Light and TidesA2290-38 1Light and TidesRelativity and AstrophysicsLecture 38Terry HerterA2290-38 Light and Tides 2Outline Metric in the Rain Frame Inside the horizon One-way motion Rain Fall Light Cones Photon Exchange Rain fall source to distance observer Distance source to rain fall frame Tides Conclusions Homework: Due today Problems 2-5 and 3-7 in Exploring Black HolesFinal Exam:2:00 - 4:30 pm, Monday, 12/14 in SS105Make-up Final:9:00 - 11:30 am, Friday, 12/11 in SS301You must have permission to take the make-up by end of class on Friday (12/04)Light and TidesA2290-38 2A2290-38 Light and Tides 3Metric in the Rain Frame (cont’d) The Schwarzschild metric is Substituting the expression for dt We get This metric works for anywhere around a non-rotating black hole but is defined for a particle which start at rest a large distance away from the black hole There are no “infinities” in the rain frame metric indicating a smooth transition through the horizon for the falling particle rMdrrMdtdtrain2122/1222222121drrMdrdtrMd 2222/1222221drdrdrdtrMdtrMdrainrainSchwarzschild metricRain frame metricA2290-38 Light and Tides 4One-way motion inside Horizon Forward light cone In a local frame we are limited by the speed of light; all motion is confined to the “forward light cone” – (+ or - radial direction) We can use the rain frame metric to look at the forward light cone of a particle inside the horizon The rain frame metric can be rewritten as: For light (d= 0) traveling radially (d= 0) there are two solutions, a “forward” (headlight) and a “backward” (taillight) Not that since 2M > r inside the horizon, dr/dtrain< 0 always, so both the headlight and taillight beams more toward the center!2222121drdtrMdrdtrMdrdrainrain2/121rMdtdrrain2/121rMdtdrrainandLight and TidesA2290-38 3A2290-38 Light and Tides 52.53.03.54.04.55.05.50123r/Mtrain/MRain Fall Light Cones Worldline of rain frame particle (free fall from infinity)  The particle emits flashes in all directions as viewed in the free float frame of the particle The light cones show the bounds of the flash Lower segment => flash sent inward Upper segment => flash sent radially outward Inside the horizon, even light aimed radial outward (in rest frame of particle) moves inward toward the center horizonv = 1A2290-38 Light and Tides 6Photons: Rain emits, Far receives Shell frame:  For a photon travelling for a shell frame to a large distance away where ffar,recand fshellare the far-away and shell photon frequencies Rain Frame – sends a light beacon with frequency, frain,emit The rain frame speed relative to the shell frame is The Doppler shift of a beacon photon relative to the shell frame Combining yields2/1,21 rMffshellrecfar2/12 rMvshell2/1,11shellshellemitrainshellvvff2/1,,21rMffemitrainrecfarThe frequency shifts to zero as the horizon is approachedPhotons climbing out are redshifted2/12/12/1,2121rMrMfemitrainLight and TidesA2290-38 4A2290-38 Light and Tides 7Photons: Far emits, Rain receives Shell frame:  For a photon traveling to a shell frame from a large distance away where ffar,emitand fshellare the far-away and shell photon frequencies Rain Frame The rain frame speed relative to the shell frame is The Doppler shift of the photon relative to the shell frame Combining yields2/1,21 rMffemitfarshell2/12 rMvshell2/1,11shellshellshellrecrainvvff2/1,,211rMffemitfarrecrainIn the rain frame, photons are actually redshifted!Shell frame sees very high energy photons2/12/12/12121rMrMfshellA2290-38 Light and Tides 8Photon propagation summary Plot summarizing frequency shift of photons traveling between  Far frame  shell frame Shell frame  far frame Far frame  rain frame Rain frame  far frame The “far frame” is at rest and located far-away from the black hole, so that space is flat Note that the only divergence occurs when the photon falls onto the shell frame For shell frames approaching the horizon, the received photons energy is unbounded.0.00.51.01.52.02.53.03.5012345shell: downshell: uprain: downrain: upr/Mfrec/femitPlot of observed (received) photon frequency relative to the emitted frequencyLight and TidesA2290-38 5A2290-38 Light and Tides 9Tides Tides occur because of “differential” gravity across an object. For the Earth the acceleration due to gravity is, in geometric units For the rain frame (setting dtrain= d), we have The acceleration in the rain frame is the change in velocity with time Where we used the formula for dr/din the last step Tides are the change in gravity across a given distance (the gradient) –that is we differentiate to get2EEEarthrMgg 2/12rMddr22drdgraindrrMdgrain32gE~ 10-16m-1ddrrM2/32/12212rMA2290-38 Light and Tides 10Feel the (tidal) pain Let’s suppose that tides become very noticeable when we feel one-g across our body, that is, our stomach is in free-fall and our head and feet each feel one-half the pull of Earth gravity Setting We have Writing this in terms of solar masses Where we assume dr = 2 meters for the last step What is the black hole size needed so that we feel 1-g at the horizon?drrMdgrain32gE~ 10-16m-1Eraingdgand using3/12EhurtgMdrr3/1242MgdrMrEhurt3/122242MMMgdrMrsunsunEhurt3/213202sunhurtMMMrsunsunMMM 000,4813202/3Light and TidesA2290-38 6A2290-38 Light and Tides 11How long does it last? How long do we feel this pull before we are “crunched”? In the rain frame, the velocity is We integrate to get the time required to travel from r1to r2 Where t1and t2are in rain frame Define thurtas the time difference to get to the singularity (r2= 0) and setting r1=