Influence of the Theory On GPS Gonçalo Filipe Andrade dos Santos Instituto Superior Técnico, University of Lisbon, Lisbon, Portugal [email protected]

Abstract — The Global Positioning System (GPS) provides both a global position and a time determination. The system utilizes stable and precise satellite atomic clocks. These clocks suffer from relativistic effects, mainly due to and gravitational frequency shifts, that cannot be neglected. Without considering these effects, GPS would not work properly because timing errors of ∆t = 1ns will lead to positioning errors of magnitude ∆x ≈ 30 cm . In the end, the total correction per day, due to relativity (both special and general theories), is approximately, 39 µs day −1 , which corresponds to an imprecision of 12kmday −1 . Figure 1.1.: Trilateration [1]. The main objective of this dissertation is to discuss the conceptual basis, founded on special relativity, that ensure a Let us consider an instant t in which the receiver receives signals proper operation of the GPS. All the consequences and effects from 3 satellites. The signal from the first satellite indicates a set of stemming from this theory, are explained based on the coordinates (x , y , z ) , that define its precise location, and the geometric approach to the hyperbolic plane which is 1 1 1 commonly known as Minkowski diagrams in the literature. c instant t1 that defines the time at which the signal was sent. If To have a correct geometric intuition on this plane, equipped is the speed of light, the radio wave signal has travelled a distance with a non-Euclidean metric, one has to make extensive use of given by c( t− t ) . As a result, the receiver knows its position two concepts: equilocs and equitemps. The main consequence 1 of the universality of the speed of light (in vacuum) is that c t t (,x y , z ) located on the surface of a sphere with a radius of (− 1 ) equitemps from different observers are not parallel, i.e. and center (x , y , z ) . In other words, we have the following simultaneity is relative. 1 1 1 Quoting Neil Ashby: “The GPS is a remarkable laboratory of equation: the concepts of special and .” r−=− r (xx )(2 +− yy )( 2 +− zz ). 2 j 1 1 1 (1.1) (xx− )(2 +− yy )( 2 +− zz ) 222 = ctt ( − ). Keywords: GPS, Relativistic effects, Time dilation, Minkowski 1 1 1 1 diagram, Hyperbolic plane, Bondi’s factor Mathematically, the receiver solves the following system of four quadratic equations 1. GPS Basic Idea (xx− )(2 +− yy )( 2 +− zz ) 222 = ctt () −  1 1 1 1 (xx− )(2 +− yy )( 2 +− zz ) 222 = ctt () −  2 2 2 2 To determine a tridimensional position of any given point on Earth,  2 2 222 . (1.2) (xx− )( +− yy )( +− zz ) = ctt () − four satellites are necessary. One satellite allows us to calculate the  3 3 3 3 −(xx )(2 +− yy )( 2 +− zz ) 222 = ctt () − position of an object placed on a sphere. The second satellite  4 4 4 4 reduces the uncertainty and the position is then defined by the intersection of two spheres, resulting in an intersection circle. The 2. Special Relativity Theory third satellite further reduces this circle into two single points. As one of these two points is located far away from Earth, by There are two fundamental postulates in the special relativity elimination, we can obtain a single precise location. The last theory: satellite is there to synchronize satellite clocks and receivers. In the  First postulate (principle of relativity): The laws of end, the GPS can locate, accurately, any given point. This physics are the same and applicable to all inertial frames calculation method is called trilateration. of reference (observers).  Second postulate (constancy of the speed of light): Electromagnetic waves travel, in vacuum and regardless of the motion of their source, with constant speed.

The conclusion of the second and most important postulate was that the speed of light is a cosmic limit, a finite value, independent of any observer and the speed of a given emitting source [2]. Additionally, any velocity that remains invariant in the different reference systems has to be equal to the speed of light [3]. This postulate demonstrates that the space and time concepts are relative. Although velocity is constant to all observers, a velocity is obtained by the quotient of space and time, concepts which vary according to the observer and hence, can have different values.

2.1 Galileu Transformation

Prior to the existence of relativistic physics, there was a fundamental incompatibility between the two postulates [4]. The Newton’s mechanics imply, wrongly, that there is no superior limit Figure 2.2.: Representation of electromagnetic signals in a to the speed of a particle. Minkowski diagram. In the end, we have to understand that the addition of velocities, from the Galileu transformation, is in direct contradiction to the There are two important concepts that we need to clarify [6]. On second postulate of the special relativity theory. The formula that one hand, the temporal axis, t, corresponds to the equation x = 0 calculates this rule is given by the algebraic sum of velocities. – all the events that belong to this line occur in the same position There is a wrong belief, consequence of Newton’s mechanics, that x = 0. For this reason, the temporal axis is called an equiloc or, time is absolute ( t= t ' ). In Galilean relativity c is just a constant. in special relativity, we can also call it a clock, because time can Thereby, the speed of light depends on the observer measuring it. be measured through this line. We can define an equiloc as a set Figure 2.1. represents Galileu transformation graphically. of events that happens at the same place (position), from the point of view of a given system of coordinates. On the other hand, the spatial axis, x, corresponds to the equation t = 0 – all the events, on this line, occur at the same time instant t = 0, they are simultaneous events. For this reason, the spatial axis is called an equitemp or an axis that is orthogonal to a clock. We can define an equitemp as a set of events that occurs at the same time (simultaneous) from the point of view of a given IFR. Finally, we can define an event as the intersection of an equitemp with an equiloc. As it was mentioned for the electromagnetic signals, when referring to a particular frame of reference (observer), all equitemps are parallel between each other, and all equilocs are also parallel between themselves. This does not necessarily mean that equillocs and equitemps are perpendicular, although they must be orthogonal. Moreover, equitemps or equilocs from

Figure 2.1.: Galilean transformation graphical representation. different frames of reference are not parallel, due to time being a relative concept dependent on the observer, consequence of the 2.2 Minkowski Diagrams second postulate.

To better understand and visualize the questions related to the 2.3 Geometry of Relative Simultaneity special relativity theory, we need to resort to the spacetime diagrams, also known in the literature as Minkowski diagrams [5]. In essence, the two events, A and B , are simultaneous for S', This geometrical representation includes the electromagnetic while occurring at different times for S . In other words, the signal as a 45 ° , with both the x and the t axis. All the straight simultaneity of the events is a relative concept that depends on the lines parallel to this signal are electromagnetic signals, while lines IFR chosen. This implies that time is neither absolute, nor the perpendicular to the signal are electromagnetic signals as well. same for different IRFs. Furthermore, the equitemps from Nonetheless, the latter propagate on the opposite sense of the different IRFs do not have to be parallel to each other – to deal former. This description of the electromagnetic signals is implicit coherently with time – as a consequence of time not being in the second postulate which explains the concept of relative absolute. simultaneity, a new concept that appears alongside relativistic physics. Figure 2.2. is a representation of the electromagnetic signals. Additionally, as event A is at a fixed distance from observer b ,

xA, then: 1 x=( t − t ). (3.2) A 2 + −

b Figure 3.1. shows how it is possible to determine the equitemp ⊥ of Alice – represented by the straight line that unites events Α and Β. By algebraically summing and subtracting the two previous t t equations, we can calculate + and − : t= t + x  + A A  . (3.3) t= t − x  − A A Figure 2.3.: Minkowski diagram of Alice’s point of view for the wagon example. When adding a new observer (Bob) c ֏ (x ', t '), defined by the equiloc P, to the last figure, as we can see in the figure 3.2. 3. Special Relativity Consequences below, we have a new equiloc containing event Α.

3.1 Bondi’s Factor and Time Dilation

Assuming that Alice sends an electromagnetic signal at time t instant −, with the direction of event Α , by the time Bob receives this signal his watch marks time instant τA. Once this signal is received by Bob, he sends back another electromagnetic signal which is perpendicular to the one sent by Alice because they have opposite senses. When Alice receives this signal, the t time instant is +. Moreover, let us also consider an event Β that represents an event simultaneous with event Α, from Alice’s point of view, as we can see in figure 3.1.

Figure 3.2.: Bondi’s radar method (2).

As previously mentioned, Bob and Alice are moving away with relative velocity β hence, the origin event O(0,0) is the only

event that belongs to both observers. Moreover, xA is linearly increasing with value β : t t xA +− − xA=β t A ⇔= β = . (3.4) tA t++ t −

Taking in to account that simultaneity is a relative concept, time is Figure 3.1.: Bondi’s radar method (1). x t not absolute and event Α has coordinates Α(A , A ), from Alice’s In this example, we assume that the light speed is c = 1, in point of view, and coordinates Α(x ' , t ' ) = Α (0,τ ) from Bob’s t A A A geometric units. Event Β must occur with equal distance to − t point of view, where τ represents Α ’s own time. With the and to +, this is a consequence of light spending the same time A t t from − to event Α and from event Α to +. In other words, coordinates of event Α well defined for each observer, we shall event Α is at a fixed distance from Alice. If we assume that event introduce Bondi’s factor κ. According to Bondi, we know that τA, Β occurs at time instant tA, we can define tA as: the time in which Bob receives the signal, is equal to a constant κ 1 tA =( t+ + t − ). (3.1) 2 t times − :

t τA = κ −. (3.5) There is reciprocity: if Alice sends a signal and Bob receives it, the 3.2 Lorentz Transformation and Spacetime Interval Invariance same behavior should occur when Bob sends a signal at instant τ A and Alice receives it at instant t : In special relativity the coordinates in a given IFR can be + transposed to another IFR through a transformation called Lorentz transformation. Lorentz transformation, in what concerns the t = κ τ . (3.6) + A interaction between algebra and geometry, represents the recognition of everything that apparently goes against our b c According to Bondi’s factor κ , t+, t − ∈ and τA ∈ hence, if intuition – Newtonian and Euclidean – due to the representation we accept the previous equations: of the new Lorentzian metric, which diverges from the Euclidean metric – our geometric intuition of distance. t   One can draw two important conclusions from the result. The first τ2 ττ κ t +  t t (3.7) A= A A =(− ) ⋅  = + − . conclusion is obtained by the ordered multiplication of both  κ    equations: With the previous result we can deduce the Minkowski theorem: ('txtx+ ')(' −=+ ')( txtx )( −⇒ ) (') t2 − (') x 222 =− tx . (3.14) τ = t t . (3.8) A + − The final equation proves the spacetime interval invariance which, consequently, establishes the Lorentzian metric. The inner This means that τ is the geometric mean between t and t , A + − product between vectors does not depend on the IFR, or where it is calculated, the spacetime interval represents a Lorentz invariant which is verifiable through Bondi’s notations – tA is the [7]. In other words, it is proved that not everything is relative, in arithmetic mean between t and t . Since the arithmetic mean is + − relativity theory. In fact, time and space are relative, but a new at least equal to the geometric mean, then: absolute arises associated with the spacetime concept: the interval is not relative, it is always the same value independently of the τA≤ t A . (3.9) IFR considered [8]. Based on this result it is possible to affirm that And, considering the result from equations 4.3. and 4.4.: the geometry of Minkowski spacetime is not Euclidian and that ttx  ttβ t  tt β the plane (x , t ) has a hyperbolic geometry. Hence, the concept of +=+AA  + =+ AA  + =+ A (1 ) ⇔  ⇔  . (3.10) ttx=−  tt =−β t  tt =−(1 β ) invariance of the distance is reviewed in special relativity theory. −AA  − AA  − A    Whereas, in an Euclidian plane the geometric place of all the points that are at a fixed distance from the same point (center), is In the end, given by a circumference. In an hyperbolic plane, the geometric place of the events that are at a fixed spacetime interval of a τ=t t ⇔= τ t 1 − β 2 , (3.11) A+ − A A certain event, is given by an hyperbole. The second conclusion is obtained by the ordered addition and with the term 1− β 2 being the slowing down factor – because it subtraction of both equations:  1  1  is smaller or equal to one –, according to the special relativity      2t ' = κ + t −  κ −  x theory: tx''− =κ ( tx − )  κ   κ  ⇒  . (3.15) τ = t s , with s ≤ 1. (3.12) tx+ =κ( tx ' + ')  1  1  A A   2x ' =κ + x −−  κ  t      κ  κ  When, from Alice’s point of view, event Α has time t , event Β A Finally, it is possible to obtain the Lorentz boost: has a smaller time as a consequence of the slowing down effect.   Alice conclusion is that as Bob is moving away from her, his time 2'22t=−γ t γβ xt  '( =− γβ tx ) ⇔  . (3.16) is slowing down. The more general approach is that moving 2'2x=−γ x 2 γβ tx  '( =− γβ xt )   clocks slowdown, which proves the existence of time dilation. Since there is reciprocity, the corresponding conclusion can be The Lorentz boost corresponds to the transformation of taken from Bob’s point of view. In other words, given Bob is coordinates from one IFR to another, and follows that can also be inside the moving train, for him, Alice is the one moving away written in matrixial form: from his IFR. t'1 −β  tt   1' β  t Finally, we can deduce Bondi’s factor κ : =γ   ⇔  = γ   . (3.17) x' −β 1  xx   β 1'  x 2       τtA s t 1− β 1 + β τ=⇔== κt κ A = A = . (3.13) A − t tx− t (1 −β ) 1 − β − A A A The explanation for Lorentz transformation is not that the moving bodies are contracting, but instead that simultaneity is a relative In case there is no movement, β = 0, and κ = 1, while if β → 1, , concept. In other words, different observers have different ways κ → ∞ . of interpreting what simultaneity is given two observers with relative motion between them have equitemps that are not parallel. In fact, only when the equitemps from different frames of reference are parallel, is time absolute and can the Galilean transformation be applied. However, different observers moving of spacetime. Comoving observers do not measure time dilation away from each other do not have parallel equitemps – equitemps or length contraction, while non-comoving observers do measure meet at some point – hence, time is relative. The spacetime clocks slowing down and objects shrinking. interval invariance, as represented through the previous equation, leads to effects such as time dilation and space contraction. 4. Special Relativity Theory: Other Effects and Paradoxes 3.4 The Minkowski Quadratic Space 4.1 Einsteinian Velocity Composition The Minkowski quadratic space is a vectoral space with a Lorentzian metric added. Figure 3.3. depicts a Minkowski The example in figure 4.1., represents the previously mentioned diagram that represents the non-Euclidean nature with the four IFRs through three different observers and their clocks unitary vectors e, e , f , f , which have the same interval { 0 1 0 1 } b֏S(,),xt c ֏ S '(','), xt and d ֏ S ''(x '', t '') , respectively. invariance:

µ()e0= µ () e 1 = µ () f 0 = µ () f 1 = 1. (3.18)

The unitary vector of time type (e0or f 0 ) defines a set of equilocs, while the unitary vector of space type (e1or f 1 ) defines a set of equitemps. By using the Lorentz boost (active transformation) the unitary vectors {e0, e 1 } are transformed into {f0, f 1 } . Additionally, the angle φ represents an angle in the hyperbolic plane and the hyperbole t2− x 2 = 1 is the geometric place of the affixes of the vectors of space type. This hyperbole is centered in the origin and has an unitary Lorentzian measurement. In contrast, the hyperbole x2− t 2 = 1 is the geometric place of the affixes from the vectors Figure 4.1.: Representation of an electromagnetic signal that of time type with center in the origin and an unitary Lorentzian passes by three different observers. measurement. Any vector located in the straight line t= x has a null Lorentzian measurement. In this figure an electromagnetic signal is sent, at time instant t0 . This signal is sent from observer b , passing through observer c ? d at instant t0 ' , and reaching observer , at instant t0 '' . Recurring to the Bondi’s factor κ radar method, it is possible to define: t t 0'= κ 1 0 . t t 0''= κ 2 0 '. t t 0''= κ 0 . (4.1)

1 + βx κx = . 1− βx

Through algebraic manipulation, we reach the expected result for Figure 3.3.: Representation of the unitary vectors with their the new velocity composition: respective different dimensions through Lorentz boost (active 1 + β 1+β 1 + β ββ + transformation). κκκ=⇔ =1 2 ⇔= β 12 . (4.2) 2 1 1−β 11 −− ββ 1 + ββ 1 2 12 3.5 Reciprocity Both time dilation and length contraction are real and reciprocal. 4.2 Causality Additionally, in these the different points of view of different observers are equally correct. In a three-dimensional space plane, The principle of causality states that a given event, A , originates the shortest distance between two points is the straight line that another event, B . In this case, A is the cause; as a result, it has to unites them. However, in a bi-dimensional spacetime plane the always occur before event B – which is the effect –, considering straight universal line (line with a clock attached) between the two all the different points of view of the different observers. events does not represent the shortest path between them, instead it represents the longest interval between the two [9,10]. Both observers are right in the exact sense that both are making correct measurements. Time dilation and length contraction really exist, they are both actual physical consequences of the lack of absolute simultaneity [11]. They are manifestations of the fabric 1 1fe 1 − β =κ ⇔=fr ⇔ f r = f e . (4.5) fr f e κ1 + β

This equation demonstrates the relativistic Doppler effect when the observers are moving away from each other, with relative velocity of β > 0 . In the end, it is possible to draw the conclusion

that fr< f e . In other words, there is a red shift of the frequency – a deviation to lower frequencies. Through this it is possible to determine that the universe is in expansion: by analyzing the spectral emissions from different stars – signals present a red shift of their frequency, it is observable that the distance between the source (star) and the receptor (Earth) is getting larger. Figure 4.2.: Representation of an impossible example where the A different scenario occurs when the observers are getting closer effect happens before the cause. to each other. In that case the relative velocity is β < 0

consequently, f> f . There is a blue shift of the frequency – a Following a mathematical approach, this example is correct r e however, from a logical point of view, it is impossible for an deviation to higher frequencies. observer that sends an electromagnetic signal to receive, as a response, another signal of the same type which has an arrival time 4.4 The Twin Paradox prior to the sending time of the first signal considered. In the end, the conclusion that can be drawn from this example is that, as There are two observers, who are twin brothers according to the explained before, all physical actions propagate with velocities paradox, Bob described by b ֏ S x t and Alice described by that are, at most, equal to the speed of light. ( , ) c ֏ S'( x ', t ') . In this case, Alice is the travelling astronaut who 4.3 Longitudinal Doppler Effect does a round trip composed by two different stages: the departure and the arrival. In both stages the relative velocity between the In this section the longitudinal Doppler effect will be deduced by observers is uniform with magnitude β = 3 / 5 . Using geometric utilizing the Bondi’s factor radar method. units – time is measured in years and distance is measured in light- years – Alice is moving away from Earth until she reaches a distance to Earth of L = 3 light-years. Once this distance is achieved Alice reverses her motion, instantaneously, going back to Earth. The only instance during the trip in which it can be considered there is acceleration is on the turning point, where Alice’s velocity passes from +β to −β – which is considered an instantaneous infinite acceleration. This moment is the most relevant given it is the point where the demystification of the paradox occurs since it breaks the reciprocity of both twins due to the impossibility of describing Alice’s round trip through a single IFR.

Figure 4.3.: Representation of an observer that sends periodically signals to another observer. Through Bondi’s factor radar method, it is possible to establish that: tκ t  '1= 1 t'=κ t ⇒  . x x t' = κ t  2 2 (4.3) 1 + β κ = = eφ. 1 − β

Which, by usage of algebraic manipulation, results in:

Tttr =−''21 =κ ( tt 21 −= ) κ T e . (4.4)

Since the relation between T and T is already known, it is also e r Figure 4.4.: Representation of both twins’ voyage possible to determine the relation between f and f , recurring to e r the previous equations: In figure 4.4., from b ’s point of view, event A( L , T / 2) belongs c arrival trip is described by the equation x=β( T − t ) ,equiloc 2, to the equiloc c and event B(0, T ) belongs to equiloc b , and the equitemp described by the equation t= −β x + t b . where L represents the distance traveled by Alice in a single Additionally, T/ 2= L / β : direction and T represents the time of one round trip done by T  T Alice. Bob’s universal line corresponds to the sequence of events 2   ta =−(1β )  =− β L = 3.2 years  2 2 O→ B while Alice’s corresponds to O→ A → B .   (4.9) T  T From b ’s point of view, and given Alice’s trip lasted T years t2   L years b =+(1β )  =+= β 6.8  2  2 for the distance L = 3 light-years: T2 L L=β ⇔= T = 10 years (4.6) On the turning point of the trip, at time instant T / 2 , there is a 2 β change of direction in Alice’s equitemps which pass, instantaneously, from time instant t= 3.2 years to time instant However, from c ’s point of view, Alice’s trip took T ' years to a be completed: tb = 6.8 years . In other words, by turning around instantaneously, T 1 5 T '= = 8 years, with γ = = (4.7) Alice loses a period of time which Bob lives [12]. This period, γ 1− β 2 4 unknown for Alice, is given by the temporal difference between both time instants: Consequently, considering Alice and Bob were 20 years old at the ∆=−ttt =β2 T = 3.6 years. (4.10) beginning of the trip, they would be, respectively, 28 and 30 years b a old when they met again at the end of the voyage. This is the reason why the turning point is so relevant: it breaks the reciprocity between the twins [13]. Another instantaneous b Bob’s IFR, ֏ S( x , t ) , is described by two constant unitary change for Alice is the IFR changing from vectors (e , e ). However, Alice’s IFR, c ֏ S'( x ', t ') , as a c ֏ ֏ c ֏ ֏ 0 1 1S'( 1 x ',') t (,f 01 f ) to 2S'(',') 2 x t (g 01 , g ) . Regardless consequence of the turning point (instantaneous infinite of these changes, Bob does not lose any time period because his acceleration) of the trip, can be described by two different IFR, universal line is described by a unique IFR b ֏S(,) x t ֏ (e , e ). with the infinite acceleration point (turning point) being 0 1 neglected. In fact, the first IFR, corresponding to the departure In the moment immediately before event A (turning point), Alice sends an electromagnetic signal described by the equation c ֏ trip, is 1S'( 1 x ', t ') – described by two constant unitary vectors t= − x + t 1 . As a result, Bob, through the signals received from (f , f ) – while the second IFR, corresponding to the arrival trip, is 0 1 Alice, only perceives that Alice turned around at instant c ֏ S'( x ', t ') – described by two constant unitary vectors 2 2 tt==+1 (1β )( T / 2) = TL / 2 += 8 years. Consequently, from

(g0 , g 1 ). As a result, it is known that: Bob’s point of view, Alice’s arrival trip should take tTt=−=−(1β )(/2) T = T /2 −= L 2 years. In the end, from f=γ e + β e 2 1  0( 0 1 )  . f=γ( e + β e ) Bob’s point of view Alice’s voyage has a duration of  1 1 0 (4.8) g=γ e − β e T= t + t = 10 years while from Alice’s point of view the same  0( 0 1 ) 1 2  . g1=γ( e 1 − β e 0 )  trip has the duration of T'= t '1 + t ' 2 = 8 years, where t'= t ' = (/2)/ T γ = 4 years. Figure 4.5. represents the unitary vectors associated with each of 1 2 the IFRs. 5. Relativistic Effects and GPS

5.1 Introduction

Both the general relativity theory and the special relativity theory, conceived by , are essential to the measurement of distances and to the phase carrier, without them the GPS system would not work properly [14]. The satellites are affected by relativity in three different ways: movement equations, signal propagation, and satellites’ clock rhythm. As a result, the important

Figure 4.5. a and b: Unitary vectors associated with each observer. effects that require correction are the gravitational deviation of the frequency of the clock satellite, time dilation, the Sagnac effect and As in Bob’s IFR, Alice’s departure trip is described by the the eccentricity effect [15-18]. c A correction is necessary given the special relativity theory equation x= β t , equiloc 1 , and the equitemp described by the predicts that a moving clock (in an IFR) is slower than a stationary equation t=β x + t . Yet, from the same point of view, Alice’s a one. As a result, as a satellite orbits Earth its clock will slow down in relation to the clock at the receiver down on Earth. This happens 1 1−β2 ≈ 1 − β 2 . (5.4) because it is considered that the signals from the satellite and from 2 the receiver are moving relatively to an IFR. Additionally, the general relativity theory predicts that clocks Taking into account the GPS satellite velocities, the error per day move faster in a higher gravitational field. Since clocks’ rhythms due to time dilation effect is: do not depend only on their velocity, there is also a deviation of 1 1 v 2 ε=− β 2 ≈−s ≈−8.3485 × 10−11 . (5.5) the frequency of the clock satellite due to the higher gravitational 2 2 c2 potential. The difference in the gravitational potential between the satellite and the receiver at the ground, translate in the receiver’s Converting this result to seconds: clock moving slower than the clock on the satellite that is further apart from the gravitating mass considered – Earth. δt= τA − t A = ετ A =− 7.2131 μs. (5.6) Although the two effects are acting in opposite directions, their magnitude is not equal, thus they do not cancel each other out. A referential clock in the equatorial plane has the same movement Moreover, these errors are cumulative, which implies that, in case as an ECI due to Earth’s rotation. However, its velocity is inferior of them not being corrected, one would not compensate for the difference in clock speeds, resulting in a wrong distance to the satellites’ by measurement and in a position estimation, potentially, hundreds −1 of meters off. These failures would render the GPS system ω ⋅re ≈ 465ms . (5.7) virtually useless. The third effect arises given the GPS’s time scale is defined This represents the velocity of Earth’s rotation, with an average through a single IFR in contrast to Earth’s NIFR associated with angular Earth velocity of ω =7.292115 × 10 −5rads − 1 . To obtain the the planet’s rotation. This difference between the characteristics of each frame of reference leads to the emergence of the Sagnac difference between a GPS satellite clock (ECI) and a reference effect. In essence, this effect is a consequence of the difference in clock in the equator (ECEF), first we need to calculate the order of location when the electromagnetic signal of the satellite is sent magnitude of the time dilation effect: when compared to when it is received. More specifically when the 1 1 (ωr ) 2 receiver on Earth suffers a finite rotation due to Earth’s rotation −β2 ≈−e ≈−1.2013 × 10−12 ≈− 103.792 ns. (5.8) motion in relation to the system of inertial time reference centered 2 2 c2 on Earth – ECI. Afterwards, it is necessary to compute the fractional frequency difference: 5.2 Effect of Time Dilation 2 2  ∆f 1v 1 (ω r )  −11 =−s −− e  =−8.2284 × 10 ≈− 7.1093 μs. (5.9) 2 2  One of the most important relativistic effects on the GPS systems f 2c 2 c   is the slowing down factor associated with the clocks that are in relative motion in relation to Earth. Finally, we can conclude that if this navigational error was not As it was already demonstrated, the time dilation is given by the accounted for, it would increase resulting in a distance error, at equation: the end of a day, of: −1 2 δx= c δ t = 2.1328kmd . (5.10) τA=t A 1 − β . (5.1) 5.3 Effect of Gravitational Deviation of the Frequency In order to calculate precisely this effect during a day, we need to of the Clock Satellite know how many seconds a day has, from the moving system of coordinates point of view (Bob): If we consider two clocks, a reference clock that stays at rest on τA =24 × 3600 = 86400 s. (5.2) Earth’s equator with radius a1 and an atomic clock orbiting the −1 Earth with radius r – satellite clock. Remembering that, the As explained before, the GPS satellite velocity is vs = 3874ms , s gravitational potential of a satellite created by a mass M (Earth’s in relation to ECI, due to Earth’s rotation. In geometric units, this mass) spherically symmetrical is given by: velocity would be β =1.2922 × 10 −5 . An observer at rest in the ECI G M Φ=− ≈−14.984979 × 106 . (5.11) s r will notice the satellite clock is affected by the slowing down s factor: Considering that GM =3.986004415 × 1014m 3 / s 2 and

r = 26600 km . However, the gravitational potential of a specific s =1 − β2 . (5.3) s point on Earth’s surface is affected given Earth is not a perfect This factor can be approximately expanded using the binomial sphere hence, the real gravitational potential is: theorem, 2   2 GM  a 12  1 Φ=−()r 1 − J1 (3sinθ −− 1) wr ()cos θ θ . (5.12)  2 2  () r()θ  r (θ ) 2 2   Φcentripetal Φstatic

With the Earth’s quadrupole moment coefficient being −3 J 2 =1.0863 × 10 , the angle θ being measured between the equator and the location of the receiver on Earth’s surface, and r(θ ) being the radius of the Earth depending on the value of the location θ . The gravitational potential is divided in two parts: the potential caused by the mass of the Earth (Φstatic ) and the potential from the centripetal forces’ consequent of Earth’s rotation – this last term of equation 5.4. compensates the time dilation effect that originates from the motion of the resting clocks on Earth (in a Figure 5.1.: Sagnac effect illustration. ECI). By considering a clock on the equator ( θ = 0 ), the equivalent gravitational potential is simplified: Considering Earth’s rotational frame, the navigation equations J  GM  2  1 6 can be described by: Φ=−(r ) 1 +− ( wa ) ≈− 62.598798 × 10 . (5.13) a   1 R 1 2  2

r()tj− r j + v × ( t − t j ) r(t ) − r Taking in to account that the time dilation effect ( ∆t/ t ) for an t= t +j = . (5.18) j c c observer at rest with potential Φ(r ) in relation to a clock with potential Φs is given by: With r(t )= 6378 km representing the receivers position at instant ∆t Φ −Φ (r ) t , v being the velocity of the receiver according to the satellite’s = s . (5.14) t c2 signal transmission time – which is small compared to the speed of light and for that reason can be neglected –, and R = 26562 km Furthermore, we can neglect the quadrupole moment, consider standing for the distance between sender and receptor at the time that the satellites are in free fall, hence neglecting the centripetal that the signal was sent. By simplification, the signal’s arrival time potential as well, and it is known that Earth’s radius is 6371 km for a fixed receptor in an IFR would be: (a ). 1 r(t j ) − r j R t= = t + . (5.19) By combining the previous equations, it is possible to calculate cj c the order of magnitude of the gravitational frequency shift of GPS satellite clocks: By applying this result, we obtain a time correction of: G M  GM J   R2 +2 R ⋅ v (t − t ) Rv Rv⋅   2   j R − −− 1 +  t=+ t ≈++≈ t . (5.20)  r  a   j j 2 2 ∆t  s   1 2    c c c c ε == ≈×5.288 10−10 . (5.15) t c2 Accounting for the fact that Earth’s rotation applies a velocity to When multiplying this result by the number of seconds in a day, the receiver of order: it is possible to derive the error per day associated with the v=w × r ( t j ). (5.21) gravitational frequency shift of GPS satellite clocks: −10 − 1 δt =5.288 × 10 × 86400 = 45688.32nsd . (5.16) Where w =7.292115 × 10 −5rads − 1 represents the mean angular velocity of Earth. As a result, the correction of the formula error Finally, we can conclude that if this navigational error was not associated with the Sagnac effect is given by: accounted for, it would increase, resulting in a distance error, at R⋅ v2w ⋅ A the end of a day, of: δt ns d −1 Sagnac =2 = 2 = 137.454 . δx= c δ t = 13.706kmd −1 . (5.17) c c 1 A= r(t ) ×= R 8.4706 × 1013m 2 . (5.22) 2 j 5.4 Sagnac Effect −1 δxSagnac = c δ t = 41.20767md .

The Sagnac effect accounts for the receiver’s movement while With the variable A corresponding to the area covered by the taking into consideration Earth’s rotation – considering a resting vector – from the rotation axis, where the signal is originally sent, receiver on the surface of the Earth, the error introduced by the until the position where the signal is received. Sagnac effect derives from the difference between the moment the signal is sent from the satellite and the moment the signal is 6. Future Work Prespectives received by the receiver. This effect is represented in figure 5.1. In the special relativity theory most of the results are possible to obtain by utilizing a hyperbolic plane with a single time dimension alongside a single space dimension. However, in Accordingly, the GPS user, even if unknowingly, has become reality, the hyperbolic plane is defined by four dimensions with dependent of the revolutionary concepts of Spacetime introduced three space dimensions plus one time dimension. This dissertation by Einstein and graphically interpreted by Minkowski. leads to future detailed works where the simplification of the spacetime diagrams cannot be applied such as: REFERENCES  Thomas rotation which allows to calculate the velocity composition of non-colinear frames of reference. [1] “Differences between Triangulation and Trilateration?”  The utilization of geometric algebra for the study of Geographic Information Systems Stack Exchange, source: motion media for non-isotropic media. gis.stackexchange.com/questions/17344/differences-between- triangulation-and-trilateration/42663.  The general relativity theory where the existence of [2] E. G. 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Conclusions [12] Anthony Philip French, Special Relativity, p. 154 – The MIT Introductory Physics Series, New York: W. W. Norton, 1968, The number of applications that use the GPS is almost 1966. [13] Tevian Dray, The Geometry of Special Relativity, p. 64, Boca uncountable. This system is worthy of a more detailed attention Raton, FL: CRC Press, 2012. from relativity researchers and specialists. Therefore, this may be [14] James B. Hartle, Gravity – An Introduction to Einstein’s General considered a source of pedagogical examples. Relativity, p. 125, San Francisco: Addison Wesley, 2003. The influence of special and general relativity on the GPS is [15] Neil Ashby, “Relativity in the Global Positioning System,” p. 260, considered in this work, through time dilation (special relativity) Chapter 10 in: Abhay Ashtekar, Editor, 100 Years of Relativity – and Einstein gravitational shift (general relativity). 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DEEC – Instituto Superior Técnico, Área Científica de Telecomunicações, Ano Lectivo: 2016/2017