The Limits of Information

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Abstract

Black holes have their own thermodynamics including notions of entropy and temperature and versions of the three laws. After a light introduction to black hole physics, I recollect how black hole thermodynamics evolved in the 1970s, while at the same time stressing conceptual points which were given little thought at that time, such as why the entropy should be linear in the black hole's surface area. I also review a variety of attempts made over the years to provide a statistical mechanics for black hole thermodynamics. Finally, I discuss the origin of the information bounds for ordinary systems that have arisen as applications of black hole thermodynamics.

Introduction

Although the citation from Einstein captures the professed attitude of many physicists, many others would today regard it as a basically sentimental statement. For we have become accustomed to regarding thermodynamics as a straight consequence of statistical mechanics and the atomic hypothesis. The usual paradigm, inferred from myriad examples—ideal gas, black body radiation, a superfluid, etc.—is that a system made up of a multitude of similar parts—molecules, electrons, phonons—with weak interactions and with no initial correlations between constituents (Boltzmann's Stosszahlansatz), will automatically exhibit thermodynamic behaviour. Thus—so goes the claim—it is statistical mechanics, not thermodynamics, which is the theory of ‘universal content’. The advent of black hole thermodynamics thirty years ago seems, however, to have turned the tables on this ‘modern’ assessment of thermodynamics’ secondary status.

In effect, black holes provide a second paradigm of thermodynamics. Black hole thermodynamics has meaning already at the classical level. It possesses a first law (conservation of energy, of momentum, of angular momentum and of electric charge), as well as a second law (in a generalised version) and a third law which delimits the kingdom of black holes. Black hole thermodynamics is no ordinary thermodynamics. Gravitation is all important in it, while in traditional thermodynamics gravitation is a nuisance which is customarily ignored. Information about a black hole's interior is not just practically unavailable, as in the garden variety thermodynamic system; rather, there are physical barriers forbidding acquisition of information sequestered in a black hole. And, again unlike everyday thermodynamic systems, a black hole is a monolithic system with no parts. Granted, as the usual description has it, a black hole is usually the product of the collapse of ordinary, thermodynamic, matter, but that infallen matter becomes invisible and thermodynamically irrelevant. To top it all, we are not sure today, despite vociferous claims to the contrary, whether there exists a statistical mechanics which reduces to black hole thermodynamics in some limit. Einstein would have been pleased: thermodynamics seems to stand by itself in the black hole domain.

And the issue is not just academic. Black hole thermodynamics seems to tell us things about mundane physical systems. For instance, a very pragmatic question in modern technology is how much information can be stored by whatever means in a cube of whatever composition one centimetre on the side. As we shall see, black hole thermodynamics has suggested upper bounds on this quantity; these are the limits of information.

Section snippets

What Is a Black Hole?

Ask what is a black hole and you will get many answers. The purist will say: a solution of Einstein's theory of gravity—general relativity—representing a spacetime which over most of its extension is like the one we are familiar with from special relativity, but which includes a region of finite spatial extent whose interior and boundary are totally invisible from the rest of the spacetime. Colloquially that invisible region is a black hole. Others will define the black hole differently: a tear

Black Hole Thermodynamics

A black hole can form from the collapse of an extremely complex mess of atoms, ions and radiation. Yet it transpires that once this object has settled down to the only stable available black hole configuration—the KNBH one—it must be specifiable by just three numbers: m, q and j. This is a paradox. But we are familiar with a similar situation. A cup of hot tea is an agglomeration of trillions of molecules of a number of species, all dancing around violently. We know, though, that from a

Hawking Radiation

Hawking had been a leader of the vociferous opposition to black hole thermodynamics. In a joint paper, ‘The Four Laws of Black Hole Mechanics’, Bardeen, Carter and Hawking (1973) argued against a thermodynamic interpretation of formulae like (2) and in favour of a purely mechanical one. Ironically, many uninformed authors still cite that paper as one of the sources of black hole thermodynamics! By his own account Hawking (1988) was trying to discredit black hole thermodynamics when he set out

Black Hole Statistical Mechanics?

In statistical mechanics, the entropy of an ordinary object is a measure of the number of states available to it, for example, the logarithm of the number of quantum states that it may access given its energy. This is the statistical meaning of entropy. What, in this sense, does black hole entropy represent? Is there a black hole statistical mechanics?

Black hole entropy is large; for instance, a solar mass black hole has SBH≈1079 whereas the sun has S≈1057. Early on I expressed the view (

The Bounds on Information

How much information can be stored by whatever means in a cube of whatever composition one centimetre on the side? Foreseeable technology making use of atomic manipulation would suggest an upper bound of around 1020 bits. But as technology takes advantage of unforeseen paradigms, this number could—and will—go up. For example, we might one day harness the atomic nucleus as an information cache. Can the bound go up without limit? Thirty years ago we would not have known what to answer. But with

Acknowledgements

I thank Avraham Mayo and Gilad Gour for many conversations; this research is supported by the Hebrew University's Intramural Fund.

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