INTRODUCTION and CONCEPTUAL ASPECTS of GENERAL...

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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4R

topology) trivial(and

isotropy andy homogeneit space with Eqs.Einstein ofsolution vacuum

0abG

Physics in Minkowski spacetime

),( 4 R

4R

Relativity Special

),(),( 3 RR

3R

spacetimeGalileu in physicsNewtonian

Physics for slow-moving systems in Minkowski spacetime

R

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,...

TIM

E

• Events, particles, strings,...

TIM

E

• Fair clocks: associate a real number to each visited event

TIM

E

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

TIM

E

Galileu spacetime

• Time is absolute

TIM

E

BA tt

BA

P

Q

BA

BA tt

Q

P

Galileu spacetime

• Time is absolute

TIM

E

BA tt

BA

P

Q

0 AtA

P1

• Simultaneity

metricEuclidean an with endowed events

ussimultaneo of classes eequivalenc ),( 3 R P1 P2 P3

TIM

E

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

E

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

),( 4 R

BABA ttDD

BD

AD

later) justified be tolines(straight congruence inertial same pertain to

shomogeneou is spacetime :fact alExperiment

TIM

E

scongruence inertial are there:spacetime Minkowski

BABA ttDD

BD

AD

congruence inertial same pertain to

shomogeneou is spacetime :fact alExperiment BA tt :fact alexperiment

TIM

E

At

Bt

linesstraight by drepresentescongruence inertial

TIM

E AtAB tt

:ATTENTION

Minkowski spacetime

ORt

O

Q

R

S

OSt

A

const ),( OSOROSOR ttttF

imply

and

equivalent

are scongruence inertialall

absolute are rayslight

Minkowski spacetime

ORt

O

Q

R

S

OSt

A

OR't

O

Q

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

R

S

O

Q

R

S


linety simultanei

O

Q

R

S

O

Q

R

S


linety simultanei

),( OSOROQ ttFL

Fair rods: associate a real number to pairs of simultaneous events.

O

Q

R

S

O

Q

R

S


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

P

0F

Q3

Q2

Q1

Q4

0F

0F

0F

conelight

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

Q

t

Minkowski spacetime

O

R

S

QS Q

)(st

0t

)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



Q Q S

Q t



S

Q

Q

QL

summary - events label toProcedure

), QQ( Lt

Minkowski spacetime

O

), QQ( LtQ

)(st

1

1)light ( sL

), PP( LtP

), RR( LtR



S

Q

Q

QL



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

)','

),

QQ

QQ

(

(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

E

AB

stD BB 1

2/1)]2/[cos( AB DD

Minkowski spacetime

A

A

B

B

O

2/1)]2/[cos( AB DD

1

1

11

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

)','

),

QQ

QQ

(

(Q

Lt

Lt

Minkowski spacetime

A

A

B

B

O

Q

1

1

11

t

L

't

'L

)','

),

QQ

QQ

(

(Q

Lt

Lt

Question: what is the relation between (t,L) and (t’,L’)?

Minkowski spacetime

A

A

B

B

O

Q

1

1

11

t

L

't

'L

)','

),

QQ

QQ

(

(Q

Lt

Lt

)tan(,1/1

)(',)('

2

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

222

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

2t

L


A

O

t

1t

2t

t

tttttL

2

4 2

2

2

1

222

2

2

1

L

A

O

t

1t

2t

L

const,s iiO


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 ''

21

QQ

QQ

12tt

A

A

B

B

O

1

1

11

t

L

't

'L

Q1

Q2

• Time dilation

.congruence blue theof membersdifferent by two measured

eproper tim gsubtractinby obtained interval time12 QQ tt

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

• 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


B

B

O 1

1

't

'L

Q1

Q2

congruenceblack theof linety simultanei



A

A O

1

1

t

L

Q3 Q2

congruence blue theof linety simultanei



A

A

B

B

O

1

1

11

t

L

't

'L

Q1

Q3 Q2

21/1,/)''( vLLLL 1232 QQQQ


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

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 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

g

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• 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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• 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


• Free test particles and light rays follow geodesics in spacetime.

General Relativity



• Massive free test particles follow timelike geodesics (geodesics which (locally) maximize the spacetime distance between events).


• 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

l del

ay o

f ab

ou

t 1

bill

ion

th o

f a

seco

nd

The time delay in gravitational fields is experimentally confirmed up to 1 in 104 parts

v

2

AB

22

B

A

/21

/1/2

/2

cghtt

cvcLt

cLt

h

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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hg

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Gravitational waves

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c

Gh

chT

c

GG t

hg

abab 21

4

2

2

2

4,

1618

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

g

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abab Tc

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4

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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






Questions:

• What was “before” the big bang?


BIG BANG






Questions:


• The big bang is a single singular event? Did it happen “somewhere”?


BIG BANG






Questions:



• How large was the universe just after the big bang?


BIG BANG






Questions:




• May distant enough galaxies move away faster from us than light velocity?


BIG BANG






Questions:





• What is dark energy? Is it a kind of ether?


BIG BANG






Questions:





• What is dark energy? Is it a kind of ether?

• Is the solar system growing up as the Universe expands?


Question:

• Why is the universe so isotropic, homogeneous and flat?


Inflationary era


common causal region

BIG BANG

• The Schwarzschild black holes

• Vacuum, static, spherically symmetric spacetime

Black holes

),( gM

g

M

abab Tc

GG

4

8

Stationary black hole


Question:

• why cannot we hold a camera with a rope down inside the hole, record some information and bring it back?


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

cT

8

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

INTRODUCTION and CONCEPTUAL ASPECTS of GENERAL...

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