Black Holes: Bizarre Cosmological Objects

Abstract

In this report, I am going to discuss about the fundamentals of black holes and their relevance to modern research scenario. We have a lot of informations about black holes but there are still several questions to be answered regarding black hole physics.

Keywords: Black hole, general theory of relativity, thermodynamics, information paradox.

Introduction

Black holes are known since long time. These are the astrophysical objects having such a strong gravity from which nothing, not even light, can escape. The boundary of the region from which no escape is possible is called the event horizon[1]. From Newton's law of gravitation, any two objects having masses m1 and m2 respectively, attract each other with the force F as:

F=G

m1m2

r2

(1)

Where r is the distance between the two objects and G is the Newton's Gravitational constant. Now, we can imagine that if the mass of one of the object is very large and its radius is very small (considering the objects to be spherical), then it can attract the other object, lying on its surface, so strongly that it can not escape the surface of the other. If a very large mass is compacted into a very small volume, the gravitational attractive force produced will be strong enough to prevent anything from escaping its surface. According to quantum physics, light is supposed to be made of particles (photons), and also feels attraction under a strong gravitational field. This is the fundamental idea behind the understanding of black holes. Black holes are called “Black” as light can't come out of it, therefore, none could see them.

John Michell, a geologist, in 1783, first gave the idea of such a massive object that even light could not escape from it[2]. The idea was ignored by the people as the light was being considered a pure wave that time. Albert Einstein proposed his theory of General Relativity in 1915. According to him, space and time are relative to each other. Three space dimensions and one time dimension jointly construct a four dimensional spacetime. A massive object causes a distortion of spacetime

around itself. Gravity is thought of as the distortion (curvature) of spacetime caused by the presence of a massive object. To understand this, let's consider the spacetime as a rubber sheet, any object placed on it will produce a deformation (dent) in the rubber sheet and another object placed on this sheet will fall towards this distortion. The more massive object produces more distorion to the sheet. Anologously we can understand the distortion in spacetime and gravitational attraction (gravity). Very soon, in 1916, Karl Schwarzschild found solution for complex Einstein filed equations for a spherically symmetric static (non-rotating) object[3]. In standard Schwarzschild coordinates, the solution (metric) takes the form

ds2=−(1− RHr )dt 2+(1− RHr )

− 1

dr 2+r 2 (dθ2+sin2θ dφ2 ) (2)

In natural units, RH = 2GM, is known as the Schwarzschild radius. When RH = r, the second term of equation (2) will become infinite and the solution becomes singular. All the mass of the spherical object could be supposed to be concentrated on a point. The solutions of Einstein's field equations for the gravitational field of an electrically charged point mass (with zero angular momentum) in empty space was obtained in 1918 by Hans Reissner and Gunnar Nordström[1][4].

Subrahmanyan Chandrasekhar, in 1931, proposed a limit for white dwarf mass, known as Chandrashekhar limit, to honour him [5]. It is the upper limit of the mass of a white dwarf, equal to 1.44 times solar masses. A star having a mass above this limit will continue to collapse to form a neutron star. In 1939, Robert Oppenheimer and others predicted that neutron stars above approximately three solar masses (the Tolman–Oppenheimer–Volkoff limit) would collapse into black holes for the reasons presented by Chandrasekhar[6]. Roy Kerr, a mathematician from New Zealand, in 1963, developed the theoretical concept of rotating black holes, known as Kerr black holes[7]. A black hole is called a Kerr–Newman black hole if it has both angular momentum and electric charge[8]. The name “Black Hole” is said to be given by an American Physicist John Wheeler, in 1967.

Until 1967, black holes and neutron stars were supposed to be just mathematical objects without having any physical existence but the discovery of pulsars in 1967 and their identification as neutron stars by 1969 [9][10], not only confirmed their physical existence but also created a great interest among experimental astrophysicist for all compct objects that might be formed due to gravitational collapse.

A massive star is stable against gravitational collapse as it has Hydrogen fuel to create an internal pressure to exacly balance the gravitational pull inwards. But at the end of their life cycle, it burns out all its fuel and can not control the gravitational collapse, resulting in a stellar mass black hole. A newly formed black hole starts absorbing mass of surroung stars and merging with other black holes to become a supermassive black hole of millions of solar masses. It is believed that supermassive black holes exist in the centres of most of the galaxies, including our own Milky Way. Astronomers have identified numerous stellar black holes in binary systems by studying the movement of their companion stars[1][5].

All black hole solutions of Einstein-Maxwell equations of gravity can be characterized by only three parameters: mass, electric charge and spin of the black hole. It means all other information (hairs) of the matter forming black holes or falling into it, will be inaccessible for external observers. This postulate was called as no hair theorem[11]. In 1965, Penrose proved that singularities could form when massive stars collapse, but later proposed that the singularity would always be hidden from us by the event horizon – the so-called “cosmic censorship” or weak cosmic sensorship hypothesis. According to Penrose, no naked singularity other than big bang singularity can exist in the universe. Also, a black hole could have been formed via a wide variety of processes. This suggests that a black hole will possess a degeneracy of states, and hence an entropy, as a function of its conserved quantum numbers[12].

Black hole Thermodynamics

The laws of classical black hole mechanics were developed by James Bardeen, Jacob Bekenstein, Carter, and Hawking in the late 1960s and early 1970s. It was found that these laws have a close remblance with the laws of Thermodynamics[13]. The analogy can be easily understood by relating mass to energy, area to entropy, and surface gravity to temperature. In support of the analogy, Hawking, in 1974, gave the famous conjecture of Hawking radiation. This semiclassical work of Hawking theoretically showed that a black hole should radiate like a black body [14][15]. Let us understand these laws in a little detail. The zeroth black hole law states that the surface gravity κ is constant over the horizon of a stationary black hole.

The first law gives a relation for the change in mass of a black hole:

dM=κ dA

2 π

+ωH dJ +Ф dQ (3)

where ωH is the angular velocity at the horizon, A is the area of the event horizon, J is the angular momentum of the black hole, Q is the total electrostatic charge on the black hole and Φ the electrostatic potential. The second law of black hole mechanics says that the event horizon area A must always be non-decreasing in any classical process.

dA≥0 (4)

The third and the last law states that it is impossible to achieve κ=0 via a physical process. It is now clear that from the laws of classical black hole mechanics, one can easily correlate them with the well established laws of thermodynamics.

Hawking showed that the temperature of a black hole at the event horizon should be

T H=

ℏ κ

2π. (5)

The entropy of a blck hole is dettermined by the famous Bekenstein-Hawking or Black Hole entropy formula. In any spacetime dimension d,

S= A

4 Gd, (6)

Where Gd is the d-dimensional Newton constant. The black hole entropy is a very huge number e.g. for a four-dimensional Earth-mass black hole which has a Schwarzschild radius of order 1cm, the entropy is S 10 66.[16]

As the black hole undergoes in the process Hawking radiation, it loses mass, and its horizon area decreases. Since the area of the horizon is proportional to the black hole entropy, it might appear that this area decrease signals a violation of the second law. On the other hand, the entropy in the Hawking radiation increases, providing a possible way out. To fix up the problem, Bakenstein defined a generalised entropy, which includes the entropy of the black hole plus the other stuff such

as Hawking radiation,

S total=S+Sother≥0 (7)

Information Paradox

Around forty years ago, Hawking wrote a paper, “Breakdown of Predictability in Gravitational Collapse” [17], in which he claimed that there would be loss of predictability of the final state if the black hole evaporated completely. This was because one could not measure the quantum state of what fell into the black hole. The loss of information would have meant the outgoing radiation is in a mixed state. This is the information paradox: How does the information of the quantum state of the infalling particles re-emerge in the outgoing radiation? This has been an outstanding problem in theoretical physics for the last forty years. Despite a large number of papers, no satisfactory resolution has been found.[18] He recently proposed that the information is stored, not in the interior of the black hole, but on its boundary, the event horizon. This is a form of holography.[19]

LIGO

Gravitational waves are ripples in the curvature of spacetime that are generated in certain gravitational interactions and propagate as waves outward from their source at the speed of light.

Gravitational-wave astronomy is a branch of observational astronomy that uses gravitational waves to collect observational data about sources of detectable gravitational waves such as binary star systems composed of white dwarfs, neutron stars, and black holes.

On February 11, 2016, the LIGO and Virgo Scientific Collaboration announced they had made the first observation of gravitational waves. The observation itself was made on 14 September 2015, using the Advanced LIGO detectors. The gravity waves originated from a pair of merging black holes. After the initial announcement the LIGO instruments detected two more confirmed, and one potential, gravitational wave events. In August 2017, the two LIGO instruments, and the Virgo

instrument, observed a fourth gravitational wave from merging black holes, and a fifth gravitational wave from binary neutron star merger[20-23].

Discussion

Black holes provide an important tool for probing and testing the fundamental laws of the universe. Although, the existence of black holes has been experimentally verified, still there are several unanswered questions related with the theory of black holes. The first one is about black hole thermodynamics. All the calculations performed by Hawking and others, were semiclassical in nature and a complete quantum mechanical approach is still missing. That is, probably, also a challenge to quantum gravity. The second one is the black hole information loss (paradox) problem. No satisfactory answer has been found till date. Black holes are also supposed to be a potential candidate to explain the early universe cosmology but a convincing explanation is still awaited. String theory has come forward to answer all these open questions and many renowned physicists are working ton hese problems. We may hope that very soon we are going to have answers of all these questions.

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