Transcript Kerr - ICRANet

Black Holes and Dark Energy
Roy Patrick Kerr
Yevgeny Lifshitz Professor
INTERNATIONAL CENTER FOR RELATIVISTIC ASTROPHYSICS
ICRAnet - Pescara & Rome, Italy; Nice, France
Kerr metric in Kerr-Schild form
The metric is rather nasty in the original coordinates but it can be put
into the Kerr-Schild form,
3
2
Mr
2
ds 2  dx 2  dy 2  dz 2  dt 2  4
(
dt

.....)
r  a2 z2
When a = 0 the metric reduces to Schwarzschild mass with mass M
DOES IT ROTATE WHEN “a” IS NONZERO?
I tell Alfred Schild, the director of the “Gravitational Research Centre” in Austin, that
I am going to my office to calculate the angular momentum of the last remaining
hope. He says “Fine, I am coming too!”
Alfred sits in an armchair smoking his pipe while I chain smoke cigarettes and
calculate. Finally, I announce
YES!!
Event Horizons for Kerr Black Hole
Inner Horizon
Singular Ring
Outer Horizon
By Fulvio Melia
Science is Like a Religion??
•
•
Eather theory:
EIH, or Post-Newtonian approximations: In 1938 Einstein, Infeld and
Hoffman wrote a seminal paper in which they used the “quasi-static”
approximation method to calculate the gravitational forces between slow
moving particles. This was originally called the EIH method but has been
renamed as the “Post-Newtonian” method.
•
fggg
G   8 T 
1905 - Special Relativity
Lorentz
Grossman
Einstein
•
The velocity of light, c, is the same for all observers.
•
The equations of physics are the same for all.
•
No signal travels faster than light.
•
Newton’s theory says gravitation acts instantaneously.
E  mc
2
Newton’s theory is inconsistent with Special Relativity
General Relativity
Albert Einstein
Marcel Grossmann
David Hilbert
“Matter and energy curve space and time”
G   8 T 
The geometry of space-time determines the motion of all bodies in it. The
quickest path between two space-time events is called a geodesic and this
is the equivalent of a straight line in Euclidean space. As the Earth moves
around the sun it thinks that it is moving on a straight line!
The Solar System
43 sec/century
perihelion
aphelion
• The place where a planet is furthest from the sun is called the “aphelion”.
• In Newton’s theory this remains the same, orbit after orbit.
• It was observed that the aphelion and perihelion of the planet Mercury
advance by 43” (43 seconds of arc) per century, and so rotate completely
around the sun every 3,000,000 years.
• When Einstein’s theory gave the same 43” per century! This result was
enough for Einstein to have complete faith in his theory.
Bending of Light
1.75”
• Einstein’ theory predicted that light passing close to a massive body
would curve towards it. This amounted to 1.75” close to the sun. Sir
Arthur Eddington organised two of the most famous scientific
expeditions in history to observe this bending during a solar eclipse
soon after the end of the first World War. He led the first of these to
Principe in Africa and sent a second to Sobral in South America.
A reproduction of one
of the negatives taken
by Eddington's group
using the 4-inch lens at
Sobral, in Brazil. The
positions of several
stars are indicated with
bars. When compared
to other photographs
taken of the same
region of the sky, it
became apparent that
those closest to the rim
of the Sun appeared to
have shifted slightly.
A modern example of light-bending . There is a Quasar (a very bright object) behind
the bright galaxy in the centre of the picture, but 5 times further away. Its light
forms an “Einstein ring”.
This bending of light is being used to study the universe. The amount of distortion of
images tells us about the total mass in any region. This is the best evidence for the
existence of dark matter clustered around galaxies formed from standard matter.
Question: Does dark matter clustering cause ordinary matter to collect around it
The Schwarzschild Solution
Within a year of Einstein proposing his theory, Professor Karl
Schwarzschild constructed a metric that was to be the most
important solution of Einstein’s equations for the next 40+ years.
It gave the gravitational field outside a “spherically symmetric”
body, i.e. one that looks the same from all directions.

1
1873-1916


2
GM
2
GM




2
2
2
2 2
2
2
2
ds

1

dr

r
(
d

sin
d
)

1

c
d
2

2

rc
rc




There was something strange happening at what is called the Schwarzschild
radius where the factor in brackets is zero. The sphere with this radius is called
the event horizon.
r
2GM
c2
At first it was thought that the metric was “singular” on this sphere, i.e. that the
curvature became infinite as one approached it.
The Dreaded “Black Hole” Appears!
•
Eddington showed that the event horizon is well behaved but that there is
something very strange happening there.
•
A spaceship can approach as close as it likes to this horizon and still escape
from the gravitational field of the central body, but if it ventures inside the event
horizon then there is no return. It is drawn rapidly to the central singularity.
•
For a normal body such as the earth or the sun, the event horizon would be
deep inside. However, it is then a meaningless concept as Schwarzschild gives
the gravity outside, and not on the inside of the physical object.
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If the Earth and the Sun were to collapse to black holes then the radii of their
event horizons would be 1cm and 3km respectively.
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The density of the Sun as it collapsed inside its event horizon is
20,000,000,000,000 kg per cubic centimetre, denser than a Neutron star.
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The Sun is 300,000 time heavier than the Earth. The density of the Earth as it
collapsed inside its event horizon would therefore be
1,800,000,000,000,000,000,000,000 kg per cubic centimetre.
It cannot happen!
Spin: Everything Spins.
The Search for
“Rotating Schwarzschild”
•
The gravitational field outside any non-rotating spherical star must be that
found by Schwarzschild. This field is constant in time, even though the
matter inside may evolve.
•
If the star collapses inside its “event horizon” it becomes a black hole. No
object or message can be sent from the inside to the outside of this sphere.
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For 40 years physicists searched for a spinning black hole solution of
Einstein’s equations.
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For simplicity, the star was assumed to be rotationally symmetric (like a
normal bottle or glass) and unchanging with time.
•
The equations were put into many elegant and beautiful forms but no
rotating solution was constructed.
Some additional assumption was needed.
•
Alexey Petrov was a Russian who studied
general properties of the curvature in an
Einstein space (such as our universe!)
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For almost all known physical solutions of
Einstein’s equation, including that of
Schwarzschild, the curvature had a special
property. They were all “Algebraically
Special”.
•
This property also seems to be true for the
gravitational field far from any source.
Alexey Z. Petrov (1910-71)
1960: Let’s look for “Algebraically Special”
solutions of the empty Einstein equations!
Ivor Robinson
Andrzej Trautman
They made a further assumption, leading to the
Robinson-Trautman metrics in 1962.
ds 2  r 2 P 2 (dx 2  dy 2 )  2dudr  [ ln P  2r (ln P),u  2m(u ) / r ]du 2
(ln P)  12m(ln P),u  4m,u  0,
STILL NOT ROTATING!!
  P 2 ( 2x   2y )
Attempts to find the most general
Algebraically Special metric
•
Ivor Robinson continued his study of the algebraically special space-times
where the special vector is not a gradient. This would be completed later.
•
In 1962-3 a group centred in Pittsburgh announced that they had solved the
complete problem and that there was there was only a fairly non-interesting
generalisation of Schwarzschild, NUT space.
•
I had been studying the same problem and was very surprised at this result.
Ivor Robinson told me later that both he and and Andrzej Trautman also
disbelieved it.
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A preprint containing the proof was sent to Alfred Schild and Alan
Thompson at the University of Texas in Austin. I was also at Texas U at that
time and was in the same small apartment building as Alan.
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Neither Alan nor Alfred could see anything wrong with the paper. Alan then
gave it to me to see why there were no interesting algebraically special
Einstein spaces.
Examining the paper
•
I thumbed through the paper to see where this surprising result came from –
which equation told them that the search was futile.
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I found a simple equation that seemed to be the key to their result,
•
I did not know what A was but this equation seemed to be the center of their
argument, so I looked back and found that it could not be true. The
coefficients must sum to zero because of the “Bianchi Identities” .
•
I rushed next door and told Alan that the conclusions were false. We
calculated the first of the three terms and found that it was incorrect. The
equation became
•
I then calculated the correct field equations for Algebraically Special spaces.
This was announced at a conference in New York. The author of the original
paper said “Yes, but the second coefficient was a misprint. The equation is”
•
I said “OK, then the third must be wrong!” Alan and I calculated it that night
and found that the correct equation was
21 A  12A  32 A  0  A
00  0!
Path to Kerr Solution
•
Assume algebraically special. This reduces the ten Einstein equations to five, and the
metrics dependence on the radial coordinate, “r”, is known. However, the equations are
much worse than those of Robinson and Trautman so something else is needed.
•
Assume independent of time. Now the equations are getting better, but they are still
intractable.
•
Assume axially symmetry. This reduces the field equations down to ordinary differential
equations which can be solved.
•
The space-time now depends on four real numbers, or parameters. These characterise
the metric completely. Getting near!!
•
Since I am looking for a physically interesting space-time I require the space to be
Minkowski space (special relativity) at large distances. Two parameters are removed.
This leaves a solution with only two parameters, (M, a).
Kerr-Schild metrics
•
Around Christmas 1963, after the First Texas Symposium in Dallas, I spend a
morning investigating whether there are other Algebraically Special spaces that have
the Kerr-Schild form. There seem to be a large class that depend on a “function of a
complex variable” and include the Kerr rotating solution but none of the others are of
physical interest so I just leave them on my desk.
•
Jerzy Plebansky, a very well-known Polish relativist visits Austin for Christmas. Alfred
Schild holds one of his excellent parties for Jerzy. During this I hear them mention
their interest in spaces of the Kerr-Schild form (that name had not been invented at
that time, of course).
•
I say “I think I know of a large group of those, but the result was not checked and
may be rubbish”.
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Alfred and I retire to his office and do a small calculation that shows that any metric
of this type has to be Algebraically Special.
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Next day we redo my original calculations, verifying that they were correct.
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We subsequently add an electromagnetic field to the problem, and find that there is a
natural charged version of Kerr, the Kerr-Newman charged black holes. This is also
discovered by Ted Newman by testing various ways that charged Schwarzschild
(Reissner-Nordstrom) and Kerr might be amalgamated!
Afterwards
•
It is proved a few years later by David Robinson, another New Zealander,
that there are no other spinning black hole solutions. All properties of the
star are lost when it collapses, except for its mass, angular momentum and
electric charge. John Archibald Wheeler coined the phrase “Black Holes
have no hair” to express this.
•
Do Black Holes really exist? Probably. We appear to be seeing millions or
more black holes in the universe. It may be that every galaxy formed around
a Black Hole that was created soon after the “Big Bang”. We do not know,
but Black Holes have something to do with the formation of galaxies.
•
Are Black Holes truly represented by the Kerr solution? Yes, but only in the
limit as they age. We can never see a Black Hole collapse inside its event
horizon. For us, it is always just on the verge of doing so.
•
The most famous example is at the centre of our own galaxy. It is called
Sagitarius A* and is around 4,000,000 times as heavy as the Sun.
Astronomers expect to be able to photograph it within the next ten years.
Quasars
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In the late 1950s hundreds of strange radio sources were discovered. In
1960 one of them, 3C 48, was shown to have an optical counterpart, a faint
blue “star” with an anomalous spectrum. John Bolton thought that it had a
large redshift, but this was not believed by others.
•
In 1962 the closest quasar, 3C 273, was occulted by the moon. Cyril Hazard
and Bolton took observations allowing Marteen Schmidt to identify it.
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When he calculated its spectrum he realised it was that for hydrogen, but
red-shifted, and so quasars were identified as being galactic objects.
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If they were as far away as their redshifts implied, then they were far too
energetic for all “reasonable” explanations.
•
Possible explanations: antimatter, white holes, ….
Quasar 3C 273
Quasar 3C 273 – (Photo by Hubble, copyright NASA)
First Texas Symposium on Gravitation and Astrophysics
•
In December 1963 a meeting was arranged in Dallas Texas to discuss these
newly discovered and highly energetic objects (subsequently called Quasars).
At least 300 astronomers/astrophysicists and 50 relativists attend.
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There were many theories presented but none that had broad appeal.
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Hoyle and Burbidge suggested a giant star with the mass of at least a million
suns. Even this did not produce enough energy to power the observed
Quasars. Black holes were mentioned but the non-rotating Schwarzschild
metric wass far too unlikely as all bodies rotate. Furthermore, where was the
energy coming from?
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About 6 months previously I had constructed a rotating generalization of
Schwarzschild. I was invited to give a 10-15 minute talk to the meeting.
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The astronomers and astrophysicists were totally uninterested and ignored my
talk.
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Papapetrou screamed at them that he and others have worked for 30 years to
find a metric like this and that they should listen. They ignored him also!
The identification of the first known
Black Hole in our galaxy: Cygnus X-1
Professor Remo Ruffini of ICRA showed
that this is an X-ray binary:
1. Supergiant,
M = 30 M
2. Black Hole,
M = 8.7 M
Black Hole Fed by Star
Black Hole passing in front of a field of stars
Black Hole in front of a spiral galaxy
•
This is the picture of a nearby Black Hole and a distant galaxy. It is a spiral
galaxy with a central bulge, just like ours, seen side on. Of course, no such
event has actually been photographed. It is just a computer simulation.
•
Notice how the light from the galaxy bends around the back of the Black Hole.
It gets very complicated as the Black Hole crosses in front of the galaxy.
Black hole at centre of our galaxy
1.5mm radio waves – Ap.J. 2000,Melia et al.