introduction and conceptual aspects of general...
TRANSCRIPT
INTRODUCTION and CONCEPTUAL ASPECTS of GENERAL RELATIVITY
George E. A. Matsas
Instituto de Física Teórica/ Unesp
Outlook
• Galileu and Minkowski spacetimes
• Apparent paradoxes of special relativity
• General Relativity
• Gravitational waves, cosmology and black holes
I. Galileu and Minkowski spacetimes
General Relativity
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theoryspacetime relativity general
General Relativity
theoryfield nalgravitatio icrelativist relativity general
Minkowski spacetime
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isotropy andy homogeneit space with Eqs.Einstein ofsolution vacuum
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Physics in Minkowski spacetime
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Relativity Special
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spacetimeGalileu in physicsNewtonian
Physics for slow-moving systems in Minkowski spacetime
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What is a spacetime theory?
• Question: what is a physical theory? • Response: physical theories must: (i) list the observables (whose values depend on system state and observer) (ii) describe how to measure them (iii) relate them (physical laws)
What is a spacetime theory?
• Question: what is a physical theory? • Response: physical theories must: (i) list the observables (whose values depend on system state and observer) (ii) describe how to measure them (iii) relate them (physical laws)
• Spacetime theory: (i) observables: space and time intervals between events (ii) measurement are (fair) clocks and rods (proper apparatus). (iii) physical laws relate space-time intervals.
observeron dependnot do which laws physical by those zedcharacteri is spacetimeA
Preliminaries
const,,scm
iiO
s/cm 232raGM
• How many dimensional constants do we need? - In general all observables may be expressed in terms of two independent dimensional constants, e.g., s and cm:
• Events, particles, strings,...
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• Events, particles, strings,...
TIM
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• Fair clocks: associate a real number to each visited event
TIM
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BA tt
BA
BA tt ?
• Fair rods: associate a real number to pairs of simultaneous events – discussion must be postponed until we define simultaneity
BA
• Inertial observers: observers endowed with accelerometers indicating null measurement
• Congruence of observers: each event of spacetime is visited by one and only one observer
con
gru
ence
Iner
tial
co
ngr
ue
nce
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Galileu spacetime
• Time is absolute
TIM
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BA tt
BA
P
Q
BA
BA tt
Q
P
Galileu spacetime
• Time is absolute
TIM
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BA tt
BA
P
Q
0 AtA
P1
• Simultaneity
metricEuclidean an with endowed events
ussimultaneo of classes eequivalenc ),( 3 R P1 P2 P3
TIM
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BA
BA tt
Q
P
• Fair rods: associate a real number to pairs of simultaneous events.
metricEuclidean an with endowed events
ussimultaneo of classes e equivalenc ),( 3 RP1 P2 P3
Question: In what sense is space absolute?
TIM
E
• Fiducial inertial congruence (arbitrary choice) represented by vertical lines
P1 P2 P3
TIM
E
other.each .rest w.r.tat lie congruence inertial same theof observers (ii)
s,congruence inertial are there(i)
:spacetimeGalileu
• Rapidity w.r.t. fiducial inertial congruence
P1 P2 TIM
E Q2
121221t /L PQPPQPv
Question: Is the distance between non simultaneous events absolute?
• Rapidity w.r.t. fiducial inertial congruence
P1 P2 TIM
E Q2
121221t /L PQPPQPv
TIM
E
P1 P2
Q2
congruence fiducial w.r.t.
cityconst velo with congruence
inertial also are scongruence inertial .city w.r.tconst velo with moving scongruence :spacetimeGalileu
Question: Is the distance between non simultaneous events absolute?
TIM
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P1 P2
Q2
equivalent are scongruence inertial all :principle Relativity
Question: may a particle, wave, etc have the same velocity w.r.t. to all congruences?
spacetimeGalilean with leincompatib iscity light velo :problem alExperiment
• The Minkowski spacetime must be considered when
• There is vacuum,
• The space is homogeneous and isotropic
• The topology is trivial
• Presence of relative velocities close to c
Minkowski spacetime
Minkowski spacetime TI
ME
source)on dependnot (do :fact alExperiment absolute are rayslight
lines)straight dashedby denoted rayslight :n(conventio
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later) justified be tolines(straight congruence inertial same pertain to
shomogeneou is spacetime :fact alExperiment
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scongruence inertial are there:spacetime Minkowski
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congruence inertial same pertain to
shomogeneou is spacetime :fact alExperiment BA tt :fact alexperiment
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:ATTENTION
Minkowski spacetime
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const ),( OSOROSOR ttttF
imply
and
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are scongruence inertialall
absolute are rayslight
Minkowski spacetime
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OSt
A
OR't
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R’
S’
S'O't
B
OS'OR'OSOR tttt
OS'OSOR'OR tttt
Minkowski spacetime
O
Q
R
S
O
Q
R
S
0),( OSOROSOR ttttF
O
Q
R
S
0),( OSOROSOR ttttF 0),( OSOROSOR ttttF
O
Minkowski spacetime
O
Q
R
S
0),( OSOROSOR ttttF
O
Q
R
S
),( OSOROQ ttFt
tionInterpreta
Question: do you see it?
O
Q
O
Q
R
S
0),( OSOROSOR ttttFtionInterpreta
Minkowski spacetime
O
Q
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S
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Minkowski spacetime
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linety simultanei
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linety simultanei
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Fair rods: associate a real number to pairs of simultaneous events.
O
Q
R
S
O
Q
R
S
0),( OSOROSOR ttttFtionInterpreta
Minkowski spacetime
P
0F
Q3
Q2
Q1
Q4
0F
0F
0F
conelight
Causal structure of Minkowski spacetime
P of planety Simultanei
Galileu x Minkowski spacetime
P
P of future Absolute
P ofpast Absolute
limbo TimeP
P of future Absolute
P ofpast Absolute
Causal structure of Minkowski spacetime Causal structure of Galileu spacetime
Minkowski spacetime
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us)simultaneo are and for whomobserver an by measured as to from (distance 11 Q PQ P F
)and visitshoobserver w inertialan by measured as to from interval (time 33 Q PQ PF
II. Apparent paradoxes of special relativity
Minkowski spacetime
O
Q
R
S
A
2/) ( OROSOQOQ Stttt
QS 2/) ( OROSQQOQ S
ttLL
OSt
ORt
choice and on depend values:Definition origincongruence inertial
Minkowski spacetime
O
Q
R
S
A
2/) ( OROSOQOQ Stttt
QS 2/) ( OROSQQOQ S
ttLL
OSt
ORt
FttLt OSOROQOQ
22
Minkowski spacetime
O
Q
R
S
B
QS
OSt
ORt
choice and on depend values:Definition origincongruence inertial
), :events Labeling QQ(Q Lt
2/) ( OROSOQOQQ Sttttt
2/) ( OROSQQOQQ SttLLL
Minkowski spacetime
O
R
S
),1S QQ( Lt
QS Q1 Q2 Q3
),2S QQ( Lt
),3S QQ( Lt
2/) ( OROSOQSS
ttttQ
B
Minkowski spacetime
O
R
S
),1S QQ( Lt
QS Q1 Q2 Q3
),2S QQ( Lt
),3S QQ( Lt
2/) ( OROSOQSS
ttttQ
0t
B congruence toaccordingty simultanei of line
B
)(st
Minkowski spacetime
2/) ( OROSOQQ ttLL
O
R
S
), QQ1( Lt
QS Q1
Q2
Q3
), QQ2( Lt
), QQ3( Lt
)(st
B
Minkowski spacetime
O
R
S
), QQ1( Lt
QS Q1
)(st
0t
)light ( sL0L
2/) ( OROSOQQ ttLL
2/) ( OROSOQQ tttt
Minkowski spacetime
O
R
S
), QQ( LtQS Q
)(st
0t
)light ( sL0L
2/) ( OROSOQQ ttLL
2/) ( OROSOQQ tttt
• Choose a congruence • Choose an origin O
• Determine simultaneity line which passes by O
• Synchronize all clocks of the congruence at this simultaneity line to mark t=0
congruence thew.r.t.
and between rodby measured distanceproper
S
Q
Q
QL
summary - events label toProcedure
andbetween congruence
theofmember by measured interval eproper tim
QQ S
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t
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)(st
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)light ( sL0L
2/) ( OROSOQQ ttLL
2/) ( OROSOQQ tttt
yaccordingltick 1
1yaccordingltick
• Choose a congruence • Choose an origin O
• Determine simultaneity line which passes by O
• Synchronize all clocks of the congruence at this simultaneity line to mark t=0
andbetween congruence
theofmember by measured interval eproper tim
Q Q S
Q t
congruence thew.r.t.
and between rodby measured distanceproper
S
Q
Q
QL
summary - events label toProcedure
), QQ( Lt
Minkowski spacetime
O
), QQ( LtQ
)(st
1
1)light ( sL
), PP( LtP
), RR( LtR
congruence thew.r.t.
and between rodby measured distanceproper
S
Q
Q
QL
andbetween congruence
theofmember by measured interval eproper tim
QQ S
Q
t
Minkowski spacetime
A
A
B
B
O
Q
Question: how do we relate ticks of congruencies A and B?
t
L
't
'L
choice and on depend values:Remember origincongruence inertial
)','
),
(
(Q
Lt
Lt
Minkowski spacetime
O
Q
R
S
AB
QS
tan/ OQOQB tLv
Question: do you see it?
2/) ( OROSOQOQQ Sttttt
2/) ( OROSQQOQQ SttLLL
then
Minkowski spacetime
stD AA 1
TIM
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AB
stD BB 1
2/1)]2/[cos( AB DD
Minkowski spacetime
A
A
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O
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1
1
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t
L
't
'L
Minkowski spacetime
A
A
B
B
O
Q
2/1)]2/[cos( AB DD
1
1
11
t
L
't
'L
)','
),
(
(Q
Lt
Lt
Minkowski spacetime
A
A
B
B
O
Q
1
1
11
t
L
't
'L
)','
),
(
(Q
Lt
Lt
Question: what is the relation between (t,L) and (t’,L’)?
Minkowski spacetime
A
A
B
B
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11
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vv
vtLLvLttations transformLorentz
Question: what is the relation between (t,L) and (t’,L’)?
• How many dimensional constants do we need? - In general all observables may be expressed in terms of two independent dimensional constants, e.g., s and light-s:
const,,s-light s
iiO
- Assuming relativity, (because of the light “absoluteness”) all observables may be expressed in terms of a single dimensional constant, e.g., s.
const,s
iiO
Question: how is it possible?
Question: may we measure distances only with clocks?
A
O
t
1t
2t
L
t
tttttL
2
4 2
2
2
1
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2
2
1
- Measuring distances only with clocks (Bill Unruh - private discussion)
A
O
t
1t
2t
t
tttttL
2
4 2
2
2
1
222
2
2
1
L
A
O
t
1t
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- Measuring distances only with clocks (Bill Unruh - private discussion)
A
O
t
1t
2t
t
tttttL
2
4 2
2
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1
222
2
2
1
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A
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t
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const,s iiO
- Measuring distances only with clocks (Bill Unruh - private discussion)
A
A
B
B
O
1
1
11
t
L
't
'L
Q1 Q2
Q3
• Simultaneity
Question: what events are simultaneous w.r.t. each other?
A
A
B
B
O
1
1
11
t
L
't
'L
Q1
Q2
• Time dilation
. and visits whocongruenceblack theofmember
very by the measured interval eproper tim ''
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• Time dilation
.congruence blue theof membersdifferent by two measured
eproper tim gsubtractinby obtained interval time12 QQ tt
A
A
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11
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• Time dilation
21/1,)''( vtttt 1212 QQQQ
A
A
B
B
O
1
1
11
t
L
't
'L
Q1
Q2
• Time dilation
21/1,)''( vtttt 1212 QQQQ
A
A
B
B
O
1
1
11
t
L
't
'L
Q1
Q2
21/1,)('' vtttt 1212 QQQQ
Twin “Paradox”
• Space contraction
B
B
O 1
1
't
'L
congruenceblack theof linety simultanei
aon rodsby measured distance 12 QQ LL
• Space contraction
B
B
O 1
1
't
'L
Q1
Q2
congruenceblack theof linety simultanei
aon rodsby measured distance 12 QQ LL
• Space contraction
A
A O
1
1
t
L
Q3 Q2
congruence blue theof linety simultanei
aon rodsby measured distance 32 QQ LL
• Space contraction
A
A
B
B
O
1
1
11
t
L
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Q1
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21/1,/)''( vLLLL 1232 QQQQ
• Space contraction
A
A
B
B
O
1
1
11
t
L
't
'L
Q1
Q3 Q2
A
A
B
B
O
1
1
11
t
L
't
'L
Q1
Q3 Q2
21/1,/)('' vLLLL 3212 QQQQ
21/1,/)''( vLLLL 1232 QQQQ
• Apparent paradox
https://en.wikipedia.org/wiki/Ladder_paradox
Question: are you able to fit a large ladder into a small garage in some sense?
https://en.wikipedia.org/wiki/Ladder_paradox
Question: can you make a large box to fall through narrowed spaced bars ?
• Apparent paradox
Question: can you make a large box to fall through narrowed spaced bars ?
• Apparent paradox
III. General relativity
General Relativity
Minkowski spacetime: vacuum, isotropic, homogeneous, and trivial topology
General Relativity
Minkowski spacetime: vacuum, isotropic, homogeneous, and trivial topology
Quite distinct from Minkowski spacetime in the presence of • compact objects : L ~ 2GM/ c2 • large pressures: P/ c2 ~ r
2-dimensional surfaces immersed in a 3-dimesnional Euclidean space
Locally, the r.h.s. surface looks like the l.h.s. one, but still there is a distinct feature. What is it?
Using 2-dimensional surfaces to create intuition on 4-dimensional spacetimes
flat space curved space
Question: how can we quantify curvature?
flat space curved space
Parallel transport of vectors on flat surface
Definition: Parallel transport of vectors along geodesics preserves vector norm and angle with geodesic
Closed loops made of geodesics pieces
Parallel transport of vectors on flat surface
Definition: Parallel transport of vectors along geodesics preserves vector norm and angle with geodesic
Closed loops made of geodesics pieces
Parallel transport of vectors on a curved surface Parallel transport of vectors on flat surface
Definition: Parallel transport of vectors along geodesics preserves vector norm and angle with geodesic
Parallel transport of vectors on a flat surface
Let us understand it better
Parallel transport of vectors on a flat surface Parallel transport of vectors on a flat surface
Other curvature effects
• General spacetimes are completely specified by stating the values of F for all events in the neighborhood of every event.
O
O
Curvature in 4D spacetime Lorentzian spacetimes
• General spacetimes are completely specified by stating the values of F for all events in the neighborhood of every event.
O
O
Curvature in 4D spacetime Lorentzian spacetimes
O
0F
Q3
Q2
Q1
Q4
0F
0F
0F
General Relativity
),( gM
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theoryspacetime relativity general
• Rather than specifying the spacetime through an infinitely large F list for all events in the neighborhood of every event, one codifies this information in a metric tensor g.
General Relativity
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theoryspacetime relativity general
• Rather than specifying the spacetime through an infinitely large F list for all events in the neighborhood of every event, one codifies this information in a metric tensor g.
• Given the spacetime matter/energy content Tab of the Universe and proper initial conditions, Einstein eqs. calculate g.
General Relativity
theoryfield nalgravitatio icrelativist relativity general
• Free test particles and light rays follow geodesics in spacetime.
General Relativity
theoryfield nalgravitatio icrelativist relativity general
• Free test particles and light rays follow geodesics in spacetime.
• Massive free test particles follow timelike geodesics (geodesics which (locally) maximize the spacetime distance between events).
• Free test particles and light rays follow geodesics in spacetime.
• Light rays follow null geodesics (geodesics which (locally) vanish the spacetime distance between events).
What is gravitational lensing?
Q1 Q1 Q1
Q2
Q3
Q4
Q2 Q4
Q3
• Free test particles follow geodesics in spacetime.
Let us understand it
• Question: which clock delays? (Remember that free test particles follow timelike geodesics (which locally maximize the spacetime distance between events.)
A B
Clocks measure the spacetime length of worldlines
he
igh
t
relative delay = height x g/c2
tota
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ay o
f ab
ou
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The time delay in gravitational fields is experimentally confirmed up to 1 in 104 parts
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Let us understand it
GPS
• GPS needs General Relativity to work out
A
B
Twin “paradox” in a space ),( 2 RSR
Question: clocks A and B are inertial and are initially synchronized w.r.t. each other; shall they be synchronized when they meet each other again?
Question: do you want to change your initial answer?
Question: how does nature is able to distinguish both clocks?
The relativistic submarine “paradox”
IV. Gravitational waves, cosmology and black holes
Gravitational waves
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Gravitational waves
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PSR 1913+16
Period of 7,75 hs, stars with masses of about 1.4 solar masses.
Agreement with theory of about 1 in 103 parts.
PSR 1913+16
Russell Hulse and Joseph Taylor Nobel prize 1993
Agreement with theory of about 1 in 103 parts.
GR 150914
Black holes masses: 29 and 36 solar masses
GR 150914
LIGO at Hanford, WA
LIGO at Livingston, LA
GR 150914
LIGO at Hanford, WA
LIGO at Livingston, LA
GR 150914
Measuring one-hundred-millionth the diameter of a hydrogen atom over the 4 kilometer length of the arm.
GR 150914
Final black hole mass: 62 solar masses
• Friedman-Robertson-Walker-Lemaitre spacetime
• The space is homogeneous and isotropic
Cosmology
),( gM
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BIG BANG
CMB – 380.000 years
First stars – 500 x 106 years
Dark energy domination– 10 x 109 years
Here and now – 14 x 109 years
comoving free observers
• Nucleosynthesis – from 10-2 s to 3min • Baryogenesis – about 10-6 s
• Inflation – about 10-32 s
proper distance increases with time
BIG BANG
CMB – 380.000 years
First stars – 500 x 106 years
Here and now – 14 x 109 years
comoving free observers
proper distance increases with time
Questions:
• What was “before” the big bang?
Dark energy domination– 10 x 109 years
BIG BANG
CMB – 380.000 years
First stars – 500 x 106 years
Here and now – 14 x 109 years
comoving free observers
proper distance increases with time
Questions:
• What was “before” the big bang?
• The big bang is a single singular event? Did it happen “somewhere”?
Dark energy domination– 10 x 109 years
BIG BANG
CMB – 380.000 years
First stars – 500 x 106 years
Here and now – 14 x 109 years
comoving free observers
proper distance increases with time
Questions:
• What was “before” the big bang?
• The big bang is a single singular event? Did it happen “somewhere”?
• How large was the universe just after the big bang?
Dark energy domination– 10 x 109 years
BIG BANG
CMB – 380.000 years
First stars – 500 x 106 years
Here and now – 14 x 109 years
comoving free observers
proper distance increases with time
Questions:
• What was “before” the big bang?
• The big bang is a single singular event? Did it happen “somewhere”?
• How large was the universe just after the big bang?
• May distant enough galaxies move away faster from us than light velocity?
Dark energy domination– 10 x 109 years
BIG BANG
CMB – 380.000 years
First stars – 500 x 106 years
Here and now – 14 x 109 years
comoving free observers
proper distance increases with time
Questions:
• What was “before” the big bang?
• The big bang is a single singular event? Did it happen “somewhere”?
• How large was the universe just after the big bang?
• May distant enough galaxies move away faster from us than light velocity?
• What is dark energy? Is it a kind of ether?
Dark energy domination– 10 x 109 years
BIG BANG
CMB – 380.000 years
First stars – 500 x 106 years
Here and now – 14 x 109 years
comoving free observers
proper distance increases with time
Questions:
• What was “before” the big bang?
• The big bang is a single singular event? Did it happen “somewhere”?
• How large was the universe just after the big bang?
• May distant enough galaxies move away faster from us than light velocity?
• What is dark energy? Is it a kind of ether?
• Is the solar system growing up as the Universe expands?
Dark energy domination– 10 x 109 years
Question:
• Why is the universe so isotropic, homogeneous and flat?
CMB – 380.000 years
Inflationary era
Here and now – 14 x 109 years
common causal region
BIG BANG
• The Schwarzschild black holes
• Vacuum, static, spherically symmetric spacetime
Black holes
),( gM
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Stationary black hole
Stationary black hole
Question:
• why cannot we hold a camera with a rope down inside the hole, record some information and bring it back?
Stationary black hole
Question:
• why cannot we hold a camera with a rope down inside the hole, record some information and bring it back?
• Why a falling astronaut cannot retract back his knees just after they entered the horizon?
Tim
e
Spherically symmetric star collapse and black hole formation
Space
Tim
e
Spherically symmetric star collapse, black hole formation and further evaporation via Hawking radiation
Hawking radiation
kGM
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4
Space
Tim
e
The NO information loss paradox
Space
Tim
e
Space
Possible(?) quantum gravity role to information loss in black holes
?????
THANK YOU