Black Holes and Information

Black holes, with their extreme gravity and ability to profoundly warp space and time, are some of the most interesting objects in the universe. However, in at least one precisely defined way, they are also the least interesting.

According to general relativity, black holes are nearly featureless. Specifically, there is a result known as the "no-hair theorem" that states that stationary black holes have exactly three features that are externally observable: their mass, their electric charge, and their angular momentum (direction and magnitude of spin). There are no other attributes that distinguish them (these additional properties would be the "hair"). It follows that if two black holes are exactly identical in mass, charge, and angular momentum, there is no way, even in principle, to tell them apart from the outside.

This in and of itself is not a problem. As usual, problems arise when the principles of quantum mechanics are brought to bear in circumstances where both gravity and quantum phenomena play a large role. At the heart of the formalism of quantum mechanics is the Schrödinger equation, which governs the time-evolution of a system (at least between measurements). Fundamentally, the evolution may be computed both forwards and backwards in time. Therefore, at least the mathematical principles of quantum mechanics hold that information about a physical system cannot be "lost", that is, we may always deduce what happened in the past from the present. This argument does not take the measurement process into account, but it is believed that these processes do not destroy information either. Black holes provide some problems for this paradigm.

However, it may seem that information is lost all the time. If a book is burned, for example, everything that was written on its pages is beyond our ability to reconstruct. However, in principle, some omniscient being could look at the state of every particle of the burnt book and surrounding system and deduce how they must have been arranged. As a result, the omniscient being could say what was written in the book. The situation is rather different for black holes. If a book falls into a black hole, outside observers cannot recover the text on its pages, but this poses no problem for our omniscient being: complete knowledge of the state of all particles in the universe includes of course those on the interior of black holes as well as the exterior. The book may be beyond our reach, but its information is still conserved in the black hole interior.

The real problem became evident in 1974, when physicist Stephen Hawking argued for the existence of what is now known as Hawking radiation. This quantum mechanism allows black holes to shed mass over time, requiring a modification to the conventional wisdom that nothing ever escapes black holes.

The principles of quantum mechanics dictate that the "vacuum" of space is not truly empty. Transient so-called "virtual" particles may spring in and out of existence. Pairs of such particles may emerge from the vacuum (a pair with opposite charges, etc. is required to preserve conservation laws) for a very short time; due to the uncertainty principle of quantum mechanics, short-lived fluctuations in energy that would result from the creation of particles do not violate energy conservation. In the presence of very strong gravitational fields, such as those around a black hole, the resulting pairs of particles sometimes do not come back together and annihilate each other (as in the closed virtual pairs above). Instead, the pairs "break" and become real particles, taking with them some of the black hole's gravitational energy. When this occurs on the event horizon, one particle may form just outside and the other just inside, so that the one on the outside escapes to space. This particle emission is Hawking Radiation.

Theoretically, therefore, black holes have a way of shedding mass (through radiation) over time. Eventually, they completely "evaporate" into nothing! This process is extremely slow: black holes resulting from the collapse of stars may take tens of billions of years (more than the current age of the Universe!) to evaporate. Larger ones take still longer. Nevertheless, a theoretical puzzle remains: if the black hole evaporates and disappears, where did its stored information go? This is known as the black hole information paradox. The only particles actually emitted from the horizon were spontaneously produced from the vacuum, so it is not obvious how these could encode information. Alternatively, the information could all be released in some way at the moment the black hole evaporates. This runs into another problem, known as the Bekenstein bound.

The Bekenstein bound, named after physicist Jacob Bekenstein, is an upper limit on the amount of information that may be stored in a finite volume using finite energy. To see why this bound arises, consider a physical system as a rudimentary "computer" that stores binary information (i.e. strings of 1's and 0's). In order to store a five-digit string such as 10011, there need to be five "switches," each of which has an "up" position for 1 and a "down" position for 0. Considering all possible binary strings, there are therefore 2⁵ = 32 different physical states (positions of switches) for our five-digit string. This is a crude analogy, but it captures the basic gist: the Bekenstein bound comes about because a physical system of a certain size and energy can only occupy so many physical states, for quantum mechanical reasons. This bound is enormous; every rearrangement of atoms in the system, for example, would count as a state. Nevertheless, it is finite.

The mathematical statement of the bound gives the maximum number of bits, or the length of the longest binary sequence, that a physical system of mass m, expressed as a number of kilograms, and radius R, a number of meters, could store. It is I ≤ 2.5769*10⁴³ mR.

This is far, far greater than what any existing or foreseeable computer is capable of storing, and is therefore not relevant to current technology. However, it matters to black holes, because if they hold information to the moment of evaporation, the black hole will have shrunk to a minuscule size and must retain the same information that it had at its largest. This hypothesis addressing the black hole information paradox seems at odds with the Bekenstein bound.

In summary, there are many possible avenues for study in resolving the black hole information paradox, nearly all of which require the sacrifice of at least one physical principle. Perhaps information is not preserved over time, due to the "collapse" of the quantum wavefunction that occurs with measurement. Perhaps there is a way for Hawking radiation to carry information. Or possibly, there is a way around the Bekenstein bound for evaporating black holes. These possibilities, as well as more exotic ones, are current areas of study. Resolving the apparent paradoxes that arise in the most extreme of environments, where quantum mechanics and relativity collide, would greatly advance our understanding of the universe.

Sources: https://physics.aps.org/articles/v9/62, https://arxiv.org/pdf/quant-ph/0508041.pdf, http://kiso.phys.se.tmu.ac.jp/thesis/m.h.kuwabara.pdf, https://plus.maths.org/content/bekenstein

The Detection of Gravitational Waves

For an introduction to gravitational waves, see here.

Before 2016, a nobel prize had already been rewarded for an observation that was consistent with, and seemed to confirm, the existence of gravitational waves. In 1974, Russell Hulse and Joesph Taylor discovered a very compact binary system of objects at a distance of 21,000 light years, consisting of two neutron stars orbiting one another. One of the bodies was also a pulsar, meaning that the radiation beams emitted from its poles periodically point toward Earth as it rotates. Since the rotation rate of a neutron star changes only very slowly over time, pulsars are fairly precise clocks. However, Hulse and Taylor detected that the pulses did not reach Earth precisely on time, but varied slightly from the expected arrival time. They were sometimes sooner, sometimes later in a regular pattern, indicating that the pulsar in question was in fact part of a binary system.

The above diagram depicts the binary system consisting of pulsar B1913+16 and its companion, another neutron star. No radiation from the companion has been observed on Earth, indicating that its poles oriented away from us. However, its presence can be inferred from the fact that the pulsar moves farther and closer to Earth in a short, regular period, indicating an orbit. The difference in arrival times is about 3 seconds, indicating that the orbit is about 3 light-seconds across. Further, the orbital period is 7.75 hours.

This discovery provided an excellent opportunity to confirm the predictions of general relativity: such a compact system with rapidly orbiting masses would radiate fairly large quantities of gravitational radiation. However, direct detection was well beyond 1970's technology. Instead, Taylor observed the pulsar system over a number of decades, and found the following:

Since the discovery of the pulsar, its orbital period had been decreasing very slowly, though steadily and measurably, by about 35 seconds over a timespan of 30 years. This is very little relative to the total period of 7.75 hours, but the data matched the predictions of general relativity almost precisely: as energy was lost to gravitational waves, the neutron stars gradually spiral inward toward one other as their orbits becoming shorter and shorter. This remarkable confirmation of a prediction of relativity won Hulse and Taylor the Noble Prize in physics in 1993.

And there the matter sat. Though detectors grew more and more advanced, no direct detections of gravitational waves were made for over 20 years. This all changed in 2015.

On September 14, 2015, at 09:50:45 UTC, shortly after LIGO (the Laser Interferometer Gravitational-Wave Observatory) resumed activity following an upgrade, the two detectors in Washington State and Louisiana picked up a transient gravitational wave signal, the first ever observed by humankind. The announcement of the discovery was made several months later, on February 11, 2016.

The above image shows the signals recorded at Hanford, Washington (left) and Livingston, Louisiana (right). The signals are also superimposed on the right to demonstrate their similarity. The horizontal axis is time, measured relative to 09:50:45 UTC on that day. The reader may notice that the event was distinguishable from the surrounding noise in the detector for only about 0.05 seconds (the third row charts the residual noise after the theoretical waveform in the second row is subtracted out). The final row shows the rapid increase in gravitational wave amplitude during the event and the subsequent silence. The vertical dimension in the first several rows is the relative strain on the detectors, or the amount by which the different arms of LIGO were stretched or compressed by the ripples in spacetime. The scale for these axes measures strain by parts in 10^-21. This corresponds to extraordinarily minute changes in length: the 4 kilometer arms of the LIGO detector changed by only about 10^-18 meters, only about one thousandth the diameter of a proton!

The theoretical wave form above was a simulation of the event that generated the gravitational waves: the final in-spiraling and ultimate merging of two black holes. The increasing frequency and amplitude of the signals corresponds to the final moments of the collapsing system as the two black holes orbit faster and faster and tighter and tighter around one another before finally combining. Further, the signals at the two detectors were separated by 6.9 ms, smaller than the light travel time between the sites of 10 ms. The delay between the arrival times allows the direction of the source to be identified.

This image shows the region in the sky from which the signals likely originated. The colors indicate the confidence that the source lay within the indicated region: purple is the 90% confidence region and yellow the 50% confidence region. The uncertainty arises from the fact that there were two detectors, and not the three required for a full triangulation.

In addition to the location of the source, the analysis of the waveform yields more. The distance of the system was roughly 1.2 billion light-years, meaning that the merger that we are just now observing occurred over a billion years ago. The two black holes had respective masses of about 36 and 29 solar masses, while the final black hole after the merger weighed in at 62 solar masses. This corresponds to a loss of about 3 solar masses, which was all converted into energy released as gravitational waves as the holes merged. The magnitude of this cataclysm can scarcely be overstated: at its peak, the rate of energy release was an estimated 3.6x10⁴⁹ W, greater than the radiation emitted from all stars in the observable universe combined!

In addition to being a resounding confirmation of general relativity, the observation was the first truly direct detection of black holes: the fact that such massive objects came within hundreds of kilometers of one another indicates that they had extremely high densities, densities only possible in black holes. But while significant, cosmologists were already nearly certain that both gravitational waves and black holes existed. However, this discovery marks the opening of a brand new field of astronomy. Gravitational waves, which pass unimpeded through nearly anything over nearly any distance, allow us to "hear" cosmic events that we could not have detected before. In theory, these waves could allow us to observe the earliest stages of the universe, before it became transparent to electromagnetic radiation. In 2016, 100 years after Einstein predicted gravitational waves, we took the first step towards seeing the universe in a new way.

Sources: http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1989ApJ...345..434T&data_type=PDF_HIGH&whole_paper=YES&type=PRINTER&filetype=.pdf, https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.116.061102, http://resources.edb.gov.hk/physics/articlePic/InterestingTopics/BinaryStars_pic04E.gif

Degenerate Matter: Black Holes

This post concerns black holes in the context of degenerate matter. For an introduction to degenerate matter, followed by a description the various stages of a stellar remnant's collapse preceding the possible formation of a black hole, see here.

White dwarfs, neutron stars, and hypothetical exotic stars are examples of objects of various masses within which some force counteracts gravity and stops collapsed stellar cores from shrinking further. However, above about 3.5-4 solar masses, no known force can counteract gravity. The collapsed star shrinks even further, until its escape velocity, the velocity necessary to leave the object from its surface, reaches the speed of light. At this stage, nothing can escape the body, not even the fastest particles in the Universe: photons. The collapsed star has become a black hole.

As far as is known, no further phenomenon can halt the contraction of the stellar remnant. Under its own weight, the black hole would collapse to become infinitely small and infinitely dense: such an object is called a singularity. Such singularities are consistent with the theory of relativity, but it is not known whether a singularity would be compatible with quantum mechanics; an infinitely dense object seems contrary to any known particle behavior.

Despite the complexities of black hole formation, the structure of a black hole is very simple. In some ways, as we shall see, it is even too simple. From long distances away, a black hole exerts gravity just the same as any other object. For example, light near a black hole undergoes gravitational lensing just as light around neutron stars does (see below).



The above is an impression (not a real image) of gravitational lensing. The gravity of a black hole is so great that it bends light, and therefore causes the view of objects behind the black hole to be distorted.

Near to the black hole, the only noticeable features are the accretion disk and the event horizon. The accretion disk is simply infalling material, sucked in by the black hole's gravity. Such matter is often accelerated to enormous speeds, and releases high energy radiation (X-rays and gamma rays) before it falls into the black hole. This is why black holes can be detected, as no radiation (with a possible exception, see below) is emitted from the hole itself.

The event horizon, which is the real "edge" of the black hole, is the boundary beyond which the velocity needed to escape the black hole exceeds the speed of light. At the center of the area bounded by the event horizon is the actual stellar remnant, the composition of which, as remarked above, is unknown.

The formation of a black hole has been discussed, but do these objects ever die? In fact, they are predicted to "evaporate" by emitting Hawking radiation, named after the physicist Stephen Hawking, who first proposed the process in 1974. Though no particles can escape a black hole, certain fluctuations of space can spontaneously create particle-antiparticle pairs near the event horizon. When this happens, one of the particles can escape by a process known as quantum tunneling. The precise nature of this radiation is unknown, but the rate of emission is so slow that the background radiation of the Universe gives more energy to most black holes than they lose through Hawking radiation. However, the amount of radiation emitted is inversely proportional to a black hole's size, so a small black hole (below stellar mass) could evaporate in our current Universe, if one existed. Black holes that are stellar remnants, however, will remain in existence for trillions of years, as the cosmic background radiation is still energetic enough to insure that they take in more mass then they give off. If the Universe's expansion continues, black holes are likely to be the last large objects in the Universe in the very far future (perhaps about 10⁴⁰ years from now).

The characteristics of a black hole are its mass, spin (or angular momentum), and electric charge. The latter two of these yield interesting phenomena. When a black hole is spinning, it provides angular momentum to the matter circling it, giving rise to a special region called the ergosphere (see below).



The ergosphere is a region outside the event horizon. Any matter in this region is subjected to not only the gravity of the black hole but also the drag of spacetime itself resulting from the angular momentum of the black hole. Therefore, even though the escape velocity in the region is lower than the speed of light, the sum of this and the additional momentum causes any matter in the region to be moved in the direction of rotation of the black hole. This occurs in such a way that a particle in the ergosphere would have to move at superliminal (over the speed of light) speeds to stay stationary (with respect to an outside frame of reference).

With regard to the electric charge of a black hole, some theories of the universe, notably string theory, acknowledge the possibility of the existence of what is called a magnetic monopole, essentially analogous to "a bar magnet with only one end". Even particles such as electrons, though possessing a net electric charge, have, due to their spin, a typical magnetic dipole called a magnetic moment, which obeys the laws of magnetism. A monopole would have to be composed of some unknown particle, as all elementary particles known to date have magnetic dipole moments. Black holes may be composed of these monopoles, as they are likely to be made of some other elementary particle. Note that these ideas are quantum mechanical. If black holes are true singularities, it may preclude this possibility.

The above are a few exotic phenomena that can arise in black holes. However, returning to the characteristics of the black hole, the three listed above are the only ones known that can be calculated by an outside observer, i.e. by the black hole's effect on its environment. From this viewpoint, one could precisely determine the nature of a black hole with only three parameters. Yet, if an object, say an apple, were to fall into a black hole, there would be no way of knowing afterward that this had occurred! One could record the mass contribution of the apple to the hole, but there is no way of recovering its shape, its color, etc., from observation of the black hole. The information that the apple carried is lost.

Or is it? Many principles of physics are contrary to this assumption. Classical physics and relativity both imply reversibility, that is, they imply that a "simulation" of the universe could be run just as well forward in time or backward. The amount of information in the universe must remain constant, or, in running time backward, one could have no idea of whether an apple, an orange, or any other object of equivalent mass had fallen into the black hole. Furthermore, an important quantum mechanical equation, the wave equation, totally determines of the probabilities of quantum states at any past or future time given the wave equation of a system at the present time. Therefore there can be no "collapse" of many states into one; the scenario with the apple and that with the orange cannot have the same outcome.

Many resolutions to this paradox have been posited. Some state that the information would be conserved in some (initially) non-observable fashion: the information would "leak out" slowly over time as black holes Hawking radiation, or that black holes, at the end of their life, would release all their stored information in a single burst, or even that the information is transported, by means of the singularity through a hole in spacetime, to another universe. However, all three of these violate some aspect of our current understanding of information conservation.

If information leaks out, there is still a time delay in which it is not known. It seems unlikely that all of the information is emitted at the end of a black hole's lifetime, as many theories put a limit on the amount of information that can be stored in a finite volume of space. Finally, if the information is transported to another Universe, the information is not conserved in any local, or obtainable, sense. There is a final alternative, however. It is possible that the fluctuations of the event horizon itself would store the impressions of the incoming (or outgoing) particles. Note that this requires the projection of the information in a four-dimensional space (three spatial dimensions plus time) onto a three-dimensional space (the surface of the event horizon is a two-dimensional surface, which again changes over time), but this poses no problem, and has sound mathematical justification; for a sufficiently "well-behaved function" on a space, the behavior of the function within a region is completely determined by the values of the function on its boundary. This result is known as Green's theorem, and its application to the projection of information onto the event horizon is known as the holographic principle.

Of course, this does not mean we could practically access this information, but simply states that it is, in theory, possible. Other obstacles prevent one from ever reaching the event horizon. One notable phenomenon is relativistic time dilation.



Any massive object creates a depression in space, and a black hole creates an especially steep depression; in fact, if singularities exist as predicted by relativity, the depression would actually be a hole in space, as illustrated above. Due to the symmetry between time and space, again predicted by relativity, time is accordingly distorted near a black hole. If one were to watch another object, or person, approach the event horizon, they would appear to slow down as they neared the edge of the black hole. In fact, from the perspective of an outside observer, they would take infinitely long to cross the horizon itself. From the viewpoint of the object, however, time does not slow, and the crossing of the horizon takes place in finite time.

Of course, no object could actually survive this crossing, but would rather be torn apart by gravitational pull. In addition, no one could actually watch the entire descent, as the radiation that renders one visible, when travelling from the object to the observer, uses energy to move "uphill" in the black hole's gravity field. Lower energy light, for example, is red, so the radiation is said to have red-shifted. By the time an object is close to the horizon, it is only visible in radio waves, and eventually, not at all.

The key concept of degenerate matter is the interplay between gravitation and quantum mechanical forces, in particular how they oppose one another. Black holes, the culmination of the process of stellar collapse, represent the crux of the differences between "large-scale" physical theories, i.e. relativity, and "small-scale" theories, i.e. quantum mechanics. Singularities are present in relativity as "pathological" points in spacetime, where density becomes infinite. However, in light of quantum mechanics, singularities are seem to be contradiction, resulting in exotic phenomena including, but not limited to, those listed above.

Solving the mysteries of degenerate matter and black holes in particular is one of the main outstanding problem in modern physics, and will continue to shape our understanding of the universe for years to come.

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Sources: http://astrofacts.files.wordpress.com/2009/07/rouge-black-hole.jpg, No-hair theorem on Wikipedia

Sizes and Structures of Black Holes

There are four main sizes of black holes: micro, stellar-mass, intermediate, and supermassive. Micro black holes can only weigh up to the mass of the moon, and all are less then a 1/10 of a millimeter in diameter. These are results of colliding cosmic waves or even bombardment of particles in particle accelerator. Stellar-mass holes are formed by collapsed giant stars (info on collapsed stars here, and giant stars here). These black holes are typically under 10 solar masses and are about 30 km across. The next type, intermediate black holes, are formed when one black hole is swallowed by another, enlarging it. These black holes weigh up to a thousand suns! Finally, supermassive black holes are found at the centers of galaxies. These black holes began as quasars and weigh from 100,000 to a billion suns. These black holes are so large that the largest, if placed in the sun's position, would extend out to the orbit of Earth!!!!

Black holes have three main parts to them. The first and outermost is called the ergosphere. This only exists if the black hole has angular momentum (see types of black holes) . In this zone, it is impossible for any object to stand still. The spin of the black hole causes a tidal force that acts upon the gravity well, shifting it. The ergosphere is in the shape of an elongated sphere that is tangent to the outer event horizon. The outer event horizon is the point of no return. Nothing, not even light can escape once beyond this point. Since no light is emitted from this part of the black hole, nothing inside can be seen. But it is believed that inside, at the very center, is the singularity. The singularity is the star's core crushed to a point (or the case of a spinning black hole, a ring) so small that it must be magnified over a trillion times to view any structure. More on black holes here and here.

Types of Black Holes

Black Holes are the result of collapsed stars that are more than approximately 4 solar masses. Although stars, planets, moons, galaxies etc. all have thousands of distinguishing characteristics, black holes only have three (this was a theory entitled "Black Holes Have No Hair" created by John Wheeler).

The three characteristics are: mass, charge and angular momentum (the last is related to spin). A Schwartzschild black hole is a black hole with only mass, and neither of the other two characteristics. These black holes have an even gravitational well and a gravitational singularity shaped as a dot. Also, these black holes are perfectly spherical, and have a fixed event horizon radius. However, these holes are rare, and usually black holes are much more complex. More info on black holes can be found here, here and here.

Quasars

A quasar is a supermassive black hole (see here, here, here and here) at the center of a young galaxy.  The farthest are 28 billion light-years away, making them that most distant objects ever observed.  They are so far that the energy they emit to make them visible most be equivalent to over 100 galaxies or a trillion suns.  Over 100,000 quasars have been detected with more on the way.  Some quasars can even be viewed with a small telescope, due to their luminosity with equals around 2 trillion suns.  An average quasar absorbs 10 solar masses (10 times the sun's mass) each year.  The biggest consumption on record being 1000 solar masses per year or 600 Earths per hour!  No quasars are in our supercluster because once they absorb all the mass in their proximity, they die out and become ordinary black holes, leaving a regular galaxy behind.  Quasars are that most powerful objects in our known Universe.

Black Hole Evaporation

An important question about Black Holes. Do they die? Hawking proposed Black hole evaporation, a theory in which Black Holes lose mass due to the separation of particle-antiparticle pairs. Vacuum fluctuations cause one particle to escape while its partner falls in. The negative particle falls in so the end equation is the Black Hole losing one particle. A massive Black Hole absorbs more Cosmic Background Radiation than it loses in mass because of Hawking Radiation. Because of this, until the Universe expands further and the Cosmic Radiation fades, a massive Black Hole will live forever. Eventually though, all Black Holes will evaporate. At the end of its life, a Black hole explodes with a temperature of over one quadrillion (1,000,000,000,000,000) degrees Fahrenheit with the force of over one billion atomic bombs.

Black Holes and Universe Budding?

Some people believe that the Universe itself is a giant Black hole, expanding and eventually contracting in a Big Crunch. Then a chain of universes follow from successive Big Crunches and Big Bangs. But if a Black hole can form a Universe, what about Black holes formed by collapsed giant stars? This opens an idea called Universal Budding. It states that a Black hole in our Universe can begin a baby universe. It would then expand into another dimension and set separate laws of physics. Thousands of universes could bud from ours.