Download gravitational waves

GRAVITATIONAL WAVES
OUTLINE
• historical introduction from Newton to Einstein :
why physicists need waves
• the revolutionary idea of General Relativity
• what are black holes?
• what generates gravitational waves?
• physical principles of gravitational wave detection
• the discovery of 14 September 2015
(announced 11 February 2016)
Newton’s theory of gravitation
Galileo Galilei (1564-1642) : two important principles
• bodies of different masses fall with uniform acceleration
MEarth
a/
r2
height of
the tower
• the laws of physics are the same in
any system moving at constant speed
in a straight line : ‘‘inertial system’’
Newton’s theory of gravitation
Isaac Newton (1642-1727) :
• the force of gravity is universal : the same force to explain falling
objects and planets motion
MEarth m2
F2 = m 2 a = G
r2
Newton second law
He found the proportionality constant!
MEarth
a=G
2
r
Newton’s theory of gravitation
• one law that worked very well to explain the motion of planets
gravitational field
generated by a
mass M
F
g(x) =
=
m
M
G 2 r̂
r
Newton’s theory of gravitation
• one law that worked very well to explain the motion of planets
• the same law in every inertial reference system
• time is absolute : it is the same in all reference systems
Poisson equation
@ @
(x) = 4⇡ G ⇢(x)
ij
@xi @xj
ALL VELOCITIES
ARE POSSIBLE !
Newton’s theory of gravitation
double derivative
in space:
how the potential
varies in space
i=j
gravitational
potential:
related to the
acceleration
(before: a)
Newton
proportionality
constant
source:
mass density
(before: M)
@ @
(x) = 4⇡ G ⇢(x)
ij
@xi @xj
EVERYBODY HAPPY…
but in the meanwhile…
in the meanwhile : ELECTROMAGNETIC WAVES
James Clerk Maxwell (1831-1879)
in the meanwhile : ELECTROMAGNETIC WAVES
James Clerk Maxwell (1831-1879)
Equations to describe the propagation of light (EM waves) :
1 @ @
A
(x,
t)
µ
c2 @t @t
@ @
4⇡
Aµ (x, t) =
Jµ (x, t)
ij
@xi @xj
c
double derivative
in space:
how the potential
varies in space
i=j
gravitational
Newton
potential:
proportionality
related to the force
constant
(before: F)
@ @
(x) = 4⇡ G ⇢(x)
ij
@xi @xj
in the meanwhile : ELECTROMAGNETIC WAVES
James Clerk Maxwell (1831-1879)
Equations to describe the propagation of light (EM waves) :
1 @ @
A
(x,
t)
µ
c2 @t @t
@ @
4⇡
Aµ (x, t) =
Jµ (x, t)
ij
@xi @xj
c
electromagnetic
potential
(before: gravitational)
BRAND NEW TERM !
double derivative in time:
how the potential varies in time
c : SPEED OF LIGHT
another
proportionality
constant
source:
charge and current
density
(before: mass density)
in the meanwhile : ELECTROMAGNETIC WAVES
James Clerk Maxwell (1831-1879)
Equations to describe the propagation of light (EM waves) :
1 @ @
A
(x,
t)
µ
c2 @t @t
@ @
4⇡
Aµ (x, t) =
Jµ (x, t)
ij
@xi @xj
c
• the EM wave propagates at a finite speed
km
c = 300 000
sec
·
• the propagation IS NOT INSTANTANEOUS
• causality : if the source changes, it takes a finite time for the
information to propagate in the electromagnetic potential
the problem of action at a distance
1. event : turn
on light bulb
3. event :
light arrives
(x1 , t1 )
(x2 , t2 )
x2
x1 = c (t2
t1 )
2. EM waves propagate…
BUT THE SAME PROBLEM IN GRAVITATION THEORY GOES :
1. event: the Sun
disappears
(sudden change in
mass)
@ @
(x) = 4⇡ G ⇢(x)
ij
@xi @xj
MISS TERM IN THE EQUATION:
NO WAVE TO PROPAGATE…
3. the Earth
feels it
immediately !
the problem of action at a distance
1. event : turn
on light bulb
Newton theory of
gravitation is not causal !
3. event :
light arrives
(x2 , t2 )
(x1 , t1 )
x2
x1 = c (t2
t1 )
2. EM waves propagate…
BUT THE SAME PROBLEM IN GRAVITATION THEORY GOES :
1. event: the Sun
3. the Earth
@ @
disappears(big disaster
feels it
(x)a= physicist)
4⇡ G ⇢(x)
for
ij
@xi @xj
(sudden change in
immediately !
mass)
MISS TERM IN THE EQUATION:
NO WAVE TO PROPAGATE…
The revolution by Einstein
1. event : two black holes inspiral
and merge somewhere in the
universe and a very long time ago
x2
t2
x1 = 1.26 · 1022 km
9
t1 = 1.3 · 10 years
3. on Earth, now, the LIGO
detector measures GW :
therefore it detects the event
‘‘black hole merger’’
2. Gravitational waves propagate
at the speed of light…
Two new theories: special and general relativity
Albert Einstein (1879-1955)
• there is a maximal speed: the speed of light c
• it is the same in all reference frames
• time is not absolute: it depends on the reference frame
• for Newton space and time were different; for
Einstein there is a unique entity, spacetime
MAXIMAL VELOCITY c !
Two new theories: special and general relativity
SPACETIME :
time
trajectory of a
light beam
space
Two new theories: special and general relativity
Albert Einstein (1879-1955)
• gravitation is no longer seen as a force, like in Newton theory,
but as a property of spacetime
• the presence of energy bends spacetime
emptiness : flat spacetime (also called metric)
Two new theories: special and general relativity
Albert Einstein (1879-1955)
mass is energy : it bends spacetime
(space is curved, time flows slower)
Two new theories: special and general relativity
Albert Einstein (1879-1955)
mass bends spacetime : new explanation of orbits of planets
objects move on trajectories determined by
the geometry of spacetime
Two new theories: special and general relativity
Albert Einstein (1879-1955)
mass bends spacetime : new explanation of orbits of planets
objects move on trajectories determined by
the geometry of spacetime
even light is deflected!
Two new theories: special and general relativity
Albert Einstein (1879-1955)
bigger masses induce bigger curvature in spacetime
Two new theories: special and general relativity
Albert Einstein (1879-1955)
Black hole: the mass is so big that spacetime is infinitely curved
what is a black hole ?
• region of space of infinite curvature: nothing can escape, not
even light (event horizon)
• for an observer far away, time becomes infinite nearby a BH
• BH arises because of the compression of mass into a tiny space
2G
M
Rs = 2 M = 2.95 km
c
M
M = 2 · 1030 kg
R = 690· 000 km
for the Earth to be a black hole :
it should be compressed into 8 mm
• a black hole can arise after the collapse of a big star : tenths of
solar masses
• or it can be formed at the center of a galaxy : millions/billions
of solar masses
SAGITTARIUS A
at the center of the Milky Way
M = 4· 000· 000 M
·
·
Rs = 12 000 000 km
Two new theories: special and general relativity
Albert Einstein (1879-1955)
gravitational waves are
SMALL PROPAGATING RIPPLES IN SPACETIME
Two new theories: special and general relativity
Albert Einstein (1879-1955)
gravitational waves are
SMALL PROPAGATING RIPPLES IN SPACETIME
1 @ @
h
(x,
t)
µ⌫
c2 @t @t
time derivative :
propagation
at speed of light
@ @
hµ⌫ (x, t) =
ij
@xi @xj
‘‘wavy’’ component of
metric - spacetime
(before : gravitational
potential)
Transverse-traceless,
spin 2
16⇡G
T
(x,
t)
µ⌫
c4
yet another
proportionality
constant
source:
time-varying energy
momentum tensor
??
Two new theories: special and general relativity
Albert Einstein (1879-1955)
gravitational waves are
SMALL PROPAGATING RIPPLES IN SPACETIME
generated by a
TIME DEPENDENT (MOVING) QUADRUPOLAR
DISTRIBUTION OF MASS
NO SPHERICAL SYMMETRY
such as a deformed
rotating neutron star
Two new theories: special and general relativity
Albert Einstein (1879-1955)
gravitational waves are
SMALL PROPAGATING RIPPLES IN SPACETIME
generated by a
TIME DEPENDENT (MOVING) QUADRUPOLAR
DISTRIBUTION OF MASS
such as two black holes orbiting around each other
MOVIE
https://www.youtube.com/watch?v=1agm33iEAuo
How to detect gravitational waves ?
it is very difficult, because GW have very small amplitude
and interact very weakly
METHOD ONE : INDIRECT DETECTION
• GW carry away energy from the binary system
• therefore the radius of the orbit of the two compact objects gets
smaller with time because of this energy loss
• one could measure the rate of shrinking of the orbit
• and then compare with what predicted by the theory
• if they agree… proof of existence of GW !
• but which astronomical object can be used for that ?
certainly not black holes…
How to detect gravitational waves ?
METHOD ONE : INDIRECT DETECTION
•
•
•
•
binary of neutron stars emitting beamed EM waves
detected by a radio telescope at very precise time intervals
from the rate of the pulses infer the rotation period
does the orbit shrink?
How to detect gravitational waves ?
METHOD ONE : INDIRECT DETECTION
method used with the
Hulse-Taylor binary pulsar
·
d = 21 000 ly
M = 1.44 M
first indirect
evidence of GW
NOBEL PRIZE IN
1993 TO HULSE
AND TAYLOR
still, it’s not the GW that directly
interacted with a detector….
prediction of
GW emission
from general
relativity
How to detect gravitational waves ?
METHOD TWO : DIRECT DETECTION
what is the effect of a GW passing perpendicularly
through a ring of test masses ?
two independent propagation modes, called polarisations
how can one measure the displacement in the masses ?
How to detect gravitational waves ?
answer : using a laser between massive mirrors
interferometer
LIGHT IN THE
DETECTOR
NO LIGHT IN
THE DETECTOR
How to detect gravitational waves ?
answer : using a laser between massive mirrors
interferometer
Advanced LIGO interferometers
4 km armlength, laser power 125 W, beam radius 6 cm, test masses 40 kg
sensitive to length differences
L ⇠ 2 · 10
18
m
size of a molecule with respect to the solar system!
pictures of the VIRGO interferometer in Pisa
comparable performance, 3 km arm length,
start operating for 2017 observation run
pictures of the VIRGO interferometer in Pisa
pictures of the VIRGO interferometer in Pisa
pictures of the VIRGO interferometer in Pisa
pictures of the VIRGO interferometer in Pisa
Gravitational wave detectors worldwide
• we need many detectors to distinguish the signal from the noise
(earthquakes, atmosphere, passing trucks… )
• the same signal should be seen after a time interval
d=c
t
about 7 msec for the two LIGO detectors
• three detectors to localize the source and measure the polarization
14/9/15 : the first direct detection by the two LIGO interferometers
(7 msec later)
14/9/15 : the first direct detection by the two LIGO interferometers
HOW DOES IT SOUND LIKE ?
https://www.youtube.com/watch?v=TWqhUANNFXw
14/9/15 : the first direct detection by the two LIGO interferometers
M1 = 36 ± 5 M
M2 = 29 ± 4 M
Mfim = 62 ± 4 M
Rs ' 210 km
x2
t2
x1 = 1.26 · 1022 km
9
t1 = 1.3 · 10 years
what next ? a new window on the universe !
• Nobel prize for this discovery
• only 16 days of data : many more events expected
• when VIRGO joins in, better sky localization : EM follow-up
with telescopes
• beginning of the era of GW astronomy : many new informations
on objects that generate GW
• new tests on general relativity
• maybe an unattended signal (from the early universe?)
• launch of a GW interferometer in the sky, called eLISA