Download Schwarzschild Radius

Rs = 2MG/c2
Schwarzschild Radius
M stands for mass
G is Newton's constant coefficient of gravity, 6.67 x 10-11
c is the speed of light, 3 x 108
What the heck does this equation mean? It's for calculating the Schwarzschild radius of an object.
What does that have to do with black holes? Well, when an object is compressed below its
Schwarzschild Radius, an event horizon forms around it. Sound familiar? The event horizon of a
black hole is just Rs = r. If you know the mass of an object, you can calculate its Rs. For a human
it is 1.5 x 10-27 meters per kilogram (for comparison a proton is 10-15 meters). Very tiny! It's tough
to make black holes out of small things!
ds2= - (1 -Rs/r)dt2 + dr2/(1 - Rs/r) + r2d
Schwarzschild Metric
2
+ r2sin2 d
2
t stands for time
r stands for the radius
is like lattitude
is like longitude
What is the Schwarzschild metric?
It is Schwarzschild's solution to Eintstein's general relativity equation set. The metric describes
the shape of spacetime outside of matter. Y'know, those cool curvey spacetime pictures in
Scientific American. Once you hit matter, be it some gas, a star, a planet, or a rock, this metric no
longer applies. The metric's kinda ... spherical. It looks a lot like an equation made for rectangular
coordinates transformed into spherical coordinates, a standard calculus problem.
Well, the ds factor tells you how space changes, what it all looks like. The dt factor tells how time
changes as spacetime changes. You can see that, if r = Rs, dt would be zero. That is to say that at
the even horizon there would be no change in time. Makes sense; you can look at the event
horizon as being the place where time "stops." The dr factor deals with how close to something
you are. You'll notice that it "blows up" when r = Rs; the Schwarzschild metric does not apply
beyond the event horizon. The d and d factors are part of the whole spherical geometry
transformation thing and aren't special to the metric. They make the math work out right, yes they
do.
Black holes by supernova
Firstly, a black hole isn't really a hole at all, but that's the easiest way to think of its effects on the
rest of the universe. Take a star that's at least thirty times larger than our sun and make it explode
(called a supernova). Stars do that at the end of their lifetime, sometimes leaving a remnant of the
violent explosion. The nature of the remnant depends on its mass. If the remnant is less than 1.4
solar masses, it will become a white dwarf, a kind of hot dead star that isn't bright enough to
visibly shine. If the remnant is roughly 1.4 solar masses, it will collapse. The protons and
electrons will be squished together, and their elementary quarky particles will recombine to form
neutrons. What you would get is small (by stellar terms) sphere of neutrons with perhaps a thin
film of electrons and other stuff at its surface. That's why it's called a neutron star. See, the
neutrons don't mind hanging around near each other; but if you get them close enough to each
other, they get anxious and resist being pushed any closer. (Yes, I'm attributing emotions to subatomic particles.) The neutrons of a neutron star are, indeed, pressed quite close to one another
and exert a certain pressure on each other. This pressure prevents the further collapse of the
neutrons star. If the remnant is larger than 3 solar masses, it becomes a black hole (well, 2 or 3
depending on who's giving you the number). I think that calculates out to a star roughly 30 solar
masses before supernova.
For example, let's pick Eta Carinæ, a superstar that is hundreds of times larger than the sun (also
8,000 ly away) and happened to have exploded around 1850ad.2 That and I have a nice current
picture of it. Very well, Eta Carinæ goes supernova! In a supernova the atmosphere of a star
collapses onto and compresses its core, and the remaining mix of stuff is blown into space leaving
a remnant of the core. During the earlier part of the star's life, it fused hydrogen into helium.
During the later part of its life, it fused helium into the heavier elements, which made their way to
the core of the star. In the last split second of its life, as waves of energy push out from the
collapsing core, the star fuses its atmosphere of helium (and a few other things) with its core of
iron. This is called nucleosynthesis. This process created ALL the atoms heavier than iron in this
universe. These heavier atoms plus millions of neutrinos are thrown out in waves as the star
collapses, leaving the remnant.
Now, since Eta Carinæ's remnant is larger than three solar masses, the pressure of collapse and
the force of gravity of all that mass squishing the core (which is very hot metal) condenses all the
matter together. No more electron shells and orbits --- the neutrons are forced together. Then, the
very neutrons themselves give up and get squished by the pressure, and the star keeps on
collapsing. See, there's this certain radius that determines whether the star can squish itself into a
black hole. It's called the Schwarzschild Radius (swar - shild). If the starstuff can collapse itself
smaller than Rs (the Radius), stand up and cheer, for it has become a black hole.
What does it look like as the star collapses?
If you were to watch Eta Carinæ becoming a black hole, you would first see the dramatic and
wonderful supernova, in which the superstar forcefully ejects most of its atmosphere very
quickly. It's a violent process but quite pretty. After enjoying this for a while, you would see the
star itself start to collapse. After watching this for a while, you would think it mighty peculiar that
the star seems to be slowing down and getting redder and dimmer as it collapses. What's all this?!
These are the effects of the gravitational field getting stronger. Okay, a brief side-note about
gravity and light cones.
Light cones --- get yer light cones right 'ere!
In the presence of a gravitational field time slows down. You don't sit there and wonder why your
watch is running slower. Everything slows down --- your watch, your brain, your cells, your very
atoms --- everything. Someone who is outside the field would watch you and wonder why you're
moving so slowly (and you would wonder why they're moving so quickly). This effect is called
time dialation.
You've probably heard many times how the speed of light is a constant in a vacuum. Suppose you
have a light ray in a gravitational field (hey, happens every day!). It can't slow down like your
watch and your brain and your cells and your atoms because it's light in a vacuum and it must go
a constant speed. So, does it just keep going regardless of gravity? No way! The speed of the light
ray can't decrease, but the frequency can. Oh, tricked you with that frequency bit, hunh? Light is
formally known as electromagnetic radiation (I shall only type that once, here, for it is a long
word and I'm liable to mis-spell it). What that means is that light is really a changing electric field
(which induces a changing magnetic field, which induces a changing electric field, blah, blah).
The important part is that light can be thought of as a wave in some respects, just like a wave in
the ocean. Just like water waves, light rays have a frequency of arrival and a wavelength (one
being the reciprocal of the other). The longer the wavelength is, the fewer waves can come in
during a certain time, thus the lower the frequency. Got it? If a yellow light ray passes through a
very strong gravitational field, its frequency will lower and it will appear reddish. Fine, fine, fine.
That explains the color shift, but what about the slowing down bit? A very useful concept for
explaining this is the light cone. Imagine that all the possible paths for light to take can be
summed up in this cone shape. Why? Uhhh ... it comes from a three dimensional interpretation of
the two light rays from a spacetime diagram. Just ... go with it. If you were on a planet, that cone
would describe what directions the light would have to shine in order to escape from the planet's
gravitational field. Normally, light can escape from planets and stars. They shine, y'see!
However, it all depends on the strength of gravity. Imagine the light rays being emitted from poor
ol' doomed Eta Carinæ. As the star collapses and approaches its Rs, the gravitational field gets
stronger because there is more matter in a smaller place. Certain light rays (such as those shot out
in a tangent to the star) will fall back down on the star because the gravity is so strong. The light
cone becomes narrower. When the star is very close to Rs, the light cone is quite narrow, and only
those light rays that travel out almost perpendicular to the star's surface will escape the
gravitational field. When Eta Carinæ reaches Rs, the light cone closes, the event horizon forms,
and light can no longer escape.
Back to what Eta Carinæ looks like as it collapses.
The star will continue shrinking, but as it neared Rs it will be so dim that you would probably
need an infrared detector to see it. Really close to Rs for all intents and purposes Eta Carinæ has
become a black hole. I've heard different sources give different times for this process. Some say it
takes forever for the last photon to escape. Some say that it will take a quite a short time. It's all
quite iffy, since we've never actually seen it happen. True that there might still be photons coming
at you that escaped before the event horizon formed and are taking their sweet time about getting
away.
Does the black hole just stay that size forever?
No way! The black hole has a certain mass and size of its event horizon when it forms, but that
changes over time. If stuff falls into the black hole like dust and light, its mass increases. If its
mass increases, its Rs gets a little larger. Therefore, if stuff falls in, the event horizon gets a
bigger. The idea that black holes could only get larger feels weird to me, so I'm quite happy to say
that there is also a reverse process, where the black hole gets smaller. It has to do with Hawking
radiation. In short it is possible for particles to form by stealing energy from the black hole. If the
black hole loses energy (and energy = (mass)c2!), it gets smaller. Black holes are rather dynamic
in that sense; you can always add charge to them, take charge away, make them rotate slower, and
change their size.
BTW, if the sun were to suddenly pop into a black hole (no, it can't actually ever be a black hole,
but just imagine), would the earth go plummeting into it? No, it would continue on its orbit and
things would get rather chilly, that's all. The black hole exerts the same gravitational force on
stuff around it as a star with the same mass would. Things just get interesting close to the black
hole.
Don't believe me? How about this argument. When calculating forces of gravity from objects on
other objects, we talk about the radius between the two of them. Does this mean the radius from
the surface of the one object to the surface of the other? No. We talk about the radius between the
two centers of mass, and, when considering stars and planets, that center is just the center of a
sphere. Why? Well, consider yourself sitting in your chair. Right now, every darned atom in the
whole earth is attracted to you (and you to each of them by Newton's Third Law, action and
reaction). Now, each bit of the earth attracts you with a certain force and has a corresponding bit
on your other side that attracts you with an equal force. Well, the force is actually to the side and
slightly down, since you're standing on a curved surface. The side-to-side force cancels out and
all you feel is the downward force, the force that points directly at the center of the earth.
And what does this have to do with the sun-turned-black-hole not yanking the earth out of orbit?
I've proven to you that you can treat the force of gravity as just a force from the center of an
object. The center of an object is just a point, it doesn't really exist. But as far as the force of
gravity is concerned, it only "sees" mass and radius. Right now, for the case of you sitting on
your chair, our calculation of the force of gravity does not care if the object doing the attraction to
you has its mass spread out the size of the earth or whether it is concentrated in a theoretical
point. We only need your mass, the mass of the earth, and the radius that separates you two.
That was lengthy! My apologies.
Black holes by coincidence
A black hole forming without a tell-tale and quite noticeable supernova to warn the neighbors --quite a frightening concept, isn't it? It is true, though. The definition of a black hole is just an
object that has no matter when you check up to its Rs. Compress something to its Schwarzschild
Radius and an event horizon forms around it. Oh, you don't belive me? Okay, take the milky way.
At the core of our galaxy is a bunch of old, massive stars. Let us say a couple of these stars
happen to wander too close to one another. *Pwoof* Event horizon.
One of the many quirks of black holes (er, things with event horizons) is that, as you consider
larger black holes, the actual density of the material needed to form said black hole decreases. For
a black hole with a mass of 2.02 x 1026 kilograms, it would have an Rs of 8.98 x 107 meters, and it
would be about as dense as water! Of course, the whole problem is in getting that much water in
one place ... then again, there was that report of water forming in the nebula in Orion's belt.
Hmmm ...
Primordial black holes
Only modern black holes need to have remnant mass greater than three solar masses to form.
Way back when, according to astrophysicists, the universe was much hotter and denser than it is
now. Due to this incredible pressure, primordial black holes could form that were smaller than
three solar masses. Since they were so small, though, most of them would have evaporated by
now. Those primordial black holes large enough to survive up until now would have the mass of
an asteroid (1012 kilograms), the size of an atom, and would emit gamma rays (the highest energy
form of light)1. When the universe was 1/10,000 of a second old, those primordial black holes had
a chance to absorb enough matter to have this mass. I recall one author saying that these little
guys would be the best power source we could find. They are too small for us to worry about
falling in, and they would put out lots of clean energy.
Why is it called a static black hole?
It has no charge and it is not rotating. This is your standard, idealistic, simple black hole. It is also
called a Schwarzschild (Swar - shild, not Swarts - child) black hole. The interesting places for this
one are the photon sphere, the event horizon, and the singularity. What happens to you near the
black hole all depends on your distance in Schwarzschild radii. Let us say that you're flying
towards a static black hole in a brand new, top o' the line spaceship. You're approaching the black
hole slowly, since only a fool would charge full-speed into one.
The black hole itself is very plain and quite difficult to actually see, presuming that you are
looking at a black hole with absolutely no accretion disk. It's the space around it that is
interesting. You see multiple images of many of the stars. That galaxy that you know is behind
the black hole appears as a ring around the black hole, commonly called an Einstein ring. Why
you see the galaxy at all, when the black hole is between you and it, and why it appears as a ring
is from the bending of light due to the strong force of gravity of the black hole. Say you have an
iron marble and a bar magnet. If you roll the ball near enough the magnet, it veers towards the
magnet. The marble ends up tracing a slightly bent path versus the straight path it would have
traced had it not encountered the magnet. Now replace the magnet with the black hole, and the
marble with a light ray, and you've got it. The light from the 'hidden' galaxy peeks around the
black hole and looks like a ring. Cool neh?
As the ship passes closer, there's a sudden flash to the rear. Backing up and hovering, you
discover that ... it's the back of your own ship? Oh, this is the photon sphere.
The Photon Sphere
Mathematically speaking, the photon sphere occurs at 3/2 the Schwarzschild Radius. It's the only
place where light rays can have (very) unstable orbits around the black hole. Why is it called
photon sphere? Photons orbit there ... and it's a sphere. You know how it takes a certain speed to
stay at a certain orbit? Satellites closer to the earth must move faster than satellites farther from
the earth or else they will fall onto the planet below. It's the same way with a black hole, just the
numbers are larger and the results more dramatic. As you approach the black hole, you can orbit
all you like. It's just that the closer you get to it, the faster you've got to go to stay in orbit and not
fly off into space or plummet into the black hole.
At the photon sphere the speed you would have to go to stay in orbit is c, the speed of light, some
3 x 108 meters per second. Say you hovered your ship in an orbit just above the photon sphere
(which would take tremendous energy, for you'd be going a significant fraction of the speed of
light) and stuck a camera down into the photon sphere (just suppose...hypothesize). The back of
the camera sends off light rays that hold temporary orbits around the black hole. A light ray can
bounce off the back of the camera, orbit around the black hole, and run smack into the front of the
camera. If you try to move the camera towards the image, the image receeds. Why? Well, think
about it. The view the camera has is like the view someone would have if they are standing
behind it. If the camera stands behind someone and they start to move forward, well, their image
receeds.
I keep saying these light orbits are only temporary, but why is that? Why can't a light ray stay in
the photon sphere forever? Firstly, the photon sphere is a tiny place --- just one precise distance!
A light ray would have just one chance to hit the right trajectory. Secondly, things aren't that
precise out there in the universe. There are all these other light rays bouncing around, making
things very unstable. A light ray may orbit the photon sphere for a long time, but, eventually, it
will encounter another light ray and its perfect orbit will be disturbed. It might go flying off into
space only delayed a thousand years, or it could fall into the black hole. You want to argue
quantum mechanics and say there is always the possibility that it would orbit forever? Fine, but
that's one possibility out of too many to count!
Okaaaay, the photon sphere is fun, but it loses its charm after a while. You take the rocket closer
to the black hole. It is now impossible to orbit the black hole, but it is still possible to stay at a
constant radius --- to hover above the black hole. Now, you approach the ... event horizon (the
astrophysical thing, not the movie). The what??
The Event Horizon
When people talk about the black hole, they're usually referring to its event horizon. The funny
thing about the even horizon is that it, just like the photon sphere, is just a mathematical
construct, a distance. If you wanted to get theoretical, it is the first sphere of light (which is, of
course, constantly expanding at the speed of light) that does not expand (due to the extreme
curvature of spacetime). Enough, enough, what is it?! Rs, the Schwarzschild radius. What makes
it special? Once something passes beyond the event horizon, it can never leave the black hole and
is doomed to a painful stretchy demise.
Ah ha, that's interesting.
Your rocket ship can come close to the event horizon and (by firing all the rockets at their
strongest and wasting a lot of fuel and, not incidentally, squishing you flat with the acceleration)
can move away from it. This all has to do with something called escape velocity. You know how,
when rockets are launched on the earth, they have to go a certain speed or else they'll just
plummet back to the earth? That's escape velocity: the speed needed to leave a gravitational field.
Want a peek at the equation for escape velocity? Black holes have escape velocities just like
anything else, it's just that the numbers are orders of magnitude bigger than the escape velocities
of planets. The closer to the black hole's event horizon, the greater the escape velocity. At the
event horizon the escape velocity is the speed of light. Things with mass can't go the speed of
light, so, if they get to or go closer than Rs, they cannot escape the black hole. Beyond the event
horizon the escape velocity is greater than the speed of light. Light rays that go beyond Rs can't
even get out!
Why is it called the event horizon?
Well, we who-are-further-away-than-Rs can't see the light rays emitted by something at Rs. Say
we had thrown something into the black hole. We could see its image get smaller and smaller as it
approached the black hole. Then it would start to move slower and slower and get dimmer and
dimmer. The slowing down and the dimming are the effects of the strong gravitational field. The
thing we threw would slow down until it seemed as if time had stopped for it, and it would dim
until we needed an infrared detector to see it. The last image we could see would be the light
emitted just before it crossed the event horizon. It's the last event we could see. Get it?
BTW, if you stuck your hand past the event horizon of a black hole, you could pull your arm
away (not really; your muscles and bones would snap from the strain, but let's just be hypothetical
for a minute), but the part of your hand that went past the horizon would forever remain there.
While moving towards the event horizon, you look out your front and back windows (high-tech
spaceships have lots of windows --- neat!) because you're curious about the effects of the
gravitational field on light (or you're just really bored). The view from the front window is
boring, for all you can see is a large black area getting much bigger. The back window is much
more interesting. From the back window you can see the view of where you came from is ...
shrinking?
Why is this? Well, it has something to do with a thought-device normally used when considering
the formation of a black hole. It's called a light cone. In short the cone describes the range of
angles that are "escape paths" for light rays released from a collapsing star. As the star gets closer
and closer to its Schwarzschild Radius, the light cone gets smaller and smaller. Eventually, it
closes up completely, when the material collapses to its Rs and the event horizon forms. It's kinda
the same thing when you move towards the event horizon and the angular size of your universe
shrinks. It's the light cone closing up. Spacetime's way of reminding you that the chances of you
getting away from the black hole are fewer and fewer the closer you get to the event horizon. At
the event horizon the cone closes and the view of your universe condenses into a single point.
Now, I'd like to clear something up. This whole business of hovering around a black hole's
photon sphere or near the event horizon is quite speculative for one major reason: acceleration. I
rather blithely spoke of hovering at the photon sphere --- if real humans were to try that, they'd be
squished by the acceleration of their rockets! We're not talking a couple G's of force --- for the
smallest possible black hole for a 100 kg person that force would be 2.9 x 1015 Newtons!! (By
comparison, the force of gravity on a 100 kg person from the earth is 1,000 Newtons.) People just
can't survive that kind of acceleration. Most things can't. Still, it's rather useful to say such things
in order to get a good feel for what happens around black holes.
That's out of my system. Onward to inside the event horizon!
What happens when you cross the event horizon?
At first....well, nothing noticeable unless you consider the closure of the light cone to be
significant. Had you been looking out the back window, as previuosly mentioned, you would
have seen the view of your universe shrink to a dot and disappear; but there is no Star Trek force
field effect to tell you when you cross. What you see and feel next all depends on your reaction to
the fact that you are now doomed to a painful stretchy death. You could fire your rockets full
blast in the hopes of surviving a little longer by braking, or you could shut off the engines to
make your last few moments as comfortable as possible. It's a choose-your-own-adventure black
hole!
Beyond and to the Singularity!
I keep saying that, when you pass the event horizon, you keep falling...but towards what? Outside
of the black hole, you always fell towards the event horizon. What's inside? The answer is a little
tricky. You'd best check out the Inside 'em section about singularities, if you want the exact
answer. Obviously, the singularity is at the "center" of the black hole, according to my own
diagram at the top o' the page. It's that little green dot. Is there really a little green dot at the center
of a black hole? No ... there's no real center and there's no little green dot. Okay, an interpretation
of gravity is the bending of spacetime. Outside and close to the black hole spacetime is bent a lot,
but the curvature is still finite. The closer you get to the singularity, the more spacetime is bent.
At the singularity the curvature of spacetime is infinite, theoretically.
There's no center and there's nothing there except a lot of bent spacetime. This sure does make
sense, Jillian. Sorry, but the singularity is a damn strange place (er, a strange time, that is). What
happens near to it and what it is depends on quantum mechanics, which I don't understand that
well. Accept this: the singularity of a static black hole is described as a point at which spacetime
is infinitly curved. I cannot give you a satisfactory physical interpretation of that sentence.
Charged black holes are rather boring without their spacetime diagrams. That's why this section
seems brief. All the fun stuff happens when time-like suddenly becomes space-like and then flips
back again, and I chose to lump that stuff all together in Inside 'em.
The Photon Sphere
You know, this photon sphere is pretty much like the photon sphere of a static black hole. Briefly,
the photon sphere has a radius of 3/2 Rs. It is where light rays can hold unstable orbits around the
black hole. It is also the last possible orbital radius, since orbiting at any smaller radius would
involved going faster than light. Hypothetically speaking, if you levelled your eye there, you
would see the back of your head. Ah, the photon sphere summed up in four sentences!
Event horizons
Whoa....horizons---plural?? And just how does this happen Schwarzschild's geometry when
before it predicted only one horizon? Simple: when considering a black hole with charge,
Schwarzschild's solution for the geometry of spacetime is no longer valid. We need something
else! We need...the Reissner-Nordstrøm solution! The two scientists with that namesake solution
discovered that, if a small charge is added to a black hole, the event horizon shrinks and a second,
inner horizon forms just above the singularity! I've heard the inner horizon called the Cauchy
horizon, and I've also heard that electrons tend to "hover" around this horizon; but I can't actually
recall where I heard these things---I'm afraid you're going to just have to forget I said 'em! What
does this mean? Well, the event horizon is where, according to a distant observer, time seems to
stop. A charged black hole just has two radii where time seems to stop.
The more charge the black hole acquires, the smaller the outer event horizon and the larger the
inner event horizon gets. If the black hole acquires a charge whose magnitude is as great as its
mass (an insanely large number in the order of magnitude of 1030 coulombs), the outer and inner
event horizons merge. Should the black hole acquire a charge with a magnitude greater than its
own mass, both horizons vanish and leave a naked (eek!) singularity. This is great. This is neat.
We can get rid of both event horizons! That means we can weave and bob around the charged
black hole's singularity all we please without worry about needing to go faster than the speed of
light.
Practicality intrudes with a catch
There's always a catch, though (you knew this was coming). The two event horizons only
disappear when the magnitude of the charge of the black hole is greater than its mass. Black holes
come in 3 solar masses or more. One solar mass is roughly 1.989 x 1030 kilograms. That's huge!!
You would be truly hard-pressed to amass a charge of that magnitude.
Let's just say, for hypothetical puposes, that you managed to charge up a black hole enough to
disperse the two horizons. Great. Now you have to carefully monitor everything that goes into
and comes out of it and must keep a great store of charge on hand. Something with that much
charge would tend to rip apart the atoms around it, pulling the oppositely charged parts into itself.
It's almost as if the black hole is actively trying to neutralize its charge like a person would try to
warm up by drinking a cup of hot liquid (mmm, hot tea!). People give electrons personalities and
desires, so I see nothing wrong with doing the same for charged black holes.
Standard Singularity
The singularity of a charged black hole is the same as that of a static black hole with the
exception that it is possible for the singularity to exist without any protective horizons. Physicists
just don't like this, Roger Penrose included; of such events he said, "thou shalt not have naked
singularities!"5 I'm inclined to agree for practical reasons, but I still think it is possible to have
such an unveiled singularity. The trouble just comes in when you try to describe such a...place?
Infinitly curved space with quantum mechanics kicking in...messy, isn't it?
Did I say there was only one difference between the singularity of a static black hole and that of a
charged one? How forgetful of me! Truly, there is one other important difference: you can avoid
the singularity of a charged black hole, whereas you must eventually encounter that of a static
black hole. Encountering a singularity (thus assuming that you surived the tidal forces from the
trip there!) is not something you want to do, I can assume. Why is this so neat? In a static black
hole, once inside the event horizon, that's it---end of your life. However, should you survive the
trip between the outer event horizon and the inner one of a charged black hole, you could in
theory turn around and leave the black hole back and return to your own universe, or you could
go into another universe...but...how?
What's a rotating black hole?
Angular velocity is one of the only three properties of a black hole, according to those-whoknow. Rotating black holes are also known as Kerr black holes. It's also called a Kerr black hole
for the same reason charged black holes are called Reissner-Nordstrøm black holes: ReissnerNordstrøm geometry describes the charged black hole. The Kerr solution for the metric for
spacetime describes the geometry of the rotating black hole. Since the static and charged black
holes are spherically symmertric, the Schwarschild and Reissner-Nordstrøm geometries are idealy
suited to polar coordinates in two dimensions and spherica coordinates in three dimensions.
What're those? Well, instead of a grid of lines, things are described by a lenth and some angles.
For the polar coords, lines of constant distance are circles; for the spherical coords, lines of
constant distance are spheres. Lines of constant angle are radial lines for both systems.
Kerr geometry uses something called oblate spheroidal coordinate system. It's similar to polar
coordinates, except lines of constant distance are ellipses, and lines of constant angle are
hyperbolas. Now, take that two dimensional grid-type-graph and rotate it about the vertical axis--that's oblate spheroidal!
Why is it called a rotating black hole? The event horizon doesn't rotate---it's just a boundary-line.
I suppose it's more descriptive (and unfortunatly more confusing) to say that everything else
revolves around the black hole. Since most stars rotate, when they become black holes, they also
rotate. As those stars are becoming black holes, they shrink---and as a rotating something gets
smaller, it rotates faster. Realistically, I suppose many black holes rotate very quickly. According
to the calculations done by Kip Thorne, most black holes would rotate with a speed that is 99.8%
their mass.6
Perhaps it would be better if I just explained how a rotating black hole is different from the
others. First stop...
Two Photon Spheres?
What's with the doubling?! Charged black holes have two event horizons instead of one, and now
rotating black holes have not only two event horizons but two photon spheres. It has to do with
the fact that the black hole rotates, and, when it rotates, it "drags" the space around it into rotating
with it. It's like jumping into a whirlpool. If you jump in the direction of the current and start
swimming, you get propelled along and go much faster than your normal speed. If you jump in
and swim against the current, you get dragged back and go much slower than your normal speed.
It is the same principle for photons as they try to orbit. Instead of just having one radius where
photon can orbit unstably like the other two black holes, a rotating black hole has two radii where
photons can orbit: one in the direction of rotating and one against.
So there are two photon spheres. How do you tell which one goes where? Well, orbiting against
the rotation of the black hole is more difficult. It's easier for me to use the example of satellites
around the earth. Take two satellites, one further from the earth and one closer to the earth. The
one closer to the earth must go much faster in order to stay in orbit and not go crashing down to
the planet below. The satellite further away can go slower and still stay in orbit. O-K. Now, back
to the black hole. Say you've got two light rays, one going in the direction of rotation and one
going against it. You can draw a not-entirely-scientific parallel between the two situations (notentirely-scientific because the speed of light is a constant). The light ray going in the direction of
rotation can kinda "go faster," so it can orbit closer to the black hole. The light ray going against
the rotation kinda "goes slower," so it must orbit further away from the black hole.
Oh, did I say it had two photon spheres? Heehee. That's not quite accurate. See, since the black
hole rotates, it's got an axis of rotation. It's no longer spherically symmetric like the other two. Its
structure depends on what angle at which you approach the black hole. If you approach in the
direction of the equator, then what I told you stands true: two photon spheres. If, however, you
approach at an angle to the equator (you know, just like lines of latitude on the earth), there is
only one photon sphere. Why? Well, if you come in from the top, there isn't really a choice of
whether you should go with or against the rotation.
What's between the two photon spheres?
A sensible question. Between the counterrotating and the corotating photon spheres are the
different photon spheres for light rays approaching the black hole not quite on the equator. This is
confusing, so perhaps I should use another example. Imagine that you're floating above the earth
in a spaceship. You can come in for a landing on the equator in Equador, further from the equator
at Syracuse, New York, or on the axis of rotation at the North Pole. Say the earth is a rotating
black hole and you are a light ray. If you came in towards "Equador," you would encounter the
counterrotating photon sphere at one point, and you could merrily and unstably orbit the black
hole against the direction of rotation. Going in further you would find the corotating photon
sphere, and you could happily orbit unstably in the direction of rotation. If you came in towards
"Syracuse" or the "North Pole," there would only be one photon sphere for ya.
The placement of the two photon spheres all depends upon the speed of rotation. For a slowly
rotating black hole the two are so close to one another (on the equatorial plane, remember) that
it's almost like the photon sphere of a non-rotating black hole. The faster the black hole rotates,
the further apart the two photon spheres are. A black hole that rotates with a speed as great as its
mass (again, the numbers we're blithely bandying about are sickeningly huge, since the mass of
the smallest possible black hole is 5.7 x 1030kg --- meaning it would have to have an angular
velocity of 5.7 x 1030 meters per second) has the greatest distance between the two photon
spheres.
Feh! Enough of the photon spheres.
The Ergosphere
The ergosphere is not a photon sphere, nor is it an event horizon. It's something very special to
rotating black holes. Firstly, what does it look like? It is a solid ellipsoid (a 3D ellipse) that
billows out from the black hole above the outer event horizon. The faster the black hole rotates,
the further it billows. This is unusual cuz it's the first thing I've mentioned that is not just a radius,
it is, in fact, a region. The photon spheres and event horizons are just distances, after all; but not
the ergosphere. By the by, the most the ergosphere can billow out is when its radius (along the
axis of rotation) is 1/2 Rs.
The outer boundary of the ergosphere is the static limit of the rotating black hole. What's a static
limit? It's where you can no longer stay still, even if you were going at the speed of light. For
static black holes the static limit is the event horizon, since after you cross that, even if you go the
speed of light, you are pulled towards the singularity. A rotating black hole is different from the
other two (once again!) in that its static limit is above its outer event horizon.
This is all very nice, but what does it mean?! It means that once you cross into the ergosphere, it
is impossible to stay still. Even light rays are dragged along in the direction of rotation. However,
you can enter and leave this region whenever you like, unlike the abandon-all-hope-ye-whoenter-here static limit/event horizon of the other two black holes. You can merrily weave in and
out of the ergosphere with no nasty side-effects. It's a place of quirky spacetime of the black hole
that we can actually visit and leave.
Familiar Horizons
The two event horizons of the rotating black hole are pretty much the same as a charged black
hole's event horizons: two radii where a distant observer would say time seems to stop. The outer
horizon switches time and space around one way, and the inner horizon flips 'em back to how
they are in the real world. Oh, I didn't explain that stuff when I talked about charged black holes,
right? It's pretty complicated and it's easier to understand if you know how to read spacetime
diagrams. I go into detail in the Inside 'em section.
Briefly, we normally go about life moving around as we please in space but being inexorably
dragged by time. Whenever you cross any event horizon, those two things...flip. That's why I
edged a lot when I mentioned the singularity. Due to this flipping, it is not a point in space; it's a
time. After a certain time after you cross a static black hole's event horizon, what's left of you
might survive to encounter the singularity. What about charged and rotating black holes and their
two horizons? That's what make's 'em special. When you cross the second horizon, space and
time flip back to what we're used to experiencing and the singularity becomes a place in space
and entirely possible to avoid. Back to the topic at hand!
The rotating black hole has an outer and an inner event horizon. They each move closer to one
another the more angular velocity the black hole has. When the speed of the black hole's rotation
equals its mass, the two event horizons merge into one. When you cross this, you don't even bat
an eyelash because there is no flipping of space and time at all. Should the black hole spin faster
than that, both horizons disappear and leave the exposed singularity. Everyone calls this a naked
singularity so often that I've sworn off using the term completely.
Ring around the Singularity
Well, that's not entirely accurate. The singularity is a ring, you see. Calculations using Kerr
geometry describe the singularity as ring-shaped. That's a curious concept. It's also much more
complicated than that! I go into detail about singularities in the Inside 'em section for a reason---I
want to keep all the wackiness in one place. Still, ring singularities are just so neat that I can't
contain myself.
You already know that a traveler can move about quite freely after crossing the inner event
horizon. Space and time are in their usual places. What you don't know is that you actually have
to make an effort to hit the singularity of a rotating black hole! If you approach it from any angle
other than equatorial, you go...(ahem)...well, you don't hit the singularity. You could turn around
and leave the black hole. Sure, you'll emerge way in the future of your own universe, but,
considering that the nearest black hole is many many light years away, you'll probably have
expected some time dialation from your method of getting to the darned thing in the first place.
Where you go if you point yourself towards to singularity depends on who you talk to. Some
physicists like to call it 'negative space.' Others (myself included) like to call it another universe
or another part of this universe.
You could just sit there and admire the view. The singularity would appear like an oval window.
If you looked at it from the axis of rotation, it would be a circular window. The closer you got to
the equator, the more oval the window would seem. At the equator, you would only see the
singularity. But...what would the oval window show you?
What's a Singularity?
Clever me hasn't quite mentioned what a singularity is, just that black holes have one. What is it?
Anything. Everything. Nothing. Infinitly curved spacetime. All of these things could be the
singularity. The whole problem is that in order to understand what happens just above and in the
singularity, you've got to have a good grasp of quantum mechanics. Much to my regret I do not
have a good grasp of quantum mechanics. I cannot explain why the things I'm going to tell you
are true, I can only repeat what I've heard and read.
First and foremost of these is that the singularity is a "seething bed of probability." What does this
mean? Well, from what I can tell, it means that trying to say what's actually *inside* the
singularity is just as bad as trying to measure the exact location and velocity of an electron (which
cannot be done with any kind of reliability, according to Herr Heisenberg). Surely, one would
think, it is possible to just poke one's head in and check. Erm...not quite. There could be whole
other simultaneous universes inside, or a crystallized fractal-shaped tree, or there could just be a
little pink kitten with creme on its whiskers. Confusing , yes. Let it stand that the singularity is
much like the early universe: extremely energetic and also quite chaotic. That's real chaos---the
inability to say what's what---versus chaotic motion and fractals. That wasn't very satisfying. Ah,
well, you'll have to wait until I take a quantum mechanics course!
A Run-through of the Singularities :
Static Singularity
The picture to the left describes the singularity of a static black hole. What are those funny graph
pictures? This is a 1-2 diagram---1 time dimension and two spatial dimensions. Okay, recall the
lines of simultaneity? The surface of these graphs describes a plane of simultaneity. This picture
shows the evolution of an eternal black hole, one that has existed from the beginning of time. The
1-2 diagrams are known as embedding diagrams, for they take a slice of spacetime and embed it
in the third dimension to show how it warps. It takes a lot of heavy calculation to create them and
make such nice pictures as this one. 7 Trust me, for I tried to draw them and failed miserably.
What does this picture mean? A, B, C, D, and E are slices of spacetime, you know that much.
This picture chronologically follows the creation of a wormhole! You know, a Star Trek thing
that allows you to travel from the Alpha quadrant to the Gamma quadrant and encounter the
Founders and wage war for a few seasons. Slice E shows spacetime beginning to pucker, D shows
the wormhole bridging the two universes, C shows the shows the wormhole at its largest possible
diameter, B shows the wormhole then pinching off, and A shows the resulting eternal spacetime.
A is the future singularity, the thing everyone falls into; and B is the past singularity, the thing
that theoretically spawned the universe. See how the bottom of the gravity well is so pointy? That
point is the singularity. But, you say, there apparently are two singularities. Yes. One for our
universe, one for another. This leads to slice B and D and C. The singularities are gone and the
spacetime is connected! This would be the actual wormhole. Two universes or two places in one
universe are "touching" in the sense that it is possible to go from one to the other. Incidentally,
you'll notice that the embedding diagram looks a lot like a dip or a well in spacetime. I'll use the
term 'gravity well' to describe the region of spacetime warped by the gravity generated by the
presence of matter.
This would be a great way to travel from place to place in the universe, except it is terribly
unstable. The whole process of the wormhole forming and then pinching off sounds like it takes a
while. Most astronomical stuff takes a while. Unfortunatly, the wormhole pinches off so quickly
that not even light can get through. A static black hole simply isn't meant for convenient space
travel. A charged black hole's singularity looks like this in the sense that the singularity is pointy
and nasty, but it has the virtue of avoidability.
Remember the Ring Singularity?
Yes, the singularity of a rotating black hole is different than that of a static or charged black hole.
The Kerr geometry describes it as a ring, whereas the other two are points. As a result, it has
some quirky properties. The oddest of which is not something I can prove to you right now. Take
my word for it that nasty calculations done by a C. T. Cunningham8 have proven that gravity
around the ring singularity is repulsive. Light actually bounces away from it! Things actually
bounce off it! Well, that's not quite accurate. It depends on your approach angle.
Remember how things were different along the equator of a rotating black hole? Well, it's the
same thing, again. Pretty much the only way to fall into a ring singularity is to approach it on the
equatorial plane. All other approach trajectories will be repelled with increasing strength the
closer the angle is to the axis of rotation. There is actually a third photon sphere right near the
singularity. If light approaches nearly parallel to the axis of rotation, the gravity and antigravity of
the singularity are balanced out; and the light traces out a funny-looking path of constant distance
(which in Kerr geometry is an ellipsoid---a three dimensional oval). This path sometimes leads
the light into another universe and back again. Pretty sneaky, huh?
That's why looking at a ring singularity is so much cooler than the other two singularities. Seen
from the axis of rotation, it is a circular window with rings of different light. The larger the angle
between your trajectory and the axis of rotation, the more oval that window becomes. Seen from
the equator, it is a sort of line-like thing glowing with only one kind of light. What you actually
see in the window is pretty complicated, so I'm going to keep it quite simple. Basically, you see
three things: light from your own universe (from which you just came) that has been bounching
around the singularity, light from the singularity itself (it is such an odd place!), and light from
the so-called negative space (however many other universes are visible from the black hole). Why
does the singularity emit its own light? It's a place where events don't always follow a logical
pattern. Random particles are created and destroyed in the area just around it. That's why it would
emit its own light. At least that's what I've been told. Yes.
What about the Other Universes in the Negative Space?
Why am I not describing those? According to one souce, it's possible to peer in and see the light
from 4 or 5 other universes and varying times. I don't quite fully understand all the possibilities.
It's easier if you look at that Penrose diagram of the black hole. Theoretically, it goes on infinitly.
You have access to an infinite number of universes from inside such a black hole. You could visit
one, two, a few, or many! You could also weave in and out of the inner event horizon, peering at
other universes and not entering them. That's why it's tough for me to give a general description
of what you see past the singularity.
Tidal Forces
Sounds like it involves water, doesn't it? In a way tides are just a side effect of the namesake idea.
This all has to do with how matter behaves in a gravitational field---and I don't just mean by
falling down! It's easier to explain this with a thought experiment. Imagine these two rocks that
are high above the earth and as far apart as a quarter of the earth's radius. They're falling towards
the planet. As they fall, the bottom rock is further down in the gravitational field, experiencing
more acceleration than the top rock. Consequently, the bottom rock falls faster than the top one,
and they start to separate.
Now, imagine the two rocks high above the earth lined
up horizontally and as far apart as a quarter of the earth's circumference. The two rocks are both
falling effectively towards the center of the earth (except there're these pesky things the crust and
the mantle in the way). As they fall, the two rocks move horizontally towards the center of the
earth. This makes them both move toward each other, since the center of the earth is kinda
between them (albeit quite below them, too).
What's this got to do with the force that ends up stretching you to infinity when you fall into a
black hole? It all has to do with the equivalence principle. An inertial frame is just like a frame in
freefall on a local scale. Why? Think about something in freefall---remember those silly physics
homework questions about how much Suzy weighs as measured on a scale in an elevator as it
falls? The answer is that her weight is zero. Normally, people are accelerating at 9.8 meters per
second upward, as a result of the force of the earth pushing on 'em. When an object is in freefall,
that so-called normal force is gone, so the total acceleration is zero.
What's the catch? Well, this only works on a local basis. It's small-scale stuff. When you fall into
a black hole, you're in freefall. You should be okay...except that the size of "local" rapidly
decreases. When "local" no longer covers your whole body, you start to deform just like those
rocks. Along the radial direction your body gets stretched apart (like the vertical rocks), and along
the tangential direction (like the horizontal rocks) your body gets squished together. This
eventually becomes so bad that your body can't take the stress and...erm...well...it painfully
breaks apart. Local keeps shrinking, smaller and smaller, until it passes atomic size and goes to
zero, as the mathematicians would say.
White Holes and Blue Sheets
Talking about black holes always involves lots of black, black, black with maybe a little red
thrown in the mix. In conjuction to hearing about black holes, most people have heard the phrase
'white hole.' It is a confusing phrase and is generally avoided by serious scientists, I've been told.
I'm not the most serious-minded person, so I'll give a once-through on the whole white hole thing,
just to clear the air and resolve any questions.
Black holes and white holes are, indeed, two sides of the same coin. In fact that metaphor
describes their relationship exactly. Whether something is a black hole or a white hole depends on
in what direction you are looking. Stand outside of the outer event horizon and you are looking
down at a fine black hole. Stand inside the inner event horizon (cuz you can't technically 'stand'
between horizons cuz there are no static observers, there) and look back where you came from
and you'll see a fine white hole. You enter the black hole and leave the white hole at the same
time. Here's another metaphor: driving on a one-way street. When you look behind you, you see
an event horizon of a white hole; when you look ahead of you, you see an event horizon of a
black hole. As per the metaphor, you cannot back up into the white hole. Just as the static
singularity is always in the future, the white hole is always in the past; any attempt to back into
the white hole is countered by insurmountable resistance.
Great, now you know about white holes. The blue sheet is pretty much the same thing. In fact you
could even look on it (not entirely accuratly) as being the 'event horizon' of the white hole ... sort
of. As stuff falls further into the gravity well of the black hole, it is losing gravitational potential
energy. Where does that energy go, you ask? It translates into kinetic energy! (The velocity of the
matter that falls in increases and the whole effect is commonly called the acceleration due to
gravity.) Light also feels the effect of gravity, but since it can't really speed up (light traveling in a
vacuum being the constant rate of 3x108 and all), its frequency increases: microwaves to visible
light to gamma rays (the whole spectrum of electromagnetic radiation...you get the point). This
process is called blue shifting, akin to red shifting, about which you've no doubt already read.
So, about the blue sheet, already! I just described what happens to a singular light ray falling into
the black hole. Imagine that happening to all of the light falling into the black hole. The wave of
high-energy radiation pouring across the black hole's horizons is known as the blue sheet
(y'know, sheet of energy, blue shifting...it makes sense). The blue sheet effect is solely from the
energy boost of the outer event horizon, just from falling into the black hole's gravity well. You
see how the blue sheet kinda is the white hole? A massive outpouring of high energy stuff? It's
one of the major hazards of traveling in a black hole (aside from the black hole, itself, that is).
Charged or Rotating Singularities: how to get from Universe A to Universe B in one piece
I've been teasing you about being able to go from universe to universe using charged or rotating
black holes.
To the right is the Penrose diagram for a charged or rotating black hole. One of the first things I
want to point out is the nature of the singualrity. I told you earlier that the singularity was a place
in time; but that was only for the static black hole! See here, the singularity is a definite place,
and places can be avoided as long as you don't have to go the speed of light to do it.
This is the road map for jumping from one universe to another. Say that purple worldline is me in
my 2085 Ford Tempo rocket (with mismatching red paint). I want to travel somewhere using the
super-massive rotating black hole right in front of me. I take the time to perch at the lip of the
gravity well and at a small angle to one of the poles of the axis of rotation of the black hole. (My
Tempo is impossibly well-shielded against radiation.) Armed with the might of relativity (and
some auto insurance), I dip the nose of my Tempo towards the outer event horizon and dive into
the 'well. As I fall, I'm trading gravitational potential energy for kinetic energy, and I end up
going quite fast as I cross the outer event horizon. The instant I reach Rs, my engines cut off just
as I preprogramed them to do.
This particular galactic black hole is rotating very quickly, so I very quickly cross the inner event
horizon. Since the two event horizons are nearly on top of one another and since I cut my engines
before I entered the realm between them, I do not experience any tidal unpleasantness. A very
curious thing happens when I cross the outer event horizon. The singularity becomes an
unavoidable place in time---it becomes my future---as the time axis and the space axis of my
spacetime diagram exchange places. As I cross the inner event horizon, time and space resume
their normal axes on my spacetime diagram, and the singularity becomes a place in space.
I should remind you that I'm rocketing along at a speed close to light. I blaze across the inner
event horizon and shoot right through the center of the ring singularity. Oooh, confusing
statement. The singularity appears to me as a round window. If the singularity emits any light on
its own, I would see that as the frame of the window. Inside that window...is reminescent of what
you see when you reflect one mirror into another: a hallway of mirrors arching into infinity. The
smaller the angle between my approach and the axis of rotation, the more mirrors I see. What I
see in the window of the singularity is the same; but, instead of mirrors, I see an infinite number
of locations.
There is only one restriction on where I may go with a rotating black hole: to enter a black hole
means to leave a black hole. Black holes are rather like subway terminals in that sense; if you
walk down the stairs to take a train, you've got to walk back up the stairs when you exit. You can
only exit at locations with those stairs. You could not use a black hole to pop out right next to
earth, 1999, because there are no black holes right next to earth right now.
I wizz through the very center of the window, nearly orthogonal to (perpendicular to) the window
(nearly because I approached nearly parallel to the axis of rotation. I recross both event horizons,
one after the other, and leave the black hole at a speed close to that of light. I gained all my speed
entering the gravitational well, now I lose it all leaving the well. I coast away from the black
hole's gravity well at the same speed I entered, the mirror-image of my worldline when I entered
the 'well---which kinda means I end up perched at the lip of the 'well, again, with the option to
fall back in or to leave and explore.
This universe-jumping is a fun thing to think about, but I always get edgy when considering the
idea of innocently wandering into a whole different universe. I mean, the only things that define
our universe are our so-called laws of physics. The speed of light in a vaccum is 3x108. Electrons
have such and such weight. Subatomic particles interact in a certain way. In another universe, the
numbers for these laws might differ somewhat---or the laws could be completely different! Recall
all that dust and gas falling into the black hole as innocent little me attempts to leave the gravity
well? Suppose the universe I just entered is one where antimatter is the dominant type of matter--and here's little matter me. Imagine my surprise as a tiny clump of anti-hydrogen atoms wisps
against my Tempo's fender. Boom! Tremendous explosion and lots of energy released, and that's
the end of my traveling days.
The other problem is that this situation is completely theoretical. The Kerr solution is very
unstable. The mere approach of a rocket to the outer event horizon (let alone one diving across
said horizon), might destabalize the black hole and make it fatal for the rocket attempting to
travel through it. Get that? It's not possible. I'm sorry (cuz it sounds like a fun way to explore),
but that's the way thing work.
Gravity Waves
Spacetime Ripples
Best if I start at the beginning (not the big bang, that's later). I'm sure you've heard the much
bandied about description of this universe wrapped up with the neat little label 'spacetime.' I'm
going to ask you to imagine that we're living in a two-dimensional world just like living in a piece
of paper. Why? See, spacetime for a three dimensional world plus the time dimension is a 4D
shape, which, understandably, humans have a tough time picturing. Spacetime for a 2D universe
like that paper would be a 3D thing, which I can describe to you rather well (if I do say so
myself!). Thus, please, dear surfer, don't tax my poor brain by asking me to describe anything
higher than three dimensions.
So, we've got this infinite 3D jello-mold of the universe, here, to serve as my analogy (wobble,
wobble). Let's take a slice---schlok! What does this slice represent? This is the shape of spacetime
at a certain time. See, over there is a star---see how the jello bends down? Over yonder is a black
hole---look at the point on that sucker! See, there, that double-bumped dip is a binary system.
These dips in spacetime are called gravity wells. The presence of matter or energy (lots and lots
of energy) warps spacetime. Anyway, when spacetime bends, it manifests as something we call
'gravity.' The more stuff, the more bend. The more bend, the more gravity. If you get enough stuff
within a certain area, you pop! right through the jello slice and get a black hole, but that's material
for another section.
For a long time people believe this was a calm, peaceful, orderly universe, where stars danced
eternally in stately paths through inky empty darkness. Now, of course, we've seen the contrary.
New stars ignite and begin to burn themselves up in their nuclear bonfire, dying stars explode and
viciously shove off skads of gas and leave sullen crisped orbs in their place, high-energy particles
scream away at fractional-light speeds from the cores of galaxies and black holes, and in between
everything is a fog of dust and gas, spiced by the flickers of virtual particles. Certainly an
interesting place!
What's this got to do with our slice o' jello-mold-turned-spacetime? It means that objects do some
interesting things over time. Take that supernova. Watch it as it happens, and you'll see a ripple
move out from the star. What was that shockwave?! Look more closely at that black hole and
you'll see it vibrate ever-so-slightly. Take the binary---it's radiating ripples the way a bobbing
soccer ball creates ripples in water. Waves...in spacetime?
Recall how warped spacetime is detected as gravity? Well, the principle holds with this wave,
too. Just like the stationary dips in spacetime are gravity wells, those travelling waves of
spacetime are also manifestations of gravity. We know what a gravity well feels like---we life
with the tug of gravity all of our lives; but what does a gravitational wave feel like when it hits?
How fast does it move? If it's a wave, what's the equivalent of the jello particles that bounce
around? These are all good questions!! Firstly, the mechanics (or, what a gravitational wave feels
like when it hits).
Mechanics
The effects of a gravity wave are seen perpendicular to the direction in which the wave is
travelling. Hunh? If a wave is moving towards you---dead on, straight atcha---you'll see the
effects of the wave left and right and up and down, not forwards and backwards. As much as
people compare gravity waves to sound waves (for some good reasons---later!), which are just
compression waves, the comparison is not complete. You always see such waves in the movies;
you know, an oil truck blows up and the heroine gets thrown away from the truck to land splat!
on the asphalt further back. That's a compression wave---it affects stuff parallel to the direction of
propagation (that is to say, it rattles stuff back and forth in the same direction in which it moves).
Gravity waves are not oil truck explosions! If you stuck gravity waves in the place of the
explosion from the oil truck, the heroine would have been rattled left and right and up and down--but she would not have moved one inch away from the truck!
I keep mentioning ripples in jello molds and sound waves and water ripples, but those are all little
atoms jiggling around like mad. Those are just compression-wave analogies to describe what a
gravitational wave does. The gravitational wave is, upon closer examination, much more like a
light ray than a sound wave. D'you get it? Electromagnetic field/gravity well --- electromagnetic
wave/gravity wave (AND transverse effects to propagating waves!) --- photon/graviton.
Graviton? Want a more detailed run-through of that? Check out the parallels section, which is
after this one.
"Rattled left and right and up and down"---what does that mean, exactly? That's just what a
gravitational wave does. Take the unfortunate heroine-near-the-oil-truck. Say there's this
extremely powerful gravitational wave whooshing towards her (at the speed that it moves,
whooshing is a woefully understated description). By the by waves this powerful aren't anything
we could experience. Before the wave hits she's 6 feet tall and 2 feet wide. As the wave hits, she
is first stretched vertically to 8 feet and squished horizontally to 1.5 feet, returned to normal,
squished vertically and stretched horizontally, and returned to normal. That was one complete
oscillation of one particular type of gravitational wave. Here's the best part: not only was the
heroine stretched in such painful ways, but everything around her was, too. I should stress that
this particular gravitational wave was extremely strong. We don't get waves that strong on the
earth. Another thing is that even though everything was stretched and compressed, it still would
have hurt a lot for the heroine. The actual stretching-and-compressing of the gravitational waves
is more complex that this, but I saved that for a part in the parallels section.
Say our heroine wished to avoid the gravity wave speeding towards her. Could she do it? How
fast does a gravitational wave move, and does it matter what it moves through? Fair enough, I
mean, compression waves travel at different speeds as they move through different consistencies.
Say I pound the bottom of a table, right underneath a glass bowl filled with jello. The wave
travels through the table at once speed, enters the glass and changes speeds, enters the jello and
changes speeds again, and pops into the air, changing speeds yet one more time. All those
different speeds! What speed is right for a gravitational wave? C, the speed of light: 3 x 10^ 8
meters per second. Ah, so, perhaps our heroine won't be able to avoid the wave. What about
putting up a jello-mold barrier so that the wave slows down enough for her to out run the wave?
Does a gravitational wave slow down as it changes media? No. Should it be through gas, star,
galaxy, black hole, or jello, gravitational waves move at the speed of light---constantly. Funny
thing is that not much affects these waves. Very few things would change a gravitational wave,
making them rather handy echo-locations for astronomers, should they detect them, which they
haven't; but it'll be really cool when they do, but they haven't. Why not? It has to do with the
strength of the signals.
Parallels to Light
The similarities (at least the surface ones) between gravity waves and light waves struck me when
I first did my research. The analogy is not complete, however, since gravity waves are not caused
by the same things as electromagnetic waves. If you know something about electromagnetic
waves, you've an easy transition to gravitational ones; if not, well, you get to learn two things for
the price of one, a deal any day.
Fields
Waves
Particles
Electromagnetic and Gravitational Fields
May my electromag teachers forgive me, but I put forth the butchered version of their wonderful
classes for those who need a brushing up on electric and magnetic fields. If you want to skip past
this part, go ahead.
Electricity and magnetism 101
I'm sure you're familiar with electricity---little electrons zipping here and there through metal
wires---and charges---which arise due to the distribution of differently charged particles in
something. There are two types or alignments of charge: positive and negative. Magnetism is that
thing that happens with magnets and compasses and the earth. There are two alignments of
magnetism: north and south poles. The cool thing is that the two things are linked. Say you've got
this wire, and there's a current going through it, which means there are lots of electrons streaming
through it in a certain direction. That wire also generates a magnetic field which looks like a
collection of rings surrounding the wire at a certain distance. Makes household electrical wiring a
pain, but it's still pretty neat.
What is the magnetic field, anyway? That's tricky. The field, be it electric or magnetic, is this
invisible thing that happens when you have electrons moving from one place to another. There
are equations that describe the field, and laws the field obeys. Should you put certain particles
(say, electrons) in the field, you could use those equations to predict what those electrons would
do. The earth itself has a magnetic field; so does the sun. They're physically large by our
standards. The earth's magnetic field streams out from one pole, curves across the length of the
planet in a high arc that looks very much like a curved handle, and streams back into the other
pole. The earth's magnetic field is actually kinda weak compared to the fields that we can
generate.
Oh, I forgot to tell you what an electric field is, didn't I? An electric field is pretty much the same
idea as a magnetic one, this invisible thing that does stuff to electrons. There is a difference in
how ya generate one, though. To make an electric field, you have to take an object---anything--and pump it full of electrons (or rip away lots of electrons). Now that object is charged and it's
radiating this electric field. Should you put something into this field, there are equations and laws
that you could use to predict what would happen.
Congratulations, you've suffered through my crash course on electromagnetic fields. Oh, since
there are both electric fields and magnetic fields, why do I keep calling 'em electromagnetic
fields. It's to honor the fact that we know that one thing can generate the other.
The similarity I'm trying to make is that both electromagnetism and gravity have a stationary
form of their effects in a field. The causes are completely different, really. Electromag fields are
caused by various arrangements of atoms and electrons, and gravity is caused by the existence of
mass or energy in one place. At second glance they don't really seem too similar. Still, they are
both this invisible thing called a field that we can predict with equations and laws. Heh, well,
mostly predict---the extremes are usually exceptions, but we don't deal with extreme charges or
gravity on a daily basis.
Electromagnetic and Gravitational Waves
Okay, so there is a similarity in that they both have stationary fields. What about the really neat
stuff, the travelling waves? How are they similar? Well, what's an electromagnetic wave do after
all? (Yes, you can skip down past this part, too) See, it took me a while to figure this one out. I've
always heard people say that a light ray, and electromagnetic wave, is an oscillating electric field
perpendicular to an oscillating magnetic field, but that didn't make sense. How could they be
perpendicular? This ties back into the electric and magnetic fields part. When I said that an
electric current generated a magnetic field, I was sneaky and didn't tell you what that field really
looked like.
Among engineers and physicists there is this love of something called the 'right hand rule.'
There's one for every situation; meaning there are skads of different right hand rules. This
particular rule tells you how to figure out which way the magnetic field is going, and, thus, what
it looks like (the two being one and the same, really). Say you've got this current flowing through
a wire, happy as can be. Take your right hand and point your thumb in the direction of the
current. Now curl your fingers around, slightly. The direction your fingers are curling in---that's
the magnetic field generated by the current in the wire. You notice something? That magnetic
field is perpendicular to the current. Aha! Now you know what I mean by perpendicular. Back to
the light ray.
Now, say you have this light ray. What really happens is that it generates an electric field in one
direction (let's say the x-axis of a traditional coordinate plane), which generates a magnetic field
perpendicular to that electric field, which in this case would be along the y-axis of that coordinate
plane. Confusing, I know. Perhaps it is simpler to say that the wave has an electric component
and a magnetic component. The eletric component pushes electrons in one direction, while the
magnetic component pushes the electrons in another direction that is perpendicular to the
direction the electric component is pushing them. Both of these components are in synch: when
the electric component is at its strongest, so is the magnetic component.
Although light rays travel in all sorts of directions and with the electric component point every
which way, there are basically two orientations of electromag rays. This is very tricky, and I'm
not quite sure if I've got it right, so bear with me. Say there is this light ray that happens to be
oriented (for my convenience) with its electric component pointing straight up. It'll make
electrons bounce around in that direction. Now, the magnetic field is pointing to the right. That is
one orientation of an electromag ray. The other is in effect inverting everything so that where
something electric was happening, now there is none, and where there was nothing doing, now
there is this electric field pushing around electrons. In a more visual description, just take the
electric component arrow that used to be pointing up and turn it 90° to the right or the left. That is
the second orientation. Told you it was tricky.
One final note: these fields the ray generates are detectable in the plane that is perpendicular to
the direction in which the wave travels. Simply put, stick a fork upright in a sheet of jello---the
fork represents the direction of the wave, the jello represents the plane in which the electric and
magnetic components are pushing the electrons around.
Right, so what does that have to do with gravitational waves? They behave in the same
perpendicular way that light rays do, except they do something slightly different. The effects of
the wave are still detectable in the perpendicular plane to the direction of the wave---the fork in
the sheet of jello is still the direction in which the wave travels and the jello sheet is still the plane
of oscillation. How the wave actually oscillates is much trickier. I admit, earlier, I gave a run
through of what the wave does, but I didn't explain it. Now is the time for the tricky details.
Just like the electromag wave has two directions of oscillation (the electric and the perpendicular
magnetic), so does the gravitational wave. However, instead of being 90° apart (perpendicular),
they are 45° apart---which is related to the type of particle that carries the wave, but that's for
further down. Okay, that's nice, but what happens with the oscillations?! I called it earlier 'left and
right and up and down.' That's...well, it's not quite accurate. Let's take 'em one at a time.
Left and right: straight forward, this one. When it strikes an object, this wave initially compresses
it horizontally and stretches it vertically. Then it reverses the treatment and stretches horizontally
and compresses vertically. It alternately stretches and compresses in perpendicular directions.
Nothing happens along the two diagonal directions. This is traditionally called the plus
polarization.
Up and down: this is the tricky one. I really lied to you by saying that it's up and down but it's
easier to say quickly. When this one strikes an object, it does the same alternating compressionperpendicular-to-expansion just like the other one, but the direction of compressing and stretching
is along diagonal lines 45° off axis. Huh?? Take the xy axis and tilt it 45°---this oscillation affects
stuff along the new x and y axes and changes nothing along the old x and y axes. If I were to be
accurate each time I described the wave, I'd have to say 'diagonal with a negative slope and
diagonal with a positive slope' every single time---which is really confusing. Food for thought:
electromag waves oscillate with electromag fields; so, if the analogy to a gravitational wave held,
that would mean that the gravitational wave oscillates with fields of stronger and weaker gravity.
Hmmm. This diagonal axis oscillation is traditionally called the cross polarization.
So, you see, the two waves are similar in that they both affect stuff perpendicular to the direction
in which the wave is going. They are also similar in that the electric and magnetic fields are
perpendicular to one another, just as the compression and expansion are also perpendicular to one
another.
Electromagnetic and Gravitational Particles
I am cheating on this section because I'm only going to introduce the fundamental particle of the
gravitational wave. I need to take a few more classes before I can explain any more.
What particle is associated with the gravitational wave? The graviton. It has spin 2 (for what
that's worth since I cannot do the explanation of spin justice). It has no mass and can therefore
travel at the speed of light. This is important---it means that gravitational waves propagate at the
speed of light. Gravitons can be measured at a range of frequencies that correspond to the source
of the wave---ah, but that's for the section on sources!
Sources
The shedding of energy in gravitational waves is much like the thing we call acceleration. Any
mass that accelerates produces gravitational waves. Increasing speed and moving in an orbit are
both types of acceleration, and there are quite a lot of examples of things that do both of these in
space already. The reason the earth is not bathed in strong and quite painful gravitational
radiation is due to the strength of gravity. Gravity is very weak on our scale compared to the other
forces of the universe; for example it's quite easy for me to acquire a charge that equals and
exceeds the gravitational attraction I have for an electron and push that electron away from me.
However, on a much larger scale (like comparing stars and galaxies) gravity is the force to
measure! Charges can cancel each other out leaving a very massive body with a very small net
charge, but gravity does not behave like charge---it cannot cancel itself out. Gravity is cumulative
and depends just on the amount of matter (or energy) present. Still, though, gravity is a subtle
force even on the scale of suns and black holes. For this reason only the really massive
accelerating bodies radiate gravitational waves, and even those produce signals that are insanely
weak and extremely difficult to detect. What would produce these waves?
Accelerating masses: irregularities and inspiralling
Well, one would look for really big accelerating masses (otherwise known as stars). Standard
stars like our sun don't really generate lots of gravitational waves. It takes a lot of matter
compacted into a small area and moving around and high speeds to generate 'em.
Let's go back to physics. A collection of masses, such as two balls at each end of a stick, can be
summed up in behavior like a single conglomerate mass at the center of gravity. If the two balls
were equal in weight, the center of gravity would be in the center of the stick. Should one ball be
heavier than the other, the center would be more towards the heavier ball. Imagine the center of
mass of a binary with two neutron stars of roughly equal size. Yep, it's roughly in between the
two stars. Now make the two neutron stars rotate about this point---that's the very simplistic
model of a binary system. The total gravitational attraction of the system can be determined by
this simple model, but a description of the actual gravitational field generated by these two stars
is quite complicated. It's actually beyond my descriptive powers right now, but I thought it would
be best to give you a rough idea of what a binary is and does.
I mentioned inspiralling, which is the gravitational wave generator for binary systems. What is it?
It's when the two stars of the binary get closer and closer to one another and it's the norm.
Everything that is gravitationally attracted to something inspirals to some degree---even if the
movement over millions of years amounts to one inch. Inspiralling ties into something in physics
called gravitational potential energy. The higher up in a gravitational field something is, the more
potential energy it has. A jello mold five feet off the ground has more gravitational potential
energy than a jello mold splattered on the ground. As something moves down into a gravitational
field (down meaning in the direction of increasing gravity), it loses gravitational potential energy.
As the stars gradually inspiral, they shed gravitational potential energy as gravitational waves.
The two stars also star moving around each other faster and faster as they inspiral. As they get
really close, such as towards the end of the binary's lifetime and just before the two stars coalesce
into one blob, they radiate waves like crazy and are spinning around each other at sizable
fractions of the speed of light. Yow!! I have a tough time wrapping my brain around something as
dense as a neutron star (which is only a few kilometers in diameter) moving that fast. As the two
stars coalesce, they're still radiating like crazy, even though they've technically lost all their
gravitational potential energy. Why? I should explain some things about spheres and gravitational
radiation.
Non-spherical Rotation with Regards to Supernovas
Well, another way of generating gravitational waves is by having a something that isn't perfectly
spherical rotate rapidly. This happens in supernovas of massive stars, when the core of the star
might become a neutron star or a black hole. Such supernovas (designated type 2) are such
violent things that the core of the star (should it survive the process) deforms into something
distinctly NON-spherical and rotating rapidly---thus generating gravitational waves. This stellar
core remnant goes through a trimming down process in which the irregularities are thrown off or
smooth down into a more spherical shape, generating more gravitational waves.
How and why does it radiate? Since the shape is non-spherical, there are some portions that are
further away from the..well, the center of the mass for lack of a better description. Since those
portions are further away from the center, they're higher up in the gravitational field of the mass.
Since they're higher up, they've still got gravitational potential energy locked in them. Gradually,
the imperfections are either thrown off due to an instability in the mass or they smooth down and
radiate away their gravitational energy, and the resulting shape becomes more and more
spherical. The refining process forms the remnant into a spherical shape and usually generates a
loud burst of gravitational waves which then fade off. Supernovas aren't really understood too
well.
Non-spherical Rotation with Regards to Binaries
When the two neutron stars coalesce, they form this distinctly NON-spherical blobby shape that's
spinning at a fraction of the speed of light. Again, some parts are higher in the gravitational field
than others, and gravitational radiation is shed as the combined lump of neutron star becomes
more and more spherical. If we're really lucky, the coalesced form will be too big and form a
black hole.
Binaries of two black holes would do much the same, except they're a lot more interesting and
radiate a lot more gravitational waves. Why would this be? Neutron stars are typically a few
kilometers in diameter. They can only get so close to each other (and therefore only go so fast)
before they touch and being to coalesce. Black holes are smaller than neutron stars by far and can
get much closer, spinning much faster, before they being to coalesce. The closer they get and the
faster they spin around each other generates a stronger wave.
The difference between a neutron star binary and a black hole binary is in the coalescence stage.
Instead of mushing lots of matter together like the neutron stars, the black holes merge their even
horizons just like two soap bubbles joining. I'm not to clear on what the singularities do, but that's
nothing new. Singularities are quite strange (<----an understatement).
Other Sources
It is very possible that certain events that occurred fractions of seconds just after the big bang (for
those of us who support the theory!) would produce a background of gravitational waves just like
the cosmic microwave background radiation detected by COBE. There are certain events in the
big bang theory that would generate gravitational waves, things called phase transitions, when
some important aspect of the nature of the universe changed. A good example of a phase shift
(although not one that would generate a gravitational wave) is when the universe became
translucent to light. See, before that the universe was such a hot soup of particles and energy that
photons just got bounced around and could barely travel in a straight line. After the phase
transition the universe had cooled enough to the point where light could, indeed, travel
respectably enough to be called light rays. There are other more exotic proposed transitions, but
I'm not too good with the details of the big bang or any other transitions (most of which probably
scream quantum mechanics, which I don't understand yet).
Signals
You will hear the analogy of gravitational waves to sound more often than not, and how when we
detect gravitational waves we're hearing the symphony of the cosmos. I think that's a beautiful
analogy, even if it does compare gravitational waves to sound (compression) waves. First of all,
light rays used to be the only way we could learn about the universe. Ancient astronomers
scanned the skies and kept records of the tiny usually still dots of light they saw. Later in life
people discovered that the skies held signals of radio waves. Later still they found microwaves
and gamma rays. Each step along the way increased our understanding of the universe because of
the new light (ye gods, a pun) shed upon it. Each type of signal corresponds to a certain object or
event, and people gradually learned about new and interesting astronomical things.
There is, however, a problem inherent in electromagnetic signals: degradation. Light rays scatter
as they pass through ubiquitous gas clouds, are bent as they encounter gravitational fields, and are
absorbed by invisible dust clouds. Light rays bring us a wealth of information about distant suns--but only if we get 'em undistorted. This is where gravity waves come in.
While an electromag wave will be scattered by a dust cloud, a gravitational wave will not.
Gravity waves pass through nebulae, stars, and gravity wells unaffected and almost nearly
undetected! Seems like they're the neutrino of wave signals. How can this be?! It has to do with
the basic idea of the wave. Kip Thorne has a wonderful turn of phrase: "Electromagnetic waves
are oscillations of the electromagnetic field that propagate through spacetime; gravitational waves
are oscillations of the 'fabric' of spacetime itself."1 In this current state of the universe nothing can
distort a gravitational wave. Pretty useful if you want a pure signal.
Electromagnetic waves are emitted by individual atoms and give information on the composition
and thermodynamic state (temperature) of the source. Gravitational waves are emitted by massive
bodies accelerating through space. They would give information on the movements of stellar
bodies and map out (as it were, anyway) the space of spacetime around those bodies. Kip Thorne
noted that the objects people observe in the electromagnetic spectrum would be largely silent in
the gravitational spectrum, and likewise the observations of gravitational wave sources would
consider objects invisible to the electromagnetic spectrum. Black holes and the until-now-hidden
cores supernovae are not the easiest things to study (hah! an understatement!),
electromagnetically, just as "light" stars of a few solar masses and nebulae would be impossible
to study.
So, according to me, there should be all these gravitational waves bouncing around and distorting
things for no good reason, right? Mmm, not quite. It's a question of frequency--- both of the
signal and of the source emitting the signal. There's an inverse relation of the frequency of the
signal to the mass of the source. An ordinary star such as our own (relatively small mass
compared to a black hole binary) would have a rather high signal. Apparently "normal" stellar
binaries have signals in the 1 microhertz (1 millionth of a hertz) range, while a neutron start
would be more in the kilohertz (1 thousand hertz) range, and a galactic core black hole would be
in the hertz range2. The greater the mass, the lower the frequency. Why? It has to do with how
fast something with that much mass can oscillate. If you've got something that's small, it can
really move around and emit high-frequency gravitational waves versus something big and slow
and ponderous.
Gravitational waves have some problems in common with electromagnetic waves, though,
mainly the faintness of distant objects and the redshifting due to expanding spacetime (our jello
mold of spacetime is stretching!). Our bowl of jello seen from two miles away is very small; this
is a normal occurrence. The curious part is the reason why. Imagine, if you will, the sphere of
photons emitted by the jello after 1 second. All those photon crowded together, just like a
firework just after it's exploded. Now imagine that sphere a couple seconds later---it's huge! Just
like the firework sparkles thin out, the photons from the jello would spread out over a greater
area. The greater the distance from the jello, the large the sphere of photons is, thus the fewer
photons-per-square meter. The fewer photons per area, the fewer that hit a sensor or an eye; thus
the smaller the source seems. For a source emitting photons---or gravitons---there is an additional
problem of how many particles are being shot out per second within that square area of the
sphere. That determines how faint the source appears. That's why distant stars are so small and
faint. Gravitational wave detectors will have this problem with gravitons, making the alreadyweak signal much weaker over distance!
Redshifting due to expanding spacetime? Not only is the frequency of electromagnetic waves
decreased by the expansion of our jello-mold-turned-spacetime, but gravitational waves are as
well. It makes sense, since gravitational waves are ripples in spacetime. As spacetime expands,
the wavelength of the signal increases. Standard physics: the wavelength is inversely proportional
to the frequency; thus, the frequency decreases.
I've mentioned that certain events in the big bang theory to generate gravitational waves. While it
would be cool and informative to have a gravitational version of COBE's results, it's not likely.
Those events would have happened a long time ago (heh, when the universe had just been
err...born), there would be few gravitons from such a signal, and there is a lot of expanding
spacetime crossed by the signal to get to us, making the already-faint signal practically invisible.
If we could detect it, chances are it might go unnoticed as static just as the microwave
background did.
In a certain Scientific American article there is a wonderful table of gravitational wave sources,
frequencies, and signal strengths. I'm going to shamelessly reproduce it here because it's quite
useful. However, all credit for work and research goes the the writers!
Source
Possible Gravitational Wave Signals
Signal Type
Frequency
Strength
Stellar Binary
Periodic
1 MegaHertz or lower
10-21
Quasi-periodic
Sweeps to 1 kiloHertz
10-22
Periodic
200 to 800 Hertz
3 x 10-27
Impulsive
1 kiloHertz
10-21
Neutron Star Binary
Accreting Neutron Star
Type 2 Supernova
Vibrating Black Hole
Galaxy Formation (by
cosmic strings)
Damped
Sinusoid
10 kiloHertz for 1 solar mass
10 Hertz for 1,000 solar
masses
Noise
Broad Band
1 Cyle per year
300 Hertz
10-14
10-24
Noise
Unknown
Unknown
Unknown
Big Bang
Funny how black hole/black hole binaries aren't in the table. I can conjecture that they would
produce signals very similar to the neutron/neutron binaries. Perhaps the frequency would be tad
higher (around 20 or more kiloHertz?) because the black holes can move more mass and get
closer to one another. Black hole/neutron star binaries? Heh, I should think they'd be somewhere
in between! I think these were neglected because people still don't quite believe in the existence
of black holes and also I imagine a binary of two black holes would be a very rare thing indeed.
Oh, yeah. I didn't mention some of these sources, did I? There are good reasons for that. Take the
galaxy formation. I've got very little experience with cosmic strings. I barely know what they are.
Take the accreting neutron star. I didn't think that would produce measurable gravitational waves--but look at that signal strength: 10-27 is very weak. The stellar binary? Well, yes, they'd make
waves, too; but neuron binaries and black hole binaries are much better at moving large masses
very quickly. The only drawback to those prime sources is their distance from us, and the more
distant the source, the weaker the signal (and they are already weak enough as it is!). Neutron
stars are usually further in towards the core of our own galaxy---or in the halo. Neutron stars are
also pretty rare. Black holes of only a few solar masses probably are, too. Their larger cousins
exist usually in the center of galaxies, which are not quite as frequent as stars are. Ah, well, no
one said detecting these waves would be easy.
So, what are these detectors, anyway?
I've mentioned gravitational wave signals and detectors, but I haven't actually described how one
would go about detecting gravitational waves. One popular method is to use something called a
laser interferometers. The LIGO project is a great example of this. Basically put, a laser
interferometer looks like two dangling mirrors at the end of two long stretches of tube arranged in
an L with lasers constantly bouncing off 'em. That's it. How does this detect a gravitational wave?
Well, there's trick to it, you see. It can detect waves that strike it head on---so that the plane of the
two tubes is perpendicular to the direction in which the waves travel.
Before the wave hits the two lasers are sending out signals that are in synch with each other. Each
light ray sent out bounces back in the same time. All's well. Then the wave hits. One leg of the L
is suddenly stretched and the other is compressed. The light rays still travel at (heh!) the speed of
light, but now there is more space to cross in one of the legs and less space to cross in the other.
The signals are no longer in synch and the wave is detected. The only drawback to this scenario is
that interferometers encounter interference and noise above 1kHz and below 10 Hz.
Another way to measure a gravitational wave is to use a resonant mass detector. There are two
basic varieties: spheres and bars. The former version has a sphere of some material with sensors
either arrayed on the sphere itself or around it. Those sensors make sure the sphere has its normal
dimensions. When a gravitational wave hits the sphere, due to the stretching and squishing effect,
the sensors detect the deformation of the sphere. Usually, though, the sensors detect the lingering
oscillation (like a jello mold that wiggles a little after it's touched) of the sphere. The latter
resonant mass detector, the bar, looks like a large soup can of some material with sensors (either
at the front and back or arrayed around the middle of the bar) worried about the front and back
"lids." It uses the same basic method of detection, looking for the oscillation of the bar, for
finding gravitational waves. The drawback to using a bar is that the detector is no longer
unidirectional. A wave striking the sphere from any direction could be detected, but only a wave
striking the bar roughly perpendicular could be detected. The laser interferometers have this same
directional problem, too.
A final note and then on to another section. A curious thing---perhaps THE most curious thing--about gravitational waves is the supporters of the theory. This is not Einstein's now-nearlyuniversally-accepted theory of relativity. (Side note: according to one of my teachers, every six
months---from the time Einstein first published his theories to today---papers are published trying
to disprove relativity for various scientific and not-so-scientific reasons.) Gravitational waves are
on shakier ground than black holes---and people are still not entirely convinced those exist.
People diligently build detectors but have had very little luck in detecting bonna fide gravitational
waves. Very, very theoretical. Actually, when gravitational waves are detected (I'm optimistic!),
they will go far in proving the existence of black holes. Stars the size of small cities people can
accept; empty bent spacetime that has a nasty peculiarity in the center takes a little more faith.