Effecs of Coordinae Curvaure on Inegraion Chrisopher A. Lafore clafore@gmail.com Absrac 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. This siuaion is hen applied o he Schwarzschild meric and geodesic equaion 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
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. Figure 3 illusraes his concep by examining an observer moving over boh fla and curved coordinaes:
T Figure 3 Moion Over Fla and Curved Coordinaes In Figure 3 our observer, Scou, is moving inerially over boh fla coordinaes (T) and curved coordinaes (). Since Scou is inerial and he T coordinaes are fla, his means ha consan inervals of T correspond o consan inervals of ime measured by he clock she is carrying wih her = cons. Bu we can see by inspecion ha as she moves, he amoun of ime icked by her clock relaive o he number of icks she passes decreases over ime = f(). In fac, we could consruc he coordinae axis such ha i exends infiniely off o he righ while each successive ick ges closer, making f() 0 as Scou moves o infinie τ. We can hen imagine shrinking he inervals of T and o an infiniesimal size such ha he coordinaes are coninuous, and we would find ha i mus be ha he sizes of he infiniesimals dt and d are differen, namely ha d becomes increasingly smaller han dt as Scou moves. We will examine observers a res and in freefall in a graviaional field o assess his siuaion in more deail. Radial Moion in he Schwarzschild Field The well-known Schwarzschild meric is given in (1) below (noe we will be using unis where he speed of ligh is 1 and we will drop he angular erm of he meric since we will only be examining radial moion): dτ = 1 d 1 dr (1) The r coordinae represens some noion of disance from he cener of he graviaional 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) [1]: = 1 Le us now subsiue (2) ino (1) o examine he proper ime of he freefalling observer: (2) = 1 (3)
Now, he freefalling observer is inerial and is herefore similar o Scou from Figure 3. I was shown in Figure 3 ha Scou s acceleraion hrough he dimension was only due o he curvaure of he coordinaes relaive o he fla coordinaes. Le us consider he coordinae acceleraion of paricles in he Schwarzschild field [2]: = A Γ (4) Equaion 4 is a re-arranged equaion for proper acceleraion. Wha equaion 4 ells us is ha he coordinae acceleraion of a paricle is he real acceleraion of he paricle minus he effecs of coordinae curvaure. The A erm can be hough of as he fla space acceleraion, resuling in a force ha an observer objecively feels as she moves. The second erm is he par of he coordinae acceleraion caused by he coordinae curvaure. This is wha is responsible for Scou s acceleraion. For he freefalling observer (A = 0), we ge he following geodesic equaions [2]: = (5) = () (6) We noice from Equaion 6 ha an inerial observer saring from any finie r > 2GM wih = 0 mus begin acceleraing hrough r due o he second erm in Equaion 6. This means ha will become non-zero and herefore will become non-zero. Thus, he inerial observer mus accelerae hrough he ime dimension, bu from Equaion 4, we know ha his acceleraion is purely a coordinae arifac as depiced in Figure 3 (A = 0 for he inerial observer). Thus he ime coordinae in he Schwarzschild meric mus be compressed relaive o he fla ime coordinae akin o he coordinae in Figure 3. Nex, le s consider an observer a res a some r in he graviaional field. In his case = = 0 for all ime and herefore = 0 for all ime (we can say ha A = 0 because a non-zero A would correspond o some change in res energy which we do no have in his case). Bu A = (we know from Equaion 1 ha an observer a res has = ) and herefore he res observer does no follow a geodesic, she objecively feels a force as if she were acceleraing in fla space. We can see why his would be he case again from Figure 3. If Scou were he res observer, she would no accelerae relaive o he coordinae (her would be consan). This means ha she would have o accelerae relaive o he fla T coordinae, and herefore she feels a force due o her acceleraion (from our perspecive, she would slow down as she crossed he screen). The even horizon a r = 2GM lies a =. The argumen being made here is ha inegraing Equaion 3 in he usual way is incorrec because i assumes a consan
magniude d. Bu from Figures 1 o 3, i has been suggesed ha d mus be reaed as a funcion of. Given ha he coordinae acceleraion for he freefalling observer comes enirely from he manifold curvaure and no from any real fla-space acceleraion, we migh re-express Equaion 4 as: = (7) Equaion 7 saes ha he coordinae acceleraion is equal o he acual rae of change of he velociy (as would be fel in fla space) minus he rae of change of he coordinae curvaure relaive o fla space. Figure 4 shows worldlines of res observers and he inerial observer ploed agains, T, and τ showing how he coordinae curvaure curves he worldlines of res observers relaive o fla space (res observers are he solid curved lines of consan, he inerial observer is he sraigh dashed line): T τ Figure 4 τ vs. for Observers a Res in a Graviaional Field If we consider he res observers from he perspecive of an inerial observer freefalling from infiniy, we can view i as follows. Firs, he res observers will appear o be moving away from he inerial observer wih a velociy proporional o heir disance from he cener of graviy (heir < 1 resuls in a graviaional redshif of he ligh hey emi observed by he observer a infiniy). As ime passes for he freefalling observer, he res observers will accelerae oward her wih a consan acceleraion proporional o heir disance from he CG. This acceleraion will iniially appear as a reducion in redshif from heir iniial velociy (hey sill appear o be moving away, bu are slowing down). A each ime in he freefaller s frame, here will be one observer whose signals will no longer be redshifed, and will herefore seem momenarily a res relaive o her (on Figure 4, his would be when a paricular res curve has a 45-degree slope. Specifically, his will happen when 1 ##$%% = 1 #$ ). When a res observer has passed ha poin, hey will appear o be moving oward he freefaller wih increasing speed and heir
signals will be blueshifed in her frame. The res observer a r will appear o pass he freefalling observer wih speed = (such ha = 1 corresponds o he speed of ligh). The res observers will pass he freefaller and heir signals will become increasingly redshifed over ime in her frame as hey move away. Given his ligh signal analysis and Figure 4, we can consruc a τ vs. r diagram for he freefalling observer: τ r = 2GM Figure 5 - τ vs. r for he Inerial Observer r In Figure 5, he inerial observer moves along he τ-axis. The solid curved lines represen curves of consan r in he inerial frame. Figures 4 and 5 emphasize he fac ha i is he res observers ha are acceleraing relaive o he inerial observer and no he oher way around. In Figure 5, he inerial observer reaches a paricular res observer when he res observer curve inersecs he τ-axis. Wha we see here is ha he r = 2GM curve is a 45- degree line (ligh-speed moion away from he inerial observer infinie redshif) ha never urns o inersec he τ-axis. Similarly, in Figure 4, he r = 2GM curve is a fla horizonal line ha never curves up. These Figures sugges ha he inerial observer will never reach he even horizon in finie ime according o her clock. If we inegrae Equaion 3 o ge he oal proper ime o he horizon, we ge a finie number. This is because alhough we are inegraing over an infinie ime, he increase in proper ime per uni ime decreases in such a way ha he inegral is asympoic. Bu given ha he decreasing derivaive comes enirely from he coordinae curvaure, we can see from Figures 3, 4, and 5 ha we could ge an infinie proper ime in spie of he derivaive. This is mos clear from Figure 3. We can see from Figure 3 ha i is he res observers ha will approach some kind of finie condiion. In his case, a given res observer (no acceleraing relaive o he coordinae) will asympoically approach some finie T as heir clock goes o infiniy. This illusraes he problem wih he radiional inegral in curved space. Because in Figure 3, if we alk abou moving o he righ in unis of, we will ineviably asympoe since he disance
beween inervals of decreases (relaive o inervals of furher o he lef). Bu Scou can jus move consanly relaive o he T coordinae which has a consan spacing (and herefore accelerae relaive o he coordinae) and will herefore move off infiniely o he righ, regardless of how compressed he coordinae ges. This idea is shown more explicily in Figures 4 and 5, where he horizon worldline never curves ino he pah of he inerial worldline. Given ha he T and axes are coninuous, we can always ake a small region of T a differen locaions and find ha he average densiy of in ha region is arbirarily large (no maer how dense he axis is in one region, i can always be denser in some oher region). Thus, in Figure 3, res observers will all appear o slow down asympoically as hey move (consan ) while he freefaller moves wih consan speed and all observers will experience infinie proper ime because he -axis exends inifiniely. Tha he size of he infiniessimals are variable over he manifold suggess ha his issue may involve counable vs. uncounable infinies, bu ha will no be analyzed here. Radial Coordinae Transformaion 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. Noe ha R as r and R as r 2GM. Making his coordinae subsiuion in (2) gives: = (9) 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 (10) 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 from Equaion 9 ha =. Therefore, raher han geing a sraigh line for he freefalling observer as was he case in Figure 4, he (recall ha r 2GM as R ): facor gives us he worldline shown in Figure 6
τ Figure 6 τ vs. R for he Freefalling Observer Combining Equaions 1 and 2, we see ha 1 as r 2GM. This is because as r goes o 2GM, 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 = 2GM (he facor 1 R ). Observaions from he Cenral Observer Furher evidence for an infinie proper ime o he horizon will be given by considering 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 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 = 2GM conracs o zero proper disance as he shell reaches is Schwarzschild radius. In oher words, in he collapsing frame, r = 2GM 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 (i is easily shown ha he relaive velociy beween he cenral and freefalling observer is V = = and he ime dilaion beween he freefalling and cenral clocks is governed by he 1 V facor which goes o zero a r = 2GM). This is ye anoher
example as o why i is nonsensical for he shell o be able o reach r = 2GM in a finie ime. 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 6. 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 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 7 below: Figure 7 Ligh Signals on -R Char Figure 7 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 7 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 7 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.
References [1] Raine, D., Thomas, E.: Black Holes: A Suden Tex. Imperial College Press, (2015). [2] Lecure Noes on General Relaiviy : hps://arxiv.org/pdf/gr-qc/9712019.pdf. Cied July 17, 2017.