Relativity in the Global Positioning System

Relativity in the Global Positioning System Neil Ashby Department of Physis,UCB 390 University of Colorado, Boulder, CO 80309-00390 NIST Affiliate Email: ashby@boulder.nist.gov July 0, 006 AAPT workshop 1

Fundamental Priniples Priniple of Inertia The laws of physis are the same in all inertial frames of referene. Constany of the speed of light The speed of light,, is a onstant independent of the motion of the soure (or of the observer); (it is in all inertial frames). Priniple of Equivalene ( weak form ) Over a small region of spae and time, the fititious gravitational field indued by aeleration annot be distinguished from a real gravitational field due to mass. July 0, 006 AAPT workshop

Changes in point of view (referene frames) In studying relativity, one must be willing to adopt different points of view-- that is, different referene frames. The physial phenomena don t hange, but our desription of them does hange. Example 1: the gulf stream urves toward the east as it flows north. For an observer fixed on the rotating earth, this is due to the Coriolis fore. To an observer in a loal, freely falling non-rotating frame attahed to earth s enter, this is due to onservation of angular momentum. Example : A pendulum in an aelerating ar points bakwards. From the point of view of someone in the ar, the fore of gravity points slightly downwards and slightly bakwards. From the point of view of someone on the ground, the pendulum bob is aelerated forwards by a omponent of tension in the string that points slightly forwards and upwards. July 0, 006 AAPT workshop 3

Question. In an aelerating vehile, if up is indiated by the diretion of the string that holds down a helium balloon, what diretion is up? Up? Up? July 0, 006 AAPT workshop 4

Relativity of Simultaneity To an observer on the ground, let two lightning strokes at the front and bak of the train be simultaneous. The moving observer at the train s midpoint finds the event at front ours first. t ' t vx July 0, 006 AAPT workshop 5

Notation-- Lab and Moving Frames y y * Event In Lab Frame: { t, x,y, z } In Moving Frame: { t, x, y, z } x,x z z Origins pass by eah other at t = t = 0. July 0, 006 AAPT workshop 6

Breakdown of simultaneity In a given inertial frame, it s OK to take differenes of veloities and obtain a veloity differene greater than. Example: Let a rod of length L=x move in the positive x-diretion with speed v. Light emitted at t=0 from the left end of the rod travels to the right end. v event at t=t =0 L v signal propagating with speed In Lab, arrival time at the rod end: In the moving frame: L L L vl t = ( v) = (1 v / ) + ; t ' = L ; t ' = t vl = t vx July 0, 006 AAPT workshop 7

Breakdown of simultaneity y y t * Event vx ' t x,x z z July 0, 006 AAPT workshop 8

x = Relation Between Doppler Effet and Relativity of Simultaneity 0 λ λ 3 λ... nλ... Moving observer says the wavefronts are marked differently: So the wavefront at x = λ ( n = 1) vλ = v λ Wavefronts are marked simultaneously by the non-moving observer: July 0, 006 AAPT workshop 9 t ' t =0 vnλ = before it gets into the right position to be marked at t = 0. To the moving observer, the wavelength is: λ v v ' = λ + (1 ). λ = + λ needs to move an additional distane

Doppler frequeny shift = f λ, ln = ln f + ln λ. So when lambda inreases, the frequeny dereases. Taking the differential of the logarithm funtion gives f f λ = = λ v. July 0, 006 AAPT workshop 10

Equivalene Priniple and Gravitational Frequeny Shifts Over a small region of spae and time, a fititious gravity field indued by aeleration annot be distinguished From a gravity field produed by mass. July 0, 006 AAPT workshop 11

Gravitational Frequeny Shift July 0, 006 AAPT workshop 1

Gravitational Frequeny Shift t = L / gl v = gt = ; f v gl Φ = = =. f GM GM J (1 + ) f r a1 = f The situation in the roket is stati. The frational frequeny differene between the loks is July 0, 006 AAPT workshop 13 f f Φ = +.

Frequeny shifts due to Gravitational Potential Differenes Let Φ be the gravitational potential on earth s geoid--at mean sea level, 0 and r be the radius of a GPS satellite. Φ = GM E r Φ 0 Φ0 inludes ontributions from earth s oblateness as well as its mass: GM 1 Φ = (1 + J ), 0 a1 where: a 1 is the equatorial radius of the earth; J.001086 is earth s quadrupole moment oeffiient. July 0, 006 AAPT workshop 14

How big are gravitational frequeny shifts in the GPS? To get a rough estimate, assume the satellite orbit is irular and the referene lok is on earth s equator. f 1 GM GM = ( ) f a a1 where GM 1 14 3 3.986 10 m /s ; a = 6,56 km; a = = 6,378 km f f 10 5 10 ; ( 13km navigation error per day) July 0, 006 AAPT workshop 15

Best IIR Rbs Navs 51, 54 July 0, 006 AAPT workshop 16

Question: If a lok makes an error in one day of 1 part in 10^14, how far would light travel in this amount of time? July 0, 006 AAPT workshop 17

Question: If a lok makes an error in one day of 1 part in 10^14, how far would light travel in this amount of time? Answer: In one day, the error is: 14 10 10 86400se = 8.64 10 se. In this amount of time, light travels a distane d = = 10 10 (8.64 10 se) 9979458m/se (8.64 10 se) = 0.6 m July 0, 006 AAPT workshop 18

Constany of r - r j = ( t t j ), j = 1,,3, 4 July 0, 006 AAPT workshop 19

Sample data SV # Transmission Epoh t i (s) Transmitter Position x i (m) Transmitter Position y i (m) Transmitter Position z i (m) 1 3739. 94 4 365 6 13004579.64 1899783.895 1346718.71 3739. 90 713 391 8 045017.566 16360459.358-4436309.875 3 3739. 95 307 870 0 098631.70 15908390.45 3486595.546 4 3739. 99 346 353 9 13799439.94-8705178.668 0959777.407 Question: Where is the reeiver and what is the time at the reeiver? July 0, 006 AAPT workshop 0

Reiproity r - r j ( ) j t = t July 0, 006 AAPT workshop 1

The onstany of the speed of light implies time dilation July 0, 006 AAPT workshop

Einstein s Light Clok v t = L v = / ; t ' = L / = 1 v / t L / 1 v / This is just Eulidean geometry and the onstany of. July 0, 006 AAPT workshop 3

How Big is Time Dilation in the GPS? 1 1 v / 1 ; v = 4000 m/s; 1 v = 3.5 10 v 11 July 0, 006 AAPT workshop 4

Aounting For Relativisti Effets Example: Time Dilation: dτ = v dt 1 / ; ( 1 / ) 1/ dt = v dτ Elapsed Coordinate time: 1 v 1 + dτ. 1 t = dτ 1+ path v Observed Proper Time July 0, 006 AAPT workshop 5

GPS Satellite in a irular earth-bound orbit Newton s law of motion: Fore toward the earth is gravitational. GmM = r r m ; 3 rɺɺ The mass of the satellite anels out (a onsequene of the Priniple of Equivalene). The orbit is nearly a Kepler ellipse. Solution of the above equation shows that the energy is onstant: where a is the semimajor axis. 1 GM GM v = r a July 0, 006 AAPT workshop 6

Frequeny shift of GPS satellite loks relative to referene lok on equator ( ω ) f Φ Φ = f 0 1 v 1 a E 1 Putting in everything that is known about these quantities and adding and subtrating some terms, f 1 1 1 3GM E GM E 1 = (1 / ) GM E + + J + a f a r a a1 GM E 1 1 10 = 4.4647 10. + a r ( ω ) E 1 The first term depends on orbital eentriity and gives rise to a periodi time error that is (E is the eentri anomaly) e trel = GM sin( ). Ea E + onst July 0, 006 AAPT workshop 7

Cloks on earth s geoid beat at equal rates July 0, 006 AAPT workshop 8

Orbit adjustments in GPS δ f 3GM E 3GM E δ a = = f a a a ; For an inrease in altitude of 0 km, the hange in frequeny is δ f f 13 1.88 10. This effet is now understood and the lok frequeny is adjusted when the orbit is hanged. (Bak-of-the-envelope alulation!) July 0, 006 AAPT workshop 9

Further development--introdue the metri Φ ds = (1 + )( dt) + dx + dy + dz ; Disuss proper time, oordinate time; show how the effets an be obtained from the metri.. END July 0, 006 AAPT workshop 30

Sagna effet July 0, 006 AAPT workshop 31

Sagna Effet on Synhronization in a Rotating System Sagna Corretion = ω A z July 0, 006 AAPT workshop 3

Priniple of Equivalene V+ V V r Loal Inertial Frame Aeleration Air Vir = = T Φir Air + Φir 1 Vir Φir T Indued potential differene/ ; Gravitational potential differene/ = ; Net potential differene/ = = + = 0; July 0, 006 AAPT workshop 33

Common-view time transfer July 0, 006 AAPT workshop 34

Why are atomi loks needed? To redue the effet of lok error to < meters, the lok error must be less than / = 6.7 x 10-9 se. Half a day = 4300 seonds, so the frational lok error must be less than: ( m)/(4300 s x ) = 1.5 x 10-13. Only atomi loks an ahieve suh stability. July 0, 006 AAPT workshop 35