AST1100 Lecture Notes

Transcription

1 AST00 Lecue Noes 5 6: Geneal Relaiviy Basic pinciples Schwazschild geomey The geneal heoy of elaiviy may be summaized in one equaion, he Einsein equaion G µν 8πT µν, whee G µν is he Einsein enso and T µν is he sessenegy enso A enso is a maix wih paicula popeies in he same way as a 4-veco is a veco wih specific popeies. This equaion is no a pa of his couse as enso mahemaics and linea algeba, no equied fo aking his couse, ae needed o undesand i. I pesen i hee anyway as i illusaes he basic pinciple of geneal elaiviy: The sess-enegy enso on he igh hand side conains he enegy conen of spaceime, he Einsein enso on he lef hand side specifies he geomey of spaceime. Thus, geneal elaiviy says ha he enegy conen in spaceime specifies is geomey. Wha do we mean by geomey of spaceime? We have aleady seen wo examples of such geomeies, Euclidean geomey and Loenz geomey. We have also seen ha he geomey is specified by he spaceime ineval also called line elemen s which ells us how disances ae measued. Thus, by inseing he enegy conen as a funcion of spaceime coodinaes on he igh side, he lef side gives us an expession fo s, i.e. how o measue disances in spaceime in he pesence of mass/enegy. Thus, in he pesence of a mass, fo insance like he Eah, he geomey of spaceime is no longe Loenz geomey and he laws of special elaiviy ae no longe valid. This should be obvious: Special elaiviy ells us ha a paicle should follow a saigh line in spaceime, i.e. a pah wih consan velociy. This is clealy no he case on Eah, objecs do no keep a consan velociy, hey acceleae wih he gaviaional acceleaion. You migh objec hee: Special elaiviy says ha a paicle coninues wih consan velociy if i is no influenced by exenal foces, bu hee he foce of gaviy is a play. The answe o his objecion is given by a vey impoan concep of geneal elaiviy: gaviy is no a foce. Wha we expeience as he foce of gaviy is simply a esul of he spaceime geomey in he viciniy of masses. The pinciple of maximal aging go back and epea i now! ells us ha a paicle which is no influenced by exenal foces follows he longes pah in spaceime, i.e. he pah which gives he lages possible pope ime. An objec falls o he gound because he geomey of spaceime aound a lage mass like Eah is such, ha when he objec follows he pah wih he longes possible pah lengh s, i falls o he gound. I does no coninue in a saigh line wih consan velociy as i would in a spaceime wih Loenz geomey. Few yeas afe Einsein published he geneal heoy of elaiviy, Kal Schwazschild found a geneal soluion o he Einsein equaion fo he geomey aound an isolaed spheically symmeic body. This is one of he vey few analyic soluions o he Einsein equaion ha exiss. Thus, he Schwazschild soluion is valid aound a lonely sa, plane o a black hole. The spaceime geomey esuling fom his soluion is called Schwazschild geomey and is descibed by he line elemen: The Schwazschild line elemen s M φ. M Thee ae wo hings o noe in his equaion. Fis, we ae using pola coodinaes, φ insead of Caesian coodinaes x, y. This is a naual choice fo a siuaion wih a well defined cene. These ae no hee dimensional coodinaes: Symmey allows us o descibe he geomey on any plane passing hough he cene of he cenal massive body. Given wo evens wih spaceime disance s as well as he posiion of he cenal mass, we have hee poins in space which define a plane on which we define he pola coodinaes. Thus, he coodinae is a disance fom he cene, we will lae come back o how we measue his disance. The φ coodinae is he nomal φ angle used in pola coodinaes.

2 Fac shee: An example of a wo-dimensional analogy of he waping of space and ime by massive objecs, ofen used in inoducoy exs on geneal elaiviy. Geneal elaiviy was poposed by Einsein in 96 and povides a unified descipion of gaviy as a geomeic popey of space and ime, o spaceime. The cuvaue of spaceime is diecly elaed o he enegy and momenum of whaeve mae and adiaion ae pesen. Some pedicions of geneal elaiviy diffe significanly fom hose of classical physics, especially concening he passage of ime, he geomey of space, he moion of bodies in fee fall, and he popagaion of ligh. Examples of such diffeences include gaviaional ime dilaion, gaviaional lensing, he gaviaional edshif of ligh, and he gaviaional ime delay. Figue: WGBH Boson The line elemen fo Loenz geomey in pola coodinaes can similaly be wien as Loenz line elemen in pola coodinaes s x φ. The second hing o noe in he equaion fo he Schwazschild line elemen is he em M/. Hee M/ mus be dimensionless since i is added o a numbe. Bu we know ha mass is measued in kilogams and disances in mees, so how can his em be dimensionless? Acually, hee should have been a G/c hee, G m 3 /kg/s being he gaviaional consan and c m/s being he speed of ligh. We have ha G c m/kg. Since M/ has unis kg/m, G/c is clealy he consan which is missing hee. We ae now used o measue ime inevals in unis of mees. If we now decide o also measue mass in unis of mees, equaion gives us a naual convesion faco. M m M kg G c, whee M m is mass measued in mees and M kg is mass measued in kg. Thus we have ha kg m. The equaion gives us a convesion fomula fom kg o m. We see ha measuing mass in mees equals seing G/c eveywhee in he fomulas. This is equal o wha happened when we decided o measue ime in mees, we could se c eveywhee. The eason fo measuing mass in mees is pue laziness, i means ha we don need o wie his faco all he ime when doing calculaions. Thus insead of wiing M kg G/c we wie M/ whee M is now mass measued in mees. All he physics is capued in he las expession, we have jus go id of a consan. Fom now on, all masses will be measued in unis of mees and when we have he final answe we conve o nomal unis by muliplying o dividing by he necessay facos of G/c and c in ode o obain he unis ha we wish. The ineial fame Figue : Two boxes in fee fall: If hey ae lage enough in eihe diecion, he objecs a es inside he boxes will sa moving. A local ineial fame needs o be small enough in space and ime such ha his moion canno be measued. In he lecues on special elaiviy we defined ineial fames, o fee-floa fames, o be fames which ae no acceleaed, fames moving wih consan velociy on which no exenal foces ae acing. We can give a moe geneal definiion in he following way: To es if he oom whee you ae siing a he momen is an ineial fame, ake an objec, leave i a es wih zeo velociy. If he objec says a es wih zeo velociy, you ae in an ineial fame. If you give he objec a velociy v and he objec coninues in a saigh line wih velociy v, you ae in an ineial fame. Clealy, a fame a oom which is no acceleaed on which no exenal foces wok is an ineial fame accoding o his definiion. Bu ae hee ohe examples? In geneal elaiviy we use he noion of a local inenal fame, i.e. limied egions of spaceime which ae ineial fames. An example of such a local ineial fame is a space caf in obi aound he Eah. Anohe example is an elevao fo which all cables have boken so ha

3 i is feely falling. All feely falling fames can be local ineial fames. How do we now ha? If an asonau in he obiing space caf akes an objec and leaves i wih zeo velociy, i says wih zeo velociy. This is why he asonaus expeience weighlessness. If a peson in a feely falling elevao akes an objec and leaves i a es, i says a es. Also he peson in he elevao expeiences weighlessness. Thus, hey ae boh, wihin ceain limis, in an ineial fame even hough hey ae boh acceleaed. Noe ha an obseve sanding on he suface of Eah is in a local ineial fame fo a vey sho peiod of ime: If an obseve on Eah leaves an objec a es, i will sa falling, i will no say a es: An obseve a he suface of Eah is no in a local ineial fame unless he ime ineval consideed is so sho ha he effec of he gaviaional acceleaion is no measuable. The only hing ha keeps he obseve on he suface of he Eah fom being in a local ineial fame is he gound which exes an upwad foce on he obseve. If suddenly a hole in he gound opens below him and he sas feely falling, he suddenly finds himself in an ineial fame wih less sic ime limis. We now need o find ou which limiaions his ineial fame has. Local means ha he ineial fame is limied in space and ime, bu we need o define hese limis. In figue we see wo falling boxes, box A falling in he hoizonal posiion, box B falling in he veical posiion. Since he gaviaional acceleaion is dieced owads he cene of he Eah, wo objecs a es a eihe side of box A will sa moving owads he cene of he box due o he diecion of he acceleaion. The shoe we make he box, he smalle his moion is. If we make he box so sho ha we canno measue he hoizonal moion of he objecs, we say ha he box is a local ineial fame. The same agumen goes fo ime: If we wai long enough, we will evenually obseve ha he wo objecs have moved. The ineial fame is limied in ime by he ime i akes unil he moion of he wo objecs can be measued. Similaly fo box B: The objec which is close o he Eah will expeience a songe gaviaional foce han he objec in he ohe end of he box. Thus, if he box is long enough, an obseve in he box will obseve he wo objecs o move away fom each ohe. This is jus he nomal idal foces: The gaviaional aacion of he moon makes he oceans on eihe side of he Eah move away fom each ohe: we ge high ides. Bu if he box is small enough, he diffeence in he gaviaional acceleaion is so small ha he moion of he objecs canno be measued. Again, i is a quesion of ime befoe a moion will be measued: The local ineial fame is limied in ime. We have hus seen ha a local ineial fame can be found if we define he fame so small in space and ime ha he gaviaional acceleaion wihin he fame in space and ime is consan. In hese fames, wihin he limied spaial exen and limied duaion in ime, an objec which is lef a es will emain a es in ha fame. The songe he gaviaional field and he lage he vaiaions in he gaviaional field, he smalle in space and ime we need o define ou local ineial fame. We have leaned fom special elaiviy ha an ineial fame has Loenz geomey. Wihin he local ineial fame, spaceime inevals ae measued accoding o s x and he laws of special elaiviy ae all valid wihin he limis of his fame. In geneal elaiviy, we can view spaceime aound a massive objec as being an infinie se of local ineial fames. When pefoming expeimens wihin hese limied fames, special elaiviy is all we need. When sudying evens aking place so fa apa in space and ime ha hey do no fi ino one such local ineial fame, geneal elaiviy is needed. This is why only special elaiviy is needed fo paicle physiciss making expeimens in paicle acceleaos. The paicle collisions ake place in such a sho ime ha he gaviaional acceleaion may be negleced: They ake place in a local ineial fame. We will now call spaceime whee Loenz geomey is valid fla spaceime. This is because Loenz geomey is simila o Euclidean geomey on a fla suface excep fo a minus sign. We know ha a cuved suface, like he suface of he sphee, has spheical geomey no Euclidean geomey. In he same way, Schwazschild geomey epesens cuved spaceime, he ules of Loenz geomey ae no valid and Schwazschild geomey needs o be used. We say ha he pesence of mae cuves spaceime. Fa away fom all massive bodies, spaceime is fla and special elaiviy is valid. We can ake he analogy even fuhe: Since he suface of a sphee has spheical and no Euclidean geomey, he ules of Euclidean geomey may no be used. Bu if we focus on a vey small pa of he suface of a sphee, he suface looks almos fla and Euclidean geomey is a vey good appoximaion. The suface of he Eah is cuved and heefoe has spheical geomey, bu since a gaden is vey small compaed o he full suface of he Eah, he suface of he Eah appeas o us o have fla geomey wihin he gaden. We use Euclidean geomey when measuing he aea of he gaden. The same is he case fo he cuved space: Since spaceime is cuved aound a massive objec, we need o use Schwazschild geomey. Bu if we only sudy evens which ae wihin a small aea in spaceime, spaceime looks fla and Loenz geomey is a good appoximaion. 3 Thee obseves In he lecues on geneal elaiviy we will use hee obseves, he fa-away obseve, he shell obseve and he feely falling obseve. We will also assume ha he cenal massive body is a black hole. A black hole is he simples possible macoscopic objec in univese: i can be descibed by hee paamees, mass, angula momen- 3

4 um and chage. Any black holes which have he same values fo hese hee paamees ae idenical in he same way as wo elecons ae idenical. A black hole is a egion in space whee he gaviaional acceleaion is so high ha no even ligh can escape fom i. A black hole can aise fo insance when a massive sa is dying: A sa is balanced by wo foces, he foces of gaviy which we no longe call foces ying o pull he sa ogehe and he gas/adiaion pessue ying o make he sa expand. When all fuel in he sella coe is exhaused, he foces of pessue ae no song enough o wihsand he foces of gaviy and he sa collapses. No foces can sop he sa fom shinking o an infiniely small poin. The gaviaional acceleaion jus ouside his poin is so lage ha even ligh ha ies o escape will fall back. The escape velociy is lage han he speed of ligh. This is a black hole. Noe ha he Schwazschild line elemen becomes singula a M. This adius is called he Schwazschild adius o he hoizon. This is he poin of no eun, any objec o ligh which comes inside his hoizon canno ge ou. A any poin befoe he hoizon a spaceship wih song engines could sill escape. Bu afe i has eneed, no infomaion can be ansfeed ou of he hoizon. The fa-away obseve is siuaed in a egion fa fom he cenal black hole whee spaceime is fla. He does no obseve any evens diecly, bu ges infomaion abou ime and posiion of evens fom clocks siuaed eveywhee aound he black hole. The shell obseves live on he suface of shells consuced aound he black hole. Also a spaceship which uses is engines o say a es a a fixed adius could seve as a shell obseve. The shell obseves expeience he gaviaional aacion. When hey leave an objec a es i falls o he suface of he shell. Thee is one moe obseve which we aleady discussed in he pevious secion. This is he feely falling obseve. The feely falling obseve caies wih him a wiswach and egises he posiion and pesonal wiswach ime of evens. The feely falling obseve is living in a local ineial fame wih fla spaceime. Thus fo evens aking place wihin sho ime inevals and sho disances in space, he feely falling obseve uses Loenz geomey o make calculaions. 4 The ime and posiion coodinaes fo he hee obseves Each of he obseves have hei own se of measues of ime and space. The fa away obseve uses Schwazschild coodinaes, and shell obseves use shell coodinaes shell, shell. Fo he feely falling obseves, we will be viewing all evens fom he oigin in his fame of efeence and we will heefoe no need a posiion coodinae since i will always be zeo using his wiswach ime which will hen always be he pope ime τ. We will now discuss hese diffeen coodinae sysems and how hey ae defined. When he shell obseve wans o measue his posiion fom he cene of he black hole, he uns ino poblems: When he ies o lowe a mee sick down o he cene of he hole o measue, he sick jus disappeas behind he hoizon. He needs o find ohe means o measue his adial posiion. Wha he does is o measue he cicumfeence of his shell. In Euclidean geomey, we know ha he cicumfeence of a cicle is jus π. So he shell obseve measues he cicumfeence of he shell and divides by π o obain his coodinae. In a non-euclidean geomey, he adius measued his way does no coespond o he adius measued inwads. We define he Schwazschild coodinae in his way. cicumfeence π The in he expession fo he Schwazschild line elemen is he Schwazschild coodinae. Now he shell obseves a shell lowe a sick o he shell obseves a shell. The lengh of he sick is shell. They compae his o he diffeence in Schwazschild coodinae and find ha shell. This is wha we anicipaed, in Euclidean geomey hese need o be equal, in Schwazschild geomey hey ae no. We have obained a second way o measue he adial disances beween shells using shell disances shell noe ha since he absolue shell coodinae shell canno be measued i is meaningless, only elaive shell coodinae diffeences shell beween shells can be measued did you undesand why?. We have obained wo diffeen measues of adial disances, he Schwazschild coodinae defined by he cicumfeence of he shell. The fa-away obseve uses Schwazschild coodinaes o measue disances. he shell disances shell found by physical measuemens beween shells.this is he disance which he shell obseves can measue diecly wih mee sicks and is heefoe he mos naual measue fo hese obseves. Wha abou ime coodinaes? Again we have wo meaues of ime, The fa-away obseve uses fa-away ime o measue ime. This is he ime eneing in he Schwazschild line elemen. Fa away ime fo an even is measued on a clock which has been synchonized wih he clock of he fa-away obseve and which is locaed a he same locaion as he even we will lae descibe how evens can be imed which such clocks in pacice. 4

5 The shell obseve uses local shell ime shell : i is simply he wiswach ime of he shell obseve, he ime measued on a clock a es a he specified shell. Noe ha shell obseves a diffeen shells may measue diffeen imes inevals shell and disances shell beween wo evens depending on which shell hey live on. Shell coodinaes ae local coodinaes. In ode o elae ime and space coodinaes in he diffeen fames we will now as we did in special elaiviy use he invaiance of he space ime ineval o line elemen s. Fis we will find a elaion beween he moe absac fa away-ime and he locally measuable shell ime shell. The shell ime is he wiswach ime, o pope ime τ of he shell obseves. We will use wo evens A and B which ae wo icks on he clock of a shell obseve. The shell obseves ae a es a shell, so, AB 0 and φ AB 0. Inseing his ino he Schwazschild line elemen equaion using ha s AB τ AB shell he ime peiod beween A and B measued on shell clocks is by definiion he same as he pope ime peiod beween A and B which we have leaned is always equal o he invaian fou dimensional line elemen beween hese evens Shell ime shell M. 3 Ae you sue you see how his expession comes abou? Fo shell obseves ouside he hoizon > M, he local ime goes slowe by a faco M wih espec o he fa-away ime. We also see ha he smalle he disance fom he cene, he slowe he shell clock wih espec o he fa-away ime. Thus, he fuhe down we live in a gaviaional field, he slowe he clocks un. This has consequences fo people living on Eah: Ou clocks ick slowe han he clocks in saellies in obi aound Eah. A he end of his lecue, we will look close a his fac. Figue : The shell obseve a shell measue he pope lengh of a sick by wo simulaneous evens A and B on eihe side of he sick. We have now found a elaion beween ime inevals measued on shell clocks and ime inevals measued on clocks synchonized wih fa-away clocks. How is he elaion beween disances measued wih Schwazschild coodinaes and disances measued diecly by shell obseves? We can measue he lengh of a sick as he spaial disance beween wo evens aking place a he same ime a eihe end of he sick see figue. Fo evens aking place wihin sho ime inevals and sho spaial exensions, he shell obseve sees fla spaceime and can heefoe use Loenz geomey: s shell shell shell we will look a a sick which is aligned wih he adial diecion, he evens heefoe ake place a he same φ coodinae so φ 0. The fa-away obseve always needs o use he Schwazschild line elemen equaion insead of he Loenz line elemen. Using invaiance of he line elemen we have fo wo evens A and B s shell AB sab aking place simulaneously on eihe side of he sick shell shell M, M whee we have se φ 0. Check ha you undesand how o aive a his expession. Now, we measue he lengh of a sick in he adial diecion by measuing he disance beween he wo simulaneous evens A and B aking place a eihe end of he sick a spaial disance. Since evens which ae simulaneous fo shell obseves a a given shell also ae simulaneous fo he fa-away obseve equaion 3, shell 0. Inseing his, we ge shell. 4 M fo sho disances close o he shell. Thus, adial disances measued by he shell obseves, loweing mee sicks fom one shell o he ohe is always lage han he adial disances found by aking he diffeence beween he Schwazschild coodinae disances. Wha abou a sick which is pependicula o he adial diecion? In his case, he obseves will agee on he lengh of his sick, check ha you can deduce his in he same manne as we deduced he elaion fo he adial sick. Thee is a pacical poblem in all his: We said ha he fa-away ime was measued by clocks locaed a he posiion of evens which can ake place close o he cenal black hole bu which ae synchonized wih he faaway clocks. How can we synchonize clocks which ae locaed deep in he gaviaional field and which heefoe un slowe han he fa-away clock? Le s imagine he clocks measuing fa-away-ime o be posiioned a diffeen shells aound he black hole. The shell obseves design he clocks such ha hey un fase by a faco M. To synchonize all hese clocks, he fa-away obseve sends a ligh signal o all he ohe clocks a he 5

6 momen when he ses his clock o 0. The shell obseves know he disance fom he fa-away obseve o he fa-away-ime clocks and hus know he ime i ook fo he ligh signal o each hei clock. They had hus aleady se he clock o his ime and made a mechanism such ha i saed o un a he momen when he ligh signal aived. In his way, all fa-away-ime clocks siuaed a diffeen posiions aound he black hole ae synchonized. Anohe pacical quesion: How does he fa-away obseve know he ime and posiion of evens. Each ime an even happen close o one of he fa-away-clocks close o he black hole, i sends a signal o he fa-away obseve elling he ime and posiion his clock egiseed fo he even. In his way, he fa-away obseve does no need o ake ino accoun he ime i akes fo he signal fom he clock o aive, he signal iself conains infomaion wih he coec fa-away-ime fo he even ecoded on he clock posiioned a he same locaion whee he even ook place. In he following we will descibe evens eihe as hey ae seen by he fa-away obseve using global Schwazschild coodinaes,, by he shell obseve using local coodinaes shell, shell o he feely falling obseve also using local coodinaes. Befoe poceeding, make a dawing of all hese obseves, hei coodinaes and he elaion beween hese diffeen coodinaes. 5 The pinciple of maximum aging evisied In he lecues on special elaiviy we leaned ha he pinciple of maximum aging ells objecs in fee floa o move along pahs in spaceime which give he longes possible wiswach ime τ which coesponds o he longes possible spaceime ineval s. We also used ha fo Loenz geomey, he longes in ems of s o τ pah beween wo poins is he saigh line, i.e. he pah wih consan velociy. We neve poved he lae esul popely. We will do his now, fis fo Loenz geomey and hen we will use he same appoach o find he esul fo Schwazschild geomey. 5. Reuning fo a momen o special elaiviy: deducing Newon s fis law We will now show ha he pinciple of maximum aging leads o Newon s fis law when using Loenz geomey. Figue 3: The moion of a paicle in fee floa in Loenz geomey. Poins x, x, x 3 as well as he imes and 3 ae fixed. Fo a paicle a fee floa, a wha ime will i pass x? Which of he possible spaceime pahs in he figue does he paicle ake? We use he pinciple of maximal aging o show ha in Loenz geomey, he paicle follows he saigh spaceime pah. Look a figue 3. We see he woldline of a paicle going fom posiion x a ime o posiion x 3 a ime 3 passing hough posiion x a ime. Say ha he poins x, x and x 3 ae fixed and known posiions. We also say ha he oal ime ineval 3 i akes he objec o go fom x o x 3 is fixed and known. Wha we do no know is he ime ineval i akes he paicle o go fom poin x o poin x. Remembe ha we do no know ha he objec will move wih consan velociy, his is wha we wan o show. Thus we leave open he possibiliy ha he paicle will have a diffeen speed beween x and x han beween x and x 3. The ime can be a any possible poin beween and 3. In figue 3 we show some possible spaceime pahs ha he objec could ake. We now assume ha he disances x and x 3 ae vey sho, so sho ha he objec can be assumed o move wih consan velociy beween hese wo poins, i.e. ha he objec is in a local ineial fame beween x and x and in a possibly diffeen ineial fame beween x and x 3. Theefoe, he ime inevals and 3 o avel hese wo sho pahs also need o be sho. The oal wiswach ime τ i akes he paicle o move fom x o x 3 is τ 3 τ + τ 3 x + 3 x 3, 5 whee we used ha τ s x fo Loenz geomey. Accoding o he pinciple of maximal aging, we need o find he pah, i.e. he, which maximizes he oal wiswach ime τ 3. We do his by seing he deivaive of τ 3 wih espec o he fee paamee equal o zeo, i.e. you look fo he maximum poin of he funcion τ 3 : dτ 3 d d. + x d 3 d 3. 3 x d 3 Since we have ha d /d emembe ha is a fixed consan and similaly fo 3. Thus we 6

7 have 3 0 x 3 x 3 o wien in ems of τ and τ 3 we have τ 3 τ 3. Check ha you undesood evey sep in he deducion so fa! This is only fo hee poins x, x and x 3 along he woldline of a paicle. If we coninue o beak up he woldline in small local ineial fames a poins x 4, x 5, ec. we can do he same analysis beween any hee adjacen poins along he cuve. The esul is ha d dτ consan, whee I have wien d insead of o 3 and dτ insead of τ o τ 3. Remembe ha we assumed hese ime ineval o be vey sho. In his final expession we have aken he limi in which hese ime inevals ae infiniesimally sho. We also emembe do you? fom special elaiviy ha d dτ γ. v So he pinciple of maximal aging has given us ha γ consan along a woldline. Bu γ only conains he velociy v of he objec so i follows ha v consan. In Loenz geomey, a fee-floa objec will follow he spaceime pah fo which he velociy is consan. We can wie his in a diffeen way. In special elaiviy we had ha E γm so we can wie γ E/m fom which follows ha γ E m consan. We have jus deduced ha enegy is conseved, o moe pecisely enegy pe mass E/m is conseved. In he lecues on special elaiviy we leaned ha expeimens ell us ha he elaivisic enegy E γm is conseved and no Newonian enegy. Hee we found ha he pinciple of maximal aging ells us ha hee is a quaniy which is conseved along he moion of a paicle. This quaniy is he same as he quaniy we call elaivisic enegy. Figue 4: The moion of a paicle in fee floa in Loenz geomey. Poins x, x 3 as well as he imes, and 3 ae fixed. Fo a paicle in fee floa, which posiion x will i pass a ime? Which of he possible spaceime pahs in he figue does he paicle ake? We use he pinciple of maximal aging o show ha in Loenz geomey, he paicle follows he saigh spaceime pah. Is i possible ha he pinciple of maximal aging can give us somehing moe? We will now epea he above calculaions, bu now we fix, and 3. All imes ae fixed. We also fix x and x 3, bu leave x fee. The siuaion is shown in figue 4. Now he quesion is which poin x will he objec pass hough?. We need o ake he deivaive of expession 5 wih espec o x which is a fee paamee. dτ 3 dx x dx 3. x dx x dx 3. x dx Again x x x so ha dx /dx and similaly fo x 3 and we have x τ x 3 τ 3, we have found anohe consan of moion Bu we can wie his as We have ha dx dτ consan dx dτ dx d d dτ vγ. vγ consan p m. Go hough his deducion in deail youself and make sue you undesand evey sep. We emembe ha p mγv, so he pinciple of maximal aging has given us he law of momenum consevaion, o acually he law of consevaion of momenum pe mass p/m. We have seen ha he pinciple of maximal aging seems o be moe fundamenal han he pinciples of enegy and momenum consevaion. I is sufficien o assume he pinciple of maximal aging. Fom ha we can deduce he expessions fo enegy and momenum and also ha hese need o be conseved quaniies. 7

8 5. Reuning o geneal elaiviy: deducing and genealizing Newon s law of gaviaion Figue 5: The moion of a paicle in fee floa in Schwazschild geomey. Poins,, 3 as well as he imes and 3 ae fixed. Fo a paicle in fee floa, a wha ime will i pass hough? We assume ha he disances and 3 ae so small ha we can assume he adial disance o be A always in he fome ineval and B always in he lae ineval. Figue 6: The moion of a paicle in fee floa in Schwazschild geomey. Which spaceime pah will he paicle ake beween poins A and B? Now, wha abou geneal elaiviy? We will see how he pinciple of maximal aging ells a paicle o move in Schwazschild spaceime. Look a figue 5. A paicle avels fom adius a ime o adius 3 a poin 3 passing hough poin a ime. We fix, and 3. We also fix he sa and end imes and 3. We leave fee. We will find a which ime he paicle passes hough poin. Again we wie he oal pope ime fo he objec fom o 3 as using he Schwazschild line elemen, equaion, fo τ τ 3 τ + τ 3 M A + M B M A 3 3, M B whee A is he adius halfway beween and. We assume ha is so small ha we can use he adius A fo he full ineval. In he same way, B is he adius halfway beween and 3 which we coun as valid fo he full ineval 3. Following he pocedue above, we will now maximize he oal pope ime τ 3 wih espec o he fee paamee. We ge dτ M 3 A d M B 3 d 3 +. d τ d τ 3 d As above, giving d /d and similaly fo 3. Thus we have ha M A M B 3. τ τ 3 We find ha M d dτ consan, 6 whee again we have aken he limi whee, 3, τ and τ 3 ae so small ha hey can be expessed as infiniesimally small peiods of ime d and dτ. In he case wih Loenz geomey we used his consan of moion o find ha he velociy had o be consan along he woldline of a feely floaing paicle. Now we wan o invesigae how his consan of moion ells us how a feely floaing paicle moves in Schwazschild spaceime. Fis we need o find an expession fo d/dτ. In special elaiviy we elaed his o he velociy of he paicle using d/dτ γ, bu his was deduced using he line elemen of Loenz geomey. Hee we wan o elae his o he local velociy ha a shell obseve a a given adius obseves. The locally measued shell velociy as an objec passes by a given shell is given by v shell d shell d shell We now use equaion 3 he equaion connecing fa-awayime and shell ime, emembe? o wie he consan of moion equaion 6 as M M M M M / dshell dτ / d shell dτ / γ shell / v shell consan. In he las ansiion we used he fac ha he shell obseve lives in a local ineial fame fo a vey sho ime. The shell obseve makes he velociy measuemen so fas ha he gaviaional acceleaion could no be noiced and he could use special elaiviy assuming fla spaceime. So using his local ime shell, he elaion d shell /dτ γ shell fom special elaiviy is valid. We have hus found a consan of moion: M / v shell consan. 8

9 Conside a paicle moving fom adius A o a highe adius B see figue 6. This ime, he disance beween poins A and B does no need o be small. As he objec moves pas shell A, he shell obseves a his shell measue he local velociy v A. As he objec moves pas shell B, he shell obseves a his shell measue he local velociy v B. Equaing his consan of moion a he wo posiions A and B we find M / M /. A v A B v B Squaing and eoganizing we find vb M v A M. A B We aleady see fom his equaion ha if B > A hen v B < v A check!. Thus if he objec is moving away fom he cenal mass, he velociy is deceasing. If we have B < A we see ha he opposie is ue: If he objec is moving owads he cenal mass, he velociy is inceasing. So he pinciple of maximum aging applied in Schwazschild geomey gives a vey diffeen esul han in Loenz geomey. In Loenz geomey we found ha he velociy of a feely floaing paicle is consan. In Schwazschild spaceime we find ha he feely floaing paicle acceleaes owads he cenal mass: If i moved ouwad i slows down, if i moved inwads i acceleaes. This is jus wha we nomally conside he foce of gaviy. We see ha hee we have no included any foces a all: We have jus said ha he cenal mass cuves spaceime giving i Schwazschild geomey. By applying he pinciple of maximal aging, ha an objec moving hough spaceime akes he pah wih longes possible wiswach ime τ, we found ha he objec needs o ake a pah in spaceime such ha i acceleaes owads he cenal mass. We see how geomey of spaceime gives ise o he foce of gaviy. Bu in geneal elaiviy we do no need o inoduce a foce, we jus need one simple pinciple: The pinciple of maximal aging. We will now check if he acceleaion we obain in he limi of lage adius and low velociies v shell is equal o he Newonian expession. We now call he consan of moion K giving M vshell K. Reoganizing his we have v shell K We wan o find he acceleaion g shell dv shell d shell M 7 ha a shell obseve measues. Taking he deivaive of equaion 7 we ge check! dv shell M d shell v shell K d d shell. Using equaion 4 and ha v shell d shell /d shell we obain g shell M M Newon s law of gaviaion is no valid close o he Schwazschild hoizon, so o ake he Newonian limi we need o conside his expessions fo M. In his limi he expession educes o g shell M, exacly he Newonian expession fo he gaviaional acceleaion. We find ha fa away fom he Schwazschild adius, geneal elaiviy educes o Newon s law of gaviaion. We can now eun o figue 6 and look a he pah maked Schwazschild pah. This is he spaceime pah beween A and B ha a feely floaing objec needs o ake in ode o ge he longes pope ime τ. Looking a he slope of his pah, we see ha he objec changes velociy duing is ip fom A o B. This is in shap conas o he esuls fom special elaiviy wih Loenz geomey whee he pah which gives longes possible pope ime is he saigh line wih consan velociy. We will now eun o ou consan of moion M / consan 8 v shell In special elaiviy we found ha a consan of moion which we obain in he same manne was jus he enegy pe mass. We will now go o he Newonian limi o see if he same is he case fo Schwazschild spaceime. We will use wo Taylo expansions, x x +... x + x, boh aken in he limi of x. In he Newonian limi we have ha M/ and v. Applying his o equaion 8 we have M + v + v M consan In he las expession we used ha since boh M/ and v ae vey small, he poduc of hese small quaniies is even smalle han he emaining ems and could heefoe be omied. Compae his o he Newonian expession fo he enegy of a paicle in a gaviaional field E mv Mm. 9

10 We see again ha he consan of moion was jus enegy pe mass E/m whee he expession now ells us how he gaviaional poenial looks like have you noiced his: you have acually deived why he fom of he Newonian gaviaional poenial is he way i is. Noe he addiional em in he elaivisic expession which is jus he es enegy m. Again he pinciple of maximal aging has given us ha enegy is conseved and i has given us he elaivisic expession fo enegy in a gaviaional field. Relaivisic enegy in a gaviaional field E m M d dτ consan. We also found ha his expession fo he enegy equals he Newonian expession fo disances fa fom he Schwazschild adius. In he execises you will use he pinciple of maximum aging o find ha angula momenum pe mass is conseved in Schwazschild spaceime and ha he expession fo angula momenum pe mass in Schwazschild spaceime is Angula momenum pe mass in Schwazschild spaceime L dφ m dτ γ shellv φ consan. he Schwazschild adius, he fis faco M sas dominaing he behaviou of v as he las faco now goes o one. In his case, he velociy is deceasing when is deceasing. A he hoizon he velociy eaches exacly zeo. Wha we see is ploed in figue 7. When he objec sas falling he velociy inceases unil a poin whee i sas deceasing. A he hoizon he objec sops. This esul was obained using Schwazschild coodinaes. Thus, his is he esul ha he fa-away obseve sees. This means ha if we le a spaceship fall ino a black hole, we, as faaway obseves, would see he spaceship sopping a he hoizon and i would say hee fo eve. Remembe also ha ime is going slowe close o he hoizon, shell M A he hoizon M, we obseve ha ime sops. Thus, looking a he spaceship we would obseve he pesons in he spaceship o feeze a he hoizon. Eveyhing sops. In he execises you will show ha ligh fom a cenal mass is ed shifed. Thus we will also see a songly edshifed ligh fom he spaceship. Using he expession fom he execises you will see ha ligh aiving fom he hoizon is infiniely ed shifed. Thus you will no see any ligh fom he hoizon. You will only see he spaceship jus befoe i eaches he hoizon and hen only as adio waves wih a lage wavelengh see poblem. 6 Feely falling Amed wih he expession fo he conseved enegy and angula momenum we will now sa o look a moion aound he black hole. Fis, we will leave an objec a es a a lage disance fom he cenal mass. We leave he objec wih velociy zeo v 0 a a disance so lage ha we can le. The enegy pe mass of he paicle is hen only he es enegy of he paicle, E m. E m M d dτ In poblem 3 you will use his fac o show ha he velociy of he objec as i falls owads he cenal mass as seen by he fa away obseve is given by v M M. 9 A lage disance he velociy goes o zeo as expeced. Wha happens when he objec appoaches he black hole? Fo lage disances he faco M/ is dominaing. This faco inceases wih deceasing, so he velociy inceases jus as expeced. When we appoach Figue 7: Schemaic plo of he vaiaion of velociy as a funcion of adial disance fom he cene fo an objec falling in fom a huge disance. Wha do he shell obseves living a shells close o he hoizon see? In poblem 3 you show ha he velociy of he falling objec as obseved by he shell obseve a disance a he momen when he objec passes he shell is given by M v shell, 0

11 We see ha shell obseves close and close o he hoizon will always obseve a lage and lage local velociy. The shell obseves on he shell jus above he hoizon M sees ha v shell, ha he velociy of he objec appoaches he speed of ligh as he spaceship appoaches he hoizon. We have seen a huge diffeence in esuls: The fa-away obseve sees ha he objec falls o es a he hoizon, he local obseve close o he hoizon sees he objec appoaching he speed of ligh. Aleady fom special elaiviy we ae used o he fac ha obseves in diffeen fames measue diffeen numbes, bu his is a eally exeme example. Wha do he feely falling obseves in he spaceship see? Fo he feely falling obseves nohing paicula a all happens when hey pass he hoizon. The feely falling obseves ae always moving fom one local ineial fame o he ohe, bu nohing special happens a M. Wha velociy do local obseves measue beyond he hoizon? Do hey measue a velociy lage han he velociy of ligh? In a coming lecue we will look a lile bi moe a moion beyond he hoizon, bu hee we will look biefly a he Schwazschild line elemen o see if we ge some hins. τ M M Exacly a he hoizon, he line elemen is singula. This is no a physical singulaiy, bu wha we call a coodinae singulaiy. By changing coodinae sysem, his singulaiy will go away and one can calculae s a he hoizon wihou poblems. One may undesand his easie by looking a he analogy wih he sphee: If a funcion on he sphee conains he expession /θ whee θ is he pola angel being zeo a he noh pole i will become singula on he noh pole. By changing he coodinae sysem by defining he noh pole a some ohe poin on he sphee, he poin of he peviuos noh pole will no be singula. In his case he funcion in iself is no singula on he poin of he pevious noh pole, i is he coodinae sysem which makes he expession singula a his poin. We will now ake a look a his line elemen when < M. In his case we can wie i as τ M M Looking a he sign, he space and ime coodinaes inechange hei oles. This does no diecly mean ha space and ime inechange hei oles, bu space does aain one feaue which we nomally associae wih ime: An ineviable fowad moion. In he same way as we always move fowad in ime, an obseve inside he hoizon will always move fowads owads he cene. No mae how song engines you have, you canno sop his moion: you canno be a es inside he hoizon, always moving fowads owads desucion a he cene exacly as we always move fowad in ime. A consequence of his is ha no shell obseves can exis inside he hoizon. You canno consuc a shell a es, eveyhing will always be moving. Inside he hoizon we canno measue a local shell velociy, so even if he shell velociy appoaches he speed of ligh a he hoizon i does no necessaily mean ha we will have a local velociy lage han speed of ligh inside he hoizon. Moe abou his lae. 7 An example: GPS, Global Posiioning Sysem Figue 8: The GPS sysem. We have seen ha geneal elaiviy becomes impoan fo lage masses and fo disances close o he Schwazschild adius M. The quesion now is when we need o ake ino accoun geneal elaivisic effecs. Clealy his depends on he accuacy equied fo a given calculaion. We will now see one example whee geneal elaiviy is impoan in eveyday life. The Global Posiioning Sysem GPS is used by a lage numbe of people, fom hikes in he mounain ying o find hei posiion on he map o aiplanes navigaing wih GPS in ode o land even in dense fog. GPS is based on 4 saellies obiing he Eah wih a peiod of hous a an aliude of abou kilomees. Each saellie sends a seam of signals, each signal conaining infomaion abou hei posiion x sa of he saellie a he ime sa when he signal was sen. You GPS eceive eceives signals fom hee saellies acually fom fou in ode o incease he pecision of he inenal clock in you GPS eceive, bu if you GPS eceive has an exemely accuae clock, only hee saellies ae sicly necessay: We will fo simpliciy use hee saellies and assume ha you GPS eceive conains an aomic clock in his illusaion. The siuaion is illusaed in figue 8. You GPS eceive conains a vey accuae clock showing he ime when you eceive he signal. This gives you GPS eceive hee equaions wih he hee coodinaes of you posiion x as he hee unknowns, x x sa c sa c, x x sa c sa c, x x sa3 c sa3 c 3.

12 The GPS eceive eceives he ime sa when a signal was emied fom he saellie. Knowing ha he signal avels wih ligh speed c and eading off he ime of ecepion of he signal on he inenal clock of he GPS eceive, he eceive can calculae he disance c ha he signal has aveled. This disance is equal o he diffeence beween you posiion x and he posiion x sa of he saellie when he signal was emied. Solving he hee equaions above, he GPS eceive solves fo you posiion x x, y, z nomally expessed in ems of longiude, laiude and aliude. Noe ha if a fouh equaion wee added using a signal fom a fouh saellie, anohe unknown could be allowed: This is how he pecision of you GPS clock is inceased: you ime is solved fom he fou se of equaions. Hee we will assume ha you GPS eceive has an aomic clock If we assume a simplified one dimensional case, i.e. ha you only have a one dimensional posiion x, he soluion would be x c ± x sa. We see ha he pecision of you calculaed posiion x depends on he pecision wih which we can calculae he ime diffeence sa. The signals move wih velociy of kilomees pe second. If hee is an inaccuacy of he ode µs 0 6 s, one micosecond, he inaccuacy in he calculaed posiion would be of he ode m/s 0 6 s 300 m. An inaccuacy of one micosecond coesponds o an inaccuacy of 300 mees in he posiion calculaed by GPS. In such a case GPS would be useless fo many of is applicaions and moe seiously, he aiplane missing he amac wih 300 mees would cash! We know ha due o special elaiviy, he clocks in he saellie and he clocks on Eah in you GPS eceive un a diffeen paces because of he elaive moions of he saellies wih espec o you. We also know fom geneal elaiviy ha he clocks in he saellie un a a diffeen pace han you clock because of diffeence in disance fom he cene of aacion cene of Eah. If he clocks in he saellies and he clocks in he GPS eceives wee synchonized a he momen when he saellies wee launched ino obi, he quesion is how long does i ake unil he elaivisic effecs make he Eah and saellie clocks showing so diffeen imes ha GPS has become useless. Relaivisic effecs ae usually small so one could expec ha i would ake maybe housands of yeas. If his wee he case, we wouldn need o woy. Bu emembe ha we equie a pecision bee han µs hee. This could make elaivisic effecs impoan. Le s check. We sa by he gaviaional effec. We conside wo shells, shell is he suface of he Eah siuaed a adial disance 6000 km appoximaely, we ae only looking fo odes of magniude hee, no exac numbes. Shell is he obi of he saellies a adial disance km. A ime ineval on he suface of he Eah is elaed o a ime ineval of he fa-away obseve by see equaion 3 M. Similaly, a ime ineval measued on he saellie clock is elaed o he fa-away ime by M. Dividing hese wo equaions on each ohe we find ha M M. This is he diffeence in clock pace beween he saellie and Eah clocks aking ino accoun only gaviaional effecs. We will fis check he ode of magniude of hese ems. Wha is he mass of he Eah measued in mees? We have M Eah kg m m. in case you have fogoen: go back and check how o go fom kg o mees. So he em M/ is of he ode 0 8 fo Eah, vey much smalle han. Thus we can use Taylo expansions, x x +... / x + x +... giving fo x M/ M + M, whee we have skipped ems of second ode in small quaniies wo x muliplied wih each ohe as hese ae much smalle han he ems of fis ode in x. We see ha < as expeced: Obseves fa away fom he cenal mass see ha clocks close o he cenal mass un slowe. Obseves fa away fom Eah will obseve ha i akes longe han one second on hei wiswach fo he clocks on Eah o move one second fowad. Inseing numbes fo and we obain We see ha afe abou one day s, he saellie clocks ae 60 micoseconds ahead of he Eah clocks. This coesponds o unceainies in posiion measuemens of he ode 0 kilomees. Thus, one day afe launching he saellies, GPS would be useless unless elaivisic effecs ae aken ino accoun! In ode o be sue abou his, we need o also look a special elaivisic effecs. Seen fom Eah, saellie clocks which send ime signals ead fom hei own clocks o

13 Eah go slowe since hey ae moving wih espec o he obseves on he suface of he Eah. We have SR γ SR, whee SR sands fo special elaiviy. In his case SR SR > opposie of he geneal elaivisic effec. We need o check whehe his effec migh be jus lage enough o cancel he geneal elaivisic effec. Fom Keple s 3d law fo he saellie we have check ha you can acually deive his, π 4π 3 v φ GM Eah,kg using convenional unis we find ha he obial speed of he saellie is v φ dimensionless velociy. In addiion an obseve a he suface of he Eah moves wih velociy due o Eah s oaion Fo low velociies and small M/ his expession educes o he appoximae expessions above. Noe ha we have no been vey caeful when measuing he angenial velociies: We did no specify angenial velociy wih espec o which ime, Eah ime, Saellie ime o fa-away ime. In uns ou ha aking ino accoun hese diffeences gives coecions o he coecion which ae so small ha hey can be ignoed. We also did no specify whehe he adial disances we used fo Eah and he saellie wee in Schwazschild coodinaes o in shell disances shell. Also hese diffeences ae so small ha hey can be ignoed in his case. v π 4 h 0.5 km/s o v φ in dimensionless unis. The velociy of he saellie elaive o he obseve on he gound is hus appoximaely v φ v φ + v φ giving γ In one day we find ha he saellie clocks un abou 0 micoseconds slow acually aboou 9, check ha you agee, by fa no enough o cancel he geneal elaivisic effec. Boh effecs need o be aken ino accoun in ode o make GPS of any use a all, and in ode o no make you aiplane cash when landing in fog. We have so fa used appoximae geneal and special elaivisic expessions sepaaely. Using he Schwazschild line elemen we may ake boh effecs ino accoun simulaneously and obain a moe accuae expession. Wiing fis he line elemen beween wo clock icks fo he obseve on he suface of he Eah, we have τ M φ, whee 0 since he obseve says a he same adial disance. We can expess his as M v φ, whee v φ is he angenial velociy of he Eah obseve, v φ dφ/d did you ge his ansiion?. Using he same agumens, we ge he same expession fo he saellie M v φ, whee v φ is he angenial velociy of he saellie. Dividing hese wo expessions on each ohe, we have M vφ M. vφ 3

14 8 Poblems Poblem 3 hous Imagine a shell obseve a shell is poining a lase pen adially ouwads fom he cenal mass. The beam has wavelengh λ. Hee we will y o find he wavelengh λ obseved by he fa-away obseve.. The fequency of he ligh emied by he lase pen is ν /. The fequency of he ligh eceived by he fa-away obseve is ν /. Hee and is he ime ineval beween wo peaks of elecomagneic waves. Show ha he diffeence in ime ineval measued by he wo obseves is given by. M Hin: Imagine ha a clock siuaed a shell icks each ime a peak of he elecomagneic wave passes.. Use his fac o show ha he gaviaional Dopple fomula, i.e. he fomula which gives you he wavelengh obseved by he fa-away obseve fo ligh emied close o he cenal mass, is given by λ λ λ λ λ M 3. Show ha fo disances M his can be wien as λ λ M Hin: Taylo expansion. 4. We will now sudy wha wavelengh of ligh ha an obseve fa away fom he Sun will obseve fo he ligh wih he wavelengh λ max 500 nm emied fom he sola suface. a Find he mass of he Sun in mees. b Find he aio M/ fo he suface of he Sun. c Find he edshif λ/λ measued by a fa-away obseve. d Is he appaen colo of he Sun changed due o he gaviaional edshif? e Fo ligh coming fom fa away and eneing he gaviaional field of he Eah, an opposie effec is aking place. The ligh is blue shifed. Find he aio M/ fo he suface of he Eah. f Find he gaviaional blue shif λ/λ fo ligh aiving a Eah noe: now he ligh is coming ino he gaviaional field insead of avelling ou ou i. Does his change he appaen colo of he Sun? 5. A quasa is one of he mos poweful souces of enegy in he univese. The quasas ae hough o be poweed by a so-called acceion disc: Ho gas cicling and falling ino a black hole. The gas eaches velociies close o he speed of ligh as i appoaches he hoizon, bu since we only see he sum of he adiaion coming fom all sides of he black hole, we expec he Dopple effec due o he velociy of he gas o cancel ou. Assume ha we find evidence fo an emission line a λ 50 nm in he adiaion fom a quasa. Assume also ha we ecognize his emission line o be a line which in he laboaoy is measued o occu a λ 600 nm. Give some agumens explaining why his obsevaion suppos he hypohesis of quasas having a black hole in he cene and find fom which disance expessed in ems of he black hole mass M fom he cene, he adiaion is emeging. Assume ha he Dopple effec due o he quasa s movemen wih espec o us has been subaced. 6. Imagine you ae a shell obseve living a a shell a.0m vey close o he hoizon of a black hole of mass M. Can you use opical elescopes o obseve he sas aound you? Wha kind of elescope do you need? Poblem 30 min. hou In his execise we will use he pinciple of maximal aging o deduce he law of consevaion of angula momenum in geneal elaiviy. In he ex you have seen hee examples of his kind of deivaion and hee we will follow exacly he same pocedue. Befoe embaking on his execise, please ead he examples in he ex caefully. Figue 9: A skech of poblem.. Use figue 9 in his execise: We will sudy he moion of an objec which passes hough hee poins, φ,, φ and 3, φ 3 a imes, and 3. We fix,, 3,, and 3 as well as φ and φ 3. The fee paamee hee is φ. We assume ha beween, φ and, φ he adius is A we assume he disance beween hese wo poins o be so small ha is consan and beween, φ and 3, φ 3 we have B see again figue 9. Use he Schwazschild line elemen o show ha he pope 4