Relativity and Astrophysics Lecture 38 Terry Herter. Rain fall source to distance observer Distance source to rain fall frame

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1 Light and Tides Relativity and Astophysics Lectue 38 Tey Hete Outline etic in the Rain Fame Inside the hoizon One-way motion Rain Fall Light Cones Photon Exchange Rain all souce to distance obseve Distance souce to ain all ame Tides Conclusions Final Exam: :00-4:30 pm, onday, /4 in SS05 ake-up Final: 9:00 - :30 am, Fiday, / in SS30 You must have pemission to take the make-up by end o class on Fiday (/04) Homewok: Due today Poblems -5 and 3-7 in Exploing Black Holes A90-38 Light and Tides A90-38

2 etic in the Rain Fame (cont d) The Schwazschild metic is Substituting the expession o dt We get d dt d dt dt ain d d Schwazschild metic d dtain dtaind d d Rain ame metic This metic woks o anywhee aound a non-otating black hole but is deined o a paticle which stat at est a lage distance away om the black hole Thee ae no ininities in the ain ame metic indicating a smooth tansition though the hoizon o the alling paticle A90-38 Light and Tides 3 One-way motion inside Hoizon Fowad light cone In a local ame we ae limited by the speed o light; all motion is conined to the owad light cone (+ o - adial diection) We can use the ain ame metic to look at the owad light cone o a paticle inside the hoizon The ain ame metic can be ewitten as: d d dtain d dtain d Fo light (d = 0) taveling adially (d = 0) thee ae two solutions, a owad (headlight) and a backwad (taillight) d dtain and d dtain Not that since > inside the hoizon, d/dt ain < 0 always, so both the headlight and taillight beams moe towad the cente! A90-38 Light and Tides 4 A90-38

3 Rain Fall Light Cones Woldline o ain ame paticle (ee all om ininity) The paticle s lashes in all diections as viewed in the ee loat ame o the paticle The light cones show the bounds o the lash Lowe segment => lash sent inwad Uppe segment => lash sent adially outwad Inside the hoizon, even light aimed adial outwad (in est ame o paticle) moves inwad towad the cente 5.5 hoizon t ain / 4.0 v = / A90-38 Light and Tides 5 Photons: Rain s, Fa eceives Shell ame: Fo a photon tavelling o a ame to a lage distance away a, ec Photons climbing out ae edshited whee a,ec and ae the a-away and photon equencies Rain Fame sends a light beacon with equency, ain, The ain ame speed elative to the ame is Combining yields v The Dopple shit o a beacon photon elative to the ame v, ain ain, v a, ec ain, A90-38 Light and Tides 6 The equency shits to zeo as the hoizon is appoached A

4 Photons: Fa s, Rain eceives Shell ame: Fo a photon taveling to a ame om a lage distance away a, whee a, and ae the a-away and photon equencies Rain Fame The ain ame speed elative to the ame is v Shell ame sees vey high enegy photons The Dopple shit o the photon elative to the ame v, ain ec v Combining yields ain, ec a, A90-38 Light and Tides 7 In the ain ame, photons ae actually edshited! Photon popagation summay Plot summaizing equency shit o photons taveling between Fa ame ame Shell ame a ame Fa ame ain ame Rain ame a ame ec / : down : up ain: down ain: up The a ame is at est and located a-away om the black hole, so that space is lat Note that the only divegence occus when the photon alls onto the ame Fo ames appoaching the hoizon, the eceived photons enegy is unbounded / Plot o obseved (eceived) photon equency elative to the ted equency A90-38 Light and Tides 8 A

5 Tides Tides occu because o dieential gavity acoss an object. Fo the Eath the acceleation due to gavity is, in geometic units g Eath g E g E ~ 0-6 m - E Fo the ain ame (setting dt ain = d), we have d d The acceleation in the ain ame is the change in velocity with time d g ain d 3/ d d Whee we used the omula o d/d in the last step Tides ae the change in gavity acoss a given distance (the gadient) that is we dieentiate to get dg ain d 3 A90-38 Light and Tides 9 Feel the (tidal) pain Let s suppose that tides become vey noticeable when we eel one-g acoss ou body, that is, ou stomach is in ee-all and ou head and eet each eel one-hal the pull o Eath gavity Setting We have dgain g E and using hut d ge Witing this in tems o sola masses 4 hut d sun g E sun dg ain d 3 4 hut d ge g E ~ 0-6 m - Whee we assume d = metes o the last step What is the black hole size needed so that we eel -g at the hoizon? 30 / sun 48,000 sun 3 hut 30 sun /3 A90-38 Light and Tides 0 A

6 How long does it last? How long do we eel this pull beoe we ae cunched? In the ain ame, the velocity is d dtain We integate to get the time equied to tavel om to t ain d t 3/ t 3 t dt 3/ Whee t and t ae in ain ame Deine t hut as the time dieence to get to the singulaity ( = 0) and setting = hut d 3/ d since t hut hut t hut 3 3 hut g g E E The time is independent o the size o the black hole! Putting in numbes thut 8 m m m t hut 0.3sec So by the time you eel it, it s all ove A90-38 Light and Tides Conclusions Relativity povides a consistent amewok that allows us to study paticles moving at high enegies and motions in stong gavitational ields. The speed o light is the limit Simultaneity is ame dependent Thee ae no paadoxes! I thee seems to be, we aen t looking at the poblem coectly (space and time ae intetwined into spacetime). Vitually all elativistic paadoxes ae misconceptions about simultaneity Thee ae impotant conseved quantities in physics Spacetime inteval, system mass (invaiant) Enegy and momentum Hopeully this couse will be a jumping point o Reading uthe e.g. you textbooks and the web Hous o un discussing elativity with you iends A90-38 Light and Tides A

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department. Problem Set 10 Solutions. r s

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Depatment Physics 8.033 Decembe 5, 003 Poblem Set 10 Solutions Poblem 1 M s y x test paticle The figue above depicts the geomety of the poblem. The position