Time Dilation in Gavity Wells By Rihad R. Shiffman Digital Gaphis Asso. 038 Dunkik Ave. L.A., Ca. 9005 s@isi.edu This doument disusses the geneal elativisti effet of time dilation aused by a spheially symmeti gavitational well. In patiula we will examine the effet the eath's gavitational field has on the ate of the passage of time on objets, both on the sufae and in obit. Fo this doument the eath is assumed to be a sphee and the satellite obits ae iula. Sine the Shwazshild eti below uses geometi units, we must onvet fom KS units to geometi units whee mass and time ae measued in metes. Veloity is a unitless atio between Vmks and, the speed of light. The Convesion fomulas ae listed immediately below. G ass. ge kg Veloity V ge V mks Time T ge. T mks The Shwazshild eti in geometi units, is one of the solutions of Einstein's field equation. This meti desibes spaetime aound a non-otating spheially symmeti ente of attation, appoximated by a non-otating sta, planet, o blak hole. Sine the tangential veloity of the Eath's sufae is so small ompaed to the speed of light, the otation of the Eath an be ignoed and we an still uses the Shwazshild eti. ds.. d... dφ sin φ. dθ Sine the above meti is spheially symmeti, we an always define the obital plane of all ou iula obits, inluding the sufae level obit, to oinide with the eath's equatoial plan. This will always keep dφ = 0 and sin φ =, allowing us to eliminate dφ fom the meti. Next though a hange of vaiables, fom θ to φ, we get the meti shown below. Sine this meti isn't positive definite, we must onside the following thee ases: Time Like Inteval dτ.. d.. Light like Inteval. 0. d.. Spae like inteval. dφ dσ. d.. dφ... dφ
The time like meti an be simplified even moe fo some speial ases. The shell model potion of this doument is based on the simplified meti below. This meti desibes a time like inteval with d =0 and dφ=0 on a fixed adial shell as obseved by a fa away obseve. The meti gives the diffeenes in ate of time between loal (pope) time on the shell and fa away time fo stationay objets.. Fo d=0 and dφ 0 dτ. The adial stething / ompession fato fo events with =0, these events ae simultaneous fo the fa away obseve. Fo =0 and dφ 0 dσ. d. Unit Definitions ae: The Speed of light in flat spaetime in KS units.9979458. 0 8. m se The Gavitation onstant in KS units G 6.676. 0. m 3 kg. se Definition of a nautial mile: nm 6076. ft seonds in a day: day se. 86400 se Nanoseond: ns. 0 9 se ioseond: us. 0 6 se Exeutable unit onvesion fomulas: ass Veloity Time G mt kg. kg Vge Vmks Vmks T ge T mks. T mks kg mt. G mt Vmks Vge. Vge T mks T. ge T ge The mass of the Eath, the Sun, and the blak hole at the ente of ou galaxy ae listed below in both kilogams an geometial units, metes. Objet ass in Kilogams ass in geometial units ( metes ) Eath ekg 5.974. 0 4. kg e mt ekg e = 4.435407 0 3 m Sun skg.989. 0 30. kg s mt skg s =.477 0 3 m ilkyway gbkg 5.. 0 36. kg mt gbkg = 3.86 0 9 m
The Shwazshild Radius alulations ae pesented fo you uiosity. As you an see, we ae nowhee nea these adii while studying satellites obiting the Eath o things on its sufae. If a mass is ompessed to sphee who's adius is equal to its Shwazshild Radius, then a blak hole is fomed. It would be wise to stay 0 o moe Shwazshild Radii away fom any blak hole. The tidal foes will destoy you befoe you get nea the Shwazshild Radius of a small blak hole. If you oss the Shwazshild Radius of a lage blak hole, you'll neve be able to etun and will beome one with the spaetime singulaity at its ente. In ode to tun a non-otating mass made fom the Eath into a blak hole, you would have to ompess it into a ball smalle than.8m in adius. If you ompessed the sun down to a adius of 3 kilometes, it would also fom a blak hole. The singulaity in the meti at the Shwazshild adius, =*, is nonessential and an be emoved by a oodinate tansfom, but the singulaity at =0 is eal and an not be tansfomed away. This means that you an oss the Shwazshild Radius in the inwad dietion of a lage blak hole with no touble o disomfot. The singulaity in the ente of the hole is of an essential natue and both time and spae as we know them stop thee. Shwazshild Radius in metes is: sh kg. mt kg Eath sh ekg = 8.87 0 3 m Sun sh skg =.953 0 3 m ilkyway sh gbkg = 7.7 0 9 m Note: If you have a mass ompessed down enough to fom a blak hole, thee is no way to measue its Shwazshild adius dietly. If you ty and dop the end of a tape measue into a blak hole, the tape will stutually fail when it osses the Shwazshild adius. Even light signals won't ome bak out of the hole. Beause of this we must find a new way to define the Shwazshild adius of a blak hole. The Shwazshild adius is defined as the edued iumfeene of the hole, whih is the event hoizon's iumfeene divided by *π, whih an be measued.
Fo alulating the time dilation, stat with the Shwazshild eti in geometi units fo a time like inteval. Then alulate the atio of time passage between the two objets at diffeent altitudes above the ente of gavitational attation. dτ.. d... dφ To do this apply the meti twie, fist to obiting satellite and then to the sufae of the eath. Then take thei atio of the two to find the time dilation. Fo the sufae of the eath and a satellite in iula obit with d 0 dτ... dφ Next solve fo: dτ whee dτ is loal wistwath time ( pope time) and is fa away time in flat spaetime. dτ... dφ.. dφ Now tangential veloity of the satellite obit and the eath is given by: v. dφ dτ. v So dτ satellite dτ eath. satellite v satellite. eath v eath and e Note: is the mass of the eath in geometial units (i.e. metes). dτ satellite dτ eath. satellite v satellite. eath v eath This is the atio of the ate of satellite time passage to the ate of eath time passage.
Case ) let v=0 and d=0 Fo the fixed shell model at the altitude of the GPS satellite's obit, whih is:.. satellite 6.6 0 6 m The atio of satellite time to eath time given by: R t dτ satellite. dτ satellite. eath eath. satellite eath. Radius of fixed shell Radius of the Eath ass of the Eath in metes satellite =.66 0 7 m.. eath 6.37 0 6 m = 4.435 0 3 m R t.. satellite eath. satellite eath. t ea day se The atio of shell time to eath time is given by: R t =.0000000005944 One Eath day in seonds is: t ea = 8.64 0 4 se Duing one eath day, the shell o satellite ages t sat. R t day se whih is: t sat = 8.64000000457438 0 4 se t t sat t ea t = 4.574 0 5 se t = 45743.85 ns The shell ages t = 45743.84553 ns moe than the eath.
Case ) GPS satellites ae in a hou eath obits and at the same altitude as the fixed shell in ase. Note: The atio of satellite time to eath time is less that the fixed shell model above. This is disussed at the end of this page. Eath Data Satellite Obital Data Radius of eath = 6.37 0 6 m satellite =.66 0 7 m.. eath π eath.. satellite π satellite Ciumfeene of eath = 4.003 0 7 m satellite =.67 0 8 m v eath eath satellite v t satellite. ea h Tangential veloity v eath = 463.3 m se v satellite = 3.869 0 3 m se Tangential veloity in geometi units v e v eath v s v satellite v e =.545 0 6 v s =.9 0 5 ass in geometi units (metes) = 4.435 0 3 m R t dτ s dτ e. satellite v s. eath v e R t. satellite v s. eath v e R t =.000000000447367 t. sat R t day se t sat = 8.64000000386554 0 4 se t t sat t ea t = 3.865 0 5 se t = 3865.535 ns In one day on the eath, whih is: t ea = 86400 se thee ae t = 3865.53595 ns moe tiks of satellite time. Eah day that passes, the eathlings gets to look 39 mioseonds into the futue of the GPS satellites. Notie that the Time dilation of ase, is less than the 45 mioseonds of ase, the shell model. This is aused by the tangential veloity of the obiting satellite. Fom the Eath's point of view geneal elativity speeds up the satellites lok and speial elativity slows it down.
Case 3) Fo a geosynhonous satellite: The obital peiod of a geosynhonous obit is 4 hous. Newtonian mehanis will be used to alulate the adius of ou geosynhonous obit. The mass of the Eath in kilogams is:.. e 5.974 0 4 kg Ciumfeene of the eath is: e. 600 nm giving a adius of: e e. π e = 3437.74677 nm The tangential veloity of the eath is: v te e. 4 h v te = 0.5 se nm v te The angula veloity of the eath is: ω e ω = e 7.7 0 5 se e To alulate the adius of the geosynhonous obit, we equate the aeleation of gavity to the aeleation equied fo iula motion. Then use ω e fo the angula veloity and solve fo the adius of the obit.. e G. ω Solving fo yields ω. G e ω 3 satellite ω e The adius of ou geosynhonous eath obit is: satellite = 809.4495 nm The tangential veloity of the geosynhonous satellite v. satellite satellite ω e v satellite =.659 nm se Convet all the geosynhonous obit paametes to geometi units fo the time atio alulation ass of the Eath in metes: = 4.435 0 3 m Tangential veloity of the satellite: v satellite v satellite Tangential veloity of the Eath: v eath v eath v satellite =.05 0 5 v eath =.545 0 6
R t is the atio of satellite time passage to eath time passage. This fomula was deived fom the Shwazshild meti with d=0 fo both objets and iula obits. R t dτ satellite dτ eath R t. satellite v satellite. eath v eath R t =.000000000447367 t. sat R t day se t sat = 8.640000004664583 0 4 se ts t sat t t sat day se t = 46.6458 us te day se Again fo one eath day, whih is: te = 86400 se, the geosynhonous satellite expeienes the passage of ts = 86400.00004664583 se. Theefoe the satellite ages by t = 46.6458 us moe than the eath lok. Fo eah eath day that passes we get to look 46 us into the futue of the geosynhonous satellite. This means that a ompute in the satellite an exeute about 46,000 moe instutions pe eath day than the same mahine on the sufae of the eath. Assuming the above omputes an exeute instution pe nanoseond. Sine the ompute in synhonous obit has 46.6 moe mioseonds to exeute Instutions pe eath day than one bak hee, should we stat obiting supe omputes to take advantage of this? Unfotunately one must onside the light tavel times fo uploading the pogam to un and downloading the esults. Theefoe, 46.6 mioseonds pe day multiplied by the un time of the omputation, in days, must be onsideably longe than twie the light tavel time fom the eath to the satellite, fo this sheme to be useful. The beak even time is the numbe of days that the omputation must un so that the 46.6us gain pe day is equal to twie the light tavel time fom the eath's sufae to the synhonous satellite. Both Newtonian mehanis and Geneal elativity will be used to alulate the beak even time and the esults will be ompaed. Sine the eath is nowhee nea beoming a blak hole and the otational veloities of the eath is muh smalle than the speed of light, the answes fom both alulations should be vey lose. Time gained pe day by satellite is: t = 46.6458 us Fom Newtonian mehanis, whee d. s t, solving fo t and multiplying yields the ound tip popagation delay. The delay of light is: pt. satellite eath pt = 0.39 se pt The Beak even time is:. day = 530.309 day t pt Numbe of yeas fo Satellite omputation to beak even is. day = 4.046 y t
Now fo the Geneal Relativisti alulation of the light tavel time between the satellite and the eath. We must use the Shwazshild meti fo a light like inteval to alulate the light tavel time in fa away time, o Shwazshild bookkeepe time. Then we must tanslate that into eath's sufae time fo ou answe.. Light like Inteval is: 0. d... dφ Fo light tavel time fom the synhonous satellite to the sufae of the eath. with dφ=0 is:.. d.. Now we solve fo in tems of d Whih has solutions... d. d Integating the expession fo fom the eath's sufae to the satellite gives the light tavel time fom satellite to the eath. this is: t i, f f. d Note: the mass of the eath in metes is: = 4.435 0 3 m i Time in metes Time in seonds t satellite, eath = 3.587 0 7 t satellite, eath m = 0.9654 se Twie the light tavel time in fa away time o Shwazshild bookkeepe oodinate time is: t. sh t satellite, eath t sh = 0.39307538937899 se
Fom the time like Shwazshild meti with d=0, the elation between Eath sufae time and Shwazshild bookkeepe time is given by. dτ. eath v eath Whee v eath is the tangential veloity of the eath and eath is the adius of the eath. In nautial miles v. eath = 0.5 se nm Twie the light tavel time given in eath sufae time is: t et. t sh. eath v eath t et = 39307.5387700 us Given the time gained pe day by satellite of t = 46.646 us t et t The beak even time is:. et day = 530.309 day o. day = 4.046 y t t Fo all patial puposes these esults ae the same as the Newtonian alulation. The atio of the light tavel times fom the satellite to the eath's sufae fo the elativisti time ove the Newtonian time is : t et t et pt = 0.9999999997704 =.958009068805 0 0 pt pt The Two times agee to appoximately.3 pat in 0^0th. Fom the Newtonian, and the Geneal Relativisti alulations we onlude that unless the ompute alulation takes moe than 4.046 eath yeas, thee is no advantage of plaing you supe ompute in a synhonous obit.. Refeenes. A.I.Boisenko & I.E. Taapov, "Veto and Tenso Analysis with Appliations", Dove Publiations, ineola, N.Y., 979. Amos Hapaz, " Relativity Theoy: Conepts and Basi Piniples", A.K. Petes, Ltd., 993 3. E.F.Taylo & J.A.Wheele, "Exploing Blak Holes Intodution to Geneal Relativity" Axddison Wesley Longman, 000 4. Rihad Shiffman, "eti Tensos", 996, (URL:= http://www.mathsoft.om/madlib/apps/mtensos.md ) 5. J.J.Callahan, "The Geomety of Spaetime An Intodution to Speial and Geneal Relativity", Spinge-Velag New Yok, New Yok, N.Y. 000, 999