BLACK HOLES

INTRODUCTION

• 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. In this theory gravity is not a force. It is the curvature of space and time, or spacetime. Planets move the way they do because of the way spacetime is warped in their vicinity.
   
• According to general relativity theory if any object, for example a star, is squeezed below a certain critical radius the gravity will overwhelm all known forces and the object will be crushed without limit to a singularity.
• A singularity can be thought of as an edge in spacetime where space and time stop. The critical radius is called the Schwarzschild radius, in honor of Karl Schwarzschild who, in 1916, found the solution, which later was interpreted as that of a stationary black hole. (Schwarzschild found this solution within months of Einstein's publication of his equations of general relativity. Remarkably, Schwarzschild did this work while a frontline soldier of World War I. Sadly, shortly after his impressive calculations, Karl Schwarzschild was killed in action at the front.)

  The Schwarzschild radius  is the radius at which the escape velocity from the star is equal to the speed of light. The escape velocity is the initial speed needed for an 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 relativity no material object can travel faster than light. Therefore, any material that approaches nearer than the Schwarzschild radius, of a collapsed object, would be trapped forever because to escape the material 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 material 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.
   
• The Schwarzschild radius, R, is calculated using
   
  • • R = 2GM/c²
       

  where G is Newton's gravitational constant, M the mass and c the speed of light. Notice that he Schwarzschild radius increases in 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. We conclude that if the Sun (whose radius is 700,000 km) were crushed into a sphere of radius less than or equal to 3 km the Sun would become a black hole. A black hole of mass 50 million solar masses would have a Schwarzschild radius of 150 million km, that is, 1 AU. Such a black hole would just fit within the Earth's orbit!
   

• 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.

DO BLACK HOLES EXIST?

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 source flickers on time scales of about one hundredth of a second. 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, to  keep the flickering in sync, in about one hundredth of a second.

This implies that  Cygnus X-1 must be smaller (probably much smaller) than  1/100 th 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 supergiant 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:

• • (m₁+m₂)p² = a³

including information about the radial velocities of the stars. The mass of the x-ray source turns out to be about 7 solar masses!  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 a black hole (bound in orbit about the supergiant) that is stripping material from the supergiant. The black hole and its partner orbit each other in about 5.6 days.

As the material is sucked into the black hole it is accelerated to enormous speeds; that is, the material heats up. The material becomes hot enough to emit the intense x-rays that we observe.

INSIDE A BLACK HOLE

• After matter has fallen into a black hole, from the point of view of an observer far from the event horizon, all traces of its former complexity is lost. The only properties that can be observed are mass, electric charge and angular momentum. From the vantage point of an external observer black holes are very simple objects, of  which there are basically two kinds:
   
  • 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 the fact than in such rotating spacetimes a spaceship could go off on a journey and return before it set off; that is, 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. Perhaps, 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. However, 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 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 the universe in a finite amount of your local time. 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.