12.1 Events at the same proper distance from some event

Transcription

1 Chapter 1 Uniform Aeleration 1.1 Events at the same proper distane from some event Consider the set of events that are at a fixed proper distane from some event. Loating the origin of spae-time at this event, the equation for this set of events is: x t = d (1.1 The parameter, d, is the proper distane of these events from the origin event. The origin event and the events on the urve are related by this distane d and thus for the set of events on the urve the origin is alled the magi point and d is the distane from the magi point to the urve. In spae-time, this is a two branh hyperbola with light ones emanating from the origin as the asymptotes. If we now onsider only the branh that has x > 0, x = d + t, we have a single urve. In Figure 1., We plot several of them for different d. Sine this equation is a form invariant under the Lorentz transformations, all inertial observers will have the same urve and Lorentz transformations will map points on the urve to points on the urve. By loating a light one on the event at (d, 0, we an see that all the events on the urve at later times are in the future; the urve is monotonially asymptoti to a light one that is later in spae-time. Thus all the events at later times on the urve are in the future of (d, 0. Similarly all the events that are before t = 0 are in the past of (d, 0. Thus the urve is time-like and is therefore a andidate for the motion of a material partile. In the next setion, we will see that this is the trajetory of the uniformly aelerated objet. 95

2 96 CHAPTER 1. UNIFORM ACCELERATION Figure 1.1: The lous of events that are at the same proper distane from the origin. 1. Uniformly aelerated motion Sine this urve is time-like, it is a possible state of motion for a material partile. It is ertainly a ase of motion that is not uniform, not a straight line in spae-time. For any observer in uniform motion, an objet following this trajetory will appear to be approahing at a very rapid rate, almost, and slowing down until at some event it is as lose as it will ever get and at rest with respet to the observer and then moving away so that at long times later it is reeding at almost. Sine the Lorentz transformations are homogeneous and linear, lines through the origin are transformed into lines through the origin and spaelike lines are transformed into spae-like lines and similarly for time-like lines. Thus if you pik an event, say (x 0, t 0, on this urve, the line through it and the origin whih is spae-like an be transformed to the spae-like line through (d, 0 and (0, 0 by the Lorentz transformation with v = t 0 x 0. This is also the transformation that brings the tangent to the urve to the vertial whih means that the instantaneous relative veloity at (x 0, t 0 is v. Or said another way, an observer with relative veloity, v = t 0 x 0, is a ommover to the this trajetory at the event (x 0, t 0. Thus we see that the instantaneous relative veloity at (x 0, t 0 is v = t 0 x 0. More signifiantly, to the respetive ommovers, the aeleration at (x 0, t 0 is the same as the aeleration at (d, 0. Therefore, as measured by ommovers, the instantaneous aeleration at any event is the same and this is the aeleration that the objet experienes in its motion. On simple dimensional grounds, the

3 1.. UNIFORMLY ACCELERATED MOTION 97 Figure 1.: The lous of events with x > 0 that are at the same proper distane from the origin for different values of the proper distane, d. aeleration at the event (d, 0 must be a = d. (1. Also note that it follows from the previous argument that the line from (x 0, t 0 to the origin is the line of simultaneity for the ommover at the event (x 0, t Details of the alulation of the aeleration The easiest way to alulate the aeleration is use alulus. dx dt = d dt ( d + t = 1 t d + t = t x (1.3 whih we already knew. The aeleration is d x dt = d( t x dt = ( 1 x t dx x dt

4 98 CHAPTER 1. UNIFORM ACCELERATION Figure 1.3: Plaing a light one at the event (1, 0 shows that the lous of events with x > 0 that are at the same proper distane, d = 1, from the origin is a timelike trajetory. = ( 1 x t x t x = ( x t x 3 = d x 3 (1.4 whih, at the event (d, 0, means that v = 0 and a = d, whih was our result from dimensional arguments in Equation 1.. If you have an aversion to alulus, you an look at the motion for small times near the event (d, 0. It must redue to the expression for the position for the onstant aeleration that we know from lassial physis, x l (t = x 0 + v 0 t + a t whih should be valid for at << 1. Expanding our x(t for small t and using the fat, (1 + x n 1 + nx for x << 1, that everyone should know from Setion 1.4., we have x(t = d + t = d 1 + d t d(1 + d t. (1.5 Comparing this with x l (t, we see that, for small times near the event (d, 0, the veloity is 0 and the aeleration is d, again our result Equation 1..

5 1.. UNIFORMLY ACCELERATED MOTION 99 Figure 1.4: The uniformly aelerated observer with the world line and the line of simultaneity of the ommover for the event (x 0, t 0. It is very important to point out that this is the aeleration that the aelerated objet feels. Consider an aelerated roket with a pair of idential springs and masses, one mass-spring system mounted on a fritionless surfae horizontally and the other mass-spring suspended vertially. Vertial in the roket is along the line from front to bak and horizontal is one of the transverse diretions. We also alibrate our springs so that we know the fore that is required to streth them a given amount, i. e. we know the spring onstant, k, of the springs. The horizontal mass-spring will have one equilibrium position and the vertial one will have a different one. If we now arefully adjust the thrust of the roket so that the streth of the springs does not hange with time, our roket when observed by someone who was initially at rest with us will register it at x(t = d + t d where d = k streth where m is the mass, k is the alibrated spring onstant, m streth is the differene in the length of the vertial and horizontal springs. The extra d is in x(t to make the roket and the original ommover oinident in spae at t = 0. At later times, the roket has moved away from the original ommover but the mass-spring system still measures the same aeleration, the aeleration that is measured by the new instantaneous ommover. This is another ase of a term whih is dimensionally the same but whose physial interpretation is different. Aeleration is generally defined kinematially as a k d x(t. Through Newton s laws, we have an equivalent dt definition in the form a s f m where f is the effet of external objets on a

6 300 CHAPTER 1. UNIFORM ACCELERATION body of mass m. It is this a s that is sensed by the aelerated system that informs it that it is not inertial. This is the essene of Galilean invariane. A free body has no aeleration. The equality of a s and a k expressed in Newton s law an be required only in the ase of a world of low relative veloities. Sine the kinemati definition is not a onstant in this motion although the sensed aeleration is onstant, we have an interpretation problem. It is required that all inertial observers of this motion agree on its sensed aeleration and from the previous disussion all events on the trajetory have the same sensed aeleration to a loal ommover and this aeleration is the same as the kinemati one as evaluated or measured for small times around the events when the objet is ommoving with that observer. For all other times, the kinemati and sensed aeleration are different. The kinemati aeleration is the aeleration evaluated by one of the ommover inertial observers for all time and it varies from d the small time value to zero at large times when the objet is distant. The kinemati aeleration is d x(t dt where both x and t are oordinates for the speifi inertial ommover. An alternative might be to all this motion not uniformly aelerated motion but uniformly effeted motion. 1.3 The proper time along the trajetory As was stated in Setion 10.3, the proper time between two events is a trajetory dependent onept. As the aelerated objet moves along its trajetory, its oordinate position and time are given by x(t = d + t. This same motion an be oneived of as both x and t both evolving as a funtion of the proper time, x(τ and t(τ. Our problem is to find these relationships. Noting that beause of the definition of the trajetory as the lous of events with the same proper distane from the origin event that for all τ the two funtions x(τ and t(τ satisfy (x(τ (t(τ = d Timelike Trajetories and Aelerated Motion Although it does not onstitute a proof, we an use aelerated motion to justify the often heard omment that there is no fore that an boost a material partile to speeds greater than the speed of light. As stated in Setion 1..1, the aeleration a that labels this trajetory is the aeleration that a material partile moving along that world line feels. In other words, the fore that aelerates the partile to move it along this trajetory is a onstant as measured by the sequene of ommovers and these are

7 1.4. EXAMPLES USING ACCELERATED MOTION 301 the suitable observers of the fore of aeleration. In this ase of onstant fore, we see that no matter how long the fore operates, the veloity of the partile that is subjeted to this fore moves relative to its initial veloity at a speed that is less than ; the trajetory remains timelike for all times. Also in any finite time interval, there is no aeleration and thus no fore that an hange the trajetory from timelike to spae like. 1.4 Examples using aelerated motion With the tools developed in the previous setions, we an now analyze all kinds of simple uniform aeleration problems. In fat, just about any of the usual uniform aeleration problems that are enountered lassial physis an be studied. In this setion, I will go through the details of three typial problem types Deeleration Sally is moving toward a wall with a relative speed of 3 5. When she is one lightyear away from the wall, she deides to deelerate. What is the minimum deeleration that she an use so that she just omes to rest at the wall? We an find the answer in the frame in whih the wall is at rest. Firstly, we should diagram the motion. Figure 1.5: Sally turning from the wall. The event (x 0, t 0 is the event at whih she deelerates. The line labeled Sally is her trajetory. The line labeled t Sally0 is her worldline before deelerating

8 30 CHAPTER 1. UNIFORM ACCELERATION From this we an see that the problem an be stated in a simpler fashion. At any event, (x 0, t 0, on the uniformly aelerated trajetory, we know the v relative veloity at that point, = t 0 x 0. For the ase shown in Figure 1.5, note that t 0 is negative and x 0 is positive so that v is negative. Thus we an ask given an aeleration, a, how far from the event (x 0, t 0 on that trajetory is the vertex of the hyperbola? In the usual oordinate system, the vertex is at (d, 0 and thus the stopping distane for that ase is δ = x 0 d. Remember the d is related to the aeleration, a, as d = a. The event (x 0, t 0 satisfies t 0 = v x 0 where v is the relative veloity at that event and x 0 = d + t 0 d or x 0 = q. Thus the general formula for the stopping distane for a 1 v given veloity and aeleration is δ = d( 1 1 = 1 v a ( 1 1 v 1. The next problem is to deide what δ is. From the problem setting, I would argue that the one light year distane is the oordinate distane in her frame at the instant that she starts the aeleration. This δ is the distane in the wall s frame. This is not Sally s distane. That distane is the proper distane between the event (x 0, t 0 and the intersetion of the line of simultaneity of the ommover at (x 0, t 0 and the worldline of the wall. The equation of the line of simultaneity is t t 0 x x 0 = v and the line of the wall is x = d. The event at the intersetion of these two lines is (d, v (d x 0 +t 0 and the proper distane between this event and the event at the start of the aeleration, (x 0, t 0, is δ, we now have a = δ (1 1 v (x 0 d. Calling her distane to the wall 1. v How does this ompare to the lassial result, stopping distane = v a? From Things, Setion 1.4., for large, 1 v 1 v. Plugging this in we have the lassial result exatly. For our speifi problem, we have v = 3 5 and δ = 1 ltyr and a = 1 ltyr 5 yr or m. s 1.4. Aelerated Roket A roket of length 1 lightyear is aelerated at a onstant aeleration of 1 lightyear. At t = 0, the roket starts to aelerate. When a lok at the year bottom reads a time τ bottom, what is the time for a lok in the top of that roket?

9 1.4. EXAMPLES USING ACCELERATED MOTION 303 Again, we have to determine what is being told to us in the problem. We have to deide where the parts of the roket are, i. e. their world lines. The top of the roket is rigidly onneted to the bottom so that as the roket aelerates the distane as measured from the bottom of the roket to the top is unhanged. Under stress but unhanged. The world line of the bottom whih is aelerating at a rate a bottom in the standard oordinate system is x(τ bottom = t(τ bottom = osh( a bottom τ bottom a bottom sinh( a bottom τ bottom (1.6 a bottom or, using d = a bottom, where d is the proper distane from the origin event, (0, 0, to any event on the world line, x(τ bottom = d osh( d τ bottom t(τ bottom = d sinh( d τ bottom. Note that the ommover to any event, (x(τ bottom, t(τ bottom, has a line of simultaneity that goes from that event through the origin event, (0, 0. A seond set of events that are all at a proper distane d + h from the origin event, (0, 0, (see Figure 1.6 would be at x(τ top = (d + h osh( d + h τ top t(τ top = (d + h sinh( d + h τ top. Also sine the lines of simultaneity are the lines through the origin event, the distane between these world lines when measured by the ommover at the bottom of the roket is h. The trajetory of the top of the roket is x(τ top = a top osh( a top τ top t(τ top = a top sinh( a top τ top (1.7 Thus these are the world lines of the top and the bottom of the roket. We see immediately that the top of the roket does not have the same aeleration as the bottom. Using d = a bottom, we get that a top = a bottom 1 + ha bottom. (1.8

10 304 CHAPTER 1. UNIFORM ACCELERATION t 3 bottom 1 top x Figure 1.6: The world lines of the top and bottom of an aelerating roket. The bottom of the roket has an aeleration of 1 lightyear. The year top of the roket is at a distane 1 lightyear from the bottom. In Figure 1.6, we also see that, sine the world lines of the top and the bottom of the roket share the same asymptotes, the hangle to the line of simultaneity to any event is the same and thus that φ = τ bottom d = τ top d + h or writing this in terms of the aelerations of the roket, or φ = a bottomτ bottom = a topτ top τ top = (1 + ha bottom τ bottom. (1.9 Thus, loks at the top and bottom of a roket run at different rates. This situation an be made a little more baffling by noting that although the top and bottom of the roket have loks that run at different rates, the top and bottom share the same lines of simultaneity. They just differ about the time of these simultaneous events John Bell s Problem The next example is the problem of two idential rokets and John Bell s Problem. Although I am not able to vouh for this story diretly, I have been told the following fasinating story about John Bell. Yes, the same John Bell

11 1.4. EXAMPLES USING ACCELERATED MOTION 305 of Bell s Theorem, see Chapter 19. When a new theoretial physiist would ome to the world famous laboratory, CERN, where Bell was employed, Bell would go to lunh room and look up the new person and as a part of the getting-to-know-you hit hat ask the new person the following question: If two idential oasting rokets were onneted by a string and the rokets then given idential uniform aelerations would the string between them break after some time? Without making a areful analysis, usually without even thinking about it arefully, the unsuspeting innoent would quikly answer that the string would not break. The quik argument being that, if the two rokets were moving at the same veloity originally and had idential aelerations, they would always stay the same distane apart. We are now enough informed about the interesting effets of relativity and partiularly uniform aeleration in speial relativity to be a little more areful. If idential loks at the top and bottom of a roket an drift apart in time, then it is plausible that idential rokets an begin to separate, see Setion 1.4. above. The proof that the string will break is easily shown graphially, see Figure 1.7. Well, at least in priniple, it is simple even if the figure is rather omplex. Two identially uniformly aelerated rokets have trajetories that are shifted from eah other. Consider two rokets that are separated by a distane h and have an aeleration a, their trajetories are x top = x bottom = ( t +, a ( t + h, (1.10 a where the top roket is the one to the side of the aeleration. The end of a string of length h suspended from the top roket has the trajetory ( x string = t + a h. (1.11 It is lear that the x string x bottom > 0 for all t. In fat, we an easily alulate the separation for small ah, physially not an unreasonable riteria for the size of the roket and the aeleration. In this limit and after a ouple of appliations of the result from Things, Setion 1.4. and some rather

12 306 CHAPTER 1. UNIFORM ACCELERATION Asymptote for bottom roket Asymptote for top roket Bottom roket End of string Line of simultaneity Top roket Figure 1.7: John Bell s Problem Two idential rokets have trajetories that follow eah other. We define bottom and top as in the earlier example, Setion 1.4., by the diretion of the aeleration. If a string is suspended from the top roket that just reahes the bottom roket at t = 0, it will have the trajetory shown. Sine the end of the string moves so that it is a fixed distane from the top roket as measured by the top roket, it shares the same asymptote as the top roket. The bottom roket has a different asymptote and, in fat its trajetory rosses the top rokets asymptote. Thus it is lear that it is further than the end of the string from the top roket. Sine the string and top roket share the same line of simultaneity, you an see along that line that at any time t to the top roket the bottom roket is further than the end of the string. The parameters for this figure were a = 1 ltyrs 3, h = 1 ltyr. yr tedious algebra, x string x bottom = h a t. (1.1 Again, it is lear that this is positive for all t. The problem is that this is not the length of interest if the question is when the string will break. Equation 1.1 is the separation of the end of the string and the bottom roket to the original ommover at some time t aording to that inertial observer s lok. We really want the distane the string realizes at any time τ to the string. Of ourse, we realize from the previous example, Setion 1.4., that different parts of the string have different times. Fortunately though, the elements of the string all share the same line of simultaneity and it is, of ourse, the same as that of the top roket. This quandary about loks along aelerated systems will be examined in more detail in the Setion 1.5 where

13 1.4. EXAMPLES USING ACCELERATED MOTION 307 we disuss the problem of allowing an aelerated observer to reate a oordinate system. It is also disussed in the development of General Relativity on the impliations of the Equivalene Priniple, see Setion Using as our time, the time τ of the top roket, we an determine the events at the end of the string and bottom roket that are simultaneous with τ on the top roket. The equation for the line of simultaneity to the top roket for any event, (x 0, t 0, and the string at a time τ on the top roket is t x = t 0 x 0 = tanh ( aτ (1.13 and the event at the end of the string simultaneous with τ at the top roket is ( ( aτ x stringτ = a h osh ( ( sinh aτ t stringτ = a h. (1.14 The event on the bottom roket trajetory that is simultaneous to the string and the top roket satisfies ( (x bottomτ + h ( = tanh aτ x bottom a τ. (1.15 Of the two roots of this equation, the physially aeptable one yields ( ( x bottomτ = ( tanh aτ ( + tanh aτ h a a h ( osh aτ with the t bottomτ given by t bottomτ ( aτ = tanh xbottomτ (1.16. (1.17 The streth of the string, δ, is the proper distane between the events at the end of the string and the bottom roket, δ = (x stringτ x bottomτ (t stringτ t bottomτ ( = (x stringτ x bottomτ 1 tanh aτ ( = (x stringτ x bottomτ osh 1 aτ. (1.18

14 308 CHAPTER 1. UNIFORM ACCELERATION Plugging in for x stringτ and x bottomτ, and doing onsiderable algebra and using the hyperboli funtion identities, δ = ( aτ (1 a h osh ( a ( aτ + h sinh (1.19 Using the same parameters as in Figure 1.7, the streth as a funtion of τ is shown in Figure 1.8 Given an elastiity and breaking tension, we ould Figure 1.8: Strethed String between Rokets The streth of a string onneted between two idential rokets as a funtion of the time of the top roket, see Figure 1.7. The parameters for this figure were a = 1 ltyrs 3, h = 1 yr ltyr. alulate the τ at whih the string breaks but that would get us into a problem in materials engineering. 1.5 The Aelerated Referene Frame Although we know that an aelerated observer does not have the same laws of physis as an inertial observer, there are often irumstanes in whih it is advantageous to make observations from an aelerating system. In addition, we will find that the General Theory of Relativity will have a very lose and important onnetion with aelerated observers and the intuition that is developed here will be valuable there, see Setion 14.. We an proeed to onstrut the referene frame for an aelerating system in the same way that we did for inertial observers, see Setion 9.1.

15 1.5. THE ACCELERATED REFERENCE FRAME 309 Immediately, there are several problems. If we use the onfederate proedures, i. e. plaing onfederates by some rule and endowing them with a lok to label events. There are atually several hoies. At some time t, we ould set at a fixed distane from eah other a set of onfederates with the same aeleration. This is not reasonable. As time goes on the onfederates would find themselves drifting apart and, worst still, they would not have ommon lines of simultaneity, see Setion Another hoie would be to plae them at a fixed distane but give them suitably adjusted aelerations so that they maintain their separations. In this ase, all the onfederates experiene different aelerations, see Setion Not only do they experiene different aelerations, If we endow them with idential loks, these loks will run at different rates, again see Setion Of ourse, we an see that sine they share the same magi point, they will agree on simultaneity. Thus ( ( gτ x h,τ = g + h osh 1 + gh g ( ( gτ t h,τ = g + h sinh 1 + gh (1.0 ould be used to label events where (x h,τ, t h,τ are the event labels provided by the inertial ommover of the origin onfederate. In Equation 1.0, h designates a position of the onfederate and τ is the time on that lok. g is the aeleration of the onfederate at the origin. These expressions are simplified if we refer all lok readings to the origin onfederate s time, i. e. the nearest onfederate reords the event time on their lok and then translates to the origin onfederate s time using Equation 1.9. This implies that one of the origin onfederate plays a speial role and is in harge. With this hange, we have ( ( gτ x h,τ = g + h osh g ( ( gτ t h,τ = g + h sinh (1.1 We an invert this system to yield the equations of h and τ in terms of the inertial oordinate labels, h = ( xh,τ + g t h,τ g

16 310 CHAPTER 1. UNIFORM ACCELERATION τ = g tanh 1 ( t h,τ x h,τ + g (1. Figure 1.9: Coordinate grid for a uniformly aelerated observer by means of onfederates. The time-like world line passing through the origin event is that of an observer that has an aeleration of 1 ltyr. This is yr the referene observer for this oordinate system omposed of onfederates at fixed distanes from the referene observer. The spae-like lines are the lous of events oordinatized at the same time in this oordinate system. Shown dotted are the lines of onstant time and plae as determined by an inertial observer that is ommoving with the aelerated observer at the initial event. This oordinate sheme still has very serious draw baks. The farthest onfederate below the referene observer is at the magi point, h = g and that onfederate has an infinite aeleration. The range in τ is < τ <. In fat, no events outside the forward elsewhere of the magi point has a nearby onfederate. The forward elsewhere from any event is all the spaelike events with positive position from that event bounded by light lines emanating from that event. An event near the magi point light trajetory although at finite times in the inertial oordinates is at plus or minus infinity in τ. This feature of not being able to over all of spae time with onfederates and bounded times will be intrinsi to aelerated oordinate systems and we will not be able to repair it. The infinite aeleration is problemati but not easy to overome exept to realize that these onfederates are hypothetial.

17 1.5. THE ACCELERATED REFERENCE FRAME 311 A simpler oordinatizing sheme whih was idential to the onfederate method in the inertial ase is ahieved by using a protool like the one in Setion 9.1 in whih there is only one observer and that observer uses a lok and reords the travel times of light to and from the event in question and then sets the oordinates as we did in the inertial ase, x = τ τ 1 t = τ + τ 1. (1.3 τ (x,t τ1 Figure 1.10: Protool for using an aelerated observer to oordinatize spae-time. The event that an inertial observer would label as (x 0, t 0 would be labeled as x = τ τ 1 and t = τ +τ 1. This oordinatizing is shown in Figure This method of oordinatizing also has the advantage of not assuming that the underlying spae is homogeneous. More will be made of this later, see Chapter 16. For a uniformly aelerated observer with aeleration g and setting the origin event at the zero veloity event of the observer, we an find the new oordinates, (x, t, in terms of the inertial observers oordinates, (x 0, t 0, by following the proedure in Setion 9..3 and Figure 9.7. The equations of the two light one lines from (x 0, t 0 are t t 0 x x 0 = ± 1. Thus τ 1 and τ satisfy ( g osh gτ1 ( g osh gτ g x 0 = ( g sinh gτ1 g x 0 = ( g sinh gτ t 0 + t 0. (1.4

18 31 CHAPTER 1. UNIFORM ACCELERATION These an be solved for τ 1 and τ and inserted into Equations 1.3 to find (x, t. [( x = g ln ( t = 1 + (x 0+t 0 g g ln ( 1 + (x 0 t 0 g 1 + (x 0 + t 0 g ( 1 + (x ] 0 t 0 g. (1.5 One again, note that these oordinates are singular on the light one boundaries, g = (x 0 ± t 0, of the forward elsewhere from the magi point, (x 0 = g, t 0 = 0. In this oordinate, the range of x is < x < and similarly for t. This looks more like a distane and a time. Despite this range in x and t, you should realize that this range of oordinates does not over the entire range of (x 0, t 0 but only the forward elsewhere from the magi point. We an get a better feel for the shape of this oordinate system by removing those pesky ln funtions. Redefining distane and time by ( gx η exp ζ exp Plugging in and doing a little algebra, ( η g ( gt. (1.6 (x 0 + t 0 g ζ 1 + (x 0+t 0 g 1 + (x. (1.7 0 t 0 g Note that, in the forward elsewhere from the magi point, η and ζ are positive with η equal to zero on both of the edges and ζ equal to zero at the lower edge and plus infinity at the upper edge. From Equation 1.7, it follows that events at the same distane, same x or η, are hyperbolas with the ommon magi point ( g, 0 in the inertial oordinate. In the new oordinate, (x, t, the magi point is at spatial minus infinity or in (η, ζ at η = 0. The events at the same time, same t or ζ, are straight lines passing through the magi point. In the (x, t oordinates, the lower edge is at minus infinity and the upper edge is at plus infinity. Thus this oordinate system

19 1.5. THE ACCELERATED REFERENCE FRAME 313 looks like the system with onfederates at fixed separation and adjusted aelerations, Figure 1.9, with just a relabeling of distanes and times. Obviously, lines of onstant time, t, are lines of simultaneity to the speial observer and the lines of fixed separation are the various suitably aelerated timelike urves. It is easy to show that this system of oordinatizing is the same as the one with the onfederates with adjusted aelerations and with orreted loks by merely reidentifying (h, τ in terms of (η, ζ or (x, t. η = g ( h + g ( gτ ζ = exp (1.8 It is interesting to note that now that, although the relevant times are the same, t = τ, the relevant distanes are not the same, h = g ( ( gx exp 1 (1.9 Confederates plaed at equal spaing as measured in h will not be equally spaed in x even though the sale of length at the origin h and x are ommensurate. At any plae labeled by either h or x, the sales of distane are related by Equation 1.9 and inrements are related by ( gx h = exp x. (1.30 This is an example of a metri relationship. We will ome upon this problem later in General Relativity, Setion Whih distane is the separation, h or x? The h was onstruted to be the proper distane between loal onfederates. The distane x is the inremental distane as measured by light travel time. Either an be used as the distane but pratially speaking the light travel time method is the one that is utilized and thus makes sense as our measure although we will have to orret for the loal distortion using the metri. This is one of the ompliations of aelerated systems. We an omplete the onstrution of our aelerated oordinate system in (x, t by inverting Equation 1.5, x 0 = g ( ( gx exp t 0 = g exp ( gx sinh ( gt osh ( gt 1. (1.31

20 314 CHAPTER 1. UNIFORM ACCELERATION Our interpretation of the distane measures an now be verified by using the metri that is provided by the inertial oordinate system. The interval, see Setion 10.6, between nearby events with differenes in their oordinates of ( x 0, t 0 is given by x prop = x 0 t 0 (1.3 where x prop is the proper distane, if the separation is spaelike, and τ prop = t 0 x 0 (1.33 where t prop is the proper time, if the separation is timelike. Using Equation 1.31, these beome ( gx { x x prop = exp t }, (1.34 if the separation is spaelike, and ( } gx τprop = exp { t x, (1.35 if the separation is timelike. These same relations in the (h, τ oordinates are x prop = h (h + g g τ, (1.36 if the separation is spaelike, and a similar expression for the timelike ase. Using the hangle, see Setion 10.5, between the magi point and the events in question, φ g τ, Equation 1.36 beomes x prop = h (h + φ. (1.37 g The similarity between this form and the usual form for the distane in polar oordinates is striking and onsistent with our interpretation of the hangle. See Figure 10.4 and Figure??. Can this system, partiularly in (x, t, generate a reasonable oordinate system? Will it? It should be obvious that that there are some serious problems here. Before we go into all the problems, lets look at how our friend the aelerated observer would indiate events. Not thinking that he or she is partiularly different, he/she would use a onventional grid for

21 1.5. THE ACCELERATED REFERENCE FRAME 315 Figure 1.11: Lines of Constant position and time in an aelerated oordinate system In Figure 1.9, the dashed lines represent events at either onstant position, vertial dashed lines, or onstant time, horizontal dashed lines, as designated by the inertial observer. In this figure these lines are the solid urves and the lines of onstant position and time as designated by the aelerated observer are shown as dashed. Again in this figure lengths are in units of g. the labels of the events that are reorded. He/she would think that his/her measures of time and spae are like those of an inertial observer and thus prepare an orthogonal grid to represent events. There is a lear and obvious distortion for the aelerated observer. Several features should be noted. It was noted above that, even though the range of position and time are the same as for the inertial observer, the events that are oordinatized are those in the forward light one from the magi point and that points on these light lines, although finite to the inertial observer are mapped to infinity in these oordinates. In partiular, note that the lines of onstant t 0 for t 0 = 1 and x 0 for x 0 = 0 never ross and move off to together. This is, of ourse, a refletion of the fat that the event (x 0 = 0, t 0 = 1 is on the light line from the magi point. Thus the aelerated observer thinks the all events are oordinatized but, as already disussed, the only events that an be oordinatized are in the forward elsewhere from the magi point. A further ramifiation is that, sine lines of onstant x 0 are inertial and ommoving with the aelerated observer at t = 0,these inertial observers experiene a finite time between the events that bound the forward elsewhere from the magi point and yet the aelerated observer says that this same observer experienes an infinite time interval between these events. Also note that, if

22 316 CHAPTER 1. UNIFORM ACCELERATION the inertial observer should hose to pursue the inertial observer by aelerating in that diretion, one the inertial observer passes the events bounding the forward elsewhere from the magi point, there is no aeleration that an aomplish this goal. This situation is very similar to the ase of the blak hole, see Setion 16.1, in whih there is an event horizon and, in fat, the underlying physis is very similar. All of these problems with the oordinatizing by the aelerated observer are also similar to those that emerge when attempting to oordinatize a urved spae with a single flat map. Atlas maps of the earth are all distorted and some points suh as the north pole are even topologially distorted, a point on the earth appears in the atlas as a line. As we will see in Setion 15.7, these similarities are not aidental.