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