Rotation of Black Holes
Hello Alyssa --
The questions you sent to Fermilab about physics didn't get lost, they just got routed to a couple of lazy postdocs. That's why it took so long to get back to you. Anyway, we thought that these were such good questions that _two_ of us decided to take a crack at answering them! Below are your questions and answers from me and from my colleague Andrew Sornborger. You'll notice that sometimes we say almost exactly the same thing, and sometimes we give totally different answers to the same question, which probably tells you more about what science is really like than anything else. I hope this helps you ask more good questions.
-- Will Kinney
- Is the rotation of a black hole the only reason that things being
pulled in form a disk instead of falling in from all directions?
Andrew: A disk forms because the matter around the black hole is
rotating, not because the black hole itself is rotating. However, the
gravitational field of the black hole extends in all directions, so
particles can still fall in from any direction. The disk forms because
of a combination of conservation of angular momentum, and friction in
the gas. If the matter falling in had no angular momentum (no
rotation), then no disk would form.
Will: An accretion disk forms not because the black hole is rotating,
but because the matter falling into the black hole is rotating (or,
using the more technical term, has angular momentum). If you had a
non-rotating black hole surrounded by a cloud of dust with zero angular
momentum, no accretion disk would form. The accretion disk forms
because of a combination of conservation of angular momentum and
friction in the gas. Friction tends to make the gas collapse, and
angular momentum makes the gas collapse preferentially in one
direction, forming a disk.
- If Hawkings radiation is virtual particles, wouldn't half of them
be antimatter? If so, wouldn't the +vps and -vps destroy each other?
Every explanation of the radiation implies that the + particles escape
and the - ones get pulled into the hole and thus lower the mass but if
the radiation is from virtual particles and not a quirk allowed by
Schrodinger's equation, they would cancel by the law of averages.
Andrew: Hawking radiation starts out as two virtual particles on the
horizon of a black hole, one particle and one antiparticle. Usually,
the virtual particles are a pair of photons. A photon is its own
antiparticle, so this doesn't contradict the fact that a
particle-antiparticle pair are produced. However, one particle falls
into the black hole, the other comes out. Once the particles separate
due to the black hole's gravitational field, they are no longer virtual
particles, but real particles. They can't destroy each other because
they are separated by the horizon. The strange thing (and this is
possibly what is confusing you) is that the particle that falls in has
negative energy. That means that it reduces the mass of the black hole.
You may have been confusing the 'anti' in antiparticle as meaning
negative energy, it really means negative charge.
It should be emphasized that the virtual particle picture of Hawking
radiation is only one way of visualizing how the radiation is created.
You can also think about the radiation as being particles which tunnel
out of the black hole horizon. In particular, if a photon or any other
massless particle tunnels out of the horizon, it can get away from the
black hole because it always travels at the speed of light.
Will: Or are you suggesting that the Hawking radiation itself is
likely to be evenly divided between particles and antiparticles? This
is likely to be true, and you're right, the particles and antiparticles
would tend to annihilate one another. But when a particle and
antiparticle annihilate, you don't get nothing, you get photons! Most
(but not all) of the Hawking radiation is in the form of photons in the
first place. And what's the antiparticle of a photon? Another photon!
The radiation, even if it is exactly evenly divided between particles
and antiparticles, doesn't cancel at all.
- It was mentioned in one of the books that the matter creation in
the Begining took a debt from gravity that would be repayed at the
End. This was never explained and makes very little sense to me.
Will: Same here. I'm not sure what they were talking about.
Andrew: The idea of taking a debt of energy is perhaps best explained
in analogy. If I have a particle that is very far away from a big mass
(for instance the sun) and not moving, it has no kinetic energy (energy
of movement) because it is not moving. However, it does feel a force
from the sun and will start to move. In terms of energy, we can
describe this system like this: The gravitational field of the sun has
'potential' energy. The potential energy is negative. When the particle
feels the potential energy, it begins to move. When it moves its
kinetic energy increases. In order to conserve total energy in the
system, the potential energy must become more negative. So, the
decrease of potential energy corresponds to an increase of kinetic
energy. We can say that the kinetic energy is gotten at the expense of
the potential energy. There is a 'debt' to the potential energy from
the kinetic energy.
The debt to gravity that is referred to in the book is probably
similar to what is described above, only what the author is probably
referring to is particle creation. In this case, what happens is
particles are created (they have energy in the form of mass) at the
expense of the gravitational field. In general relativity, the
gravitational field corresponds to properties of spacetime. Thus, when
you form a spacetime, i.e. a universe, a debt is owed to gravity from
the particles that are created in the spacetime.
- If all 'matter' is just a bunch of point particles with forces
between them, how do they 'take up space'.
Andrew: Matter is just a bunch of point particles with forces between
them. They 'take up space' because they interact with each other. For
instance, you can't put your hand through a wall because of the
electromagnetic interaction of the atoms in your hand and the atoms in
the wall. The particles 'feel' each other via their interactions.
Often physicists will say that one particle is 'bigger' than another.
What this means is just that the interactions of one particle with
other particles are stronger than the interactions of a different
particle with other particles.
Will: What you perceive as an object "taking up space" is a result of
the forces holding the object together (along with the forces holding
YOU together!). A table, for instance, is mostly "empty space", and so
is your hand, but you can't pass your hand through the table because
the atoms in your hand and the atoms in the table interact through
electromagnetism. Your hand feels a force from the table, and can't go
through.
- What happens to photons trapped at the event horizen at the moment
that it becomes the event horizen? Assume the hole is not rotating;
the photons have to go the speed of light but can not move.
Andrew: That is correct. Photons at the horizon have the speed of
light, but gravity is pulling them in so fast, that they don't move.
There are a few things to remember, however. First, the horizon is
'infinitely thin', that is, it has no width, it is a surface. So, the
chances that a photon will be directly on top of it is very, very
small. Furthermore, if other matter falls into the black hole, the
horizon will move outwards.
For your edification, I will also point out a few changes in our
thinking due to the fact that matter on small scales is quantum
mechanical. When we talk about photons, we are talking about quantum
particles. That means that we can only talk about the probability that
they are in any particular location. Also, that probability gets
smeared out with time, so the likelihood that a photon would stay at
the horizon, even if it were formed there, is very small. The horizon
is also a quantum mechanical object. That means that it's location will
also smear out with time.
Will: There are two answers to this question which might at first seem
to contradict one another. But the important thing to remember in
relativity is that you can't ask what happens without saying from whose
point of view you are asking. I'll describe what happens from two
points of view. First look at the photon from the point of view of
somebody collapsing along with the black hole, right at the event
horizon. From this person's point of view, absolutely nothing happens
to the photon! It zips by at the speed of light, just like always. Now
look at the photon from the point of view of someone a long distance
from the black hole, watching it collapse from outside. The photon is
not detected at all: it never arrives.
I've changed your question into a different, but more precise question:
how do we consistently reconcile these two points of view? My answer to
the puzzle is that the photon must also lose energy as it climbs away
from the event horizon. A photon that loses energy doesn't go slower --
it always travels at the speed of light -- but it "redshifts", or gets
longer in wavelength. A photon climbing up from the event horizon is
infinitely redshifted, and is never seen by the outside observer.
- How do exchange particles exchange information? For example, when
the question of how gravty worked came up, gravitions were needed.
I've never found anything that actually explained what they do.
Will: I'm going to steal an analogy from the Feynman Lectures on
Physics, Volume III. A force created by exchanging particles is very
much like the force felt by two atoms when they bind to each other by
"sharing" an electron. In this case, the electron is real, but it must
make "virtual" jumps, or tunnel. It works roughly like this: take the
simplest case of a neutral hydrogen atom (a proton and an electron) and
an ionized hydrogen atom (just a proton).
(p e) p
The electron is bound to the first proton. It doesn't have enough
energy to escape and bind to the second proton. But quantum mechanics
says it has a probability of tunneling over to the second proton -- a
"virtual" process, because the electron must momentarily (have) _negative_
energy:
The electron can then tunnel back to the first proton, and so on. This
exchange is forbidden by classical physics, but allowed by quantum
mechanics. The closer the protons are to each other, the easier it is
for the electron to jump from one to the other. Because of this
electron jumping back and forth, the two protons feel a force that
depends on the distance between them. This process is very similar to
charged particles exchanging virtual photons when they interact through
electromagnetism, or massive particles exchanging virtual gravitons.
The process is the same, it's just the particles involved that vary.
Andrew: Before quantum mechanics was discovered, people thought that
fields, like the electromagnetic field, exchanged information. That is,
a charged particle felt another charged particle's electromagnetic
field and reacted to it. Einstein was the first one to realize that
energy in fields came in discrete packets called quanta. Every particle
is one of these quanta. So, when you combine the two ideas, the fields,
which before were thought of as continuous, are no longer continuous,
but made up of quanta, or particles. The quanta that 'mediate' the
electromagnetic field are called photons. The quanta that mediate the
gravitational field are called gravitons.
So, after the advent of quantum mechanics, the language changed from
fields conveying information to particles conveying information. The
basic idea is the same, a particle interacts with other particles due
to a field. Only due to quantum mechanics it was realized that the
field was not continuous, but made up of many particles (quanta).
- Do the universal constants (speed of light and planck constants)
exist before Planck time? Basically, did the universe pop into
existance as a point which was actually a sphere with a dimeter of the
plank length?
Will: Good question. Nobody knows the answer. One way to think of the
Planck length is that it is the scale at which nobody has the foggiest
idea what happens. Not yet, anyway. String theory is an attempt to
understand Planck scale physics, but it has a long way to go before it
can answer these kind of fundamental questions.
Andrew: Not much is known about what things are like at lengths below
the Planck length or times before the Planck time. There are some
people who think the universe popped into existence as a sphere with
the diameter of the Planck length. Or, at least, they think that this
is a good approximation to what might have happened and they can
calculate various properties of such a universe. But, right now, it's
an open question. Some theorists think superstring theory may provide
answers to these kind of questions.
- In the system where the speed of light is defined as 1 (so meters
are seconds) how do you get kg equal to a number of meters? This was
in a book by Wheeler so it should work, but the only explanation was
that the sun was equal to about 1400 meters of mass.
Andrew: There are a number of different interesting coordinate systems
that one can use which are convenient when solving equations in
physics. For instance, in quantum mechanics, people often set Planck's
constant h equal to the speed of light and both equal to 1: h = c = 1.
In this coordinate system mass is equal to frequency, which is
proportional to one over the wavelength, or inverse length. This is
because the energy E of a photon is hf, where h is Planck's constant
and f is the frequency of the photon. Since E = mc^2, hf = mc^2. In
units with h = c = 1, f = m. Since f is in units of inverse length, m
is also in units of inverse length. Physically, what these units are
saying is that a given mass can be formed by a photon of a given
frequency.
However, Wheeler is using units where G, the gravitational constant
is 1, and so is c. G = c = 1. In these units, distances are all
proportional to mass. The reason for this is that for each mass, there
is a Schwarzschild radius (the radius to the horizon) that a black hole
would have if the black hole had the mass m. Therefore, what Wheeler is
saying is that the Sun has a Schwarzschild radius of 1400 meters.
Will: Setting c = 1 isn't enough! When you set c = 1, length (meters)
and time (seconds) are in the same units. That is, if you measure time
in seconds, you are choosing to measure distance in units of "light
seconds", or the distance light travels in one second, which is about
300,000,000 meters. Then
c = 1 light second per second = 1.
Equivalently, you can measure length in meters, and measure time in
units of how long it takes light to travel one meter. Again, c = 1.
Same difference.
But what about mass? When c = 1, Einstein's famous equation E = m c^2
just becomes E = m. Mass and energy are in the same units, but
_neither_ can be expressed entirely in units of length or time, so we
haven't gained much yet. In order to express mass (or equivalently,
energy) in units related length, you need to do more. One way to do it
is to set Newton's constant G equal to 1. The energy of a particle of
mass m from its gravitational attraction to another mass M is
E = G M m / r = (mass)^2 / length = mass
so that mass and length must be in the same units. You are measuring
energy (kilograms) in meters.
A different choice is to use units where Planck's constant h is equal
to 1. Planck's constant relates the energy E of a photon to its
wavelength l:
E = h / l
Now, when c = 1, E is in units of kilograms. The wavelength l is in
meters. Then Planck's constant h has to have units of kilograms times
meters. When you set h = 1, you are measuring energy (kilograms) in
_inverse_ meters. Weird, huh?
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