Class9:BlackHoles
InthisclasswewillstudyanexactsolutionofGeneralRelativityknownastheSchwarzschildMetric,whichdescribestheunusualspace-time
propertiesaroundaBlackHole
Class9:BlackHoles
Attheendofthissessionyoushouldbeableto…
• … describethespace-timearoundanobjectusingtheSchwarzschildmetric
• … understandtheeffectofthemetriconclockrates andgravitationalredshifting oflightemittedneartheobject
• … solveforthemotionoflightraysandfreely-fallingobjectsonradialandcircularpathsnearablackhole
• … describeeffectstakingplaceattheSchwarzschildRadius
Strong-fieldgravity
• WehaveseenhowGeneralRelativityrecoversNewtoniangravityinthecaseofa“weakfield”,suchasneartheSun
• Whathappensinastrongerfield,whereNewton’sLawsdon’thold?
• Inthisclasswewillconsiderabeautifulexactsolutionofsuchacase– theSchwarzschildmetric
http://scienceblogs.com/startswithabang/2015/06/30/astroquizzical-how-does-gravity-escape-from-a-black-hole-synopsis/
Schwarzschildmetric
• Weknowthatspace-timecurvatureiscompletelyspecifiedbythemetric 𝑔"#,suchthat 𝑑𝑠& = 𝑔"#𝑑𝑥"𝑑𝑥#
• Schwarzschildfoundthemetricfortheemptyspacearoundaspherically-symmetric,static,matterdistribution
K.Schwarzschild (1873-1916) https://www.physicsoftheuniverse.com/scientists_schwarzschild.htmlhttp://www.skyandtelescope.com/astronomy-resources/are-black-holes-real/
Schwarzschildmetric
• TheSchwarzschildmetricisexpressedintermsofspace-timeco-ordinates(𝑐𝑡, 𝑟, 𝜃, 𝜙),andis:
• Wecanseethenon-zerocomponentsofthemetric𝑔"# are𝑔22 = − 1 − 56
7,𝑔77 = 1 − 56
7
89,𝑔:: = 𝑟&,𝑔;; = 𝑟 sin 𝜃 &
• 𝑅@ = 2𝐺𝑀/𝑐& istheSchwarzschildradiusintermsofthetotalmassenclosed,𝑀
𝑑𝑠& = − 1 −𝑅@𝑟 𝑐𝑑𝑡 & +
𝑑𝑟&
1 − 𝑅@𝑟+ 𝑟& 𝑑𝜃& + sin 𝜃 𝑑𝜙 &
• Thismetrictellsushowanobjectcurvesthespace-timearoundit,andcanbeusedtocompute…
• … theorbitsofplanets(whichareinfree-fallaroundthecentralobject)
• … thegravitationaltimedilation aroundblackholes
• … thedeflectionoflight byagravitationalfield http://www.astronomy.com/magazine/ask-astro/2014/11/frozen-stars
https://www.quora.com/What-is-gravitational-deflection-of-light-rayshttps://www.sciencenews.org/article/einsteins-genius-changed-sciences-perception-gravity
Schwarzschildmetric
• Notethatthespace-timearoundtheblackholeisempty,butstillcurvedbythenearbymass(e.g.,particleswillmoveoncurvedorbitsasseenbyadistantobserver)
Schwarzschildmetric
Schwarzschildmetric
• Let’sbreakdowntheSchwarzschildmetricinmoredetail,bycomparingittotheMinkowski metric:
• Notethat𝑟 isaradialco-ordinate,butnotadistance– theproperdistancemeasuredbyalocalobserveris𝑑𝐿 = G7
9856/7�
• Space-timeco-ordinatesare“likestreetnumbers”– thatis,notsufficienttodeterminedistanceswithoutthemetric!
𝑑𝑠& = − 1 −𝑅@𝑟 𝑐𝑑𝑡 & +
𝑑𝑟&
1 − 𝑅@𝑟+ 𝑟& 𝑑𝜃& + sin 𝜃 𝑑𝜙 &
Radialdistortionsin𝑔22 and𝑔77Normalsphericalco-ordinatesfor𝑔:: and𝑔;;
𝑑𝑠& = − 𝑐𝑑𝑡 & + 𝑑𝑟& + 𝑟& 𝑑𝜃& + sin 𝜃 𝑑𝜙 &Minkowski:
Schwarzschild:
Gravitationaltimedilation
• Consideratickingclock,fixedinplaceatdifferentradii𝑟
• Since𝑑𝑟 = 𝑑𝜃 = 𝑑𝜙 = 0,theco-ordinatetime𝑑𝑡 betweentwoticks(recordedbythereferenceframe)isrelatedtothepropertime𝑑𝜏 (recordedbytheclock)as𝑑𝑡 = GK
98L6M�
• Atlarge𝑟,𝑑𝑡 = 𝑑𝜏 (thereisnodistinctionbetweentimes)
• As𝑟 decreasestowards𝑅@,theco-ordinatetime𝑑𝑡 betweentheticksincreases
• At𝑟 = 𝑅@,𝑑𝑡 = ∞ andinco-ordinatetime,theclockstops!!
• SomethingweirdishappeningattheSchwarzschildradius!
Blackholes
• Foralmostallobjects(e.g.theEarth,Sun),𝒓 = 𝑹𝒔 liesinsidethesurface,sotheseeffectsneverapply(N.B.theSchwarzschildmetricdescribesemptyspace outsidetheobject)
• AnobjectwhosesizeislessthanitsSchwarzschildradiusiscalledablackhole – andtheyexistinastrophysics!!
e.g.(1)ablackholeisproducedattheendofastar’slife,whennuclearfusionisover;
(2)supermassiveblackholesdevelopatthecentres ofgalaxies
https://www.nasa.gov/audience/forstudents/k-4/stories/nasa-knows/what-is-a-black-hole-k4.html
TheSchwarzschildRadius
• Wecangainmoreintuitionfor𝑅@ usingtheresultfromClass4relatingtheclockrate,𝐶 = 𝑑𝜏/𝑑𝜏S, totheproperacceleration𝛼 thatisequivalenttothegravitationalfield:𝛼 = 𝑐& G
GUln 𝐶
• Here,𝐶 = 1 − 567
�andproperdistance𝑑𝐿 = G7
98L6M�
• Wefind:𝛼 = − 56WX/7X
& 9856/7�
• Farfromtheblackhole,theequivalentproperaccelerationis𝛼 = −𝐺𝑀/𝑟&,justasNewtonwouldhavepredicted
• At𝑟 = 𝑅@,𝛼 = ∞:ittakesaninfiniteproperaccelerationtoremainstatic,i.e.theclock“feelsinfinitelyheavy”
Lightraysnearablackhole
• Considershiningatorchradiallyoutwardsnearablackhole!
https://warwick.ac.uk/newsandevents/pressreleases/black_hole_kills/
Lightraysnearablackhole
Considertwosuccessivecrestsofaradiallightray:
∆𝑡
∆𝑡
• Theco-ordinatetime∆𝑡 betweenthecrestsremainsthesame,butthepropertime
betweenthecrestsvariesas∆𝜏 = 1 − 567
� ∆𝑡
• Thefrequency 𝑓 ofthelightscalesas1/∆𝜏 –lightemittedfrom𝑟 willexperiencea
gravitationalredshift 1 + 𝑧 = \M\]= 1 − 56
7
8^X
http://archive.ncsa.illinois.edu/Cyberia/NumRel/EinsteinTest.html
• Let’ssolvefortheworld-lineoftheradiallightray𝒓(𝒕)
• Alightrayconnectsspace-timeeventsseparatedby𝑑𝑠 = 0
• Foraradially-movinglightray,𝑑𝜃 = 𝑑𝜙 = 0
• Themetricisthen𝑑𝑠& = − 1 − 567
𝑐𝑑𝑡 & + G7X
98L6M
• Hence𝑑𝑠 = 0 implies9WG7G2= 1 − 56
7
• Wecansolvethisequationtogive𝑐𝑡 = 𝑅@ ln 𝑟 − 𝑅@ + 𝑟 + 𝐾
Lightraysnearablackhole
• AttheSchwarzschildradius𝑟 = 𝑅@,G7G2= 0 – whichimplies
thatallradiallightraysremainatthehorizon!
• Alightrayatthehorizondirectedalittlebitsidewaysmustmovetosmaller𝑟 – therearenopathstoescape𝒓 < 𝑹𝒔
Lightraysnearablackhole
http://www.wired.co.uk/article/what-black-holes-explained
Radialplungeintoablackhole
• We’llnowlookatamaterialobjectfallingintoablackhole!
https://www.nature.com/news/astrophysics-fire-in-the-hole-1.12726
Radialplungeintoablackhole
• Freely-fallingobserverswithworldlines𝑥"(𝜏) follow
geodesicsGXbc
GKX+ Γef
" Gbg
GKGbh
GK= 0,wheretheChristoffel
symbolsaregivenbyΓef" = 9
&𝑔"# 𝜕f𝑔#e + 𝜕e𝑔f# − 𝜕#𝑔ef
• Wecanusethe𝜇 = 𝑡 geodesicequationtoshowthat,foranobjectinradialfree-falltowardsablackhole,G2
GK= k
98L6Mand
9WG7GK
&= 𝐾& − 1 − 56
7,where𝐾 = constant
• Iftheobjectstartsfromrestat𝑟 = ∞,then𝐾 = 1
• Wededuce9WG7GK= − 56
7�
,orplungetime∆𝝉 = 𝟐𝒓𝟑/𝟐
𝟑𝒄𝑹𝒔𝟏/𝟐
Radialplungeintoablackhole
• Theco-ordinatetimealongthepathisG2G7= G2/GK
G7/GK= 7/56
�
W 987/56
• Cansolvetoobtain𝑡 = 56W2 7
56
^X − &
q756
rX + ln 7/56
� s97/56
� 89
• 𝑡 = ∞ at𝑟 = 𝑅@??
• Thepropertimeintervalfortheobjecttoreach𝒓 = 𝟎 (or𝒓 = 𝑹𝒔)isfinite,buttheco-ordinatetimeintervalisinfinite!!
• Thedefinitionof𝑡 intheSchwarzschildmetrichasaproblem!
Fallingintoablackhole
Let’scomparethedifferentperspectivesofaspacecraftplungingintoablackholeof…
• spacecrafttravellers(e.g.students!)
• a distantobserver(e.g.professorsippingcocktails!)
Thespacecrafttravellersaresendingbackregularlightsignalsintheirframe
http://scienceblogs.com/startswithabang/2009/11/20/believe-it-or-not-a-black-hole/
Fallingintoablackhole
• Fromtheperspectiveofthespacecrafttravellers…
• Theclocksonthespacecraftareusingpropertime𝜏 (i.e.arepresentatallevents)
• Thetravellersnoticenosingularityat𝑟 = 𝑅@,andcomplete
theirplungefrom𝑟 = 𝑅u to𝑟 = 0 intime𝜏 = &5vr/X
qW56^/X
• As𝑟 decreases,thetidalgravitationalforce(stretchingforce)onthespacecraftincreases
• At𝑟 = 𝑅@,theproperaccelerationthatwouldberequiredtoholdastaticpositionbecomesinfinite
Fallingintoablackhole
• Fromtheperspectiveofthedistantobserver…
• Theco-ordinatetimeintervalbetweenthelightsignalsincreases,since𝑑𝑡 = 𝑑𝜏/ 1 − 𝑅@/𝑟
� and𝑑𝜏 = constant
• Thespacecraftwillseemtoslow,andstopas𝑟 → 𝑅@
• Thelightsignalsfromthespacecraftarebeinggravitationallyredshiftedas𝑓S/𝑓7 = 1 − 𝑅@/𝑟
�
• Theimageofthespacecraftredshiftsandfadesfromview
• Thedistantobserveriseffectivelyagingmorequicklyrelativetothespacecrafttravellers
Thephotonsphere
• Consideralightrayinacircularorbitaroundtheblackhole,suchthat𝑑𝑟 = 0.Wechoosetheorbitsuchthat𝜃 = 90°
• 𝑑𝑠 = 0 impliesthat9WG;G2= 9856/7
�
7
• The𝜇 = 𝑟 geodesicequationimpliesthat9WG;G2= 56
&7r�
• Wehencefindthatlighthasacircularorbitaroundablackholeatthephotonsphere𝒓 = 𝟑
𝟐𝑹𝒔
• Anycircularorbitcloserthanthisisspace-like(impossible)
Natureofthesingularity
• Aco-ordinatesingularityisaplacewherethechosensetofco-ordinatesdoesnotdescribethegeometryproperly
• AnexampleisattheNorthPoleofasphericalco-ordinatesystem,whereallvalues0 < 𝜙 < 2𝜋 correspondtoasinglepointinspace!
• Similarly,thepoint𝒓 = 𝑹𝒔 oftheSchwarzschildgeometryisaco-ordinatesingularity.𝒓 = 𝟎 isatruesingularity
• Wecantransformtoanotherradialco-ordinatesysteminwhichthissingularityisremoved