Physics Notes: Cosmology and Black Holes

Lecture by Professor Leonard Susskind

 

 

Lecture 3: Jan 24, 2011                                                                  Back to PHY33

Topics:  Black hole metric near the horizon, temperature and lifetime of black holes.

 

Q:   The question was about the details of building a black hole one bit at a time.    I arrived in the middle so I think the question had to do with the real information content of photons or gravitons.   In the derivation of the area/entropy relationship we assumed that the photons carried one bit of information.

A:  There are actually two bits of information.   Is the photon there, and if so, then what is its polarization?  [I suppose that there are really because there are only 3 states.   The state Ňnot presentÓ does not include polarization sub states.]

Gravitons could also be used to build the black hole.

Lecture Start

 

A black hole that is not spinning damps out fluctuations very quickly and becomes flat.   The classical statement is that Ňblack holes have no hairÓ.   In contrast, if they carry entropy, then they must carry degrees of freedom with tiny fluctuations.   These fluctuations are a consequence of quantum mechanics.

The horizon is not a ŇdangerousÓ place.   An infalling observer would observe nothing special about it.  Points just outside and inside would appear the same.   This is not clear from the metric (and coordinates) we used for the space-time around a black hole.

            where 

 

[Note: The radius here is not the distance from the center, since that is not well defined, it is the value that would yield the horizon area in the standard equation for the area of a sphere.]

To show this we will change to a coordinate system that makes it clear that  is not a physical discontinuity.   [It is a coordinate system discontinuity for our first coordinates.]   To start, we will restate the metric in terms of the proper distance instead of proper time.

           

 

One advantage of using proper distance is that for space-like event separations that the proper distance will be real.    We tend to choose the formulation that is simpler for our problem.

Now letŐs express the proper distance in flat x-y space.

  

We can convert to polar coordinates

Notice that if , then changes in  donŐt change the proper distance, much like what happens to  when  in the black hole metric.

Now letŐs change to Minkowski space-time.

There is an equivalent to polar coordinates where we use a hyperbolic angle.

[           A little background on hyperbolic functions É

           

           

So the hyperbolic sinh and cosh functions are closely related to the normal sin and cos functions.   Changing to sinh/cosh compensates for the sign change in Minkowski space-time. ] 

 

The hyperbolic angle  expresses the position along the line of constant , and ranges from –infinity to +infinity.     It is a kind of time variable, which can see in the sign of the contribution to the proper distance.

A particle following one of the lines of constant in flat space is under constant acceleration.   The dotted lines are the light paths from the origin.

 

Now letŐs go back to the Schwarzschild metric for the black hole.     The first thing is to reorganize so that the places we are taking differences between r and  are explicit.

 

Now we restrict ourselves to the region near the horizon.   In this region r is very close to .    To a very good approximation, we can replace by  when we arenŐt taking a difference from .    We will also ignore the angular changes for now.   The modified metric is:

           

 

Now we make a variable substitution.

           

So

           

 

We integrate (we donŐt need the constant of integration – just a constant offset in coordinate).

           

This is useful because we can also substitute for the coefficient for

           

 

Substituting into the modified metric (valid near ) we have

           

Now we make one last variable substitution

           

Yielding

           

Which is very familiar.   This is the same metric we had using hyperbolic coordinates for flat Minkowski space-time.  [I think the point is that if your metric matches a metric for Minkowski space-time, then your space is Minkowski space-time.  In this case there was an approximation that , but this approximation is good to first order.   ]

What this means is that if you fell through the horizon of a large black hole [sufficiently bigger than you], that there would be no tidal forces and you would notice nothing special [other than the increasing blue shift of the universe].

Suppose we have a couple, Alice and Bob, who are near the horizon.      Alice lets go and free falls toward the horizon.    Bob maintains his proper distance from the horizon, following one of the hyperbolic trajectories.

In the case of Minkowski space-time, Bob is simply constantly accelerating while Alice remains at the original location – in free fall.   Eventually, there is no light path, one with proper time 0, from Alice to Bob.   Bob can always send messages to Alice.  

This same diagram also describes what happens near the horizon of a black hole.  In order to hover at a given proper distance from the horizon, Bob must be undergoing constant acceleration, following one of the hyperbolic trajectories in the diagram.   Alice is in free-fall, just as in Minkowski space-time.   The same signaling restrictions also apply.

[One comment from the audience (Michael Peeler) was that Bob can only see finitely far into AliceŐs future.   That is why Bob sees signals from Alice that show Alice slowing down (the rate of a clock carried by Alice) as she approaches the horizon.   Bob never observes Alice crossing the horizon even though from AliceŐs point of view nothing special happens as she crosses it.]

Back to Black Hole Thermodynamics

 

In the last lecture we built a black hole with photons that carries one bit of information.     Each bit added the same area to the horizon, indicating that the entropy of a black hole is proportional to its horizon area.

We are also interested in the temperature of a black hole.

                      E is energy, S is entropy

The change in energy is the energy of the photon we add.

           

We chose lambda to be proportional to  so that no location information was carried by the photon.    This isnŐt quite the right constant, but we are looking for the form of the temperature.

So, for a change of 1 bit of entropy we add

               (dropped the 2)

What does this mean?   Big black holes are cold.    The temperature – energy relationship is backwards from our intuition.  Normally, when we add energy, the temperature goes up.   In this case it goes down.

With the right constants the expression is:

[Adding correction here – forgot Boltzmann constant]

           

What is the temperature of a solar mass black hole?

            Mass of the sun = 1.98892 x 1030 kg

           

 

Which is much colder than empty space, which is filled with a microwave background.   Therefore, heat will flow from empty space to the black hole, making it even colder.

 

Suppose that the black hole was warmer than the surroundings, then heat flows out of the black hole, which makes it hotter!    This increases the rate of heat flow and you have a runaway.    A black hole in thermal equilibrium is unstable.

How fast do black holes evaporate?    LetŐs calculate luminosity in Plank units where h=c=G=1.

For reference:

           

           

           

The rate at which energy is lost by a black body in these units is

                    Where A is the area of the surface.

But energy and mass are the same thing, so:

           

We can use our formula for temperature  and area 

[I added the 8pi]

           

 

We can solve this by integrating

           

 

So if at time 0, the black hole was gone, then at time –t it has mass M.   This expression gives the negative of the evaporation time.

So how long does a solar mass black hole take to evaporate [It wouldnŐt really evaporate until the universe expanded enough to cool the microwave background.   For fun we can assume that empty space is at 0 degrees.]

This is quite a long time.

Exercise:  find the mass and size of a  black hole.

so

           

Which is about half the mass of the earth.

 

Wrap Up

 

Standard General Relativity is like fluid dynamics.   It misses the discreteness of energy and the microscopic structure and degrees of freedom of space.   More is needed to explain the information stored on the horizon of a black hole.

Conservation of information is probably the most fundamental physical law of all.

An example:    We associate translation invariance with momentum conservation.  Suppose that when we slide an eraser across a table that it came to a stop because of the equations of motion instead of through frictional interaction with the tabletop.   There is no problem with translational invariance, so what happened to momentum conservation?   Notice that one could start the sliding with distinct initial conditions, but have precisely the same outcome.   Information is lost.   Conservation of momentum and energy depend on information conservation.

In classical physics information conservation is expressed in LiouvilleŐs theorem, which requires that volume in phase space is conserved.

In quantum mechanics, the equivalent information conservation idea is expressed by unitarity.