with System experiments combination Positioning comparison in clock Global analyzed the from is data (GPS) letter, this In h tmctasto rqec sdi naoi clock field, atomic gravitational an a in in used at frequency order transition first atomic to the LPI of violations results constant. both Planck’s use of we invariance the Finally estimate dilation. to time determine for we to limits experiments Then LPI clock clocks. of atomic results for the limits incorporate LPI determine to data observa- recent [8]. of neutrinos review super-luminal the of to tions relevant be our approach and may relativity, This results general of transitions. check consistency atomic a provides of energy ef- proper the observable variable a The of fects phenomena. herein, correctly relativity used macroscopic general is describes that approach assumption broad the less on fundamental A based in 7]. variations 6, possible [1, constants study to partly oped necessary. is ance ayeprmnsadosrain aetse possi- tested have observations and experiments Many hsaayi rvdsats f‘oa oiininvari- position ‘local of test a provides analysis This sn h tnadcneto eg,[] o expressing for [9]) (e.g., convention standard the Using GPS use we First, steps: three in proceeds analysis The devel- been has (SME) Extension Model Standard The x n tsm rirr eeec point reference arbitrary some at and h ASnmes 62.r 48.c 06.30.Ft 04.80.Cc, 06.20.Jr, numbers: violation. PACS LPI of extent the represents co withi that clock constant parameter that Planck’s terrestrial of indicate of (LPI) results results invariance the position with local combination in used hs xeietleiec fisinvari- its of evidence experimental Thus, . ulcyaalbecokcreto aafo h lblPo Global the from data correction clock available Publicly α P eto h oa oiinivrac fPac’ consta Planck’s of invariance position local the of test GPS vrteps ilo er costhe across years billion 8 past the over h r eue rmteivrac of invariance the from deduced are α 1.Tog otrslshave results most Though [1]. eateto hsc n srnm,Clfri tt Univ State California Astronomy, and Physics of Department α eet aya suggested as vary to were h otrde otrde aiona93086,USA 91330-8268, California Northridge, Northridge, sivratwti ii of limit a within invariant is h . α = .Kentosh J. e O 2 / uha a as such – 4 Dtd eray2,2012) 29, February (Dated: πε o ¯ hc [3]. ∗ n .Mohageg M. and ersra aoaoy–cnb eae sfollows: as related be can – laboratory terrestrial hntecoki at is clock the when where point where nryprui asand mass unit per energy rm at frame iesols parameter dimensionless h P aelts ti ett h srt aclt and calculate to user the to left is It satellites. of eccentricities GPS the the for corrections relativistic general the 0.02. to 0.01 from that ranged for lowest eccentricities rates the Satellite clock clock in in had satellite. range changes satellites theoretical their the Those of to deviation corrections 11, 32. standard 2, the and of clocks: 28, ratios stable 26, most 21, the 18, and orbits eccentric more and stations, [14]. control stations GPS ground by 12 350 collected day. satellites, approximately data GPS 32 on clock the one based and from is for IGS files position satellites the includes in GPS file information 32 The to SP3 for precise single are data A corrections correction Positions clock [13]. and ps time. mm GPS 1 1 of to are in minutes precise corrections are 15 used, clock every was and for format positions published SP3-c satellite in [12]. GPS product (IGS) which ‘Final’ Service IGS GNSS International The the in by made Internet are 11]. effect [10, relativity this general for on clock Corrections based a than GPS, slower faster ground. and a the perigee has at on than its satellite faster cause run a which to potential, apogee, clock gravitational At higher and navigation [10]. speed generally the 0.02 eccentricities have of than orbits part less satellite as GPS broadcast is signal. correction continuous satellite clock a Universal each a drift, Coordinated clock for time, for to correct GPS To related (UTC). on be Time operated can is that GPS timescale The bit. frequencies. clock atomic for violation h lc orcin ulse yISd o include not do IGS by published corrections clock The had that chosen were seven satellites, GPS 32 the Of the on available made data satellite GPS analyzed We or- in and ground the on clocks atomic uses GPS The | β h O | h otx fgnrlrltvt.The relativity. general of context the n U f < o i hntecoki at is clock the when prsneprmnst ofimthe confirm to experiments mparison O Φ = stefeunymaue ya bevrat observer an by measured frequency the is 0 iinn ytmwsaaye and analyzed was System sitioning h oeta ieec s∆ is difference potential The . . 0,where 007, † f i xo − /f v o i 2 / 1+ (1 + 1 = ,Φ 2, x β smaue rmtereference the from measured as ersity, h β i f v sadimensionless a is stegaiainlpotential gravitational the is i ersnsteetn fLPI of extent the represents O stecoksvlct.The velocity. clock’s the is and , β f )∆ nt f U/c xo stefrequency the is 2 , U = U x − U (1) o , 2 add those effects [15]. Since we did not have to subtract per unit mass and velocity of a satellite at apogee, respec- it out, the missing relativistic component to the clock tively; and ΦGPS and vGPS are the same for a satellite corrections simplified the analysis. in an ideal, circular GPS orbit corresponding to a radius To find βf the changes in the IGS-published clock cor- of 26,561.75 km, representing GPS time. (GPS satellites’ rections of the GPS satellites over each 15 minute inter- clocks are adjusted to run more slowly pre-launch so that val were analyzed. One year of such data was reduced, when they reach orbit, they run at GPS time.) Tep is the spanning the time period from April, 2010 through May, ‘epoch time,’ which is the 15 minute interval between 2011, including extra days used for averaging. For each clock corrections. satellite, 95 changes in its clock correction during each The coordinates in the SP3 files are published in an day were plotted as a function of its distance from the earth-fixed reference frame. A satellite’s velocity rela- earth’s center. (The 96th spanned two SP3 files and was tive to the earth’s non-rotating frame was interpolated omitted for simplicity.) A representative plot is shown in from its instantaneous radial distance, using Newtonian Fig. 1. gravity. For an ideal, circular GPS orbit, this method reproduces the standard velocity vGPS of 3,873.83 m/s, used as a baseline in Eq. (3). 2.5 The mean offset of the changes in clock corrections from zero in Fig. 1 indicates how much faster or slower 2 a satellite’s clock is advancing than GPS time. That has no bearing on the current analysis. Of interest is how 1.5 the changes in clock corrections might vary with radial distance from the earth’s center. 1 The best fit slope and the corresponding daily values for βf were calculated for the seven satellites, for each day 0.5 in the test period, with 2,749 plots generated. (The clock Change in Clock Correction (ns) data was partially missing or corrupted on some days.) 0 26,400 26,600 26,800 Results for three representative satellites are shown in Radial Distance (km) Fig. 2(a-c).

FIG. 1: Changes in IGS-published clock corrections every 15 minutes for Satellite No. 18 on 6/10/10, plotted against dis- 0.05 Satellite 2 0.05 Sat 18 tance from the earth’s center. The uncertainty in the ordinate is σ, as published in the SP3 files. The least-squares linear fit βf has a small slope m. 0 0

Since the relativistic eccentricity effect is not included (a) (b) −0.05 −0.05 in the published clock corrections, changes to those cor- 100 200 300 100 200 300 rections should not vary with distance from the earth’s center. Standard theory predicts the slope m of the linear fit in Fig. 1 to be zero. The small slope evident in Fig. 0.05 Sat 32 1, extracted by a linear least-squares regression fit to the βf data, might be explained by a non-zero βf . A non-zero 0 (d) slope could also be caused by any number of effects on (c) satellite clock corrections, including random errors and −0.05 atmospheric effects. 100 200 300 −0.005 0 0.005 To estimate conservatively large bounds for βf , all Day of test year βf nonzero slopes were attributed to βf . The slope m in

Fig. 1 is related to a possible non-zero value of βf by FIG. 2: Daily and filtered values of βf for one year for (a) Sat 2, (b) Sat 18, (c) Sat 32. Sat 2 had the least annual oscillation; βf = m(ra − rp)/(2∆Tmax), (2) Sat 32 had the greatest range in βf . (d) Combined probability distribution of filtered βf for 7 satellites for one year. where ra is the apogee radius, rp is the perigee radius and ∆Tmax is the maximum theoretical clock correction The plots of βf exhibit significant daily variations, and of a satellite at apogee, given by the following, adapted an annual oscillation that varied among satellites. The from [16]: daily variations were consistent in magnitude among the 2 2 2 satellites. The annual oscillations are likely caused by ∆Tmax = Tep Φa − va/2 − ΦGPS + vGPS/2
/c . (3) atmospheric effects. The maxima of these oscillations In this equation, Φa and va are the gravitational potential roughly coincide with a semi-major axis aligned with 3

Eurasia and the Pacific Ocean, an orientation that shifts Cryogenic optical resonator clocks are another type of 360 degrees over a year. clock used in tests of fundamental constants. Unlike Time-series and Fourier analysis of the data was used atomic clocks, the rates of these clocks depend on the to develop appropriate digital filters to remove yearly, bi- speed of light rather than Planck’s constant. An invari- monthly and daily variations driven by nongravitational ant c means that the locally-measured (proper) rates of effects. To remove the annual oscillation, a least-squares suitably configured resonator clocks will not vary with fit was computed using y = A+B cos(2πt/1yr+φ), where position, even if h varies. (Some resonator clocks are A, B and φ are fit parameters. The B cos(2πt/1yr + φ) locked to an atomic frequency and would depend on h term was subtracted from the daily values. Based on the [18].) Therefore, a resonator clock measures proper time frequency distribution in the Fourier analysis, we then and can be used to determine βt. used a 51-day moving average to filter out short-term Comparisons between atomic clocks (hydrogen masers) fluctuations in βf . The filtered data for three satellites and cryogenic sapphire oscillators (CSO) have been per- (2, 11, 32) exhibited a sawtooth-shaped oscillation with formed at the Paris Observatory [4] and other laborato- a 120-day period, of unknown origin. That oscillation ries (e.g., [19]). Tobar et al. summarize that work and was filtered out for those three satellites in the same way show that little measurable difference is observed as the as the annual oscillations. The limits on βf were most earth moves in its elliptical orbit within the gravitational sensitive to filtering out the annual oscillations, but less potential of the sun, within a limit of βH maser − βCSO = sensitive to the duration of the rolling average or to re- −2.7(1.4) × 10−4 for annual variations [4]. The rela- moving the 120-day oscillations. The residual daily val- tive rates of atomic and resonator clocks differ little with ues of βf for three satellites are shown with solid lines in gravitational potential. Fig. 2(a-c). Once a GPS satellite reaches orbit, the clock experi- The residual values of βf from seven satellites exhibit ments [4] indicate that a resonator clock and an atomic the bell-shaped distribution shown in Fig. 2(d). A small clock on the satellite would advance at the same rela- residual mean of βf = 0.0003 was present. We cannot tive rates. Therefore, the variations in clock corrections rule out GPS systemic errors as its source. Based on should apply equally to resonator clocks, and can be used a 95% confidence level, the following limits were found: to determine a limit on βt. In other words, within a GPS −0.0027 <βf < 0.0033. The results for the seven satel- orbit we can treat GPS atomic clocks as a surrogate for lites were weighted equally. Finding these limits on βf resonator clocks, so that βt ≈ βf . is the first step in setting limits on the invariance of In the clock comparison experiments, βH maser − βCSO Planck’s constant. corresponds to βf −βt, and their limits were added to the The well-behaved annual oscillations in Fig. 2 demon- limits on βf to determine limits for βt. Combining the strate some statistical significance for the linear fits for maximum uncertainties, the limits on LPI violation for 2 −5 m. However, values of R for the fits ranged from 10 time dilation were found to be −0.0031 <βt < 0.0037. to 0.6 with a mean of 0.04, suggesting a low to moderate To prove that h is invariant in the context of SME statistical significance for m [17]. theory is probably not possible with existent experimen- To study this effect, a statistical model was developed tal evidence. However, in general relativity the locally- in which the standard deviations of the GPS clock correc- measured (proper) rest energy of a particle is indepen- tions were used to create a randomly-generated data set dent of gravitational potential. Based on arguments sim- to mimic the GPS data. Ninety-five randomly generated ilar to Nordtvedt’s [20], the proper energy emitted by a points were used to generate fits for m, repeated 2,555 specific atomic transition should also be invariant. For times and then filtered to model our analysis. The results example, the locally measured energy emitted by the indicate that about 2/3 of the limits on βf arise from ran- transition between the two hyperfine levels of the ground dom variations in the GPS clock corrections. However, state of Cs133 – the isotope used in atomic clocks – should some GPS satellites exhibit more consistent clock correc- be the same at all points and time in the universe. tions and the bounds could be reduced by analyzing only With a locally-invariant emission energy Eo, the those satellites. proper frequency of emission fx of a given atomic tran- If h were to vary, general relativistic time could differ sition measured at any elevation x would then be given from atomic time. Possible LPI violations of gravita- by tional time dilation can be expressed to first order as fx = Eo/hx, (5) 2 dtx/dto =1+(1+ βt)∆U/c , (4) where hx is the locally measured value of h at x. If h where dtx/dto is the ratio of the time rates at x and O, satisfies LPI, then hx would be constant and the proper and βt is a dimensionless parameter that represents the frequency of an atomic clock would also be invariant. extent of LPI violation for time dilation. If βt = 0, the However, if hx varies with position then the proper fre- formula reduces to general relativity. quency of an atomic clock would also vary. 4

LPI violations for h can be written as those constraints we find, based on experimental evidence

2 from the GPS and clock comparison experiments, that hx/ho =1+ βh∆U/c , (6) h satisfies local position invariance to within a limit of 0.007. The data also supports the gravitational time di- where ho is the locally measured value of h at reference lation of general relativity for geometrodynamic clocks point O, hx is its locally measured value at x, and βh is within a limit of 0.004. the parameter for Planck’s constant. The work reported in this letter was carried out at Let fo be the proper frequency of an atomic clock when California State University, Northridge. it is at O. Since Eo = hofo, it follows from Eq. (5) that at x,

fx = hofo/hx. (7) ∗ Electronic address: [email protected] Due to time dilation, the frequency fxo of the clock at † Electronic address: [email protected] x measured by an observer at O will be [1] J.-P. Uzan, Rev. Mod. Phys. 75, 403 (2003). [2] J. K. Webb, M. T. Murphy, V. V. Flambaum, V. A. fxo = (dtx/dto)(hofo/hx). (8) Dzuba, J. D. Barrow, C. W. Churchill, J. X. Prochaska, and A. M. Wolfe, Phys. Rev. Lett. 87, 091301 (2001). A formula relating the three β′s to first order can be [3] K. A. Bronnikov and S. A. Kononogov, found by substituting Eqs. (1), (4) and (6) into (8), which ArXiv:gr-qc/0604002. yields β = β − β + O(µ2), where µ = ∆U/c2. Second [4] M. E. Tobar, P. Wolf, S. Bize, G. Santarelli, and V. Flam- h t f baum, Phys. Rev. D 81, 022003 (2010). order relativistic effects in the earth’s field are negligible, [5] C. M. Will, Theory and experiment in gravitational so higher order terms can be ignored, leaving physics (Cambridge University Press, Cambridge, 1993). [6] D. Colladay and V. A. Kosteleck´y, Phys. Rev. D 55, 6760 βh ≈ βt − βf . (9) (1997). [7] V. A. Kosteleck´y, Phys. Rev. D 69, 105009 (2004). The traditional interpretation of general relativity pre- [8] T. Adam et al., arXiv:1109.4897. dicts that βt = βf ≡ 0, yielding βh = 0, which would [9] A. Godone, C. Novero, and P. Tavella, Phys. Rev. D 51, signify that h is invariant. 319 (1995). [10] N. Ashby and J. J. Spilker, in Global positioning system: The estimated limits on βt and βf were used in Eq. (9) to determine LPI limits for Planck’s constant. Combin- theory and applications, Vol. I, edited by B. W. Parkinson and J. J. Spilker (American Institute of Aeronautics and ing the limits on those two parameters and accounting for Astronautics, 1996), pp. 623-697. their signs, our final results, accurate to one significant [11] E. D. Kaplan, editor, Understanding GPS, principles and digit, are summarized in Table I. applications (Artech House, Boston, 1996). [12] J. M. Dow, R. E. Neilan, and C. Rizos, J. Geodesy 83, 191 (2009). TABLE I: Estimated limits on LPI violation, based on this [13] S. Hilla, The extended standard product 3 orbit format study unless otherwise noted. (SP3-c), (National Geodetic Survey, August 17, 2010). Type of LPI Limits found Basis [14] A. El-Rabbany, Introduction to GPS, the Global Posi- tioning System (Artech House, Boston, 2006). Atomic clocks −0.0027 < βf < 0.0033 GPS data [15] Navstar GPS Space Segment/Navigation User Interfaces, −4 Clock comparisons βf − βt = −2.7(1.4) × 10 Ref. [4] Interface Specification IS-GPS-200, Rev. E (Global Posi-

Time dilation −0.0031 < βt < 0.0037 GPS & [4] tioning System Wing, June 8, 2010). [16] N. Ashby, Living Rev. Relativity 6, 1 (2003). Planck’s constant |β | < 0.007 Eq. (9) h [17] J. R. Taylor, An introduction to error analysis (Univer- sity Science Books, Mill Valley, California, 1997). To illustrate the effects of the filtering, we obtain [18] L. Maleki, V. S. Ilchenko, M. Mohageg, A. B. Matsko, A. |βh| < 0.01 if we use a 1-year filter but substitute an A. Savchenkov, D. Seidel, N. P. Wells, J. C. Camparo, and B. Juduszliwer, in Frequency Control Symposium, 11-day moving average of βf (smaller than the 28-day 2010 IEEE International, pp. 119-124. lunar cycle) for the 51-day average and omit the 120-day [19] J. P. Turneaure, C. M. Will, B. F. Farrell, E. M. Mattison, filters. and R. F. C. Vessot, Phys. Rev. D 27, 1705 (1983). In conclusion, general relativity requires the local in- [20] K. Nordtvedt, Phys. Rev. D 11, 245 (1975). variance of the speed of light and emission energy. Within