Introduction to Modern Physics 2018/19 (SEF038) Black Hole And

Introduction to Modern Physics 2018/19 (SEF038)

Black hole and compact objects

Keywords

Compact objects: White Dwarf, Neutron Star, and Black Hole Schwarzschild radius, the event horizon Coordinate singularity and true singularity Inside of the black hole? Ergosphere

Compact objects, dead stars

3 special types of stars, White Dwarf, Neutron Stars, and Black Hole are generally called compact objects. They share the following common features

They are small and heavy, namely, the density is very high They stop nuclear fusion (=dead stars)

1 of 9 White Dwarf

If stars run out fuel to burn, they stop nuclear fusion and start to cool down and becomes a White Dwarf (WD). This dead star is roughly ~10% of all stars, and it has

a size of Earth (radius~1R⊕ ), with the mass of Sun (~1M⊙). This makes the density to be around ~106g/cm3 , yes, sugar cube made by White Dwarf has 1 ton!

Because of this high density, it needs a high pressure to support the star from the gravitational force to collapse. This pressure is generated by the electron Fermi gas (Lecture 15: Fermions and Bosons), where electrons cause Fermi-Dirac degeneracy and generate a large pressure.

Type Ia supernova

Compact objects often make a binary stars, where 2 stars are orbiting each other. If one star is a large giant and the other is a white dwarf, then the white dwarf can suck the gas and infalling gas makes a heat, and eventually, it causes a nuclear explosion. This is the mechanism of the type Ia supernova. Type Ia supernova is very bright, also its spectrum is well known and it is often used as a standard candle to measure the distance in the universe.

2 of 9 Neutron Star

If stars are large (>> 1M⊙), they can evolve to the ﬁnal stage of stars, where possible heaviest elements are burning in the core. However, eventually stars run out all elements to burn, and they cannot produce the pressure any more to support the core. Such star causes the type II supernova (core-collapse supernova). In this phenomenon, the core of a giant star implodes and collapses to a Neutron Star (NS). Neutron stars are gigantic nuclei ﬂoating in the universe. This is the highest density object in the universe and made by neutron Fermi gas, neutrons with the

state of Fermi-Dirac degeneracy. Typical neutron stars have 1 solar mass (1 M⊙) with 10 km radius, and the density reaches to the nuclear density (~1014g/cm3 ), the density of a typical nucleus.

Neutron stars themselves are too faint to see. However, typical neutron stars are surrounded by gas, and gas falling into the stars emits strong beams of light from both poles of the star. Since the rotation axis and this emission axis are slightly misaligned, from the observers on the Earth it looks like the star is blinking, called pulsar. In fact, the fast period of the pulsar (=the star must be small) and the strong light emission (=star must be heavy) indicate the star is a neutron star.

SN1054

The Type II supernova in 1057 was observed by many civilizations, including China (Song dynasty). The location recorded by Chinese astronomers coincides to the location of the crab nebula and the crab pulsar, so this is a direct evidence that nebulas and pulsars were made by the core collapse supernova!

SN1987A

There are only 8 supernovae in the Milky Way galaxy in the last 1000 years. The latest one is 1987. For this SN1987A, Kamiokande-II and IMB detectors also observed neutrinos, which also conﬁrmed supernova caused nuclear fusion (Nobel prize 2001). We have not found a pulsar at the supernova remnant of SN1987A, yet.

Hulse-Taylor binary

Neutron stars also make binary stars. PSR1913+16, so-called Hulse-Taylor binary was discovered by the Arecibo Observatory, Puerto Rico (1973, the famous large dish from "James Bond"). It is famous since scientists also measured how much the binary slows down due to the emission of the gravitational wave. The calculation from general relativity agrees well with the observation, conﬁrming the correctness of general relativity (Nobel prize 1993).

3 of 9 Black Hole

A Black Hole (BH) is believed to be made by the implosion of a really large star. Some sense, this is just a star with a higher density of a neutron star. But physics of black hole is beyond all physics we know. Imagine, an object with mass m is trapped by the gravity of an object M. The kinetic energy for this object to leave the system is,

1 mM mv2 = G 2 r

What if, the gravity is really strong? then only really fast objects can escape. For maximumly strong gravity, even light cannot escape

M GM 1 2 2 c = G or rs = . 2 rs c2

This rs is called Schwarzchild radius, and generally used as the radius or the event horizon of a black hole. Once an object, or light, across this radius, they cannot escape anymore from the gravity of the black hole.

Schwarzchild metric is one of the solutions of Einstein equation. This describes space-time curvature of the spherically symmetric system (Lecture 14: Structure of Atoms) such as space-time surrounding a heavy star.

rs 1 c2dτ2 c2dt2 dr2 r2 sθ2 sin2θdϕ2 = 1 − − rs − ( + ) ( r ) 1 − r

The solution suggests physics changes at the Schwarzschild radius. At the Schwarzschild radius, many crazy things happen...

4 of 9 Concept of space and time

If r < rs (inside of the black hole), the ﬁrst term of the right side and the second term of the right side switch signs. This implies time and space swap roles. What does this mean?

In Lecture 6: Lorentz transformation, we learned faster moving object has a trajectory t', closer to the light path. At the speed of light, time coordinate and space coordinate overlap, this is the path of light in the space-time diagram. However, inside of the black hole, it accelerates an object faster than the light, this means time coordinate and space coordinate swaps. So inside of the black hole, time and space switch roles. For example, in our world, time goes only one direction but we have a choice in space. However, inside of the black hole, once you pass the horizon, you have no choice to go anywhere, except you go to the centre of the black hole, called singularity. Thus, we lose the choice about the space, and space behaves more likely time in our world. Does it mean, we have a choice on time? We can go back and forward freely in time?

Coordinate singularity and true singularity

Near the black hole, every object look to take inﬁnite time to fall in, because of strong time dilation by the strong gravity, everything moves so slowly near the black hole, and everything is eventually frozen. Of course, if you are in a rocket and fall into a black hole, you can pass the horizon in a ﬁnite time. This means our choice of the coordinate is not correct to describe the physics. The event horizon is a vertical line in the space-time diagram (r = rs) and in principle, you can pass it any time. But it behaves like a point (r, t) = (rs, ∞), namely you can pass it only when t = ∞. This is a type of a coordinate singularity, where strange things show up due to a bad choice of the coordinate. An example is a Mercator map. In this projection, The north and south poles are lines, even though they are points. Thus, we need to choose a correct coordinate to describe a physics of the black hole correctly.

5 of 9 Kruskal-Szekeres coordinate is such a coordinate system designed to describe black holes correctly. Here, 2 axes v and u are combinations of t and r, time and space. Singularity is a line, and a trajectory of falling object is a continuous line from outside of the horizon, then pass the horizon, then reach the singularity. Although we successfully describe a falling object, now we give up the concept of time and space...

Penrose diagram

Penrose diagram is an analogous diagram of Kruskal-Szekeres coordinate. Here, the entire universe is just a square, then horizons. The singularity of the black hole is a horizontal straight line. You can imagine Penrose diagram can be made by stretching the Kruskal-Szekeres coordinate system. A diﬀerence is Penrose diagram has another side, which is naturally deﬁned to be "other universe". What is another universe? Anyway, this is a very handy way to write the whole universe!

If the black hole is rotating and or having electric charge, even more, a strange thing could happen. Notice now the singularity is a vertical line. Then, let's describe a trajectory of the object fall into this black hole (=pass the horizon). There is a trajectory where the object passes the horizon but not reach the singularity. Instead, such an object could pass the horizon again and arrive outside of the black hole. Where is it? Is it our universe or diﬀerent universe? Can we travel time and space in this way?

6 of 9 There are 3 parameters of a black hole; mass, electric charge, and angular momentum. These 3 parameters completely ﬁx the "style" of the black hole (no-hair theorem of black hole). Depending on these parameters, the style of the black hole looks diﬀerent in the Penrose diagram. For example, it is possible to have singularities in vertical lines, not horizontal lines. In this case, once an object passes the event horizon, there is a chance that it can reach another horizon and come back to the universe, without reaching to singularities. Is this the same universe? or another universe? how about space and time? This implies a possibility of time travel and space teleportation (not quantum teleportation, but the real one from SciFi movies)...

Ergosphere

Rotating black hole also has 2 horizons. The second horizon is the real event horizon, where you cannot come back once you pass. The region between the ﬁrst horizon to the second horizon is called ergosphaere. An object in this region can exchange angular momentum with the black hole through space, or stationary object can start rotating without any physical contact. By using this, one can extract energy from the black hole. Imagine, you fall into the ergosphere and come back, but in the way to start to spin your device. You can rotate the turbine and generate electricity with free! It is speculated that if there were very advanced civilizations in the universe, such an alien could build their cities around a black hole. By throwing their devices from one side to the other, passing the ergosphere but not the event horizon, an alien can enjoy extracting an inﬁnite amount of angular momentum (=energy) of the rotating black hole.

7 of 9 This phenomenon is not limited to the black hole. In fact, any heavy rotating object has ergosphere. The effect is small but the Earth also has a small effect by the ergosphere which causes a tiny rotation for an object near the Earth. This effect, called frame dragging was measured recently by a satellite experiment (Gravity Probe B). To do this experiment, scientists prepared a gyroscope made by the most precise sphere on the Earth.

Useful data (exam format) Planck constant h 6.626 × 10−34J ⋅ s Speed of light c 2.998 × 108 m/s ℏ ⋅ c 197 MeV ⋅ f m Electric charge e 1.602 × 10−19 C Calorie cal 4.184 J 23 Avogadro number NA 6.022 × 10 −23 Boltzman constant kB 1.380 × 10 J/K 2 Mass of electron me 0.511 MeV/c 2 Mass of proton mp 938.3 MeV/c −27 mp 1.673 × 10 kg 2 Mass of neutron mn 939.6 MeV/c −27 mn 1.675 × 10 kg 24 Mass of the Earth M⊕ 5.972 × 10 kg 30 Mass of the Sun M⊙ 1.989 × 10 kg 8 Radius of the Sun R⊙ 6.961 × 10 m

8 of 9 Useful formula (exam format)

Lorentz contraction and time dilation

v ′ 1 ′ 1 L = ⋅ L, and t = γ ⋅ t, where γ = ⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ , β = γ √1 − β2 c

de Broglie wave length and uncertainty principle

h ℏ ℏ λ = , ΔE ⋅ Δt ≥ , Δp ⋅ Δx ≥ p 2 2

Periodic table

H, He, Li, Be, B, C, N, O, F, Ne, Na, Mg, Al, Si, P, S, Cl, Ar, K, Ca, Sc, Ti, V, Cr, Mn, Fe, Co, Ni⋯ A Radius and total mass of nucleus Z XN

1/3 2 R ∼ RoA , where Ro ∼ 1.2fm, and M = Z ⋅ mp + (A − Z) ⋅ mn − B(A, Z)/c

9 of 9