AST2000 Lecture Notes

Transcription

1 AST000 Lectue Notes Pat C Geneal Relativity: Basic pinciples Questions to ponde befoe the lectue 1. What is a black hole? how would you define it?). If you, situated in a safe place fa away fom the black hole, see somebody falling into a black hole, what will you see? 3. If you instead ae situated close to the black hole, will things look diffeent? 4. In Geneal Relativity the idea of a foce of gavity is eplaced by the cuvatue of spacetime. Based on what you leaned in the lectues on special elativity, can you imagine what cuved spacetime might mean? 5. A fee-float fame in Geneal Relativity is defined as a place whee an object which is left with no velocity will continue with zeo velocity. We know that this will happen in an intetial fame which is not affected by extenal foces. But can you imagine a situation whee this is also the case in an acceleated fame? 1

2 AST000 Lectue Notes Pat C Geneal Relativity: Basic pinciples 1 Schwazschild geomety The geneal theoy of elativity may be summaized in one equation, the Einstein equation G µν = 8πT µν, whee G µν is the Einstein tenso and T µν is the stess-enegy tenso A tenso is a matix with paticula popeties in the same way as a 4- vecto is a vecto with specific popeties). This equation is not a pat of this couse as tenso mathematics and linea algeba, not equied fo taking this couse, ae needed to undestand it. I pesent it hee anyway as it illustates the basic pinciple of geneal elativity: The stess-enegy tenso on the ight hand side contains the enegy content of spacetime, the Einstein tenso on the left hand side specifies the geomety of spacetime. Thus, geneal elativity says that the enegy content in spacetime specifies its geomety. What do we mean by geomety of spacetime? We have aleady seen two examples of such geometies, Euclidean geomety and Loentz geomety. We have also seen that the geomety is specified by the spacetime inteval also called line element) s which tells us how distances ae measued. Thus, by inseting the enegy content as a function of spacetime coodinates on the ight side, the left side gives us an expession fo s, i.e. how to measue distances in spacetime in the pesence of mass/enegy. Thus, in the pesence of a mass, fo instance like the Eath, the geomety of spacetime is no longe Loentz geomety and the laws of special elativity ae no longe valid. This should be obvious: Special elativity tells us that a paticle should follow a staight line in spacetime, i.e. a path with constant velocity. This is clealy not the case on Eath, objects do not keep a constant velocity, they acceleate with the gavitational acceleation. You might object hee: Special elativity says that a paticle continues with constant velocity if it is not influenced by extenal foces, but hee the foce of gavity is at play. The answe to this objection is given by a vey impotant concept of geneal elativity: gavity is not a foce. What we expeience as the foce of gavity is simply a esult of the spacetime geomety in the vicinity of masses. The pinciple of maximal aging go back and epeat it now!) tells us that a paticle which is not influenced by extenal foces follows the longest path in spacetime, i.e. the path which gives the lagest possible pope time. Aound a massive object, spacetime as alteed such that the path which gives the longest possible path length s, is the path which leads to the cente of the massive object. This is why things on Eath fall towads the gound and do not continue in a staight line with constant velocity as it would in a spacetime with Loentz geomety. Few yeas afte Einstein published the geneal theoy of elativity, Kal Schwazschild found a geneal solution to the Einstein equation fo the geomety aound an isolated spheically symmetic body. This is one of the vey few analytic solutions to the Einstein equation that exists. Thus, the Schwazschild solution is valid aound a lonely sta, planet o a black hole. The spacetime geomety esulting fom this solution is called Schwazschild geomety and is descibed by the

3 line element: The Schwazschild line element s = 1 ) t ) φ. 1 1) Thee ae two things to note in this equation. We ae using pola coodinates, φ) instead of Catesian coodinates x, y). These coodinates ae in the plane defined by the two events and the cental mass. Thus, the coodinate is a distance fom the cente, we will late come back to how we measue this distance. The φ coodinate is the nomal φ angle used in pola coodinates. The line element fo Loentz geomety in pola coodinates can similaly be witten as Loentz line element in pola coodinates s = t x y = t φ. The second thing to note in the equation fo the Schwazschild line element is the tem 1 /. Hee M/ must be dimensionless since it is added to a numbe. But we know that mass is measued in kilogams and distances in metes, so how can this tem be dimensionless? Actually, thee should have been a G/c hee, G = m 3 /kg/s being the gavitational constant and c = m/s being the speed of light. We have that G c = m/kg. ) Since we ae dealing with gavity and M/ has units kg/m, G/c must be the constant which is missing hee. We ae now used to measue time intevals in units of metes. If we now decide to also measue mass in units of metes, equation ) gives us a natual convesion facto. M m M kg = G c, whee M m is mass measued in metes and M kg is mass measued in kg. Thus we have that 1 kg = m. The equation gives us a convesion fomula fom kg to m. We see that measuing mass in metes equals setting G/c = 1 in evey fomula o if you wish, G = 1 and c = 1). This is equal to what happened when we decided to measue time in metes, we could set c = 1 eveywhee. The eason fo measuing mass in metes is pue laziness, it means that we don t need to wite this facto all the time when doing calculations. Thus instead of witing 1 kg G/c ) we wite 1 / whee M is now mass measued in metes. All the physics is captued in the last expession, we have just got id of a constant. Fom now on, all masses will be measued in units of metes and when we have the final answe we convet to nomal units by multiplying o dividing by the necessay factos of G/c and c in ode to obtain the units that we wish. The unit system whee c = 1, G = 1 and eventually, when you do quantum physics, also Planck s constant h = 1, is called natual units and is widely used in seveal banches of physics. The inetial fame In the lectues on special elativity we defined inetial fames, o fee-float fames, to be fames which ae not acceleated, fames moving with constant velocity on which no extenal foces ae acting. We can give a moe geneal definition in the following way: To test if the oom whee you ae sitting at the moment is an inetial fame, take an object and thow it with velocity v. If the object continues in a staight line with velocity v, you ae in an inetial fame. Clealy, a fame a oom) which is not acceleated and on which no foces ae acting is an inetial fame accoding to this definition. But ae thee othe examples? In geneal elativity we use the notion of a local inetial fame, i.e. limited egions 3

4 Fact sheet: An example of a two-dimensional analogy of the waping of space and time by massive objects, often used in intoductoy texts on geneal elativity. Geneal elativity was poposed by Einstein in 1916 and povides a unified desciption of gavity as a geometic popety of space and time, o spacetime. The cuvatue of spacetime is diectly elated to the enegy and momentum of whateve matte and adiation ae pesent. Some pedictions of geneal elativity diffe significantly fom those of classical physics, especially concening the passage of time, the geomety of space, the motion of bodies in fee fall, and the popagation of light. Examples of such diffeences include gavitational time dilation, gavitational lensing, the gavitational edshift of light, and the gavitational time delay. Figue: WGBH Boston) of spacetime which ae inetial fames. An example of such a local inetial fame is a space caft in obit aound the Eath. Anothe example is an elevato fo which all cables have boken so that it is feely falling. All feely falling fames can be local inetial fames. How do we know that? If an astonaut in the obiting space caft takes an object and leaves it with zeo velocity, it stays with zeo velocity. This is why the astonauts expeience weightlessness. If a peson in a feely falling elevato takes an object and leaves it at est, it stays at est. Also the peson in the elevato expeiences weightlessness. Thus, they ae both, within cetain limits, in an inetial fame even though they ae both acceleated. Note that an obseve standing on the suface of Eath is in a local inetial fame fo a vey shot peiod of time: If an obseve on Eath leaves an object at est, it will stat falling, it will not stay at est: An obseve at the suface of Eath is not in a local inetial fame unless the time inteval consideed is so shot that the effect of the gavitational acceleation is not measuable. The only thing that keeps the obseve on the suface of the Eath fom being in a local inetial fame is the gound which exets an upwad foce on the obseve. If suddenly a hole in the gound opens below him and he stats feely falling, he suddenly finds himself in a local inetial fame with less stict time limits. Figue 1: Two boxes in fee fall: If they ae lage enough in eithe diection, the objects at est inside the boxes will stat moving. A local inetial fame needs to be small enough in space and time such that this motion cannot be measued. Local means that the inetial fame is limited in space and time, but we need to define these limits. In figue 1 we see two falling boxes, box A falling in the hoizontal position, box B falling in the vetical position. Since the gavitational acceleation is diected towads the cente of the Eath, two objects at est at eithe side of box A will stat moving towads the cente of the box due to the diection of the acceleation. The shote we make the box, the smalle this motion is. If we make the box so shot that we cannot measue the hoizontal motion of the objects, we say that the box is a local inetial fame. The same agument goes fo time: If we wait long enough, we will eventually obseve that the two objects have moved. The inetial fame is limited in time by the time it takes until the motion of the two objects can be measued. Similaly fo box B: The object which is close to the Eath will expeience a stonge gavitational foce than the object in the othe end of the box. Thus, if the box is long enough, an obseve in the box will obseve the two objects to move away fom each othe. This 4

5 is just the nomal tidal foces: The gavitational attaction of the moon makes the oceans on eithe side of the Eath move away fom each othe: we get high tides. But if the box is small enough, the diffeence in the gavitational acceleation is so small that the motion of the objects cannot be measued. Again, it is a question of time befoe a motion will be measued: The local inetial fame is limited in time. We have thus seen that a local inetial fame can be found if we define the fame so small in space and time that the gavitational acceleation within the fame is constant. In these fames, within the limited spatial extent and limited duation in time, an object which is left at est will emain at est in that fame. The stonge the gavitational field and the lage the vaiations in the gavitational field, the smalle in space and time we need to define ou local inetial fame. We have leaned fom special elativity that an inetial fame has Loentz geomety. Within the local inetial fame, spacetime intevals ae measued accoding to s = t x and the laws of special elativity ae all valid within the limits of this fame. In geneal elativity, we can view spacetime aound a massive object as being an infinite set of local inetial fames. When pefoming expeiments within these limited fames, special elativity is all we need. When studying events taking place so fa apat in space and time that they do not fit into one such local inetial fame, geneal elativity is needed. This is why only special elativity is needed fo paticle physicists making expeiments in paticle acceleatos. The paticle collisions take place in such a shot time that the gavitational acceleation may be neglected: They take place in a local inetial fame. We will now call spacetime whee Loentz geomety is valid flat spacetime. This is because Loentz geomety is simila to Euclidean geomety on a flat suface except fo a minus sign). We know that a cuved suface, like the suface of the sphee, has spheical geomety not Euclidean geomety. In the same way, Schwazschild geomety epesents cuved spacetime, the ules of Loentz geomety ae not valid and Schwazschild geomety needs to be used. We say that the pesence of matte cuves spacetime. Fa away fom all massive bodies, spacetime is flat and special elativity is valid. Figue : A visualization of how spheical geomety can look like Euclidean geomety at sufficiently small scales. We can take the analogy even futhe: Since the suface of a sphee has spheical and not Euclidean geomety, the ules of Euclidean geomety may not be used. But if we focus on a vey small pat of the suface of a sphee, the suface looks almost flat and Euclidean geomety is a vey good appoximation. The suface of the Eath is cuved and theefoe has spheical geomety, but since a gaden is vey small compaed to the full suface of the Eath, the suface of the Eath appeas to us to have flat geomety within the gaden. We use Euclidean geomety when measuing the aea of the gaden. The same is the case fo the cuved space: Since spacetime is cuved aound a massive object, we need to use Schwazschild geomety. But if we only study events which ae within a small aea in spacetime, spacetime looks flat and Loentz geomety the local inetial fame) is a good appoximation. 3 Thee obseves In the lectues on geneal elativity we will use thee obseves, the fa-away obseve, the shell obseve and the feely falling obseve. We will also assume that the cental massive body is a black hole. A black hole is the simplest possible macoscopic object in univese: it can be descibed by thee paametes, mass, angula momentum and chage. Any black holes which have the same values fo these thee paametes ae identical in the same way as two electons ae identical. A black hole is a egion in space whee the gavitational acceleation is so high that not 5

6 even light can escape fom it. A black hole can aise fo instance when a massive sta is dying: A sta is balanced by two foces, the foces of gavity which we no longe call foces) tying to pull the sta togethe and the gas/adiation pessue tying to make the sta expand. When all fuel in the stella coe is exhausted, the foces of pessue ae not stong enough to withstand the foces of gavity and the sta collapses. No foces can stop the sta fom shinking to an infinitely small point. The gavitational acceleation just outside this point is so lage that even light that ties to escape will fall back. The escape velocity is lage than the speed of light. This is a black hole. Note that the Schwazschild line element becomes singula at =. This adius is called the Schwazschild adius o the hoizon. This is the point of no etun, any object o light) which comes inside this hoizon cannot get out. At any point befoe the hoizon a spaceship with stong engines could still escape. But afte it has enteed, no infomation can be tansfeed out of the hoizon. The fa-away obseve is situated in a egion fa fom the cental black hole whee spacetime is flat. He does not obseve any events diectly, but gets infomation about time and position of events fom clocks situated eveywhee aound the black hole. The shell obseves live on the suface of shells constucted aound the black hole. Also a spaceship which uses its engines to stay at est at a fixed adius could seve as a shell obseve. The shell obseves expeience the gavitational attaction. When they leave an object at est it falls to the suface of the shell. Thee is one moe obseve which we aleady discussed in the pevious section. This is the feely falling obseve. The feely falling obseve caies with him a wistwatch and egistes the position and pesonal wistwatch time of events. The feely falling obseve is living in a local inetial fame with flat spacetime. Thus fo events taking place within shot time intevals and shot distances in space, the feely falling obseve uses Loentz geomety to make calculations. 4 The time and position coodinates fo the thee obseves Each of the obseves have thei own set of measues of time and space. The fa away obseve uses Schwazschild coodinates, t) and shell obseves use shell coodinates shell, t shell ). Fo the feely falling obseves, we will be viewing all events fom the oigin in his fame of efeence and we will theefoe not need a position coodinate since it will always be zeo) using his wistwatch time which will then always be the pope time τ. We will now discuss these diffeent coodinate systems and how they ae defined. If you look back at the Schwaschild line element equation 1) you should see that it only contains the measuements of the fa-away obseve. In othe wods, the Schwaschild line element contains no subscipts such as t shell ), and is theefoe only valid with mesuements made by the fa-away obseve. When the shell obseve wants to measue his position fom the cente of the black hole, he uns into poblems: When he ties to lowe a mete stick down to the cente of the hole to measue, the stick just disappeas behind the hoizon. He needs to find othe means to measue his adial position. In Euclidean geomety, we know that the cicumfeence of a cicle is just π. So the shell obseve measues the cicumfeence of the shell and divides by π to obtain his coodinate. In a non-euclidean geomety, the adius measued this way does not coespond to the adius measued inwads. We define the Schwazschild coodinate in this way: = cicumfeence π The in the expession fo the Schwazschild line element is the Schwazschild coodinate. Now the shell obseves at shell lowe a stick to the shell obseves at shell. The length of the stick is shell. They compae this to the diffeence in Schwazschild coodinate and find that shell =. This is what we anticipated: in Euclidean geomety these need to be equal, in Schwazschild geomety they ae not. We have obtained a second way to measue the adial distances between shells using shell distances 6

7 shell note that since the absolute shell coodinate shell cannot be measued, this is a meaningless quantity: only elative shell coodinate diffeences shell between shells can be measued did you undestand why?)). We have obtained two diffeent measues of adial distances, the Schwazschild coodinate defined by the cicumfeence of the shell. The fa-away obseve uses Schwazschild coodinates to measue distances. the shell distances shell found by physical measuements between shells.this is the distance which the shell obseves can measue diectly with mete sticks and is theefoe the most natual measue fo these obseves. What about time coodinates? two meaues of time, Again we have The fa-away obseve uses fa-away time t to measue time. This is the time t enteing in the Schwazschild line element. Fa away time fo an event is measued on a clock which has been synchonized with the clock of the fa-away obseve and which is located at the same location as the event we will late descibe how events can be timed which such clocks in pactice). The shell obseve uses local shell time t shell : it is simply the wistwatch time of the shell obseve, the time measued on a clock at est at the specified shell. Note that shell obseves at diffeent shells may measue diffeent times intevals t shell and distances shell between two events depending on which shell they live on. Shell coodinates ae local coodinates. In ode to elate time and space coodinates in the diffeent fames we will now as we did in special elativity) use the invaiance of the space time inteval o line element) s. Fist we will find a elation between the moe abstact fa away-time t and the locally measuable shell time t shell. The shell time is the wistwatch time, o pope time τ of the shell obseves. We will use two events A and B which ae two ticks on the clock of a shell obseve. The shell obseves ae at est at shell, so, AB = 0 and φ AB = 0. Inseting this into the Schwazschild line element equation 1) using that s AB = τ AB = t shell the time peiod between A and B measued on shell clocks is by definition the same as the pope time peiod between A and B which we have leaned is always equal to the invaiant fou dimensional line element between these events) we get Shell time t shell = 1 ) t. 3) Ae you sue you see how this expession comes about?) Fo shell obseves outside the hoizon > ), the local time goes slowe by a fac- 1 ) to with espect to the fa-away time. We also see that the smalle the distance fom the cente, the slowe the shell clock with espect to the fa-away time. Thus, the futhe down we live in a gavitational field, the slowe the clocks un. This has consequences fo people living on Eath: Ou clocks tick slowe than the clocks in satellites in obit aound Eath. At the end of this lectue, we will look close at this fact. Figue 3: The shell obseve at shell measue the pope lenght of a stick by two simultaneous events A and B on eithe side of the stick. We have now found a elation between time intevals measued on shell clocks and time intevals measued on clocks synchonized with fa-away clocks. How is the elation between distances measued with Schwazschild coodinates and distances measued diectly by shell obseves? We can measue the length of a stick as the spatial distance between two events taking place at the same time at eithe end of the stick see figue 3). Fo events taking place within shot time intevals 7

8 and shot spatial extensions, the shell obseve sees flat spacetime and can theefoe use Loentz geomety: s shell = t shell shell we will look at a stick which is aligned with the adial diection, the events theefoe take place at the same φ coodinate so φ = 0). The fa-away obseve always needs to use the Schwazschild line element equation 1) instead of the Loentz line element. Using invaiance of the line element we have fo two events A and B s shell AB) = sab)) taking place simultaneously 1 on both sides of the stick t shell shell = 1 ) t ), 1 whee we have set φ = 0. Check that you undestand how to aive at this expession. Now, we measue the length of a stick in the adial diection by measuing the distance between the two simultaneous events A and B taking place at both ends of the stick at spatial distance. Since t shell = t = 0, we get fa-away-time fo the event ecoded on the clock positioned at the same location whee the event took place. In the following we will descibe events eithe as they ae seen by the fa-away obseve using global Schwazschild coodinates, t), by the shell obseve using local coodinates shell, t shell ) o the feely falling obseve also using local coodinates. Befoe poceeding, make a dawing of all these obseves, thei coodinates and the elation between these diffeent coodinates. 5 The pinciple of maximum aging evisited shell = 1 ). 4) fo shot distances close to the shell. Thus, adial distances measued by the shell obseves, loweing mete sticks fom one shell to the othe is always lage than the adial distances found by taking the diffeence between the Schwazschild coodinate distances. What about a stick which is pependicula to the adial diection? In this case, the obseves will agee on the length of this stick, check that you can deduce this in the same manne as we deduced the elation fo the adial stick. Anothe pactical question: How does the faaway obseve know the time and position of events. Each time an event happens close to one of the fa-away-clocks close to the black hole, it sends a signal to the fa-away obseve telling the time and position this clock egisteed fo the event. In this way, the fa-away obseve does not need to take into account the time it takes fo the signal fom the clock to aive, the signal itself contains infomation with the coect In the lectues on special elativity we leaned that the pinciple of maximum aging makes objects in fee float to move along paths in spacetime which give the longest possible wistwatch time τ which coesponds to the longest possible spacetime inteval s. We also used that fo Loentz geomety, the longest in tems of s o τ) path between two points is the staight line, i.e. the path with constant velocity. We neve poved the latte esult popely. We will do this now, fist fo Loentz geomety and then we will use the same appoach to find the esult fo Schwazschild geomety. 5.1 Retuning fo a moment to special elativity: deducing Netwon s fist law We will now show that the pinciple of maximum aging leads to Newton s fist law when using Loentz geomety. 1 Since events which ae simultaneous fo shell obseves at a given shell also ae simultaneous fo the fa-away obseve equation 3), t shell = t = 0. 8

9 Fact sheet: Thee is a pactical poblem in all this: We said that the fa-away time was measued by clocks located at the position of events which can take place close to the cental black hole) but which ae synchonized with the fa-away clocks. How can we synchonize clocks which ae located deep in the gavitational field and which theefoe un slowe than the fa-away clock? Let s imagine the clocks measuing fa-away-time to be positioned at diffeent shells aound the black hole. The 1 ) shell obseves design the clocks such that they un faste by a facto. To synchonize all these clocks, the fa-away obseve sends a light signal to all the othe clocks at the moment when he sets his clock to t = 0. The shell obseves know the distance fom the fa-away obseve to the fa-away-time clocks and thus know the time t it took fo the light signal to each thei clock. They had thus aleady set the clock to this time t and made a mechanism such that it stated to un at the moment when the light signal aived. In this way, all fa-away-time clocks situated at diffeent positions aound the black hole ae synchonized. In a simila fashion, we should be able to imagine how the fa-away obseve is able to measue the cicumfeence of a shell: since φ is uneffected by the cental mass, he/she can place ules in a pefect cicle aound the cental mass. This way, he/she will be able to measue the cicumfeence of the shell. the object is in a local inetial fame between x 1 and x and in a possibly diffeent) intetial fame between x and x 3. Theefoe, the time intevals t 1 and t 3 to tavel these two shot paths also need to be shot. The total wistwatch time τ it takes the paticle to move fom x 1 to x 3 is Figue 4: The motion of a paticle in fee float in Loentz geomety. Points x 1, x, x 3 as well as the times t 1 and t 3 ae fixed. Fo a paticle at fee float, at what time t will it pass x? Which of the possible spacetime paths in the figue does the paticle take? We use the pinciple of maximal aging to show that in Loentz geomety, the paticle follows the staight spacetime path. Look at figue 4. We see the woldline of a paticle going fom position x 1 at time t 1 to position x 3 at time t 3 passing though position x at time t. Say that the points x 1, x and x 3 ae fixed and known positions. We also say that the total time inteval t 13 it takes the object to go fom x 1 to x 3 is fixed and known. What we do not know is the time inteval t 1 it takes the paticle to go fom point x 1 to point x. Remembe that we do not know that the object will move with constant velocity, this is what we want to show. Thus we leave open the possibility that the paticle will have a diffeent speed between x 1 and x than between x and x 3. The time t can be at any possible point between t 1 and t 3. In figue 4 we show some possible spacetime paths that the object could take. We now assume that the distances x 1 and x 3 ae vey shot, so shot that the object can be assumed to move with constant velocity between these two points, i.e. that τ 13 = τ 1 + τ 3 = t 1 x 1 + t 3 x 3, 5) whee we have used that τ = s = t x fo Loentz geomety. Accoding to the pinciple of maximal aging, we need to find the path, i.e. the t, which maximizes the total wistwatch time τ 13. We do this by setting the deivative of τ 13 with espect to the fee paamete t equal to zeo, i.e. you look fo the maximum point of the function τ 13 t ): d dt τ 13 = t 1 t 1 x 1 + ) d t 1 dt t 3 t 3 x 3 d dt t 3 Since t 1 = t 1 t ) = t t 1 we have that d/dt ) t 1 = 1 emembe that t 1 is a fixed constant) and similaly fo t 3. Thus we have t 1 t 3 t 1 x 1 t 3 x 3 = 0 It can be shown that this gives the maximum of the pope time, not the minimum, we will not show this hee) Witten in tems of τ 1 and τ 3 we have t 1 τ 1 = t 3 τ 3. ) 9

10 Check that you undestood evey step in the deduction so fa! This is only fo thee points x 1, x and x 3 along the woldline of a paticle. If we continue to beak up the woldline in small local inetial fames at points x 4, x 5, etc. we can do the same analysis between any thee adjacent points along the cuve. The esult is that dt dτ = constant, whee we wite dt instead of t and dτ instead of τ. Remembe that we assumed these time intevals to be vey shot. In this final expession we have taken the limit in which these time intevals ae infinitesimally shot. We also emembe do you?) fom special elativity that dt dτ = 1 1 v = γ time dilation!). So the pinciple of maximal aging has given that γ = constant along a woldline. But γ only contains the velocity v of the object so it follows that v = constant. In Loentz geomety, a fee-float object will follow the spacetime path fo which the velocity is constant. We can wite this in a diffeent way. In special elativity we had that E = γm so we can wite γ = E/m fom which follows that γ = E m = constant. We have just deduced that enegy is conseved, o moe pecisely enegy pe mass E/m is conseved. In the lectues on special elativity we leaned that expeiments tell us that the elativistic enegy E = γm is conseved and not Newtonian enegy. Hee we found that the pinciple of maximal aging tells us that thee is a quantity which is conseved along the motion of a paticle. This quantity is the same as the quantity we call elativistic enegy. Figue 5: The motion of a paticle in fee float in Loentz geomety. Points x 1, x 3 as well as the times t 1, t and t 3 ae fixed. Fo a paticle in fee float, which position x will it pass at time t? Which of the possible spacetime paths in the figue does the paticle take? We use the pinciple of maximal aging to show that in Loentz geomety, the paticle follows the staight spacetime path. Is it possible that the pinciple of maximal aging can give us something moe? We will now epeat the above calculations, but now we fix t 1, t and t 3. All times ae fixed. We also fix x 1 and x 3, but leave x fee. The situation is shown in figue 5. Now the question is which point x will the object pass though?. We need to take the deivative of expession 5) with espect to x which is a fee paamete. ) dτ 13 x 1 d = x 3 dx t 1 x dx 1 ) x d x 3 = 0 t 1 x dx 1 Again x 1 = x 1 x ) = x x 1 so that d/dx ) x 1 = 1 and similaly fo x 3 ) and we have x 1 = x 3, τ 1 τ 3 we have found anothe constant of motion But we can wite this as We have that dx dτ = constant dx dτ = dx dt dt dτ = vγ. vγ = constant = p m. Go though this deduction in detail youself and make sue you undestand evey step). We e- 10

11 membe that p = mγv, so the pinciple of maximal aging has given us the law of momentum consevation, o actually the law of consevation of momentum pe mass p/m. We have seen that the pinciple of maximal aging seems to be moe fundamental than the pinciples of enegy and momentum consevation. It is sufficient to assume the pinciple of maximal aging. Fom that we can deduce the expessions fo enegy and momentum and also that these need to be conseved quantities. 5. Retuning to geneal elativity: deducing and genealizing Newton s law of gavitation Figue 6: The motion of a paticle in fee float in Schwazschild geomety. Points 1,, 3 as well as the times t 1 and t 3 ae fixed. Fo a paticle in fee float, at what time t will it pass though? We assume that the distances 1 and 3 ae so small that we can assume the adial distance to be = A always in the fome inteval and = B always in the latte inteval. t fee. We will find at which time t the paticle passes though point. Again we wite the total pope time fo the object fom 1 to 3 as using the Schwazschild line element, equation 1, fo τ) τ 13 = τ 1 + τ 3 = 1 ) t 1 1 ) A 1 A + 1 ) t 3 3 ), B 1 B whee A is the adius halfway between 1 and. We assume that 1 is so small that we can use the adius A fo the full inteval. In the same way, B is the adius halfway between and 3 which we count as valid fo the full inteval 3. Following the pocedue above, we will now maximize the total pope time τ 13 with espect to the fee paamete t. We get d dt τ 13 = 1 A ) t 1 τ 1 + ) d t 1 dt 1 B ) t 3 τ 3 ) d t 3 dt As above, t 1 = t t 1 giving d/dt ) t 1 = 1 and similaly fo t 3 ). Thus we have that ) ) 1 A t 1 1 B t 3 =. τ 1 τ 3 Figue 7: The motion of a paticle in fee float in Schwazschild geomety. Which spacetime path will the paticle take between points A and B? Now, what about geneal elativity? We will see how the pinciple of maximal aging tells a paticle to move in Schwazschild spacetime. Look at figue 6. A paticle tavels fom adius 1 at time t 1 to adius 3 at point t 3 passing though point at time t. We fix 1, and 3. We also fix the stat and end times t 1 and t 3. We leave We find that 1 ) dt dτ = constant, 6) whee again we have taken the limit whee t 1, t 3, τ 1 and τ 13 ae so small that they can be expessed as infinitesimally small peiods of time dt and dτ. In the case with Loentz geomety we used this constant of motion to find that the velocity had to be constant along the woldline of a feely floating paticle. Now we want to investigate how this constant of motion tells us how a feely floating paticle moves in Schwazschild spacetime. Fist we need to find an expession fo 11

12 dt/dτ. In special elativity we elated this to the velocity of the paticle using dt/dτ = γ, but this was deduced using the line element of Loentz geomety. Hee we want to elate this to the local velocity that a shell obseve at a given adius obseves. The locally measued shell velocity as an object passes by a given shell is given by v shell = d shell dt shell We now use equation 3 the equation connecting fa-away-time and shell time, emembe?) to wite the constant of motion equation 6) as 1 = = = ) ) 1 1/ dtshell dτ ) 1/ dt shell dτ ) 1/ γ shell ) 1/ 1 1 v shell = constant. In the last tansition we used the fact that the shell obseve lives in a local inetial fame fo a vey shot time. The shell obseve makes the velocity measuement so fast that the gavitational acceleation could not be noticed and he could use special elativity assuming flat spacetime. So using his local time t shell, the elation dt shell /dτ = γ shell fom special elativity is valid. We have thus found a constant of motion: 1 ) 1/ 1 = constant. 1 v shell Conside a paticle moving fom adius A to a highe adius B see figue 7). This time, the distance between points A and B does not need to be small. As the object moves past shell A, the shell obseves at this shell measue the local velocity v A. As the object moves past shell B, the shell obseves at this shell measue the local velocity v B. Equating this constant of motion at the two positions A and B we find 1 A ) 1/ 1 1 v A = 1 B ) 1/ 1 1 v B. 1 Squaing and eoganizing we find 1 vb) 1 ) = 1 v A) A 1 B ). We aleady see fom this equation that if B > A then v B < v A check!). Thus if the object is moving away fom the cental mass, the velocity is deceasing. If we have B < A we see that the opposite is tue: If the object is moving towads the cental mass, the velocity is inceasing. So the pinciple of maximum aging applied in Schwazschild geomety gives a vey diffeent esult than in Loentz geomety. In Loentz geomety we found that the velocity of a feely floating paticle is constant. In Schwazschild spacetime we find that the feely floating paticle acceleates towads the cental mass: If it moved outwad it slows down, if it moved inwads it acceleates. This is just what we nomally conside the foce of gavity. We see that hee we have not included any foces at all: We have just said that the cental mass cuves spacetime giving it Schwazschild geomety. By applying the pinciple of maximal aging, that an object moving though spacetime takes the path with longest possible wistwatch time τ, we found that the object needs to take a path in spacetime such that it acceleates towads the cental mass. We see how geomety of spacetime gives ise to the foce of gavity. But in geneal elativity we do not need to intoduce a foce, we just need one simple pinciple: The pinciple of maximal aging. We will now check if the acceleation we obtain fo lage adius that is, small gavitational effects) and low velocities v shell is equal to the Newtonian expession. We now call the constant of motion K giving 1 ) Reoganizing this we have v shell = 1 1 K 1 1 v shell = K. 1 We want to find the acceleation g shell = dv shell dt shell ) 7)

13 that a shell obseve measues. Taking the deivative of equation 7 Since K is a constant we have dk/dt shell = 0), we get check!) dv shell = 1 dt shell v shell K 1 ) d dt shell. Using equation 4 and that v shell = d shell /dt shell we obtain g shell 1 ) M Newton s law of gavitation is not valid close to the Schwazschild hoizon, so to take the Newtonian limit we need to conside this expessions fo. In this limit the expession educes to g shell M, exactly the Newtonian expession fo the gavitational acceleation GM ). We find that fa away fom the Schwazschild adius, geneal elativity educes to Newton s law of gavitation. We can now etun to figue 7 and look at the path maked Schwazschild path. This is the spacetime path between A and B that a feely floating object needs to take in ode to get the longest pope time τ. Looking at the slope of this path, we see that the object changes velocity duing its tip fom A to B. This is in shap contast to the esults fom special elativity with Loenzt geomety whee the path which gives longest possible pope time is the staight line with constant velocity. We will now etun to ou constant of motion 1 ) 1/ 1 = constant 8) 1 v shell In special elativity we found that a constant of motion which we obtain in the same manne was just the enegy pe mass. We will now go to the Newtonian limit to see if the same is the case fo Schwazschild spacetime. We will use two Taylo expansions, 1 x 1 1 x x x, both taken in the limit of x 1. In the Newtonian limit we have that / 1 and v 1. Applying this to equation 8) we have 1 M ) ) v 1+ 1 v M = constant In the last expession we used that since both / and v ae vey small, the poduct of these small quantities is even smalle than the emaining tems and could theefoe be omitted. Compae this to the Newtonian expession fo the enegy of a paticle in a gavitational field E = 1 mv Mm. We see again that the constant of motion was just enegy pe mass E/m whee the expession now tells us how the gavitational potential looks like have you noticed this: you have actually deived why the fom of the Newtonian gavitational potential is the way it is). Note the additional tem in the elativistic expession which is just the est enegy m. Again the pinciple of maximal aging has given us that enegy is conseved and it has given us the elativistic expession fo enegy in a gavitational field. We will theefoe edefine enegy accoding to ou findings such that enegy is indeed a conseved quantity also at high velocities and in lage gavitational fields. Definition of elativistic enegy in a gavitational field E m = 1 ) dt dτ = constant. We also found that this expession fo the enegy equals the Newtonian expession fo distances fa fom the Schwazschild adius. In the execises you will use the pinciple of maximum aging to find that angula momentum pe mass is conseved in Schwazschild spacetime. As with enegy, you will find that we need to define the angula momentum in Schwazschild spacetime in ode to get a conseved quantity: 13

14 Definition of angula momentum pe mass in Schwazschild spacetime L m = dφ dτ = γ shellv φ = constant. 6 Feely falling Amed with the expession fo the conseved enegy and angula momentum we will now stat to look at motion aound the black hole. Fist, we will leave an object at est at a lage distance fom the cental mass. We leave the object with velocity zeo v = 0 at a distance so lage that we can let. The enegy pe mass of the paticle is then only the est enegy of the paticle, E = m. E m = 1 ) dt dτ = 1 In poblem C.4 you will use this fact to show that the velocity of the object as it falls towads the cental mass as seen by the fa away obseve is given by v = 1 ). 9) At lage distance the velocity goes to zeo as expected. What happens when the object appoaches the black hole? Fo lage distances the facto / is dominating. This facto inceases with deceasing, so the velocity inceases just as expected. When we appoach the Schwazschild adius, the fist facto ) 1 stats dominating the behaviou of v as the last facto now goes to one. In this case, the velocity is deceasing when is deceasing. At the hoizon the velocity eaches exactly zeo. What we see is plotted in figue 8. When the object stats falling the velocity inceases until a point whee it stats deceasing. At the hoizon the object stops. This esult was obtained using Schwazschild coodinates. Thus, this is the esult that the fa-away obseve sees. This means that if we let a spaceship fall into a black hole, we, as faaway obseves, would see the spaceship stopping at the hoizon and it would stay thee fo eve. Remembe also that time is going slowe close to the hoizon, t shell = 1 ) t At the hoizon, we obseve that time stops. Thus, looking at the spaceship we would obseve the pesons in the spaceship to feeze at the hoizon. Eveything stops. In the execises you will show that light fom a cental mass is ed shifted. Thus we will also see a stongly edshifted light fom the spaceship. Using the expession fom the execises you will see that light aiving fom the hoizon is infinitely ed shifted. Thus you will not see any light fom the hoizon. You will only see the spaceship just befoe it eaches the hoizon and then only as adio waves with a lage wavelength see poblem C.1). Figue 8: Schematic plot of the vaiation of velocity as a function of adial distance fom the cente fo an object falling in fom a huge distance. What do the shell obseves living at shells close to the hoizon see? In poblem C.4 you show that the velocity of the falling object as obseved by the shell obseve at distance at the moment when the object passes the shell) is given by v shell =, We see that shell obseves close and close to the hoizon will always obseve a lage and lage local velocity. The shell obseves on the shell just above the hoizon = sees that v shell 1, 14

15 that the velocity of the object appoaches the speed of light as the spaceship appoaches the hoizon. We have seen a huge diffeence in esults: The fa-away obseve sees that the object falls to est at the hoizon, the local obseve close to the hoizon sees the object appoaching the speed of light. Aleady fom special elativity we ae used to the fact that obseves in diffeent fames measue diffeent numbes, but this is a eally exteme example. What do the feely falling obseves in the spaceship see? Fo the feely falling obseves nothing paticula at all happens when they pass the hoizon. The feely falling obseves ae always moving fom one local inetial fame to the othe, but nothing special happens at =. What velocity do local obseves measue beyond the hoizon? Do they measue a velocity lage than the velocity of light? In a coming lectue we will look a little bit moe at motion beyond the hoizon, but hee we will look biefly at the Schwazschild line element to see if we get some hints. τ = 1 ) t ) 1 Exactly at the hoizon, the line element is singula. This is not a physical singulaity, but what we call a coodinate singulaity. By changing coodinate system, this singulaity will go away and one can calculate s at the hoizon without poblems. One may undestand this easie by looking at the analogy with the sphee: If a function on the sphee contains the expession 1/θ whee θ is the pola angle being zeo at the noth pole) it will become singula on the noth pole. By changing the coodinate system by defining the noth pole at some othe point on the sphee, the point of the peviuos noth pole will not be singula. In this case the function in itself is not singula on the point of the pevious noth pole, it is the coodinate system which makes the expession singula at this point. We will now take a look at this line element when <. In this case we can wite it as τ = 1 1 t Looking at the sign, the space and time coodinates intechange thei oles. This does not diectly mean that space and time intechange thei oles, but space does attain one featue which we nomally associate with time: An inevitable fowad motion. In the same way as we always move fowad in time, an obseve inside the hoizon will always move fowads towads the cente. No matte how stong engines you have, you cannot stop this motion: you cannot be at est inside the hoizon, always moving fowads towads destuction at the cente exactly as we always move fowad in time. A consequence of this is that no shell obseves can exist inside the hoizon. You cannot constuct a shell at est, eveything will always be moving. Inside the hoizon we cannot measue a local shell velocity, so even if the shell velocity appoaches the speed of light at the hoizon it does not necessaily mean that we will have a local velocity lage than speed of light inside the hoizon. Moe about this late. 7 An example: GPS, Global Positioning System Figue 9: The GPS system. We have seen that geneal elativity becomes impotant fo lage masses and fo distances close to the Schwazschild adius. The question now is when we need to take into account geneal elativistic effects. Clealy this depends on the accuacy equied fo a given calculation. We will now see one example whee geneal elativity is impotant in eveyday life. The Global Positioning System GPS) is used by a lage numbe of people, fom hikes in the mountain tying to find thei position on the map to aiplanes navigating with GPS in ode to land even in dense fog. GPS is based on 4 satellites obiting the Eath with a peiod of 1 hous at an altitude of 15

16 about kilometes. Each satellite sends a steam of signals, each signal containing infomation about thei position x sat of the satellite at the time t sat when the signal was sent. You GPS eceive eceives signals fom thee satellites actually fom fou in ode to incease the pecision of the intenal clock in you GPS eceive, but if you GPS eceive has an extemely accuate clock, only thee satellites ae stictly necessay: We will fo simplicity use thee satellites and assume that you GPS eceive contains an atomic clock in this illustation). The situation is illustated in figue 9. You GPS eceive contains a vey accuate clock showing the time t when you eceive the signal. This gives you GPS eceive thee equations with the thee coodinates of you position x as the thee unknowns, x x sat1 = ct t sat1 ) c t 1, x x sat = ct t sat ) c t, x x sat3 = ct t sat3 ) c t 3. The GPS eceive eceives the time t sat when a signal was emitted fom the satellite. Knowing that the signal tavels with light speed c and eading off the time of eception of the signal on the intenal clock of the GPS eceive, the eceive can calculate the distance c t that the signal has taveled. This distance is equal to the diffeence between you position x and the position x sat of the satellite when the signal was emitted. Solving the thee equations above, the GPS eceive solves fo you position x = x, y, z) nomally expessed in tems of longitude, latitude and altitude. Note that if a fouth equation wee added using a signal fom a fouth satellite, anothe unknown could be allowed: This is how the pecision of you GPS clock is inceased: you time t is solved fom the fou set of equations. Hee we will assume that you GPS eceive has an atomic clock) If we assume a simplified one dimensional case, i.e. that you only have a one dimensional position x, the solution would be x = c t ± x sat. We see that the pecision of you calculated position x depends on the pecision with which we can calculate the time diffeence t = t t sat. The signals move with velocity of kilometes pe second. If thee is an inaccuacy of the ode 1 µs = 10 6 s, one micosecond, the inaccuacy in the calculated position would be of the ode m/s 10 6 s = 300 m. An inaccuacy of one micosecond coesponds to an inaccuacy of 300 metes in the position calculated by GPS. In such a case GPS would be useless fo many of its applications and moe seiously, the aiplane missing the tamac with 300 metes would cash! We know that due to special elativity, the clocks in the satellite and the clocks on Eath in you GPS eceive) un at diffeent paces because of the elative motions of the satellites with espect to you. We also know fom geneal elativity that the clocks in the satellite un at a diffeent pace than you clock because of diffeence in distance fom the cente of attaction cente of Eath). If the clocks in the satellites and the clocks in the GPS eceives wee synchonized at the moment when the satellites wee launched into obit, the question is how long does it take until the elativistic effects make the Eath and satellite clocks showing so diffeent times that GPS has become useless. Relativistic effects ae usually small so one could expect that it would take maybe thousands of yeas. If this wee the case, we wouldn t need to woy. But emembe that we equie a pecision bette than 1 µs hee. This could make elativistic effects impotant. Let s check. We stat by the gavitational effect. We conside two shells, shell 1 is the suface of the Eath situated at adial distance 1 = 6000 km appoximately, we ae only looking fo odes of magnitude hee, not exact numbes). Shell is the obit of the satellites at adial distance = km. A time inteval t 1 on the suface of the Eath is elated to a time inteval t of the fa-away obseve by see equation 3) t 1 = 1 ) t. Similaly, a time inteval t measued on the satellite clock is elated to the fa-away time t by t = 1 ) t. 1 16