The Global Positioning System (GPS) provides an excellent educational example
as to how the theory of general relativity is put into practice and becomes
part of our everyday life. This paper gives a short and instructive derivation
of an important formula used in the GPS, and is aimed at graduate students and
general physicists.
  The theoretical background of the GPS (see \cite{ashby}) uses the
Schwarzschild spacetime to deduce the {\it approximate} formula, $ds/dt\approx
1+V-\frac{|\vv|^2}{2}$, for the relation between the proper time rate $s$ of a
satellite clock and the coordinate time rate $t$. Here $V$ is the gravitational
potential at the position of the satellite and $\vv$ is its velocity (with
light-speed being normalized as $c=1$). In this note we give a different
derivation of this formula, {\it without using approximations}, to arrive at
$ds/dt=\sqrt{1+2V-|\vv|^2 -\frac{2V}{1+2V}(\n\cdot\vv)^2}$, where $\n$ is the
normal vector pointing outward from the center of Earth to the satellite. In
particular, if the satellite moves along a circular orbit then the formula
simplifies to $ds/dt=\sqrt{1+2V-|\vv|^2}$.
  We emphasize that this derivation is useful mainly for educational purposes,
as the approximation above is already satisfactory in practice.Comment: 5 pages, revised, over-over-simplified... Does anyone care that the
  GPS uses an approximate formula, while a precise one is available in just a
  few lines??? Physicists don'

Bergmann P G

M Matolcsi

Matolcsi T

Matolcsi T Matolcsi M

Pauli W

Sachs R K

T Matolcsi

Weinberg S

English

arXiv

Matolcsi, T

Matolcsi, M

ELTE Digital Institutional Repository (EDIT)

arXiv:math-ph/0611086v3  8 Oct 2008Coordinate time and proper time in the GPST. Matolcsi∗, M. Matolcsi†October 30, 2018AbstractThe Global Positioning System (GPS) provides an excellent educa-tional example as to how the theory of general relativity is put into prac-tice and becomes part of our everyday life. This paper gives a short andinstructive derivation of an important formula used in the GPS, and isaimed at graduate students and general physicists.The theoretical background of the GPS (see [1]) uses the Schwarzschildspacetime to deduce the approximate formula , ds/dt ≈ 1+V − |v|22, for therelation between the proper time rate s of a satellite clock and the coordi-nate time rate t. Here V is the gravitational potential at the position of thesatellite and v is its velocity (with light-speed being normalized as c = 1).In this note we give a different derivation of this formula, without us-ing approximations, to arrive at ds/dt =q1 + 2V − |v|2 − 2V1+2V(n · v)2,where n is the normal vector pointing outward from the center of Earthto the satellite. In particular, if the satellite moves along a circular orbitthen the formula simplifies to ds/dt =p1 + 2V − |v|2.We emphasize that this derivation is useful mainly for educationalpurposes, as the approximation above is already satisfactory in practice.1 IntroductionThe most significant application of the theory of General Relativity in everydaylife, arguably, is the Global Positioning System. The GPS uses accurate, stableatomic clocks in satellites and on the ground to provide world-wide positionand time determination. These clocks have relativistic frequency shifts whichneed to be carefully accounted for, in order to achieve synchronization in anunderlying Earth-centered inertial frame, upon which the whole system is based.For educational purposes it is very instructive to see how a highly abstract theorysuch as the general theory of relativity becomes a part of our everyday life.∗Department of Applied Analysis, Eötvös University, Pázmány P. sétány 1C., H–1117Budapest, Hungary, matolcsi@cs.elte.hu, supported by OTKA-T 048489.†Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda utca13–15., H–1053, Budapest, Hungary, matomate@renyi.hu, (also at BME, Dept. of Analysis,Egry J. u. 1, Budapest, Hungary) supported by OTKA-PF 64061, T 049301, T 047276.1Throughout this note we will normalize universal constants for simplicity, sothat light speed and the gravitational constant are the unity, c = 1, γ = 1.Let us briefly sketch here how the GPS works, including the major ingredi-ents for the reader’s convenience (a full and detailed description is available in[1]). In order to determine your position at the surface of Earth you must beable to see some (at least four) GPS-satellites simultaneously, and know theirposition and your distance to them (with this much information at hand, it isthen elementary geometry to determine your position). To make this possible,every GPS-satellite emits signals continuously, describing its position and itslocal time tsat. If your GPS-device measures its own local time, tdev, then youcan use the constancy of the speed of light c to calculate your distance d to thesatellite asd = (tdev − tsat)c = tdev − tsat (1)keeping in mind the normalization c = 1. (We remark here that in practice theGPS-device is unable to measure time with sufficient accuracy, therefore tdevalso has to be deduced from the data sent by the satellites. However, this detailis not important for the purposes of this paper.)This looks simple enough, but the problem is that formula (1) is only valid ifall clocks are synchronized in some special relativistic inertial frame. Therefore,for the purposes of the GPS one imagines that an inertial frame is attached tothe center of Earth, and we try to synchronize all clocks such that they measurethe time t of this ideal inertial frame. This means that one should imagine’ideal’ clocks placed everywhere in the vicinity of Earth, measuring the time t,and thus tsat and tdev should be the read-outs of these ideal clocks at the placeof the satellite when the signal originates and at the place of your device whenthe signal is received. However, what we actually can measure is the propertime tE of stationary clocks on the surface of Earth (which is measured in time-keeping centers throughout the world), and the proper time s of satellite clocks.Therefore, we need to establish a relation between the time rates tE , s and t.Fortunately, the time rate tE measured on Earth is independent of where youare (i.e. the same manufactured clocks beat the same rate in London or Tokyoor New York) and differs from the ’ideal’ time rate t only by a multiplicativeconstant,dtdtE= 1− Φ0, (2)where Φ0 is a constant corresponding to the Earth’s geoid (see [1]). This re-lation is very convenient because the ideal time rate t can be replaced by tE ,something we can actually measure, and an equation of the form (1) still re-mains valid after rescaling by the factor 1−Φ0. This leaves us with the task ofestablishing a relation between s and t (or s and tE , whichever turns out to bemore convenient).To determine the relation of the proper time rate s of the satellite clocks andthe time rate t of ideal clocks measuring the coordinate-time of the underlyingEarth-centered inertial frame, the customary theoretical framework [1] is to use2Schwarzschild spacetime, and arrive at the formulads/dt ≈ 1 + V −|v|22, (3)after several first-order approximations in the calculations. Here V denotes thegravitational potential at the position of the satellite, and v is the velocity ofthe satellite measured in the underlying non-rotating Earth-centered inertialframe. Formula (3) is the internationally accepted standard relating the clockfrequencies, as described in [1, 3] and references therein.We remark here that the derivation of formula (2) is somewhat more involvedthan that of formula (3). In fact, for the purposes of deriving (2) one needs tomodify the Schwarzschild metric by a small term, taking into account multipolecontributions corresponding to the Earth’s geoid. However, for formula (3) themodifying term is disregarded and the standard Schwarzschild metric is used,the argument being that the modifying term becomes negligible far enough fromthe Earth surface, where the satellites orbit (cf. [1]). In this paper we do notinclude the derivation of formula (2) and the modifying term which is used there.In general, in physics it is justified to use approximate formulae for twodifferent reasons. One is that in some cases the derivation of an exact result isanalytically not possible. The other is that in certain cases an exact analyticderivation, even if it exists, would lead to involved and lengthy calculations thusconcealing the important and possibly very simple aspects at the heart of theissue. In calculations involving the theory of relativity (in particular, generalrelativity) there is a tendency to turn to approximations automatically, due tothe involved nature of the theory. However, in some rare cases a modified pointof view and an adequate choice of coordinate system can lead to exact results.In this note, we adhere to the standard theoretical framework used in [1, 3],but we point out that an exact formula can be derived in a very simple mannerfor the clock frequency rate (3) in question. Instead of using the customaryisotropic coordinates we will treat Schwarzschild spacetime with Schwarzschildcoordinates (an entirely coordinate-free derivation of the same formulae can befound in [5] but it is rather cumbersome). This treatment of Schwarzschildspacetime is motivated by a similar account of special relativity in [4] and thatof general relativity in [7]. The coordinate-free point of view often has the ad-vantage of conceptual clarity and, in this particular case, brevity of calculations.Our results here are mostly of educational and theoretical interest as theexisting formula (3) provides good approximation to the desired precision inthe GPS (see [3]).2 Schwarzschild’s spacetimeSchwarzschild’s spacetime describes the gravitational field of a pointlike inertialmass m. It is a well-known model of general relativity, but we include its shortdescription here for convenience.3Let us introduce some notation. Let E be a three-dimensional Euclideanvector space, the inner product of x,y ∈ E being denoted by x·y. For 0 6= x ∈ Ewe putn(x) :=x|x|(4)for the outward normal vector at x.Consider R × E as a spacetime manifold with its usual special relativisticmetric, i.e. the Lorentz form given by(−1 00 I), (5)where I is the identity matrix. For a world point (t,x) ∈ R × E one shouldthink of t as the synchronization time corresponding to the center of Earth, andx as the space vector pointing to the world point from the center of Earth. TheLorentz form (5) means that the Lorentz length-square of a four-vector (s,q)is given by the usual formula |(s,q)|2Lor= −s2 + |q|2. This special relativisticspacetime will be called the Earth Centered Reference Frame (ECRF). Thisis an ’ideal’ special relativistic frame, and the task in the GPS is to achievesynchronization of clocks in this underlying inertial frame.Introduce the potential V (x) := − m|x| on E, and restrict your considerationto world points (t,x) ∈ R × E for which 1 + 2V (x) > 0 (i.e. for world pointsoutside the Schwarzschild radius). Then it is easy to see (cf. [2, 6, 8]) that forsuch world points, the standard form of the Schwarzschild metric in R×E (i.e. asmooth collection of Lorentz forms g(t,x) depending on the world points) takesthe form:g(t,x) =(−(1 + 2V (x)) 00 I − 2V (x)1+2V (x)n(x)⊗ n(x).)(6)In particular, when we measure the length-square of any space vector q atthe space point x in Schwarzschild’s metric we obtain|q|2Sch := |q|2 −2V (x)1 + 2V (x)(n(x) · q))2. (7)Similarly, when we measure the Schwarzschild length-square of a four-vector(e.g. a four-velocity) (s,q) at the point x we get|(s,q)|2Sch = −(1 + 2V (x)))s2 + |q|2Sch. (8)3 A satellite in Earth’s gravitational fieldIn this section we derive the formula relating the proper time s of satellitevehicle-clocks to the ideal time t of the underlying inertial frame.A material point in spacetime is described by a world line functionR → R× E, s 7→(t(s), x(s))(9)4where s is the proper time of the material point. The four-velocity,(ṫ(s), ẋ(s)),of the material point always satisfies|(ṫ(s), ẋ(s))|2Sch = −1. (10)Therefore, using equation (8) we obtain−(1 + 2V (x(s)))ṫ(s)2 + |ẋ(s)|2Sch = −1 (11)The time function t(s) can be inverted, with s(t) denoting the proper timeinstant s corresponding to the ECRF-time t. Let v(t) := dx(t)dtdenote therelative velocity of the material point with respect to the center of the Earth.Note that by the chain rule we have v(t) = ẋ(s(t))dsdt. Now, rearranging (11)we obtain(dsdt)2+ |ẋ(s(t))|2Sch(dsdt)21 + 2V (x(s(t)))= 1 (12)which yieldsdsdt=√1 + 2V (x(s(t))) − |v(t)|2Sch(13)Finally, applying (7) we obtain the desired relationdsdt=√1 + 2V − |v|2 −2V1 + 2V(n · v)2 (14)Using a series expansion, assuming that all the terms on the right-hand sideare much less than 1, we get back the approximate formula (3)dsdt≈ 1 + V −|v|22. (15)Of course, there is still some work to be done before one can apply equation(1) in the GPS. Instead of dsdtwhat we really need is the function t(s), becausethe signal emitted by the satellite contains the proper time instant s measuredby clocks on board, and we would like to replace it by the coordinate-timeinstant t which it corresponds to. However, having deduced formula (14) (orits approximation (3)) it is possible to obtain the function t(s), or at least agood enough approximation of it. This is described very well in full detailin [1], and we feel it inappropriate to repeat those calculations word-by-wordhere. Nevertheless, we warmly recommend that the reader turn to [1] for theinteresting details.References[1] N. Ashby: Relativity in the global positioning system, in 100 years of rela-tivity, 257–289, World Sci. Publ., Hackensack, NJ, (2005). (also at Livingre-views in Relativity, http://relativity.livingreviews.org/Articles/lrr-2003-1/)5[2] P.G. Bergmann, Introduction to the theory of relativity, Dover, New York(1978), ch.XIII, pg.203, eq(13.25).[3] J. Kouba: Improved relativistic transformations in GPS, GPS Solutions(2004), 8, 170-180.[4] T. Matolcsi: Spacetime without Reference Frames, Akadémiai Kiadó Bu-dapest, 1993.[5] T. Matolcsi, M. Matolcsi: GPS revisited: the relation of proper time andcoordinate time, arXiv:math-ph/0611086v1[6] W. Pauli, Theory of relativity, Dover, New York (1981), pg.166, eq(421a).[7] R. K. Sachs, H. Wu: General Relativity for Mathematicians, Springer, NewYork, (1977).[8] S. Weinberg, Gravitation and Cosmology, J. Wiley & Sons Inc., New York,(1972), pg.181.6

Coordinate time and proper time in the GPS

Crossref

The global positioning system (GPS) provides an excellent educational example of how the theory of general relativity is put into practice and becomes part of our everyday life. This paper gives a short and instructive derivation of an important formula used in the GPS, and is aimed at graduate students and general physicists. The theoretical background of the GPS ( see [ 1]) uses the Schwarzschild spacetime to deduce the approximate formula, ds/dt approximate to 1 + V - \v\(2)/2, for the relation between the proper time rate s of a satellite clock and the coordinate time rate t. Here V is the gravitational potential at the position of the satellite and v is its velocity ( with light-speed being normalized as c = 1). In this paper we give a different derivation of this formula, without using approximations, to arrive at ds/dt = root 1 + 2V - \v\(2) - 2V/1+2V(n.v)(2), where n is the normal vector pointing outwards from the centre of Earth to the satellite. In particular, if the satellite moves along a circular orbit then the formula simplifies to ds/dt = root 1 + 2V - \v\(2). We emphasize that this derivation is useful mainly for educational purposes, as the approximation above is already satisfactory in practice

Matolcsi, T.

Matolcsi, Máté

Repository of the Academy's Library

https://edit.elte.hu/xmlui/bitstream/handle/10831/49573/Coordinate%20time%20and%20proper%20time%20in%20the%20GPS.pdf?sequence=1&amp;isAllowed=y

Coordinate time and proper time in the GPS

Abstract

Similar works

Full text

Available Versions

ELTE Digital Institutional Repository (EDIT)

Crossref

Repository of the Academy's Library