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8/10/2019 10 BlackHoles
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10 - Relativity & Black Holes
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Problem:
Some aspects of electromagnetism dont ork rig!t "nless
t!ere is a preferred frame of reference - i#e# of motion
$s t!ere% $s it t!e material' t!at lig!t propagates in - t!e
aet!er'%
1887 Michelson &
Morley Experiment
speed of light in a
vacuum is independentof the motion of the
source!
Consequence:all observers
measure the
same speed
of light,
regardless oftheir motion
w.r.t. the light
source!!!!
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Ramification:
(lt!o"g! relativevelocities
add)s"btract for most
common ob*ects+
THIS IS T T"#$
%" &I'HT! ,!e
speed of a lig!t
so"rce coming
toard yo" is t!e
same as one going
aay from yo"#
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ot!er points
$f p!ysical las' are
"niversal+ everyone
s!o"ld be able toderive t!e same sets
of las+ even if t!ey
dont see e.actly t!e
same t!ing#
/vents' t!at
appear
sim"ltaneo"s
in one frame of
reference may
not appear so
in anot!er#
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1905 Einstein published his first paper on
the special theory of relativity. In it are 2
postulates:
1. The speed of light in a vacuum is constant for
all observers
2. Every observer derives the same physical
laws
/instein realied t!at eton !ad not really defined time'
in a consistent ay: it is not an absol"te' 2"antity+ b"t m"st
be defined in terms of observable 2"antities#
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Consequences of Relativity
Time Dilation
Suppose you flew a rocket ship parallel to a perfectly
reflecting mirror, and shown a flashlight beam out the
widow so that it came right back to the flashlight.
=2
( )
2
=42
2,ime for lig!t to !it mirror & ret"rn:
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Now suppose you were stationary on the ground where the
mirror is mounted. What would you see?
t=2
=
2
2 +
2
2
2 =42
2 +
22
2
so t2 = ( )
2
+ 22
2 2 1
2
2
?
??
?
??
= ( )2
2 =( )
2
1 2
2?
??
?
??
=
1 2
2
,!e rate t!at time
passes for t!e to
observers of t!e
same event is
different33
,ime dilation' - time
passes more sloly!en traveling fast+
compared to a
stationary'
observer
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Observer in the Rocket:
Outside Observer:
The outside observer will see the clock time of the ship advance 1
sec for every 2.3 sec that his own clock ticks. The stationary
observer sees the ships clock running slower!
B"t t!e observer on t!e rocket sees t!e person on t!e gro"nd
travel past at a !ig! speed+ and the observer on the ship must
see the same thing occurring to the clock of the outside observer!
= 1
t= 1
1
2
= 0.9=
1
10.81 =
1
0.192.3
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(not!er conse2"ence: 4engt! 5ontraction
6b*ects appear to be compressed along t!eir direction
of motion33
(nd+ as it! time+ t!e observer on t!e train see t!e
orld aro"nd t!em compresses along t!e same line ofmotion33
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1
1 ( )2
= (ll t!ese effects get
bigger)smaller it!
0#0 1#0
0#1 1#0070#7 1#177
0#8 1#99
0#; #;strictly speaking it is t!e
moment"m+ not t!e mass?@
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6t!er eird t!ings:
,o on o"tside
observer+ a traveling
lig!t so"rce seems to
beam' its lig!t
forard
View of
space for astationary
observer
View of
space for a
movingobserver -
sees a
starbow
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$n m"c! t!e same manner t!at t!e time dilation can be
derived+ anot!er direct conse2"ence is t!at matter at rest
!as b"ilt in' energy:
/Amc
ost importantly+ it! t!ese ne clarifications' t!e lasof electromagnetism can be derived correctly3
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So far - applies to v A constant - inertial motion'
Special ,!eory of Relativity >for a specialframe ofreference it! acceleration A 0@
C!at abo"t acceleration% e live in a universe
premeated b+ gravit+, which causesacceleration-. In general, a .
Dor t!e more generalcase e need a more'eneral Theor+ of "elativit+
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6bserved Problem
,!e orbit oferc"ry:
Newtonian Gravity Predicts: 5557.62 arcsec/century
Observed Value: 5600.73 arcsec/century
Difference: 43.11 0.45 arcsec/century too fast!!
This seeming small inconsistency led Einstein to completely change
the way we look at the universe.
(lso tr"e for all planetary orbits+ b"t less so f"rt!er from S"n
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The Principle of Equivalence
The force of gravity is, in most instances, indistinguishablefrom the force due to accelerated motion.
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Deflection of Starlight & Gravitational Lensing
Eistant gala.ies lensed)arped
in appearance by close gala.y
masses
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4ig!t travels along straig!t' lines in a c"rved space-time'
$f t!is ere a soccer
field+ !o o"ld a
soccer ball roll' on it%
4ig!t be!aves similarlytraveling t!ro"g! c"rced
=E space
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Fravitational Reds!ift:
4ig!t loses energy as it travels aay from a so"rce of gravity
/2"ivalent viepoint: time r"ns more sloly
t!e closer yo" are to a so"rce of gravity3
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/lac0 Holesake t!e gravity stronger by cramming
t!e mass into a smaller & smaller vol"me:
event"ally even lig!t cannot escape3
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$ts a case of e.treme c"rvat"re
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on-rotating
>Sc!arsc!ild@
black !ole:
/vents inside t!e
Schwar1schild
"adiuscannot be
seen by a distant
observer - t!is is t!e
$vent Hori1on
Rs =2
2
Rs
= 3
$n "sef"l' "nits:
Every object in the universe has a Schwarzschild radius. But
only if their mass is contained within this limiting distance
do they become a BH.
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Rotating >Gerr@
BH:
6b*ects insidet!e ergosp!ere
co"ld
>s"pposedly@ be
transported toot!er
places)times3
/ven o"tside+
frame dragging'
s!o"ld occ"r
A rotating mass has a tendency to pull space-time along with it
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Frame Dragging at Earth being measured using Gravity Probe B,
launched April 20, 2004. Frame dragging occurs because a
rotating mass has a tendency to pull space-time along with it.
http://einstein.stanford.edu/http://einstein.stanford.edu/8/10/2019 10 BlackHoles
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6bserving Black Holes
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(ccreting Black Holes
(s materialsflos don
toard t!e
black !ole in
t!e form of an
accretion disk+
it does t!e
energy
conversion
t!ing':
Fravitational potential energy kinetic >motion@
energy t!ermal energy radiant energy
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C!ite darf+ ne"tron star+ or black !ole% Ho do e kno%
Size R - arying I-ray emission - cannot occ"r more 2"ickly
t!an it takes lig!t to cross t!e diameter of t!e ob*ect >or elset!e b"rst signal' is as!ed o"t@
Mass - "se binary systems & Geplers =rd 4a3
$s R J RS%
HE 9898 A 5yg I-1 as first knon' BH
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Black !oles s!o"ld also radiate Haking Radiation'
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Travel into a black hole? ot me!
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