 | INTRODUCTION |
| DO BLACK HOLES EXIST? |
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INSIDE A BLACK HOLE
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In 1783 John Michell, a Cambridge professor, pointed out that a star that was sufficiently massive and compact would have such a powerful gravitational field that even light would not be able to escape from it.
In 1915 Albert Einstein introduced a new theory of gravity called General Relativity.
Einstein's great insight was to recognize that objects like planets move the way they do, not because of a mysterious force, called gravity, which acts at a distance (as Newton suggested), but because the space and time in which they move is warped in their vicinity by the presence of matter and energy.
Spacetime
Space and time are so tightly bound together, in fact, that we prefer to refer to them as a single 4-dimensional entity called spacetime.
Planets orbit the Sun simply because they are forced to move in a spacetime that has been warped by the Sun's mass.
It is important to note that not only is space warped but also time. Indeed, clocks run slowest where the warping of spacetime is greatest. Since the gravity at the Sun's surface is stronger than that at the Earth's surface a clock at the Sun would run slower than one on Earth.
Critical Radius
The general theory of relativity predicts that if any object, for example a star, is squeezed below a certain critical radius the gravity will overwhelm all known forces. If this happens the inescapable conclusion is that the object will be crushed without limit to a point of infinite density, called a singularity.Not all singularities are points. A better way to think about a singularity is as an edge to spacetime where space and time stop.
Karl Schwarzschild
In 1916, Karl Schwarzschild found the first exact solution to Einstein's general relativity equations. This solution was interpreted as describing the spacetime geometry around a stationary mass.
Later it was understood as also describing the spacetime geometry of a stationary black hole. That solution provided an exact formula for the critical radius.
Remarkably, Schwarzschild found his solution within months of Einstein's publication of his equations of general relativity. Even more astonishing, he did this work while a frontline soldier in World War I. Sadly, shortly after his impressive calculations, Karl Schwarzschild died at the front.
The critical radius is called the Schwarzschild radius, in his honor.
Schwarzschild Radius
The Schwarzschild radius is
the radius at which the escape velocity from the star is equal to the speed of light.
Recall that the escape velocity is the initial speed needed for an object to escape all the way to infinity from another object, for example a star. On Earth you would need an initial speed of at least 11 kilometers per second (7 miles/second) to escape to infinity.
According to the theory of relativity no material object can travel faster than light. Therefore, any material object that approaches closer than the Schwarzschild radius, of a collapsed object, would be trapped forever because to escape the object would have to travel at a speed greater than that of light, which, according to relativity theory, is impossible.
Such a collapsed object from which no object can escape is called a black hole.
We believe that if no anti-gravity forces exist then any star more massive than about 10 solar masses will collapse to a black hole.
Schwarzschild Radius Formula
The Schwarzschild radius, R, is calculated usingwhere G is Newton's gravitational constant, M the mass of the collapsed object and c the speed of light.RS = 2GM/c2
Notice that the Schwarzschild radius increases in direct proportion to the mass of the collapsed object. Therefore, an object that is twice as massive as another will have a Schwarzschild radius that is twice as big.
Example: The Sun's Schwarzschild radius is about 3 km. That is, if the Sun (whose radius is 700,000 km) were crushed into a sphere of radius less than 3 km the Sun would become a black hole.
A black hole of mass 50 million solar masses would have a Schwarzschild radius 50 million times bigger than that of the Sun, that is 150 million km, or 1 AU. Such a black hole would just fit within the Earth's orbit!
Event Horizon
The Schwarzschild radius defines a one-way boundary (a membrane) that separates the inside of the black hole from the rest of the universe. This boundary, which is not made of any material, is called the event horizon.All events that occur inside the boundary are forever hidden from us. Therefore, in a real sense the inside of a black hole is disconnected from the rest of the universe in both space as well as in time.
Gravitational Time Dilation
Gravity slows time down.
Suppose that you are far away from a black hole, but a friend of yours decides to spend a long time near its event horizon and is able, somehow, to return to you. You would find that she had aged less than you. How much less depends on how close to the event horizon she traveled.
We can get an idea of how much time slows down in a gravitational field from the formula
t = t0Ö(1 - RS/r)
where t is the time she has spent at a radius r from the singularity and t0 is your elapsed time.
Example: Suppose she goes to within 1% of the event horizon (that is, r = 1.01* RS) and stays there for t0 = 10 years, according to your clock. Upon her return you would find that she had aged only one year!
Black holes are so strange, that for a long time they were thought to be just a theoretical curiosity with no relevance to our world. But in the 1970s a powerful x-ray source, called Cygnus X-1, was discovered lying about 8000 light years from us.
This x-ray source flickers on time scales of about one hundredth of a second. Moreover, observations suggest that every part of the source changes its brightness at the same time. That can only happen if the source is small enough for some influence to travel from one part to the other in about one hundredth of a second to keep the flickering in sync.
This implies that Cygnus X-1 must be smaller (probably much smaller) than 1/100 of a light-second across; that is, smaller than the size of the Earth!
We have some idea of the mass of the x-ray source because if forms a binary with a blue super giant star (HDE 226868) whose mass is expected to be about 30 solar masses. Information about the masses in a binary system can be had by using (a more exact form) of Kepler's 3rd Law:
including information about the radial velocities of the stars. m1 and m2 are the masses of the two stars. The mass of the x-ray source turns out to be about 7 solar masses!(m1+m2)p2 = a3
With such a large mass squeezed into so small a volume (less than the size of the Earth) the best explanation for Cygnus X-1 is that it is a black hole (bound in orbit about the blue super giant) that is stripping material from the super giant star. The black hole and its partner orbit each other in about 5.6 days.
As the material is sucked into the black hole the material is accelerated to enormous speeds; that is, the material heats up. The material can become so hot that it is able to emit x-rays. It is these x-rays that we believe we are observing.
Schwarzschild black holes
These black holes are formed from non-rotating matter. They have a point singularity at the center, which is shrouded by an event horizon.Kerr black holes
In 1963 Roy Kerr (a New Zealander) worked out the structure of a black hole formed from rotating matter. He showed that there is a region outside the event horizon, called the ergo-region, that drags space and time around with the rotating black hole, rather like a vortex. Because of the rotation the singularity at the center of a Kerr black hole is a ring, rather than a point.Time travel
In 1949 Kurt Godel (famous for his proof of the impossibility of proving all true statements in any logical system that includes the rules of arithmetic) discovered a solution of Einstein's equation that described a rotating spacetime.Einstein was very disturbed by Godel's solution because it predicted that in such rotating spacetimes a spaceship could go off on a journey and return before it set off! Einstein was shocked that his theory of general relativity allowed the possibility of time travel.
We do not know whether or not such spacetimes can exist in our universe. Some, like Hawkings, argue that because we have no compelling evidence of visitors from our future, who presumably would have had enough time to develop time travel, we can assume that no such spacetimes exist in the part of the universe to which, in principle, we have access.
Nonetheless, such strange time-warping spacetimes are predicted to exist within Kerr black holes!
Wormholes
In 1935, Einstein and Nathan Rosen discovered solutions of Einstein's equations that they interpreted as describing bridges between different parts of spacetime. Today we call these Einsten-Rosen bridges wormholes.
The theory predicts that the ring singularity inside a Kerr black hole could be a gateway to another part of spacetime, via a wormhole.
Unfortunately, passage through this ring would be truly a one-way ticket to oblivion: for as you passed through the ring you would witness the entire future history of (a part of ) the universe in a finite amount of your local time. So from your point of view the universe outside the black hole would, in a short time, cease to exist.
More likely, however, you would be vaporized by the infinitely blue-shifted radiation with which you would be bathed, coming from the ever more rapidly evolving universe.
Either way, you have a ticket to oblivion!