C. V. VISHVESHWARA (VISHU) ON THE BLACK HOLE TREK

Though he proscribed, black hole to have no hair, he himself had many hairs indicating his multifaceted interests and concerns. Besides science of which we would talk later in some detail, he had keen interest and appreciation of literature in all forms and arts in all its presentations. He was culturally accomplished person, connoisseur of good music (Indian and Western) classical as well as modern, theatre and painting. In particular, he was quite accomplished in drawing and sketching, and had a keen observation and eye of a cartoonist. His collections of cartoons laced with subtle and tickling humour bear testimony to his skill and accomplishment in this art.

On the other hand his book, Einstein’s Enigma or Black holes in my bubble bath, demonstrates his prowess in prose writing of high quality and tenor. It reads like a travelogue novel with generous sprinkling of humour, thoughtful perceptions and conversations. It is thoroughly absorbing and engaging, and at the same time leaves one thinking and wondering. This literary bug, he has inherited from his father who was a well known Kannada writer and scholar.

The first work involves consolidating the concept of event-horizon in the black hole spacetimes. There are three intimately related ideas associated with a black hole, and they are: (a) One-way membrane or event-horizon, (b) Static or Stationary limit and (c) Infinite redshift surface.

The event-horizon: The first concept is the idea of a one-way membrane as the definition of event-horizon, popularly known as a black hole. In all relativistic theories, the speed of light is the upper limit for communication between any two points. This constraint divides the interval between any two events in the spacetime into timelike, lightlike or null and spacelike. This idea further develops into the concept of the light-cone. The collection of all lightlike paths starting from an observer’s location $\mathcal{O}$ gives the light-cone, as shown in the figure-1.

A massive particle can carry a signal or message from an observer at $\mathcal{O}$ to all the events inside the light cone. The light-cone events are moving at the speed of light, and there is no way to reach them from the observer at $\mathcal{O}$ without moving faster than light. There is no way to communicate with events outside the light-cone. A timelike curve, $\gamma$ sneaking outside light-cone can no way come back inside again without speeding up faster than light. As one goes closer to horizon, observer’s light-cone gets tilted inwards and at the horizon it is entirely pointing inwards (Fig.2). Similarly, surfaces can be classified into timelike, null or spacelike depending on their normal vector is timelike, null or spacelike, as shown in the figure-3. Only timelike surface having its timelike normal can be crossed both ways, going in and out. The surfaces we generally come across, like a class room, are two way crossable – an observer can go in and can come out.

Most fascinating of them all is the null surface. A wavefront of light is a classic example of a null surface. The normal vector to a wavefront is the direction vector along which light propagates, which is null or lightlike by definition. An observer can cross a wavefront, or a wavefront crosses an observer only once without violating the speed limit. This crossing once is the property of all null surfaces, and hence they are also known as one-way membranes. There is a one way surface we encounter at every epoch, $t=const.$, which we all cross only once and one way. It is different that many of us would like to cross it other way too, but can’t! It is however not bounded in space.

The precise definition of a black hole is an inward or future pointing null surface that is finite and closed. Finite and closed because an observer will not be able to sneak out from any direction. The direction is along inward such that an observer would be able to go in but not come out, as shown in the figure-2.

However there is a common sense indicator of horizon where a freely falling massive particle attains the velocity of light. That is, as in the Newtonian gravity, velocity is given by $v^{2}=2\Phi(R)$ where $v=\frac{\sqrt{g_{rr}}dr}{\sqrt{g_{tt}}dt}$ is the proper velocity relative a local observer and $\Phi(R)=M/R$ is the gravitational potential. This is so because the inverse square law remains unaltered in general relativity with $3$-space being curved rather than flat as is the case for the Newtonian gravity. In fact, we can say, Einstein is Newton with space curved [9].

Static limit: The second concept associated with the black hole is called the static limit. An observer is said to be static in spacetime if his/her spatial velocity is zero. When we stay at rest on the earth’s surface, the gravitational force acts downwards towards the centre of the earth, while the floor’s reaction force (so-called Newton’s third law) acts upwards to balance the gravity. Without ground to support, one needs to put on a rocket suit to give an upward acceleration to remain at rest and avoid falling. Similarly, an observer can remain at rest around a black hole by providing an outward acceleration or rocket suit to counter the radial pull towards the hole. These are the static observers in static spacetimes, which are spherically symmetric, such as the Schwarzschild solution (Fig. 4).

Stationary limit: In the case of rotating black hole described by the axially symmetric Kerr solution presents a new situation. Since it is rotating about an axis, a direction gets identified and hence the spacetime has to be axially symmetric. Here a particle is subjected to pull in two different directions, one in the radial as in the static case but in addition also in the tangential direction to carry it around. This is because there is inherent rotation, indicated by the frame dragging angular velocity $\omega=-g_{t\phi}/g_{\phi\phi}$ at every point in space surrounding the hole. That is, even a particle with zero angular momentum has to move with this angular velocity.

It turns out that first the angular pull becomes irresistible and that defines the stationary limit; i.e., below this limit an observer cannot remain stay put at a location, he or she has to rotate around. That is, she or he can though remain stationary at fixed $R$ by countering the radial pull but has to rotate around with the angular velocity $\omega$. At the stationary limit radial pull could be resisted but not the angular pull. As one goes further down when radial pull also becomes irresistible, that is when the event-horizon is defined. For the static black hole both event-horizon and stationary limit are coincident which separate out for the rotating Kerr black hole. The region separating the two is called the ergo-region (Fig. 4). It is this that lends to rotating black hole the most exciting and interesting physical phenomena.

The most remarkable feature of the ergo-region is that a particle can have its total energy negative relative an observer at infinity. It turns out that in this region, spin-spin interaction energy, which would be negative for counter-rotating particle, could become dominant and thereby making the total energy negative. Using this property, Penrose in 1969 [10] proposed an ingenious process of energy extraction (known as Penrose process) from a rotating black hole. It is envisaged that a particle of energy, $E_{1}$, falls from infinity and splits into two fragments in the ergo-region having energies $E_{2}<0$ and $E_{3}$. Then the fragment with negative energy, $E_{2}<0$, falls into the hole and the other, $E_{3}=E_{1}-E_{2}>E_{1}$, comes out with enhanced energy. This is how black hole’s rotational energy could be extracted out (Fig. 5). This doesn’t happen for static black hole because there is no ergo-region there. It is rotation that causes the ergo-region and hence the extracted energy is rotational.

Infinite redshift surface: The third concept is the infinite redshift surface. When light travels from a stronger gravitational field to a weaker gravitational field, it experiences a redshift called gravitational redshift. As the source of light approaches the black hole, the observer at infinity sees a more significant shift in light frequency. When the emitter reaches $R=2M$ or $g_{tt}=0$ in general, and light will experience infinite redshift. This limit is called the infinite redshift surface.

In spherically symmetric black holes such as the Schwarzschild solution, the event-horizon, static limit, and infinite redshift coincide at the location $r=2M$ or the Schwarzschild radius. However, rotation introduces considerable complexity and richness of phenomena. Vishveshwara, in his work, highlighted the differences and similarity in the geometry of rotating and non-rotating black holes.

The black holes as astrophysical objects need to fulfil another important criterion, i.e. they need to be stable under perturbations. When black holes are formed in a stellar collapse or by the merging of two stars, often subjected to extreme perturbations, and they need to be stable at least on the life spend of a galaxy or the Universe itself. To examine stability of black hole is the problem assigned to Vishu by his supervisor Charles Misner. Finally, it turned out to be a major topic by itself, so much so that Chandrasekhar had to write an over 500 pages book, The Mathematical Theory of Black holes, entirely devoted to black hole stability and perturbations.

To state the problem in another way, can one destroy a black hole? Let us try to understand the subject from an example of the ringing of a bell. When one strikes a bell with a small hammer, a small amount of energy is transferred to the bell and that distorts it slightly. The bell will start ringing with notes depending on the shape and the material of the bell. Designed to ring for a long time, they will eventually stop ringing when all the excess energy imparted is converted into sound wave. Finally, it settles back to its original state. These types of perturbation are linear, and bells are stable under linear perturbations. While hammering, if energy transferred is considerable, the bell might get distorted permanently or even get obliterated.

Vishu’s seminal work includes perturbing a black hole with a small energy field. When disturbed, he first discovered that black holes ring pretty much like a bell by emitting gravitational waves. These characteristic modes of black holes are called Quasi-Normal Modes (QNMs)

Since a black hole can harbour only mass, charge and angular momentum, objects that fall in can only add to these three parametrs alone. All other modes get evaporated away before the object reaches the null horizon. In other words, it is a property of the closed null surface that it cannot sustain any other parameter than mass, spin and charge. This is what is popularly known as the No-Hair Theorem.

In an amusing, but probably legendary, experiment, Galileo is said to have publicly demonstrated that two balls made up of different materials fall similarly when dropped from the leaning tower of Pisa. If the balls were to be dropped into a black hole, they would be converted into a pile of mass and angular momentum with no trace of their material structure. So do the waves emitted by the spacetime, namely QNMs, depend only on black hole’s fundamental entities; i.e., mass, charge, and angular momentum [figure-6].

The QNMs signals are one of the unambiguous signatures of black holes. If detected directly, they can be a powerful tool in understanding black hole physics. With QNMs, one may be able to distinguish between black holes and other compact objects mimicking them. One can verify the no-hair theorem by analysing multiple QNM modes whether they carry any other signature than mass, spin and charge. If any other parmeter black hole have would leave an imprint on QNMs. QNMs are the only messengers we have from black holes, the only source of information. Since existence of QNMs is prediction of general relativity, their existence hence also marks its test. [figure-6].

The first detection of gravitational waves from a black hole merger, GW150914, by the LIGO and VIRGO collaboration is a significant step in the direct identification of QNMs signals. The final phase of merger of black holes leaves a highly distorted black hole, emitting QNMs. It then settles down as QNMs damp out to a stationary state of rotating Kerr black hole. This phenomenon is called the ringdown phase. The observed signal from GW150914 confirm the ringdown phase [figure7]. Vishu was also one of the first to use then available computing facility for analysing the QNMs as shown in the figure, and it is remarkable that the curve is pretty much alike the one in the gravitational wave discovery observation. Work is underway for identifying and characterising QNMs frequencies. For GW150914, the signal strength is weak in the ringdown phase and is not adequate enough to draw any clean inference in the context of no hair theorem. What is needed is a closer and more massive black hole merger to get a sufficiently loud signal. In the coming years, QNMs will be one of the powerful tools to probe black hole physics directly.