Types of black hole

• Observational evidence for the existence of intermediate mass black holes is far from clear.
• Primordial black holes might have been formed in the early Universe.

Black holes

• A black hole is an object which has all its mass inside its event horizon.
• The event horizon is a boundary within which nothing can escape, not even light.

Event horizon radius

• The simplest type of black hole is the Schwarzschild black hole.
• It is a non-rotating black hole that has a spherical event horizon.
• The radius of the event horizon is called the Schwarzschild radius and is proportional to the object's mass.
• If M is the mass in solar masses, then the radius is equal to:

M × 2.95 km

Concentration of mass

• At distances much larger than the event horizon, the gravity exerted by a black hole is the same as for any object of the same mass.
• For example, if we replaced the Sun with a black hole of equal mass, the Earth would carry on moving in exactly the same orbit.
• It is only when you get close to a black hole - within a few times its Schwarzschild radius - that unusual effects are noticed.

Dropping into the event horizon

Thought experiment: You are located many Schwarzschild radii away from a black hole, and you release an object with a blue flashing light on it from rest, so that it falls directly towards the black hole. You would observe:

• At first the object behaves as you'd intuitively expect.
• As it gets closer to the black hole, you notice the time between flashes lengthens.
• You also start to notice that the light is looking more red and less blue.
• The object then seems to dim and then vanish - it now emits only infrared radiation.
• At the moment it appears to reach the event horizon, the light (which had begun to emit only radio waves!) stops flashing all together.

Theory of black holes

• Black holes cannot be understood using the classic physics laid out by Newton's laws.
• Instead we must use Einstein's theory of General relativity.
• It is easiest to begin with Special relativity.

Special Relativity

This theory was published in 1905 by Albert Einstein. It is based on two postulates:

1. The laws of physics are the same to all observers travelling at constant speed relative to each other (often called inertial frames).
2. The speed of light in a vacuum is measured to be the same by all observers.

The speed of light is denoted by c, and is 299,792,458 m/s. It is commonly approximated to 300,000 km/s.

It is the second postulate that allows us to predict the counter-intuitive effects that become noticeable for objects moving close to the speed of light.

Special relativity effects

• Moving clocks tick more slowly.
• Moving objects are reduced in size along their direction of motion.
• If you view two spaceships moving in opposite directions at 0.9c, the occupants of each spaceship will see the other moving away at less than c.
• Time travel into the future is possible by travelling close to the speed of light.
• Two observers might not agree on length and time measurements, but cause and effect are always preserved.

The twin paradox

• Two identical twins Jack and Jill are born on Earth.
• At age 20, Jill departs on a space ship to the nearest star (4 light years) travelling at 0.8c.
• It takes 5 years to cover this distance at 0.8c, so the round trip is 10 years.
• Jack is 30 when Jill returns; Jill is only 26. Time has passed at different rates for each twin.
• But, from Jill's point of view the Earth with Jack on it moves at 0.8c; why isn't Jack 4 years younger at the end? This is the paradox.
• The answer is that we have ignored acceleration and special relativity cannot be simply applied to this situation. But what is described above is accurate - Jill will be approximately 4 years younger on her return.

General relativity

• General relativity deals not only with accelerating observers, but also with gravity.
• Einstein published it in 1916.
• It includes Special relativity as a special case.
• The postulates of special relativity are still held, but the equivalence principle is introduced.

The equivalence principle and mass

Newton's laws stated:

• An accelerating object must be experiencing a force equal to the its acceleration multiplied by the object's inertial mass.
• The gravitational force on an object is proportional to its gravitational mass.
• The equivalence principle states that these two masses are the same.

But why are those two masses equal?

Equivalence principle and orbits

• If the Earth were replaced by a pea moving at exactly the same velocity, it would orbit exactly as the Earth does.
• The orbit of the Earth, or any object, does not depend on its own mass, nor on any other of its own properties.

Equivalence principle - in a lift!

If you were inside a lift (or some windowless, sealed container), there is no experiment you can do to distinguish between the following:

• The lift is in free fall towards the Earth.
• The lift is in orbit around the Earth.
• The lift is in space, far from any large mass.

In all three cases you would feel weightless.

Gravity vs spacetime curvature

So, instead of the Earth, or a pea, or any object feeling a mysterious force of gravity that just happens to be proportional to its mass, we imagine that the motion of an object is influenced by the curvature of spacetime:

• flat spacetime: objects move with constant velocity.
• curved spacetime: objects have their velocity changed, and follow certain curves dictated by the shape of spacetime.

Spacetime

• space: three dimensions - up/down, left/right, forward/back.
• time: one dimension - we can only move forward in time.
• But we live in a four dimensions, e.g. if I arrange to meet you at a particular place (specifying three co-ordinates), that is of no use if I fail to specify the time too.
• So, just as we have a point in space and a moment in time, we can talk of an event in spacetime, which is four dimensional.

Rubber sheet analogy

Source: Johnstone CC-BY SA 3.0

Curvature of space time

• The rubber sheet analogy shows that spacetime is deformed by the presence of mass, and the curvature is greatest nearest the mass.
• Imagine flicking a marble onto the rubber sheet - given the correct velocity, and if there were no friction, the marble could be made to orbit. This extends the analogy to explain the orbit of planets.
• The situation can be pithily summarised as follows:

Mass tells spacetime how to curve and spacetime tells masses how to move.

Some General Relativity effects

• At speeds much less than light and if you are many times the Schwarzschild radius away from a compact mass (black hole or neutron star), Newton's laws hold and our intuitive understanding applies - this corresponds to a nearly flat spacetime.
• If we are in a nearly flat spacetime region and observe an object close to a compact, massive object, we will see the effect of gravitational time dilation (clocks tick more slow) and the closely related phenomenon of red shift.
• The path that light takes, though not its speed, is affected by the curvature of spacetime, giving rise to gravitational lensing.

Evidence for General Relativity

The theory is now very well supported by evidence:

• Positions of stars during a solar eclipse were observed to have their apparent positions altered by gravitational lensing.
• Discrepancies of Mercury's orbit that could not be accounted for by Newton's laws were explained by General relativity.
• Gravitational time dilation has been observed in atomic clocks in the Earth's gravitational field, and requires corrections to be made in GPS satellite signal processing.

Hawking radiation

• By applying quantum physics to a location just outside the event horizon, it can be shown that black holes do emit a tiny amount of energy.
• Unlike stars, the luminosity is inversely proportional to mass.
• Very low mass primordial black holes will emit significantly and will evaporate completely, becoming very luminous in their final moments.
• Larger black holes emit so little that they will absorb more energy, e.g. from cosmic rays, than they emit at present.

Worm hole - schematic

Source: AllenMcC CC-BY SA 3.0

Worm hole

• One region of spacetime is connected to another region of spacetime.
• This means that it might be possible to travel from one place in the Universe to another in short times without travelling faster than the speed of light.
• It may even possible to travel between times in this way, possibly even back in time.
• It remains to be seen if this is physically possible; quantum physics may prohibit it.