Download Black Holes - University of Oregon

Black Holes – are you Kidding Me?
A star more massive than about 20 solar masses is highly likely to leave a
remnant core with a mass that exceeds 3 solar masses. In this case,
theoretically, gravity would achieve its final victory as neutron degeneracy
would fail and nothing would oppose gravitational collapse. The core would
therefore collapse into an object known as a singularity. A singularity is an
object with a radius of zero and therefore, since it has a finite mass, its
density is infinite. Well, that’s just not right, is it?
First, let’s consider the hypothetical case were the sun suddenly turned into a
Black Hole (like in the bad Walt Disney Movie). At out position on the
Earth, well yeah, we would be exterminated because of the lack of sunlight,
but the Earth itself would still be orbiting and object that has a mass of 1
solar mass. This is an important point – to an external observer a black hole
is merely a point with mass. As long as you are a long way from that point,
you will not notice anything strange – it’s only when you get near the black
hole, that strange things start to happen – some of which are detailed below.
The concept of the black hole was first espoused in 1798 by the French
mathematician Laplace, who coined the term corps obscures. Laplace
reasoned that if light is made up of particles (as Newton thought) then if a
star was sufficiently massive the escape velocity of particles from that star
might be larger than the speed of light. In this case, no light emitted by the
object could ever escape the object and therefore it would be black. As we
will see, this is a relatively naïve characterization of a black hole. The
escape velocity from any object can be expressed as a function of its density
and therefore Laplace, knowing the speed of light at that time, was able to
calculate the mass and radius a star must have in order for it to be black.
The resulting calculations were so absurd that Laplace thought that no such
configuration of matter could exist in the Universe.
Curved Space-time:
The early 20th century brought many new and profound developments to our cosmology
and for the first time showed that Newtonian theory was not a complete specification of
gravity. In the late 19th century, precision observations of the position of Mercury
showed that it did not agree with the predictions from Newtonian theory. This is similar
to the case of Mars, where Kepler used positional discrepancies to show that planetary
orbits had to be elliptical in shape. The resolution for Mercury isn't quite so simple. It
turns out that Newton never explicitly considered the "shape" of spacetime but implicitly
assumed that space is completely flat. The resolution of the positional discrepancy of
Mercury requires that space be "curved" in the vicinity of Mercury so that Mercury orbits
inside this curvature. Such an orbit will differ slightly from an orbit in purely flat space.
This is Shown in Figure 2.1.
Figure: Visualization of the orbital precession of Mercury as it is orbiting
in curved space near the Sun. The curved space causes the orbit to precess
over time thus tracing out a different orbital path with each new orbit.
But, why would space be curved in the vicinity of Mercury? The answer lies in Mercury's
proximity to the Sun. Einstein postulated that gravity is really the manifestation of curved
space and that very massive objects cause a greater degree of curvature and hence have a
greater gravitational influence. Mercury is sufficiently close to the Sun that it orbits in
curved space. The rest of the planets are far enough away from the Sun that the degree of
curvature caused by the mass of the Sun is negligible. While a full description of the
General Theory of Relativity is well beyond our scope, we can summarize it qualitatively
as follows:
Space communicates with matter and instructs it how to move and, in turn, matter
communicates with space and instructs it how to curve.
In this way, the distribution of mass determines the overall curvature of the Universe as
well as the particular pathways that objects, including light, must follow. We can then
use General relatively and the curvature of space time to better characterize a black hole.
Shown below in some relative scale are the local curvatures of space time that exist near
certain kinds of objects. The most severe curvature is clearly for the case of the black
hole.
In terms of space-time curvature, a black hole can be though of as a region
of space time which has locally infinite curvature. Thus once an object
enters into a black hole (i.e. crosses the event horizon – see below) – that
object effectively disappears from the Universe because now it is
disconnected from space-time and in a very real sense is stranded inside its
own, little, mini-Universe. Of course, the accretion of this object by the
black hole has allowed its mass to grow and so now the external universe
interacts with a larger mass black hole.
The Event Horizon:
Black holes can be described as objects with a mass and a “radius” as long
as we use a physically sensible mechanism for defining a radius. Such a
definition was first provided by Karl Schwarzschild in 1916 who simple
asked the question – at what radius in an object of mass M does the escape
velocity equal the speed of light. The answer to that is the Schwarzschild
radius:
G is the universal gravitational constant of Newton.
The meaning of Rs is
that light can not escape if it is emitted from a region with R < Rs. For an
object of 1 solar mass, Rs is 3 km. Compare this with a typical 0.6 Msun
White Dwarf, which would have a radius of about 1 Rearth (6370km), and a
1.4 Msun neutron star, which would have a radius of about 10km.
Events which occur in the Universe are transmitted via photons at the speed
of light. Therefore, if say an event occurs at a distance of 1 km inside a 1
solar mass black hole, that event can never be transmitted to the rest of the
Universe since there is no way the information (e.g. the photons) can escape
from that region of the black hole.
RS therefore defines the "Event Horizon" surrounding
the black hole's singularity:



Events occurring inside RS are invisible to the
outside universe.
Anything closer to the singularity than RS can
never leave the black hole
The Event Horizon hides the singularity from the
outside universe (this is known as the Law of
cosmic censorship – there can be no naked
singularities, all of them are hidden by an event
horizon).
The Event Horizon marks the "Point of No Return" for
objects falling into a Black Hole. In principle, this location
is ideal for the disposal of unwanted roommates, if only
you could find one when you needed it!
Journey to a Black Hole:
So what do we know so far about black holes?
a) they are totally collapsed objects
b) gravity is so strong that not even light can escape within some
distance from the point of total collapse (e.g. the singularity)
c) they represent local regions of the Universe where the curvature is
infinite
d) from a large distance away they act merely as point masses
Now let’s imagine what might happen if we have a cloud of gas falling
towards a black hole. When the gas cloud is far from the black hole it feels
a uniform gravitational acceleration. However, when the gas cloud gets
within a few Schwarzschild radii (3-5 Rs) it stars to experience differential
gravitational forces. The part of the cloud that is closest to the event horizon
is accelerated more rapidly than the part farther away. This differential
gravitational force is known as a tidal force and it’s the same as the earthmoon system. The side of the earth that is closest to the moon feels a
slightly stronger force than the side opposite the moon and this causes a
small deformation in the shape of the Earth. That deformation manifests
itself via rising and falling ocean levels (e.g. the tides).
But the tidal forces near the event horizon are enormous and they have the
effect of tidally compressing and heating the gas. In fact the gas becomes
so hot that it starts to emit X-rays and this is the way in which black holes
have been discovered. The image below shows an artists conception of this
process. Far from the black hole, the gas is cold but as it nears the event
horizon an accretion disk develops (blue disk in the image below). That
accretion disk becomes rather dense and it blocks the flow of radiation
through it, so the radiation can more readily escape at perpendicular to the
disk and so jets of X-ray emission occur. The inner radius of the accretion
disk is the event horizon (Rs ) and the outer radius is 3—5 Rs. So black
holes in our galaxy have been discovered by closely observing binary
systems that emit X-ray. The fact that the system is a binary star also allows
for masses to be measured. The first such system discovered is called
Cygnus X-1 where x-ray emission is emanating from an unseen companion
star to a much bigger star yet the mass of that unseen companion is about 6-9 solar masses. It therefore has to be a black hole, with X-ray emission
coming from his accretion disk. Hence, black holes really can not escape
detection. When matter falls into them, it heats up and emits X-rays.
https://publicaffairs.llnl.gov/news/news_releases/2005/images/NASA_bla
ck_hole.jpg
Journey to a Black Hole: A Thought Experiment
A concept that we have not yet introduced in this course but what that is now
needed is the gravitational redshift. This effect theoretically existed and has
been observed for some white dwarfs. As we have seen, there is
equivalence between energy and mass and this is the physical basis for
thermonuclear fusion and energy generation in stars. This means that
photons have an energy related mass and therefore are affected by gravity
(this is another prediction of Einstein’s theory of General Theory of
Relativity). Hence, photons that are emitted from an object with a very
strong surface gravity (like a white dwarf) lose part of their energy in
escaping from the surface. That energy loss means that the wavelength of
the photon increases (towards the red – hence the term redshift) to longer
wavelengths. Remember, the energy of a photon is inversely proportional to
its wavelength; long wavelength photons have less energy than short
wavelength photons. For white dwarfs stars, the gravitational redshift is
equivalent in velocity to about 100 km/sec. In term of wavelength this
represents approximately a 3 parts in 10,000 shift in wavelength – very small
but still measurable. Hence, if a 10,000 angstrom photon were emitted by
some process from the surface of a white dwarf, that photon would arrive to
an external observer at a wavelength of 10,003 angstroms because a small
amount of energy was lost in escaping the surface gravity of the white
dwarf. We will see below that another way to understand the event horizon
is that it represents the surface of infinite gravitational redshift.
Imagine two friends, Batman and Robin, whom decide to take a journey to a
black hole. Batman, being the macho hero, decides that he will travel
towards the event horizon and signal Robin with information about his
condition as he approaches the event horizon. Robin will stay at a safe
distance to monitor the progress of Batman. Batman has developed a
special bat-laser beacon that emits a blue laser pulse once a second. Robin
is monitoring this signal and they have agreed in advance, that if Batman is
in trouble, he will send two pulses a second and Robin will use the BatShip
to come and rescue him. Unfortunately, this isn’t going to work, as we
will see below. Here is the sequence of events that will occur.
 Batman initially is gravitationally pulled towards the Black hole but
still is relatively far away from the event horizon
 Robin is orbiting the black hole in a stationary circular orbit, in the
BatShip, at a safe distance and is recording Batman’s laser pulses and
receiving one every second.
 Batman observes the BatShip getting increasing far away but there is
no cause for concern.
Sometime later the situation changes:
 Batman is still sending laser pulses, according to this Bat watch, at
one per second.
 Robin, however, notices that the laser flashes are taking longer to
arrive than they did before and that furthermore, the laser pulses were






transitioning from blue to green. Robin dismissed this as yet another
one of Batman’s practical jokes.
A few more minutes has passed and Robin now beings to concern.
Laser pulses are now coming once every 3 seconds and they are deep
red in color. Robin knows that Batman doesn’t have a red laser
device (because those are no longer cool) and suddenly becomes very
confused as to what is going on.
In the meantime, Batman is approaching the event horizon and
accelerating. He has trouble seeing the BatShip and the outside world
is beginning to look wildly distorted as the positions of the stars
appear to have changed from when he first started. He is also
beginning to feel a slight tidal tug on his Bat Cape, but his Bat Suit
was built to withstand severe tidal forces.
By now Robin is getting a laser flash once very hour and those are
arriving at radio wavelengths (fortunately, Robin brought the Bat
Radio). Robin is wondering what the Batman is doing? Why is he no
longer sending pulses at the rate of one per second – why did he rather
suddenly switch to radio pulses every hour?
Another hour passes and no pulse is observed. And then another hour
pasts and then another hour pasts and then a few months past and
finally an extremely long wavelength pulse is observed.
Meanwhile Batman has finally arrived at the Event Horizon and
realizes that he is in a lot of trouble. He starts to frantically signal
Robin, at the rate of 10 pulses per second, but alas, all of those
photons had to expend all of their energy just to leave from the event
horizon and therefore Robin received no message.
Sadly, Batman slips across the event horizon to be heard from no
more while Robin is just left waiting for another pulse to signify that
the Batman is still alive. Alas, Robin now waits forever.
In summary, the powerful gravity of a black hole warps space and time
around it:



Time appears to stand still at the event horizon as seen by a distant
observer. This is the meaning of infinite gravitational redshift.
Time flows as it always does as seen by an in falling astronaut.
Light emerging from near the black hole is Gravitationally Redshifted
to longer (red) wavelengths.
Take a Virtual Trip to a Black Hole or Neutron Star.