Black Holes

Black Holes from Cambridge
The Light Cone (an introduction to relativity)
falling into a black hole (by Andrew Hamilton - more advanced than this class)
virtual trips to black holes and neutron stars

Outline

black hole structure
orbits
falling in

The Structure of a Black Hole (non-rotating)

First, the structure of of flat space-time

Draw a light-cone in 2 spatial and 1 time dimension
up = future; down = past
point N is here and now
the past lightcone consists of the trajectories of all possible light rays that can hit us here and now
the future lightcone consists of the trajectories of all possible light rays that travel outward from here and now

point P is in our past
point F is in out future
point R is on our past lightcone
point G is on our future lightcone
points S and Z have a "spacelike" separation from us: we have no possible contact with those points here and now at point N (although we can see them in the future).
if point G occurs 1 sec in the future, then it is 300,000 km away.
the nearest star, alpha centari is has a spacelike separation from point N from 4 years ago to 4 years in the future because it is 4 lightyears away.

since a light ray moves at the speed of light it must travel at a 45^o angle, along the lightcone
massive objects like us move on lines that are more vertical than 45^o because we move slower than light.

Here's a diagram of a black hole space-time showing little lightcones at different space-time points:

Note that inside the event horizon, the lightcones all point inward
This indicates that not even light can escape from inside the event horizon.
Inside the event horizon, the time and radial distance coordinates switch roles

the radial distance coordinate becomes a time coordinate
the singularity = your future (if you are inside the event horizon)
avoiding the singularity is like avoiding Monday

Note that this diagram depicts a collapsing black hole
The event horizon forms before the singularity indicating that infalling observers can be doomed to hit the singularity even before it forms.

outside the event horizon (see diagram in GFA p. 12)

R_Sch= Schwarzschild radius = event horizon radius
at r = 1.5 R_Sch is the "photon sphere" where photons go in circular orbits
at r = 2 R_Schyou can orbit at v = 0.75c, but the orbit is unstable

if you go too fast, you escape
if you go too slow, you fall in
for r > 3 R_Sch= circular orbits are stable

Here's what an isolated black hole would look like:

left panel: no black hole; right panel, the same view with a black hole

Here's another view:

Note the dark circle has a radius of 1.5 R_Schbecause that is as close as a photon from far away can come to the black hole and still escape

Black Hole Tourist

suppose you could visit a black hole in a spaceship

the spaceship will pass by the black hole in a close approach orbit
the distance of closest approach is 2 R_Sch
the spaceship will protect you from any radiation, but it can't shield you from gravity
the tour operators do not go with you
you have two choices of destinations:

a typical black hole from stellar collapse

M = 10M_sun, R_Sch= 30 km

the black hole at the center of our Galaxy

M = 2.5 million M_sun, R_Sch= 7.5 million km

Where should you go?

at 2 R_Sch from the stellar black hole

tidal force = 600,000 g

at 2 R_Sch from the Galactic center black hole

tidal force = 10^-5g

massive black holes have much smaller tidal forces

Falling in to a Black Hole

There is a large gravitational redshift when we watch something fall in to a black hole

At the event horizon (R = R_Sch), the red shift becomes infinite

light waves from the moment of horizon crossing shift to infinite wavelength
the outside observer sees the last moments before horizon crossing stretched out forever.

The infalling observer sees something much different

he's doomed once she crosses the horizon
but he doesn't know it
he'll hit the singularity in a finite (and short) time
tidal forces will rip him apart before he hits the singularity.

Observable Properties of Black Holes

black holes don't suck in every thing in sight

most objects would orbit - angular momentum conservation

no stable stars with M > 3 M_sun

after the nuclear fuel is gone, collapse can't be stopped
if we see a dark star with M > 3 M_sun it is probably a black hole

what else could it be

black holes are efficient energy sources

nuclear reactions yield only 0.8% of mc²

burn lots of Hydrogen to Iron and the total mc² decreases by 0.8%

matter falling in to a black hole can lose > 10% of its energy to radiation

infalling matter must lose energy to fall in

a very compact high energy source is likely a black hole

if its brightness varies every day, then it is probably less than a light-day in diameter

Black holes of ~10 solar masses can be observed as x-ray binaries:

a normal star + a black hole
doppler shift of normal star's spectral lines => lower limit on mass
x-rays or gamma rays from the accretion disk
sometimes gamma ray jets as in SS433
possible detection via gravitational microlensing

Most Galaxy are thought to have Super Massive Black Holes in their centers

Quasars: brightest objects in the Universe

vary on daily timescales
black holes of ~100 million solar masses swallowing lots of matter

Active Galactic Nuclei

like quasars but less mass or fuel

White Holes and Wormholes

white holes and wormholes are solutions of Einstein's Gravity equations
they are closely related to black holes
they are likely not to actually exist
wormhole link

Here are some 2-dimensional embedding diagrams

2 of the spatial dimensions are plotted

The diagram above is for a black hole

The radial lines turn blue inside the event horizon where the radial distance coordinate becomes a time coordinate.

The diagram above is the non-rotating black hole again

the squiggly lines show photon (or light ray) trajectories
one trajectory leaves the black hole, and is redshifted

Einstein's equations have the same form if we travel backwards in time

so, a white hole is a solution to Einstein's equations
a white hole is just like a black hole seen backwards in time
i.e. with lots of stuff spewing out of the singularity
they are thought not to exist in nature, but they can exist in theory

It is possible to connect a white hole solution to a black hole solution:

Just connect them at the event horizon and throw away the blue stuff
This is called a wormhole as shown above.
Since the top and bottom connect onto normal, almost empty space, a wormhole might connect to very distant parts of the Universe

If so, perhaps one could take a shortcut through the wormhole and travel somewhere very far away.

But, here's the catch:

we've left out the time variable in the plot above
and the solution is not static

Here's the time dependent solution:

The white and black holes start out separate and then attach
Then, they quickly break off again.
a space traveler cannot pass through the wormhole moving at slower than light speed

he would be doomed to hit the black hole singularity if he tried.

Kip Thorne has suggested that it might be possible to stabilize the wormhole with some form of very exotic matter: