Integration Over Manifolds with Variable Coordinate Density

Transcription

1 Inegraion Over Manifolds wih Variable Coordinae Densiy Absrac Chrisopher A. Lafore In his paper, he inegraion of a funcion over a curved manifold is examined in he case where he curvaure of he manifold resuls in a varying densiy of coordinaes over which he funcion is being inegraed where he upper bound of he of inegraion is infiniy. I is shown ha when he coordinae densiy varies in such a case, he rue area under he curve is no correcly calculaed by radiional echniques of inegraion, bu mus accoun for he varying coordinae densiy. This inegraion echnique is hen applied o he Schwarzschild meric of General Relaiviy o examine he proper ime aken for a freefalling observer o reach he even horizon of a black hole. Inegraion and Coordinae Densiy Consider a velociy defined by some funcion ha is parameerized by a variable r where r increases (or decreases) as increases such ha = f(r). We will begin by inegraing his funcion beween and some finie. In Figure 1, we see his funcion ploed in wo differen cases: on he lef, we have a fla manifold where he ime coordinae densiy is consan along he lengh of he funcion, and on he righ we have a curved manifold which causes he densiy of he ime coordinaes o increase as r increases. dx/d dx/d f(r) f(r) Figure 1 Velociy vs. Time on Fla (lef) and Curved (righ) Manifolds (Finie Upper Bound) On boh he lef and righ sides of Figure 1, he numerical beween ick marks on he ime axis is he same beween any wo adjacen ick marks. Now le us examine wha happens when we approximae he inegrals by summing he areas of he recangles (where each recangle has area A = (Δ)). When comparing he lef and righ images in Figure 1, we see ha since he beween ick marks in boh images is he

2 same, he approximae inegral on he righ side will give a larger value han he inegral on he lef as a resul of he increasing densiy of he ime coordinae as r increases (we have more recangles of equal on he righ side). Now, i is no enirely surprising ha he inegral on he righ would be larger han ha on he lef since we are essenially inegraing he funcion over a larger ime inerval on he righ side (we could jus srech he coordinaes on he righ side such ha hey have equal densiy and hen inegrae normally over a larger ime inerval). The purpose of Figure 1 is o inroduce he effec of increasing coordinae densiy on an inegral o emphasize ha when we approximae he inegral by summing recangles when he densiy increases, we ge a larger value for he inegral because we are summing more recangles of equal. Now consider he same ype of velociy funcion inegraed from some finie ime o infinie ime in fla and curved spaceime. Figure 2 shows boh of hese cases for a velociy funcion ha decreases o zero as goes o infiniy. dx/d dx/d Figure 2 Velociy vs. Time on Fla (lef) and Curved (righ) Manifolds (Infinie Upper Bound) Jus as was he case in Figure 1, we can see ha when we approximae he inegrals in Figure 2 by summing he areas of recangles, he inegral on he righ will give a larger han he inegral on he lef. However, in his case, since he upper bound of is infinie in boh cases, we can aribue he increase in area o an increase in he ime inerval. If we suppose ha f(r) decreases in such a way ha he inegral from o on he lef side (fla manifold) gives a finie value, we can see ha he inegral on he righ side will give a value greaer han ha and if he coordinae densiy goes o infiniy as goes o zero, he inegral can even be infinie. This idea of coordinae densiy can be hough of as being analogous o a dynamic uni change. For insance, in he fla manifold case, suppose x and were measured in he same unis and we muliply he inegral by a consan. Tha would essenially be a change of unis (minues o seconds or mm o meers). Bu a change of unis is really jus a rescaling of he axes. So since he coordinae densiy is describing how he coordinaes are scaled over he manifold, i is as if he coordinae unis are being changed as you move along he manifold. This is essenially wha lengh conracion and ime dilaion in General Relaiviy is, a relaive sreching or squeezing of he coordinae axes.

3 This concep will nex be applied o he worldline of a freefalling observer in Schwarzschild spaceime o examine he limi of proper ime of he freefaller she approaches he even horizon. Freefall in he Schwarzschild Field The well-known Schwarzschild meric is given in (1) below (noe we will be using unis where he Schwarzschild radius is 1 and we will drop he angular erm of he meric since we will only be examining radial freefall): dτ = 1 d 1 dr (1) These coordinaes are quie useful for describing he spaceime for observers a res in he graviaional field, paricularly he observer a infiniy in asympoically fla spaceime. The r coordinae represens some noion of disance from he cener of he graviaional source, where he unis of r are in unis of Schwarzschild radius of he source. Thus, his radial coordinae gives circles around he source where, in a op-down view of he source, he circle radii increase linearly as one moves away from he cener. Le s now consider he coordinae speed of a freefalling observer (who sars o fall from res a infiniy) in he frame of an observer a infiniy in Schwarzschild coordinaes [1]: = 1 (2) Le us now subsiue (2) ino (1) o examine he proper ime of he freefalling observer: dτ = 1 d (3) I is conjecured here ha Equaion 3 is he case described in he firs secion of his paper where he coordinae densiy increases as one moves oward he even horizon. If one inegraes (3) in he usual way saring from some finie disance from he horizon o he horizon, he inegral will yield a finie proper ime, bu i will be argued ha when accouning for he increasing coordinae densiy near he horizon, he acual ime measured by he freefalling observer will be infinie. In order o demonsrae his, we mus firs make a change of variables for he radial coordinae. Radial Coordinae Transformaion I is conjecured ha he freefalling observer will fall for infinie proper ime before reaching he even horizon, and his means ha he freefalling observer mus raverse an infinie amoun of space while falling o he horizon. Bu he Schwarzschild radial coordinae r is defined such ha if someone begins falling from some finie disance from he horizon, hey will raverse a finie r. This is depiced graphically in Figure 3 below.

4 = r = 1 - Figure 3- Relaionship Beween Schwarzschild Coordinaes and he Curved Manifold In Figure 3, we see our inrepid explorer, Scou, freefalling along a radial geodesic in he Schwarzschild graviaional field in he frame of observers a res in he field. The infinie observer would be off o he righ on his diagram where he geodesic (he dark black line) would be horizonal. Since, in his paricular depicion, he angen o he manifold is horizonal a he infinie observer who is inerial in fla space, he acceleraion needed for an observer o remain a res a a given poin is proporional o he slope of he angen a ha poin. I is desirable a his poin o make a coordinae change for he radial coordinae such ha i is beer able o capure he curvaure near he horizon in he same way he ime coordinae does. We will choose coordinae R such ha =. This coordinae varies idenically o he r coordinae for large r (his is good because r is a good physical coordinae a large r) and hen diverges from i a he horizon. Inegraing he expression gives: R = r + ln r 1, r = W(e ) + 1 (4) Where W is he produc-log funcion. Noe ha R as r and R as r 1. R is zero in he region of he elbow of he geodesic picured in Figure 3. Making his coordinae subsiuion in (2) gives: = = (5) This coordinae choice is also useful because he speed of ligh in hese coordinaes is 1 independen of R and. The Schwarzschild meric wih he new coordinae becomes: dτ = ( ) ( ) d dr = d dr (6) A porion of he worldline of a freefalling observer ploed on he -R plane is shown in Figure 4 below:

5 R Figure 4 vs. R The slope of he worldline is close o bu less han 1 in he upper righ quadran for all finie R and. If we subsiue Equaion 5 ino Equaion 6, we ge he expression for proper ime of he freefalling observer: dτ = ( ) W ( ) e dr = (r 1) dr (7) This funcion decreases o zero as R and if i is inegraed direcly as-is, we find ha here is a finie proper ime o reach he horizon from any finie R. Bu we know ha he coordinae densiy increases as he freefaller approaches he horizon and herefore, as discussed in he firs secion of his paper, a ypical inegraion will no give he correc quaniy of proper ime elapsed. Inerial Moion in General Relaiviy Suppose we have wo observers in fla spaceime where for some period of ime, heir imes are relaed by he expression d = d. Then, his relaionship changes such ha heir imes are relaed by d = 0.5d. We migh depic his as shown on he graph on he lef side of Figure 5 below:

6 Figure 5 Acceleraion as a Change of Unis The lef side of Figure 5 suggess ha a some poin, observer 1 acceleraed relaive o observer 2, changing he slope of he worldline. Bu when we ask boh observers, hey each say ha hey never fel any acceleraion (i.e. hey were a res relaive o one anoher he enire ime). How can his be? Afer inspecing boh clocks, i is found ha in he firs half of he graph, boh clocks were measuring seconds. Bu in he second half, observer 1 s clock icked off seconds while observer 2 s clocks was icking off halfseconds. The problem wih he graph on he lef of Figure 5 is ha i was consruced using he derivaives given earlier, wihou accouning for he fac ha he change in he derivaive from 1 o 0.5 was no caused by acceleraion, bu was simply he resul of a uni change in 2. Therefore, we can see he correc depicion of he siuaion on he righ side of Figure 5. The coordinae marks represen he acual clock icks. Since hey were boh a res relaive o each oher he whole ime, we know ha if boh of heir clocks had icked off seconds he whole ime, hey would have agreed ha he same amoun of ime had passed. Therefore, we should draw he relaionship as a 45-degree sraigh line on he graph. Bu since observer 2 s clock icked off half-seconds in he second half of he graph, he ick marks mus be wice as dense on he second half relaive o hose on he firs half. This allows us o preserve he derivaives relaing he imes. This scenario is wha we have when considering an observer freefalling in a Schwarzschild field. Boh he observer a infiniy and he freefalling observer remain inerial as he freefaller falls. The apparen acceleraion beween hem is coming from he curvaure of he freefaller s ime coordinae relaive o he infinie observer s ime coordinae. This curvaure manifess iself as an increase in ime coordinae densiy for he freefaller as she falls. I is analogous o a coninuous uni change of he freefaller s clock relaive o he infinie observer s clock. The meric in Equaion 6 is useful because i is essenially he Minkowski meric wih a variable muliplicaive facor. This muliplicaive facor is analogous o a funcion keeping rack of he unis of he freefaller s clock relaive o he infinie observer s clock. Since he meric is quasi- Minkowskian, we can compare he worldline of he freefalling observer, defined by Equaion 5, o he same worldline for an observer acceleraing in Minkowski space. This comparison is shown in Figure 6 below:

7 τ τ Figure 6 τ vs. for he Acceleraing Observer in Fla Space (lef) and τ vs. for he Freefalling Observer (righ) In Figure 6, we see he worldline of Equaion 5 in boh Minkowski space and in he Schwarzschild field. The graph on he righ side of Figure 6 can be consruced by laying he fla grid from he lef graph on i and hen adding ick marks o he axis beween he fla icks. The number of icks o add beween wo fla icks will be proporional o. As we can see in Figure 6, since he axis of he fla grid goes off o infiniy, he axis on he Schwarzschild grid will do he same wih an ever-increasing coordinae densiy. The resul of his is ha here mus be an infinie amoun of proper ime elapsed when falling oward he horizon from any finie disance. If he freefaller acceleraes a any ime or sars falling wih a non-zero velociy relaive o he infinie observer, hen he sraigh worldline on he lef of Figure 6 will be curved or will have a differen slope depending on ha acceleraion/iniial velociy. So in he Minkowski case, he observer is in an acceleraing reference frame, causing her o approach a maximum τ asympoically as. Figure 6 herefore makes plain he difference beween freefall in a Schwarzschild field and acceleraion in fla spaceime when boh observers have he same. I shows ha for he freefalling observer, he underlying coordinae curvaure is he cause of he acceleraion whereas in fla space i is a real acceleraion. This real acceleraion manifess iself as a curvaure in he worldline iself as shown in Figure 6. Figure 6 is herefore an effecive graphical represenaion of he Equivalence Principle a work. As will be shown in he conclusion of his paper, he freefalling observer will see he res observer s clock slow as she falls. This is jus like he symmery of Special Relaiviy where inerial observers moving relaive o each oher each claim ha i is he oher person who is moving. In order o acually compare heir clocks, one of he observers mus accelerae oward he oher, breaking he symmery and resuling in less ime having passed for he acceleraing observer when he wo observers mee (so in order o mee and compare clocks, eiher he freefaller has o accelerae away from he cener of graviy or he res observer mus accelerae oward i in order o cach up o he freefaller).

8 In erms of he R coordinae, he τ-r grid will look very much like Figure 4 because he and R coordinaes are curved by he same facor. However, we see ha we can express Equaion 7 as = r. Therefore, raher han geing a sraigh line as was he case in Figure 6, he r facor gives us he worldline shown in Figure 7 (recall ha r 1 as R ): τ R Figure 7 τ vs. R for he Freefalling Observer Combining Equaions 1 and 2, we see ha 1 as r 1. This is because as r goes o 1, boh and for he freefalling observer go o zero, where goes o zero for he reasons discussed above, and goes o zero because of he exreme curvaure of space near r = 1, which will be examined in furher deail below. Given ha he freefalling worldline is a geodesic in curved spaceime, we can use he grid on he righ side of Figure 6 o show he τ vs. relaionship for observers a res in he graviaional field. This is depiced in Figure 8: τ Figure 8 - τ vs. for Observers a Res in a Graviaional Field For observers a res, is consan. Bu he observers a res are also in an acceleraed reference frame, hus we should expec ha he worldlines be curved in he curved spaceime. This is wha we see in Figure 8, where he curved lines are lines of consan r. The lines of consan r are curved such ha he change in τ per change in is consan,

9 bu since he ime dimension is curved, he worldline becomes curved relaive o he worldline of he inerial observer, which reflecs he fac ha he res observer is in an acceleraed frame. The dashed line in Figure 8 is he freefalling observer. The res observer a he even horizon would jus be a horizonal line a he boom of Figure 8. The freefalling worldline will never inersec ha line and herefore we see again ha i mus ake an infinie amoun of proper ime o reach he horizon. Noe ha if we plo curves of consan R coordinae on Figure 8, we would see ha curves of equal inervals of R would ge closer and closer ogeher as R goes o negaive infiniy in he same way ha he coordinae spacing decreases. This is wha we would expec from Equaion 6 where he R and coordinaes are boh curved graviaionally by he same facor. By manipulaing Equaion 6, we can see ha he sraigh line over he curved of he ime coordinae in Figure 8 is he resul of wo facors: dτ = 1 V d (8) Where V =. The facor in Equaion 8 is he graviaional ime dilaion caused by he spaceime curvaure (he Minkowski ime coordinae is relaed o he Schwarzschild ime coordinae by dt = d). This facor is wha governs he ime coordinae spacing in Figure 8. The facor 1 V is an addiional ime dilaion caused by he relaive velociy beween he freefaller and he res observers. We see ha as he worldline moves o increasing, he facor 1 V ges closer o zero, reducing he amoun of proper ime elapsed for each inerval of coordinae ime passed. On Figure 8, his reducion is wha keeps he worldline sraigh. Noice ha when a res observer worldline inersecs sraigh line, is apparen slope on Figure 8 is greaer han one. These observers have =, he graviaional ime dilaion facor. For he inerial worldline o remain sraigh, his slope mus be decreased, and ha is wha he 1 V facor does. This is why he inerial observer acceleraes relaive o he spaceime coordinaes in order o remain inerial. Nex, le s consider he proper disance of a spacelike slice of he Schwarzschild meric. From Equaion 1, we see ha he relaionship beween proper disance and he r coordinae in his case is given by =. In Minkowski space, his relaionship would be = 1 and he difference beween he wo is caused by he manifold curvaure. So Figure 9 shows a spacelike s vs. r graph for he Schwarzschild case:

10 s Figure 9 s vs. r for Spacelike Slice of he Schwarzschild Meric Since he fla space relaionship beween s and r is a 45-degree line on he graph, we draw ha firs. Then we dynamically srech he axis wih he radial coordinae as r goes o 1 o mach he derivaive given by he Schwarzschild meric (in Figure 9, r sars a some finie disance far from he horizon on he lef side and hen approaches 1 as we move o he righ). So alhough here will only be a finie number of oal r icks on Figure 9, as we ge closer o r = 1 he coordinae marks ge increasingly sreched unil hey ge infiniely sreched as r goes o 1. Thus, we see ha saring from any finie coordinae disance from he horizon, here will be an infinie proper disance o he horizon. The r coordinae curvaure in Figure 9 for he worldline of he inerial observer can be deduced from he meric as was done wih he ime coordinae. Namely, he relaionship dτ = rdr for he inerial observer can be expressed as: r dτ = r 1 dr = 1 dr (9) Equaion 9 shows he relaionship beween he ime dilaion caused by he inerial observer s velociy and he curvaure of he r coordinae, jus as Equaion 8 did he same wih he coordinae. In his case, we see ha he r coordinae becomes increasingly sreched near r = 1 as opposed o compressed as is he case wih he and R coordinaes. Comparing Equaion 9 o Equaion 7, we see ha he shape of he freefall worldline on a τ vs. r plo should be he same as he one shown in Figure 7, he only difference in he plos being ha he r coordinae will be sreched (as in Figure 9) insead of he R coordinae being compressed. I is noable ha for he freefalling observer, = V. Therefore, when he freefalling observer is a some radius r, he res observers will be moving relaive o her wih velociy = V such ha as she approaches r = 1, he res observers will appear o be moving closer and closer o he speed of ligh relaive o her. Nex, imagine an observer a res a he cener of a collapsing spherically symmeric shell. According o Birkhoff s heorem, he space inside he shell, where he cenral observer is, will be fla. Therefore, according o he clock of an observer a infiniy, ligh wihin he shell will ravel jus like i does a infiniy. Therefore, as he collapsing shell approaches

11 is Schwarzschild radius (say 1 ligh-second), he observer a infiniy will find ha according o her clock, i will ake jus over 1 second for a signal o ravel from he cenral observer o an observer on he shell. Bu he clocks of boh he cenral observer and shell observer will slow o a near sop relaive o he observer a infiniy. Thus in he frames of he cenral and shell observers, signals exchanged beween hem will be received almos insanly as he shell approaches is Schwarzschild radius. Thus, in heir frame, i will appear as hough he space beween r = 0 and r = 1 conracs o zero proper disance as he shell reaches is Schwarzschild radius. In oher words, in he collapsing frame, r = 1 will correspond o he cener of graviy (here will be nowhere else o fall afer ha in he freefall frame). I is also noable ha he clock of he cenral observer icks a he same rae as an observer a res a he locaion of he shell. Therefore, if he shell were acually able o reach he horizon, he cenral observer s clock would sop icking and signals from i would be infiniely blueshifed when received by he collapsing shell (we can also see his from Equaion 8, where he ime dilaion beween he freefalling and cenral clocks is governed by he 1 V facor which goes o zero a r = 1). This is ye anoher example as o why i is nonsensical for he shell o be able o reach r = 1 in a finie ime. Figure 10 shows he freefall worldline depiced wih all relevan quaniies such ha all he Schwarzschild differenial relaionships are capured: = - r =... è R = r = 1 τ = -... R = - τ è Figure 10 Freefalling Geodesic Ploed agains Muliple Coordinaes Conclusion I has been shown ha when accouning for curved spaceime while inegraing he freefall geodesic, he freefaller experiences an infinie amoun of proper ime before reaching he horizon. We also know ha he freefalling worldline approaches a null geodesic asympoically, as can be deduced from Figure 4. This means ha here will be a final ligh signal receivable by he freefaller from res observers. Therefore, we mus conclude ha in he frame of he freefalling observer near he horizon, when she looks

12 ou o signals coming from he res observers, hose observers will appear o her o be slowing down since she experiences infinie proper ime in her frame while receiving a finie number of ligh signals from he res observers. Wha we find is ha he res observers will see he freefalling observer slow exponenially as heir imes go o infiniy, while he freefaller will see he res observers slow asympoically as her ime goes o infiniy. This means ha in he res observer frame, he freefaller will have an open fuure, unfolding a an exponenially slower rae over ime, while in he freefalling frame he res observers will have a closed fuure, where he res observers will appear o evolve oward a finie fuure ime a an asympoically slower rae. These feaures are shown in Figure 11 below: Figure 11 Ligh Signals on -R Char Figure 11 is a -R char ha shows a single infalling signal represening he signal o which he freefall worldline is asympoic. The freefalling observer will receive his signal afer an infinie proper ime and will receive no signals lying above ha one on he char. If a any ime he freefaller acceleraes in a direcion away from he black hole, he will receive more fuure signals from he res observers beyond his asympoic signal since his worldline will curve upwards on Figure 11 as a resul of his acceleraion. Then if he sops acceleraing and begins freefall again, here will be a new ligh signal o which his worldline will be asympoic. The dos in Figure 11 represen inervals of equal proper ime along he worldline and we can see ha since he worldline is infinie (wih angens always below he speed of ligh) on his char, here will be an infinie number of dos on he line spaced increasingly far apar and res observers will receive an infinie number of signals from he freefalling observer a longer and longer inervals. Noe ha in his case we showed ha given he curvaure of coordinaes near he even horizon, a finie-looking inegral acually has an infinie resul. I should also be rue ha in differen applicaions, where he coordinae densiy decreases raher han increases as i does in he presen paper, an inegral ha, when inegraed in he radiional manner, gives an infinie answer may in fac give a finie value when he mehods demonsraed here are applied. References [1] Raine, D., Thomas, E.: Black Holes: A Suden Tex. Imperial College Press, (2015).