Download Professor Emeritus, University of Canterbury Yevgeny Lifshitz

Cracking the Einstein Code
Roy Patrick Kerr
Professor Emeritus, University of Canterbury
Yevgeny Lifshitz Professor of Physics
International Centre for Relativistic Astrophysics
Network, Pescara & Rome, Italy
E-Mail received on 21Jul 2009
• To: [email protected]
•
Subject: omg you're still alive :)
• How old are you? I thought you have joined
Schwarzschild in afterlife academy of physics long time
ago... but now I've found your email address and
correspondence at some crank site dated by xxi century.
Amazingly cool. Too bad, however, that the progress on
GR is so slow.
Andromeda Galaxy: The “Little Cloud” has been known since 905
but it was only in 1923 that it was known to be extragalactic.
Crab Nebula: 1054
Quasars = “Quasi-stellar radio sources”
After the end of world war II radio telescopes were constructed
around the world. Many strange radio sources were observed but
the sensitivity of the telescopes was very poor and they were
unable to locate the sources accurately. In 1960 3C48 was shown
to have an optical counterpart, a faint blue “star” with an anomalous
spectrum. John Bolton, an Australian radio astronomer, thought that
it had a large redshift and that it was therefore outside the milky way,
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. He realised that
the spectrum of the optical counterpart was red-shifted and so quasars were
identified as galactic objects.
Parkes Telescope
Taken with backyard telescope
Quasar 3C-273
Optical Picture
Optical and Radio
Newton’s Theory of Gravitation
The inverse square law: The force between two bodies is proportional
to the product of their masses and inversely proportional to the
square of the distance between them.
GM
1M
2
F
R2
If one of the bodies moves then an immediate change is felt
across the universe. This means the speed of gravitational
signals is infinite. Newton thought that this was ridiculous, but he
could not think of a good alternative
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.
Second Test: Bending of Light
1.75”
• Einstein’ theory predicted that light passing close to a massive body
would curve towards it. This amounts to 1.75” close to the sun.
The only time that photographs can be taken successfully near the
sun is during a solar eclipse so the observations had to wait for a
suitable moment. This did not occur until after World War I.
Bending of Light around the Sun
• Light bending by a strong gravitational field was not a
new idea. In 1801 J. Soldner had pointed out that
Newtonian gravity predicts that starlight will bend around
a massive object, but the effect is only half that predicted
by General Relativity and calculated by Einstein.
• Sir Arthur Eddington organised two of the most famous
scientific expeditions in history to observe this bending
during a solar eclipse in 1919. He led the first of these to
Principe in Africa and sent a second to Sobral in Brazil,
in case it was raining at the first site. In fact, the weather
was bad in Principe but Eddington was still able to take
some useful photographs.
Agreement with Relativity, disagreement with the Newtonian alternative
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 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.
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
 2M 
 2M
2
2
2
2
2
ds 2  1 
dr

r
(
d


sin

d

)


1 
r 
r


 2
dt

There was something strange happening at what is called the Schwarzschild
radius where the factor in brackets is zero, the event horizon.
r  2M
At first it was thought that the metric was “singular” on this sphere, i.e. that
the curvature became infinite as one approached it.
“Black Holes” appear!
•
Eddington showed that the event horizon is well behaved but that there is still
something strange happening there.
•
A spaceship can approach as close as it likes to this surface 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.
•
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. However, the
Schwarzschild metric gives the gravity outside and not on the inside of the
physical object at the centre. There is no singularity inside the earth.
•
The density of the Sun as it collapsed inside its event horizon would be
20,000,000,000,000 kg per cubic centimetre, denser than a Neutron star.
•
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!
The search for “rotating Schwarzschild”
• Physicists wondered whether a spinning object could form a Black
Hole or whether the spin would make the event horizon disappear.
• For 40 years Relativists searched for any rotating generalisation of
Schwarzschild, without any success.
• The obvious approach was to assume the exterior field to be
rotationally symmetric and time independent. This eliminates two out
of the four coordinates in Einstein’s equations.
• The equations where put into many elegant and beautiful forms
(particularly by Papapetrou) but no rotating solution was found.
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!)
•
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 its source.
Alexey Z. Petrov (1910-71)
1960: Let’s look for “Algebraically Special”
solutions of the empty Einstein equations!
Attempts to find the most general
Algebraically Special metric
• Ivor Robinson and Andrzej Trautman simplified the equations further
and found a rather nice but non-spinning solution of the Einstein
equations.
• In 1962-3 a group centred in Pittsburgh announced that they had
solved the problem and that there were no spinning solutions
• I had been studying the same problem and was very surprised at
this result. Ivor Robinson told me later that he and Andrzej also
disbelieved it!
• 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 at
that time and was in the same small apartment building as Alan.
• Neither Alan nor Alfred could see anything wrong with the paper.
Alan then gave it to me to review.
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.
•
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 centre 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 paper was wrong. 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 symmetric. This reduces the field equations 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 metric in Kerr-Schild form
The metric is rather nasty in the coordinates originally used to find it, but I
realise that it can be put into the simple Kerr-Schild form,
3
2
Mr
2
ds 2  dx 2  dy 2  dz 2  dt 2  4
(
dt

.....)
r  a2 z 2
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!!
NO!!
Event Horizons for Kerr Black Hole
Inner Horizon
Singular Ring
Outer Horizon
By Fulvio Melia
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 Quasars. Three hundred
astronomers/astrophysicists and fifty relativists attended.
• There were many theories presented but none that had broad appeal.
•
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 was far too unlikely as all bodies rotate.
• About 6 months previously I had constructed “Rotating Schwarzschild”.
I was invited to give a 10-15 minute talk to the meeting.
• The astronomers and astrophysicists were totally uninterested and
ignored the talk.
• Papapetrou screamed at them that he and others have worked for 30
years to find this metric and that they should listen. They ignored this!
Afterwards
•
It is soon realised that matter close to a Black Hole forms an accretion disk. As this
spirals closer and closer to the BH it heats up. This supplies the enormous flux of
radiation observed from Quasars
•
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. It may be that every galaxy formed around a
Black Hole that was created soon after the “Big Bang”. We do not know precisely
how, 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.
Stars circling Black Hole at Galactic Centre
This is a series of real photographs of the stars circling the Black Hole at the
centre of our very own galaxy. The star coming in from the top left on a
cometary orbit takes 13 years to circle the Black Hole. It has now been
observed for one complete orbit. From these orbits the mass of the central
object has been calculated to be almost 4,000,000 times the mass of our Sun.
•
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.
“Cracking the Einstein Code”
by Fulvio Melia,
with an Afterword by Roy Kerr
University of Chicago Press
October 2009