The Schwarzschild Metric

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1 The Schwazschild Metic Relativity and Astophysics Lectue 34 Tey Hete Outline Schwazschild metic Spatial pat Time pat Coodinate Fames Fee-float Shell Schwazschild bookkeepe Pinciple of Extemal Aging Consevation of Enegy A90-34 Schwazschild Metic A

2 Schwazschild Metic In the pesence of a spheically symmetic massive body, the flat spacetime metic is modified. This metic is found by solving Einstein s field equations fo geneal elativity Kal Schwazschild found the solution within a month of the publication of Einstein s geneal theoy of elativity This metic is the solution fo cuved empty spacetime on a spatial plane though the cente of a spheically symmetic (nonotating) cente of gavitational attaction The timelike fom of the solution is: whee M d d M 1 1 d = angula coodinate with the same meaning as in Euclidean geomety = the educed cicumfeence. t = the fa-away time which is measued on clocks fa away fom the cente of attaction. A90-34 Schwazschild Metic 3 Schwazschild Metic (cont d) The spacelike fom of the solution is: M d d M 1 1 d Evey (non-quantum) featue of spacetime aound a spheically symmetic non-otating unchaged massive body is descibed by the Schwazschild metic. Note clealy something stange happens at = M (the Schwazschild adius o event-hoizon adius) moe late Why this looks ight The cuvatue facto (1 - M/) appeas in the and d tems and depends on but not. This is expected fo spheical symmety. As gets vey lage, the metic becomes flat (spacetime) As M becomes small, the metic becomes flat (no cuvatue without mass) A90-34 Schwazschild Metic 4 A90-34

3 Spatial pat of metic Fo fixed t, we have the spatial pat of the metic d d 1 M d = 0 The facto fo the d tem is 1/(1 - M/) which is geate than one fo > M, thus fo fixed, d > d. Think of a od extending diectly (d = 0) between to concentic s. Two fiecackes go of at each end at the same time, = 0. The pope distance between the explosions is d d d 1 M 1/ = 0, d = 0 This says that the distance between s is geate that the diffeence in -values. Thus we have spatial cuvatue. A90-34 Schwazschild Metic 5 Pictuing the Space Pat Using an embedding diagam we can attempt to visualize the spatial pat of the Schwazschild metic The limitation hee is that is that the vetical dimension is not an exta dimension of space The only the paabolic suface epesents cuved-space geomety (you have to stay on the suface since locations off the suface don t exit!) d d Hoizon Spacetime geomety fo a plane sliced though the cente of a black hole d d > d A90-34 Schwazschild Metic 6 A

4 Time pat of metic Fo fixed and, the metic is d 1 M 1/ Clock at est on a at adius The tem is the fa-away time (ephemeis time) and d is the pope time (tick occu on the same clock). Time Dilation Conside two successive ticks (events) of a clock on a. The facto fo the tem is (1 - M/) which is less than one so that d <. This means that the time between ticks ( ) is smalle at emission than thei value () when eceived at a geat distance. Send out a light pulse fom to a distant obseve with each clock tick. The emote obseve will measue a longe time () between ticks than. A90-34 Schwazschild Metic 7 Gavitational Redshift Instead of using pulses of light, use the light wave itself. The peiod of the light wave is a fom of ticking. A the distant obseve measues a longe time between ticks (wave cests) than someone at. The light is ed-shifted (longe time between cests => lowe fequency) to longe a wavelength Switching the oiginato & eceive gives a gavitational blue shift. Example: edshift between two adii Suppose light is emitted at 1 = 4M and absobed at = 8M. By what faction is the peiod of the light inceased?,1, 1 M 1/ 1 M 1/ 1 The atio of the peiods is then,,1 1/ / ,,1 1/ 1 M 1 M 1/ 1 So that the wavelength shifts long wad (to the ed) by a facto of 1. as the photon climbs fom = 4M to = 8M, e.g. yellow goes to ed. A90-34 Schwazschild Metic 8 A

5 Choice of Refeence Fames Thee a numbe of efeence fames we can choose fom to examine the Schwazschild metic. Fo instance, A fee-fall (fee-float) fame Standing on a at given adius Use,, and fa-away time t (Schwazschild bookkeeping) Each of these obsevations equies a diffeent set of coodinates and povides a diffeent way of examining spacetime aound a black hole. Can a peson exist in these fames? If so, what kind of existence is it? How do we elate one fame to anothe? A90-34 Schwazschild Metic 9 Fee-float fame Conside an unpoweed spaceship falling into a black hole As measued by obseves the ship inceases in speed as the ship plunges in. Inside the ship, we have a special-elativity capsule equivalent to being in open space Howeve, this fame is only local as tides will poduce acceleations between sepaated paticles => it is not a fee-float (inetial) efeence fame Need to make the spatial and tempoal extent of the fame smalle (easy nea the Eath o fa fom a gavitating body) But the cente of a black hole tides become vey stong, able to ip apat any physical object This fame is the only one in which humans could exist nea a black hole. Fo a lage enough black hole tidal foces could be toleated by humans. A90-34 Schwazschild Metic 10 A

6 Local Shell Fame We live on a (nealy) spheical Eath s suface This suface foces us away fom the natual motion of a fee paticle This is the foce of gavity we expeience which is diect towad the cente of the Eath What is the fom of the metic fo a obseve? We have d 1 M M 1 1 d d And substituting ou pevious expession fo d and d gives d 1 M 1/ 1/ d 1 M d d d which looks like flat spacetime but it is not since, d and d ae all functions of. Notice that this gives = 0 fo light on the since the distance along the suface is given by ds d d A90-34 Schwazschild Metic 11 Local Shell Fame (cont d) This fame has to deal with the gavitational foce (since it is not a fee float fame) Special elativity (SR) descibes bief, local expeiments Shell vs. fee-float fame SR will wok well fo a longe time in a fee-float fame by making the spatial extent smalle (always have gavity in fame) Fee-float obseve can coss the hoizon and continue expeiments (until tidal foces ip he apat) Inside the hoizon neithe no obseves can exist Shell and fee-float obseves can compae local measuements using special elativity (including the Loentz tansfomation) A90-34 Schwazschild Metic 1 A

7 Schwazschild coodinates The Schwazschild coodinates t,, and povide a global desciption of events These events can be fa apat The Schwazschild obseve is a bookkeepe who aely makes measuements heself She examine epots fom local and fee-float obseves and combines these to descibe events that span spacetime aound the black hole The local obseves convet thee coodinates to Schwazschild coodinates and send them back to the Schwazschild bookkeepe No one lives in these coodinates. They ae an accounting system Bookkeepe coodinates have univesality but most of the data enties ae isolated fom diect expeience A90-34 Schwazschild Metic 13 Tajectoies Fa-away time, t t = Schwazschild map Table kept by bookkeepe of coodinates sent to he by (o fee float) obseves along the tajectoy of a satellite No one diectly obseves this tajectoy This is not the way the obit would look to an obseve Thee ae time delays and gavitational bending of light The dots ae close togethe at the beginning and end of the tajectoy Why? A90-34 Schwazschild Metic 14 A

8 Schwazschild lattice & obseve Given knowledge of the metic, we can constuct a set of lattice points with clocks akin to what we did fo flat spacetime Clocks ae placed at the coodinates and, and ae stamped at each point Shell and fee-fall obseves can ead off these coodinates and the clock eading fo events Why use local obseves? Not equied, but allows us to use local physics (e.g. atomic clock uns just fine locally) O plunge into a black hole A90-34 Schwazschild Metic 15 Applicability The Schwazschild metic applies only outside the suface of an object Also, slowly spinning object like the Eath and Sun ae okay A black hole has no suface so the Schwazschild metic can apply to almost = 0 (the singulaity). A90-34 Schwazschild Metic 16 A

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