Einstein's General Theory of Relativity

Important thing to remember #1 - GR is a theory of gravity.

The basic idea is that instead of postulating a mysterious, invisible force between masses, GR says that

In this picture, the reason the planets orbit the Sun is not because of a central force, but rather because the planets are tracing out straight lines in curved space.

• The usual analogy of tossing a shotput on a bed and then rolling marbles along works pretty well.

• G.R. makes some predictions.
  1. Mercury's orbit will differ from the Netwonian predictions.

  2. Clocks will slow down in the presence of large gravitational fields.

  3. About 20 other tests.

  4. The deflection of starlight by large masses.

     Einstein predicted that the position of a star whose light just grazed the edge of the Sun would be deflected by 1.75 arcseconds. This is easy to measure, but the problem is the Sun is too bright...

     In 1919 astronomers made measurements during the total eclipse of the Sun and verified the prediction.

     The distortion of light when passing by large masses also causes "gravitational lenses", first identified in the 1980s, and gravitational redshifts that are routinely measured in the spectra of White Dwarfs.

• Now - this is leading towards Black Holes.

  With Newtonian gravity, photon's are massless and therefore don't feel gravitational forces. But, under GR, photons also just follow straight lines in curved space.

• Let's think about escape velocity.

  Imagine tossing rocketships up in the air from the surface of the Earth. A feeble toss and the rocketship goes up a little way then falls back to Earth because its kinetic energy did not exceed its gravitational potential energy.

  However, toss it with a larger and larger velocity and it goes higher and higher before falling back to Earth.

  There is a velocity above which the rocket ship will not return to Earth.

  This is the escape velocity and it is calculated by setting the gravitational potential energy equal to the kinetic energy of any object.

  

  where M is the mass of the object from which you want to escape and R is the radius from which you want to escape. G is the universal gravitational constant.

  For Earth, V_escape is 11 km/sec, or about 25,000 miles/hour.

  This is independent of the mass of the object you are tossing up, the only mass in the formula is the mass of the Earth.

• Now, suppose you decrease the radius of the Earth by a factor off 9. The escape velocity is proportional to .

  So, for a 1/9 SMALLER Earth (that is, smaller radius, same mass, higher density) the escape velocity goes up by a factor of 3.

• Now imagine shrinking the Earth down to about the size of a grape (1 cm).

  The Escape Velocity will have increased to

  3 x 10⁵ km/sec -> "c"

  With our new theory of Gravity, where photons are affected by "gravitational fields", if the escape velocity of an object equals or exceeds the speed of light, that object will no longer be observable as no information can escape from it.

  This is a Black Hole.

• There is a name for the critical radius for which a given mass has an escape velocity of "c" -- the Schwarzschild Radius and this is also the Event Horizon.

  So, any object that is within its Schwarzschild Radius is a Black Hole.

• This is all vaguely interesting, but it turns out that it is very, very, very difficult to shrink objects down to their Schwarzchild Radius.

• For the Sun, you have to somehow overcome Thermal Pressure, then e^- degeneracy, then Neutron degeneracy and there is no known cosmic vice that can do that for a 1M_Sun object.

• But! if we go back to a neutron star, we are starting to put together a pretty big vice.

  Thermal pressure has already been overcome, e^- degeneracy has been exceeded and it is neutron degeneracy supporting the object's mass against gravity.

• There is a limit to the mass that can be supported by neutron degeneracy

  - this is hard to calculate but it is probably between 2M_Sun and 3M_Sun

• OK, add mass to a SNII core up to say 3M_Sun and what happens? The core collapses to a ZERO radius "singularity"

  For a 3M_Sun object:

  

  So, this core would be a 3M_Sun Black Hole with a Schwarzchild Radius of 8.9km. (!).

• Q. What if the Sun collapsed into a Black Hole? Would the Earth and other planets be dragged in an disappear from the Universe? No, the gravitational attraction of the Sun would not change at all.

• Q. What happens when a Black Hole DOES absorb some mass? As M increases, the radius of the event horizon increases.

• Q. Where does all that 3M_Sun go? Good question, no answer.

• Is the Event Horizon a physical boundary? No, it is just the radius inside of which photons can not escape.

• Is there any reason to believe that Black Holes exist? Yes!

• The best candidates may be observed in some X-ray binaries. Cygnus X-1 is one of the brightest X-ray sources in the sky. It was identified as a hot, blue main-sequence star with mass of around 30M_Sun. These stars are not usually x-ray sources but, it was soon discovered that it was a Spectroscopic Binary with a relatively short period of 5.6 days.

  The companion was determined to be someplace between 5 and 10 M_Sun. What is the companion? It is completely invisible at all wavelengths (although remember the x-rays).

  1. If it was a Red Giant, it would be easily seen.

  2. If it was a main-sequence star, it would be seen with some difficulty.

  3. It can't be a WD because M > 1.4M_Sun,

  4. It can't be a Neutron star because M > 3.0M_Sun.

  5. Not much left except Black Hole.

• The X-ray emission is also highly suggestive of a system where mass is streaming from the companion onto a Black Hole. For an accreting white dwarf, the temperature of the gas when it crashes to the surface is such that these are UV radiators. For accreting Neutron stars, the temperature is high enough for "soft" x-rays. In Cyg X-1 we see "hard" x-rays indicative of a much depper gravitational potential well, consistent with a Black Hole.

• Among all the x-ray binary systems in the Galaxy and in nearby galaxies, there are 4 good candidates for black holes and another 4 or 5 pretty good candidates.

• Q. Do Neutron stars exist? For sure.

• Q. Do Black Holes exist? Maybe, there is not yet an airtight case.

• There are also some excellent candidates for Supermassive black holes in the centers of some and perhaps all big galaxies.