Download Answers

Day 8.12 Black Holes
( 6.3)
1) In 1783 Rev. John Michell recognized that a large enough mass in a small enough space would result in
a dark star – an object whose escape velocity was greater than the speed of light. He used Newton’s
law of gravity. What radius would our sun have if it turned into a black hole?
Conservation of energy:
GMm/r = ½ mv2
r = 2GM/v2 = 2x 6.67x2.0/9 x 10-11+30-16= 3.0 km
2) In 1919 Karl Schwartzschild used Einstein’s equations of general relativity to find the radius at which a
star would become a black hole. His answer was exactly the same as the calculations made 136 years
earlier! This radius is called the Schwarzschild radius and not the Michell radius because Michell got
the right answer for the wrong reason. Which is the wrong reason? He assumed that light particles
A) will be slowed down by gravity, but light always travel at c.
B) have mass. However, light has no mass.
C) have energy given by E = ½ mv2, but light’s energy is given by E = hf.
D) all of the above
His calculations treated light as if it was made of particles with mass that would slow down and return if
they weren’t moving fast enough. His answer just happens to be the same answer that you get from GR,
and this is only true if the black hole is not rotating. (Newton’s laws of gravity also predicted that light
will bend its path when it passes near a mass – but the calculated amount is half of what GR predicts and
experiments show.) As light moves against gravity, it doesn’t slow down but it does get red-shifted. The
frequency and therefore the energy decreases. At the event horizon the wavelength and period become
infinitely large.
3) Suppose the sun became a black hole. What would happen to Earth?
A) spiral rapidly into the sun. B) spiral slowly into the sun. C) fall straight in. D) orbit as usual.
The formula F =GMm/r2 doesn’t say anything about how the mass is concentrated. Nothing changes.
4) If the sun became a black hole what would we see in the sky?
A) a regular night sky
B) a black circle
C) a black circle with ring of starlight around it
The Schwartzchild radius is only 3 km. We wouldn’t be able to see it or its lensing without a telescope.
5) Spacetime around a black hole is strongly curved. The Schwarzschild radius defines a region known as
the event horizon. What would the sky inside and outside the event horizon look like? Alice and Bob:
Can we travel through time? http://www.youtube.com/watch?v=HHYECUcfC1Y
Outside of the event horizon, you will see light from the stars behind the black hole and you will see
them multiple times forming a ring. Within the event horizon it will be completely black.
6) If you are near a black hole and have your back to it so that you are looking away from it, you will see
A) more stars and they will be red-shifted B) more stars and they will be blue-shifted
C) fewer stars and they will be red-shifted D) fewer stars and they will be blue-shifted
Light from beside and behind the black hole will be bent so that you can see it. As this light is falling
into the black hole it gains energy and shifts to the bluer colours.
7) What would it feel like to be on this planet orbiting the black hole?
http://jila.colorado.edu/~ajsh/insidebh/lensearth_640x480.gif
It looks like the planet is getting squished, but it isn’t. It would feel pretty normal.
8) What will it be like inside a black hole?
We don’t know. The equations say that the mass keeps getting more and more dense until it is all in a
singularity of infinite density. We can never find out either. No light and no information can escape.
9) What is the most significant way that black hole is different from a dark star? The black hole
A) won’t let light escape
B) is bigger C) mostly empty
D) sucks in nearby matter
Not A! Most popularizations of black holes use the first reason. However, a dark star can also do this
and is not nearly as interesting. Not D! Most people think of black holes as cosmic vacuum cleaners. It
is possible to orbit a black hole. Not C. Black holes come in a variety of sizes and there may be very tiny
ones formed during the Big Bang and which in particle physics experiments. The reason why black
holes are so bizarre and were rejected as ridiculous by Einstein is that within the Schwartzchild radius,
matter collapses into an infinitely dense point. This means that the rest of the space within is empty.
10) What is the most significant way that a black hole is different from a hole in a box? The black hole
A) won’t let light escape
B) is bigger C) is darker D) looks like a hole in all directions
A 1-cm hole in a well-sealed box can look very black, even when a bright flashlight is pointed into the
hole. A GR black hole is much stranger than the classical dark star that Michell proposed. It looks like a
hole in every direction. Consider the saucer model or the stretchy fabric model that reduced space to a 2D surface that is warped into a third dimension. The reality is a 3-D space warped into a fourth.
11) The strongest candidate for a black hole is the supermassive black hole Sagitarius A* at the centre of
our galaxy. The closest measured approach is 1.8 x 1013 m and the star is moving at 4.9 x 106 m/s.
http://www.aei.mpg.de/einsteinOnline/en/spotlights/milkyway_bh/
a) How much mass is needed to cause the star to orbit like that? How many suns is this?
M = rv2/G = 6.6 x 1036 kg = 3.3 million suns
b) How big is the Schwarzschild radius for this mass? Which planetary orbit is it similar to?
A) Mercury (~1010 m) B) Mars (~1011 m) C) Uranus (~1012 m) D) Eris (~1013 m)
r = 2GM/c2 = 3.6 x 1011 m. We have 3.3 million suns inside the inner solar system.
c) Is this definitely a black hole?
We know that there is a huge concentration of mass, 3.3 million suns within 1.8 x 1013 m (the radius of
the orbit of Eris which is past Pluto). To be a black hole, we need to know that it is 50 times smaller in
radius. Other techniques have shown that this is the case and in the next five years we should be able to
have images of the event horizon.
p. 293 #12, p. 294 #9, 11, p. 300 #18, 19