black holes lecture ucsd physics 161

8/21/2019 Black holes lecture UCSD physics 161

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Black Holes (Ph 161)

An introduction to General Relativity- 2015

ecture !!


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“Natural Units”

In this system of units there is only one fundamental dimension, energy .

This is accomplished by setting Planck’s constant, the speed of light,

and Boltzmann’s constant to unity, i.e.,

By doing this most any quantity can be expressed as poers of energy,

because no e easily can arrange for

To restore ! normal " units e need only insert appropriate poers of

of the fundamental constants abo#e


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It helps to remember the dimensioof these quantities . . .

for example, picking convenient units


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!geometrized" or "natural" units for

spacetime

"e#ne the $lanck %ass

. . . and no& the 'ravitational constant is ust . . .


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It turns out that in a weak gravitational field the time-timecomponent of the metric is related to the Newtonian gravitational

potential by . . .

Where the Newtonian gravitational potential is

dimensionless !


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A convenient coordinate system for A convenient coordinate system forweak & static no time dependence gravitational fieldsweak & static no time dependence gravitational fieldsis given by the following coordinate system"metric is given by the following coordinate system"metric $

This ould be a decent description of the spacetime

geometry and gra#itational effects around the earth,

the sun, and hite darf stars, but not near the surfaces

of neutron stars.


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"B#$%&"B#$%& 'A'A(solar asses)

RA*!+RA*!+ (c)

,etonianGravitational

Potential

earth 3 x 10-6 6.4 x 108 ~10-9

sun 1 6.9 x1010 ~10-6

white

dwarf

~1 5 x 108 ~10-4

neutron

star ~1 106

~0.1

to 0.2

%haracteristic &etric 'e#iation


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)avendish expt.

U*+N$


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-and /acts0 1olar 1stem

radius of earth2s orbit around sun


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-and /acts0 the Universe


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3he essence of 'eneral 4elativit0

There is no gravitation0 in locall inertial coordinate s&hich the 5quivalence $rinciple guarantees are al&as t the e6ects of gravitation are absent

3he 5instein /ield equations have as their solutionsglobal coordinate sstems &hich cover big patches of sp


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#he $%uivalence rinciple

• (ot#os experiments

• meaning for freely falling bodies

• geometric implications

• geodesics


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Galileo drops diferent size balls of the eaning 3o&er . . .

Jinavie.tumblr.com


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#he $'#(') $*periment

torsion balance

see .npl.ashington.edu)eotash

&agnitude of torque on fiber$


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$otWash lab+s results, sensitivity for long range forces is at about part in /

.npl.ashington.edu)eotash)


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5$ experiment

U*+N$


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012 what does this mean3 012 what does this mean3

$verything falls at the same rate!

*pollo + astronaut 'a#id -. cott drops a hammer

and a feather . . . /uess hat happens0

.hq.nasa.go#) . . .)1istory)P234+3)co#er.html

http://www.hq.nasa.gov/http://www.hq.nasa.gov/http://www.hq.nasa.gov/


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0ne begins to get a creepy feeling that the acceleration produced by 4gravity5 has nothing to do with what the bodies

in %uestion are made out of2 but rather is a property of spacespacetime itself !


21/76cse.ssl.berkeley.edu)bmendez)ay+5)4554)notes)pics

equi#alence of inertial and gra#itational mass$


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elevators in free fall . . . & the $. .elevators in free fall . . . & the $. .

g 6 7.8 m s-9

If we make the elevator small enough2 it looks to us as if #:$;$ I) N0 annot tell the difference betweenan elevator in free fall and theabsence of gravitation.

)omeone cuts your elevator+s cable and you release the two balls

that you have in your hands . . . What happens3


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)tatement of the

$%uivalence rinciple,

In a sufficiently small regionof space & time we can find a freely falling locally ?inkowskicoordinate system in which theeffects of gravitation are absent

- the laws of physics arethe same as they are in a?inkowski coordinate systemwith no gravitation.

)tatement of the fundamental theorem

of differential geometry for 9-@ surfaces,

In a sufficiently small region on any 9-@surface2 the geometry is locally flat and >artesian. We can pass a tangent planethrough any point on the surface. In asufficiently small region around wherethis tangent plane touches the surface2the geometry will be flat2 like a>artesian plane.


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>oordinate #ransformations>oordinate #ransformations6ollos from the chain rule$ #ie coordinates in one system

as functions of the coordinates in the other frame.

e.g., consider these four functions$

chain rule gives,

>oordinate transformation is these functions,


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In locally inertial ?inkowski coordinates the particle is unaccelerated2 andmoving on a straight line

multiply both sides by and sum,


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so in a sense the basis vectors transform “oppositel” fr

the vector components in order that the contraction of t (the vector geometric obect! be frame invariant


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7ne+/orms linear functions of vectors into real numbers

ust like vectors are linear functions of one+forms into rea


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National 'eographic – Everest Region

A 1-orm is a geometric obe

-!t is li"e a contour map#i.e.# a set o nested $surace

&onstant value o some 'uaon each $surace%.

)he undamental# rame invariant# operationbet*een vectors +arro*s,and 1-orms +nested suraces, is the CONTRACTION

ay the vector . on the 1-/or P

and count ho any

sur/aces are ierced

&hat is a /rae-invariant real nuer 3


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3he metric tensor turns vectors into 9+formsand the inverse metric does the opposite


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>onnection >oefficients>onnection >oefficients >hristoffel symbols>hristoffel symbols

These are ob#iously related to ho the locally &inkoski

coordinates differ from our lab coordinates x


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9o, in considering these to coordinate systems, the locally

&inkoski coordinates , and the !lab" coordinates ,

e ill demand that the spacetime inter#al is alays the same$


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Bength2 Area2 and (olume and the ?etric Bength2 Area2 and (olume and the ?etric

Watch out! Actual physical or proper lengths2 areas2 and volumesare not the same as coordinate values of the same %uantity.

It must be kept in mind that the spacetime interval is preserved under coordinate transformations. #hink about the invariant interval

corresponding to an infinitesimal coordinate increment,

&ore (xamples . . .


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The components of the metric tensor in freely falling,

locally &inkoski coordinates are

The components of the metric tensor in the !lab"

coordinate system are


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#he ?etric #ensor and#he ?etric #ensor and 4oordinates4oordinates

. . . this is how the 4components5 of the metric transformunder a coordinate transformation2 where the transformationmatri* elements are for a so-called 4coordinate basis5

for the two different coordinate systems and

The (qui#alence Principle says that at any e#ent in spacetimeThe (qui#alence Principle says that at any e#ent in spacetime

it is alays possible to find a transformation to locally it is alays possible to find a transformation to locally

&inkoski coordinates.&inkoski coordinates.

Th l t i t fi ld i


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The general metric tensor field is

It is symmetric$

Therefore, there are +5 independent functionsat any e#ent 7point8 in spacetime.

:hy0

The metric tensor defines a coordinate system

and #ice #ersa through the line element

3he 5quivalence $rincipal


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With this the Christofel symbols canWith this the Christofel symbols can

written in terms o the inverse metricwritten in terms o the inverse metric

partial derivatives o the metric as . . partial derivatives o the metric as . .

&here the inverse metric is so+named because . . .

e qu a e ce c paguarantees that

in general true onl for coordinate bases


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(xamples in ;2'$ flat %artesian coordinates

and spherical polar coordinates.


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4flat space5 means ?inkowski coordinates t, x, y, z

the metric for which is Cust

:hat about coordinates 7 t, r, θ, ϕ 8 0

D0#: >00;@INA#$ )=)#$?) D0#: >00;@INA#$ )=)#$?)

@$)>;ID$ #:$ )A?$ ;ID$ #:$ )A?$


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:ow do we tell the difference between:ow do we tell the difference betweenbetween coordinates that imply between coordinates that imply

curvaturecurvature gravitational effects and gravitational effects and Cust plain old Cust plain old flat spaceflat space mas%uerading mas%uerading itself withitself with 4curvilinear coordinates53 4curvilinear coordinates53

#he answer can be found in the $%uivalence rinciple whichsays that physics is invariant under coordinate transformations2

all coordinate transformations. We are free to choose a coordinatetransformation any way we want,


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The (.P. gi#es us enough freedom to choose coordinates at

any e#ent 7point8 to transform the metric components

to be those of the &inkoski metric and the first deri#ati#es

of the metric to be zero, thereby making the %hristoffel symbols zero as ell.

(xpand our !lab" coordinates in a Taylor series about point in the desired

coordinates 7hich ill of course be the locally inertial, &inkoski coordinates8 . . .

imilarly expand the metric functions and use


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We would like to transform the metric to the flat space2 ?inkowski metric and we would like to get rid of as many derivatives of themetric functions as possible . . . What can we do with our freedom to choose the coordinate transformation3

9 second derivatives of the metric which cannot in general be9 second derivatives of the metric which cannot in general beset to Eero with the coordinate freedom given by the $. .set to Eero with the coordinate freedom given by the $. .

Crvatre ! Riemann Tensor


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"e#ne intrinsic curvature as the di6erence bet&een an initial vect and the same vector parallel+transported around an in#nitesimal l

these terms give the nonlinear i.e., products of #rst derivativemetric components

It t t th t i k it ti l fi ld th ti ti


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It turns out that in a weak gravitational field the time-timecomponent of the metric is related to the Newtonian gravitational

potential by . . .

Where the Newtonian gravitational potential is

dimensionless !


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"B#$%&"B#$%& 'A'A(solar asses)

RA*!+RA*!+ (c)

,etonianGravitational

Potential

earth 3 x 10-6 6.4 x 108 ~10-9

sun 1 6.9 x1010 ~10-6

white

dwarf

~1 5 x 108 ~10-4

neutron

star ~1 106

~0.1

to 0.2

%haracteristic &etric 'e#iation

A convenient coordinate system forA convenient coordinate system for


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A convenient coordinate system for A convenient coordinate system forweak & static no time dependence gravitational fieldsweak & static no time dependence gravitational fieldsis given by the following coordinate system"metric is given by the following coordinate system"metric $

This ould be a decent description of the spacetimegeometry and gra#itational effects around the earth,

the sun, and hite darf stars, but not near the surfaces

of neutron stars.

:e ill explore this metric ith #ariational principles later.

#h ) h hild ? i


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#he )chwarEschild ?etric #he )chwarEschild ?etric spherically symmetric2 static spacetime

charzschild coordinates

6unctions of radial coordinate r to be determined

by particular spherically symmetric, static distribution

of mass2energy$


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charzschild metric in #acuum outside a spherical, static distribution

of mass M is

M

:hat is the physical 7proper8

distance along this radial line0


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-emember that the metric functions are dimensionless$

9etonian Potential$


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)chwarEschild ;adius


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charzschild &etric$ conser#ed quantities

9ote that none of the metric functions depend on the timelike coordinate t

This means that the timelike co#ariant component of the four2momentum

of a freely falling particle ill be conser#ed along this particle’s orld line7a geodesic8.

co#ariant components$

timelike co#ariant component$

Photon emitted at r 1 ith energy E em . :hat is its energy hen it gets to r 2 0


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M

In freely falling coordinates$

But this inner product could be e#aluated in any coordinate system and you ill alays get the same result.


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This is the gra#itational redshift$

-edshift is defined as $


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&osmolog(

riedmann-e/aitre-Robertson-0al"er m


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A 4uick and dirty tour o/ all o/ the hole universe

- the lare scale structureevolution o/ sacetie7


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Hule (H&)

+ltra *ee 8ield

oe o/ the /irst

ala9ies to /or


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George Gamow

George LeMaitre

A. Friedmann

Albert Einstein

Homogeneity and isotropy of the universe:


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Homogeneity and isotropy of the universe:

implies that total energy inside a co-moving spherical surface is constant with time.

total energy = (kinetic energy of expansion) + (gravitational potential energy)

mass-energy density = ρtest mass = m

≈ −

' 43π a3 ρ [ ]ma

≈1

2ṁa

2

total energy > 0 expand forever k = -1

total energy = 0 for ρ = ρcrit k = 0

total energy < 0 re-collapse k = +1

= ρ ρcrit

≈ 0.3

a

(k=0)

ȧ2+ k =

8

3

π G ρ a2


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The key point in our aru!ent was sy!!etry"

spe#ifi#a$$y% a ho!oeneous and isotropi# distri&ution

of !ass and enery'

(hat e)iden#e is there that this is true*

+ook around you. This is !anifest$y ,T true on

s!a$$ s#a$es. The os!i# /i#rowa)e a#kroundadiation / represents our &est e)iden#e that

!atter is s!ooth$y and ho!oeneous$y distri&uted

on the $arest s#a$es.

The sate$$ite - the !i#rowa)e &a#kround radiation


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$a#k&ody radiation

ried!an-+e/aitre-o&ertson-(a$ker +( #oordinates


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ried!an +e/aitre o&ertson (a$ker +( #oordinates

defined throuh this !etri# . . .


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k = -1 k = 0 k = +1

Ho /ar does a hoton travel in the ae o/ the universe:


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onsider a radia$$y-dire#ted photon

photons tra)e$ onnu$$ wor$d $ines so ds20

#ausa$ hori7on

%ausal (Particle) Hori;on


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%ausal (Particle) Hori;on

radiation do!inated

!atter do!inated

)a#uu! enery do!inated

!n every case the hysical (roer) distance a liht sinal travels oes

to in/inity as the value o/ the tielike coordinate t does

,ote< hoever< that /or the vacuu doinated case there is a /inite

liitin value /or the 8R= radial coordinate as t oes to in/inity

;$@):IF#;$@):IF#


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;$@):IF# ;$@):IF# 9ote that ith the 6


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y

(theronuclear e9losions)

serve as >standard candles


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B suernova cosoloy esite


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B suernova cosoloy esite


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='AP cosic icroave ackround satellite

$u#tuations in / te!perature i)e

nsiht into the #o!position% si7e% and ae

of the uni)erse and the $are s#a$e #hara#ter

of spa#eti!e.

Ae 3 1C Gyr

acetie 3 >/lat? (eanin k30)

%oosition 3 2D unknon nonrelativistic

atter< CD unknon

vacuu enery (dark enery)<

ED ordinary aryons


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o&ser)ationa$ #onstraints

on the #ontent of

of nonre$ati)isti# !atter and )a#uu! enery

dark enery in the

uni)erse

2 f r a # t i o n o f # r i t i # a $ d e n

s i t y # o n t r i & u t e d & y ) a # u

u ! e

n e r y 3

fra#tion of #riti#a$ density #ontri&uted &y nonre$ati)isti# !atter


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(vacuum energy)

W li i k 0 iti ll l d i


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We live in a k = 0, critically closed universe.


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photon decoupling T~ 0. 2 eV

vacuum+matter dominated

at current epoch

black holes lecture ucsd physics 161

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