Lebesgue Measure

The idea of a "measure", intuitively a rule that determines the size of an object, is very important to mathematics. It is used as a baseline for comparing mathematical "objects", usually sets. However, what consistent way is there to assign sizes to sets that varies over such the wide variety of sets found in mathematics?

One approach is called the Lebesgue measure, named after the mathematician Henri Lebesgue, who introduced the concept around 1902. With this measure, the size of a set A is denoted λ(A). The "objects" or sets that can be measured are considered to reside within the a real Euclidean space Rⁿ for some positive integer n. For each dimension n, Rⁿ is the typical flat (Euclidean) space. For n = 1, it is the line, n = 2, the plane, etc. An intuitively easy way to deal with many sets in real space is simply to measure their volumes. For such sets where this is easily done, the Lebesgue measure coincides with what we traditionally calculate as "volume".

The basis for area in n dimensions is found in one dimension, where "volume" is simply length. To a closed interval [a,b] in R¹ (the set of real numbers from a to b, including a and b, see below), the Lebesgue measure assigns the value b - a.

In more than one dimension, we define the Cartesian product of two closed intervals [a,b] and [c,d] in R¹ as the rectangle in R² bounded by the lines x = a, x = b, y = c, and y = d. Its volume, as per the usual practice, is the product of the lengths of the respective intervals; in this case, it is (b - a)(d - c). In equation form, with the Cartesian product indicated by x, λ([a,b]x[c,d]) = (b - a)(d - c).

This approach is extended to three dimensions by considering the Cartesian product of 3 (non-degenerate) closed intervals, which takes the form of a rectangular prism, the volume of this set being the length times the width times the height, as is normal. In n-dimensions, the Cartesian product of n closed intervals I₁, I₂,..., I_n forms an n-dimension rectangular prism with volume |I₁|·|I₂|·...·|I_n|, where |I_i| is the length of the respective interval.

This rectangular prisms in n-dimensions, or rectangular n-prisms, are the building blocks of volume in Rⁿ. Another basic property of the Lebesgue measure is that, if two sets A and B are disjoint, i.e. do not overlap, or more formally, their intersection is empty, then λ(A∪B) = λ(A) + λ(B). This expression holds as long as the measures are defined. More generally, if n disjoint measurable sets are involved:

This formula simply states that the volume of the union of the sets is the sum of their individual volumes. Remarkably, this even applies to infinite unions of sets, as long as the number of sets is countable, or, equivalently, if the collection of sets can be put into a one-to-one correspondence with the natural numbers (for more, see here).

To define volume for a more general set, one approximates the set as an addition of smaller sets, usually n-prisms. To be more precise, for a set A, if a set of disjoint rectangular n-prisms are contained completely within A, then the sum of their volumes is an lower approximation to the volume of A. If a set of rectangular n-prisms (not necessarily disjoint, i.e. there could be some overlap) completely covers the set A in Rⁿ, then it serves as an upper approximation to the volume of A. This method of finding area is called integration.

An illustration of how rectangular n-prisms can provide upper and lower approximation to the volume of a more general set A.

Another property of the Lebesgue measure is invariance under translation and rotation. In other words, if a set A is rotated or moved around in Rⁿ to another set B, then λ(A) = λ(B). On the other hand, if a set is dilated, or expanded, by a factor d, where d is a positive real number, and the dilated set is denoted dA, then

λ(dA) = dⁿλ(A),

where n is the dimension of the ambient space (Rⁿ).

Now, one can consider various sets in Rⁿ and their Lebesgue measures. First, any point, or more precisely, the set containing a single point P, {P}, has measure 0. This is because it can be viewed as a product of intervals each with length zero. Now, any finite set, that is, including a finite number of points, has Lebesgue measure 0, as such a set can be expressed as the union of many sets, each containing one point, which we known to have measure 0. A set with measure 0 is called negligible.

Similarly, the set of integers, or, for higher dimensions, the set of ordered n-tuplets of the form (a₁,a₂,...,a_n) where the a_i are integers, is negligible, as these sets are countable (this is proven here).

Returning to R¹, the set Q of rational numbers, or those numbers expressible as a quotient of two integers, is countable, and therefore negligible, with λ(Q) = 0. This is the first example of a dense set that is negligible. The higher dimension analog of the rational numbers in R¹ is, for each dimension n, the set of n-tuplets (a₁,a₂,...,a_n) where the a_i are rational numbers. For every n, this set is negligible, with Lebesgue measure 0.

However, the Lebesgue Measure does far more than distinguish between countable and uncountable sets. More properties and measures of sets are presented in the next post.

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Sources: Advanced Calculus of Several Variables by C.H. Edwards Jr., Lebesgue Measure on Wikipedia

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

Degenerate Matter: Exotic Stars

This post deals with hypothetical "exotic" stars and their composition. For an introduction to degenerate matter followed by descriptions of its "first two" stages, see here.

Neutron stars, as with white dwarfs, can only exist at a certain range of masses, before gravity overcomes the neutron degeneracy pressure. Above a certain density, neutron degenerate matter can no longer exist.

Here one enters the realm of speculation, as it is not known exactly where this upper limit is, and what stages a degenerate object undergoes immediately after. By most estimates, neutron degenerate matter cannot exist in an object weighing more than 3 solar masses. Since, by the Pauli Exclusion Principle, no two neutrons can occupy the same location at any amount of pressure, it is reasonable to assume that they, under the pressure of gravity, eventually break down instead into their constituent particles: quarks and gluons.

The quark structure of a neutron. The neutron contains two down quarks and one up quark, and connecting particles called gluons (the wavy lines) that bind quarks together.

As the mass of a degenerate object exceeds three solar masses, the matter at the core will collapse from neutron degenerate matter to quark-degenerate matter, better known as quark-gluon plasma. Such an object would then be called a quark star.

Gluons can effectively be ignored in considering the properties of a quark star; since they have a mass of 0, they only contribute in binding quarks into larger particles. In quark stars, however, quarks are free rather than bound in nuclei, and gluons would therefore have little effect.

Therefore, typical (if any such matter can be called "typical") quark-gluon plasma would contain up and down quarks, the lightest and most common quarks. However, at the extremely high densities of a quark star, the energies may be high enough for a third type of quark, the strange quark, to spontaneously form. Strange quarks, the next-heaviest quarks, are not commonly found but, under normal circumstances, decay into up quarks. Strange quarks, however, do compose more exotic particles that have been synthesized in particle accelerators. Quark stars which include strange quarks in their composition are sometimes called strange stars.

At the present time, there is no conclusive evidence in favor of the existence of quark stars, though they have been proposed as an explanation for certain celestial phenomena. For example, there are a few known stellar remnants whose masses have appeared to those of heavy neutron stars (around 2 or 3 solar masses), but whose radii are only 5 miles! Usual neutron stars, with diameters usually twice that figure, are not as dense as these strange objects. As of now, the measurements are not completely certain, and revisions are possible.

Perhaps the most compelling evidence for quark stars are certain types of gamma-ray sources that defy explanation due to their tremendous brightness. One such event was recorded in 2007, and was classified as a supernova, receiving the designation SN 2006gy.



An X-ray image of SN 2006gy (the bright spot in the upper right). The source is 238 million light years away in the galaxy NGC 1260. At its peak, SN 2006gy outshined its entire home galaxy (bright spot in lower left) and was one of the most luminous objects in the universe.

Though initially thought to be a supernova, the explosion released ten times more energy than other similar supernova events. Therefore, some have hypothesized that the event was a so-called quark nova. A quark nova would mark the collapse of a neutron star into a quark star. The transition of the quarks from bound state (in the neutrons) to a free state (quark-gluon plasma or strange matter) would release tremendous amounts of energy: up to a thousand times a "typical" supernova explosion. Another indicator is the concentration of high-energy radiation; SN 2006gy released an unusual amount of radiation in the X-ray and gamma ray areas of the electromagnetic spectrum.

Despite claims of the existence of quark stars, all evidence thus far, including SN 2006gy, is ambiguous and does not confirm nor deny their existence. When a neutron star's gravity overcomes neutron degeneracy pressure and collapses, the resulting stellar remnant does pass through a quark stage, but it is not known whether such a state is stable, or, if it is, what range of masses a quark star can assume. This uncertainty primarily stems from our ignorance of the properties of such super-dense matter.

However, even if quark stars exist, they are only kept stable by quark degeneracy pressure. Quarks, as with neutrons, resist being forced to occupy the same location. But, if the mass exceeds about 3-3.5 solar masses, gravity (theoretically) overcomes quark degeneracy pressure and the stellar remnant collapses. The next step in the life of the collapsed star is, if anything, even more mysterious. One plausible theory is the formation of yet another type of star, termed an electroweak star.

Following the pattern of a neutron star, the quarks in a quark star, when subjected to enough gravitational pressure, break down. However, quarks have no known component particles, so they may instead undergo a decay, related beta decay. The decay only happens in a small area of the core of an electroweak star, and the resulting situation is a striking analog of normal stellar fusion (see below). The use of fuel in this case is called electroweak burning, rather than fusion.



The above diagram is a visualization of a hypothetical electroweak star (not to scale). Such a star, as with a quark star, would probably have an outer layer of neutron degenerate matter, and would appear from the outside as an overdense neutron star. It would probably be on the order of 2-5 miles in diameter, and would have a miniscule core, only the size of an apple! However, packed into this core would be approximately two earth masses. Here, quarks are converted into leptons (including electrons). In the process of this conversion, very small neutrinos (or antineutrinos) are released. The outward pressure provided by these particles stops the electroweak star from collapsing under the force of gravity.

Another curiosity of electroweak stars is the property that gives them their name. At certain extreme temperatures and densities the electromagnetic force (the force which describes the charge of particles, and whose large scale effects are electricity and magnetism) and the weak nuclear force (the force which controls beta decay) "unify". The unification of these forces at extraordinarily high temperatures and densities (on the order of a billion billion degrees, for instance) means that these forces, under such circumstances, are no longer distinguishable and rather behave as manifestations of a single force, called the electroweak force. The center of an electroweak star is one of the only places in the Universe that the conditions necessary for this unification occur. Such conditions notably occurred a trillionth of a second after the Big Bang. Insight into electroweak stars would correspondingly elucidate aspects of the Big Bang.

One might think that such an electroweak star, since it emits a wealth of radiation in the form of neutrinos, would be easily detectable. However, neutrinos are notoriously elusive particles; they hardly interact at all with normal matter, but rather pass right through it. Efforts to detect neutrino sources in the cosmos, therefore, are difficult, and, to this day, only very low-resolution "neutrino-images" have been achieved.

The key parameter in determining how many electroweak stars are out there is stability. Early models seemed to suggest that electroweak burning would only last for a very short time, maybe only a matter of seconds. However, some more recent research has indicated that electroweak stars may have enough fuel to last millions of years or more. In the latter case, there is likely to be a small, but detectable, population of such stars throughout the cosmos.

It is possible, in addition, that electroweak burning would shed enough mass (through neutrinos) from the stellar remnant to render it stable, i.e. supportable by quark degeneracy pressure. However, in other cases, after production of leptons ceases, electroweak stars would collapse further, powerless against gravity, eventually transforming into the most mysterious objects in the universe (see the next post).

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Sources: http://www.sciencedaily.com/releases/2008/06/080628224224.htm, Quark Star at Wikipedia, http://news.discovery.com/space/exotic-electroweak-star-predicted.html, http://www.sciencedaily.com/releases/2009/12/091214131132.htm

Degenerate Matter: Neutron Stars

This post deals with neutron stars and their composition. For an introduction to degenerate matter and the quantum mechanical principles involved, see here. For a description of white dwarfs, the "first" state of degenerate matter, see here.

If only electron degeneracy is taken into account, one would predict that white dwarfs, after reaching a certain threshold, would contract to nothing, the electron degeneracy pressure not being enough to hold of the force of gravity, partly due to relativistic effects. However, this is not the case. For stars that leave stellar remnants in excess of 1.44 solar masses, the rapid shrinking of the core under the force of gravity that occurs after fusion ceases forms a new type of object: a neutron star.

In a neutron star, the nucleons (protons and neutrons) are pushed into such close proximity that the Pauli Exclusion Principle comes into play, forcing the particles involved to stay separate and not assume the same quantum state. Neutrons, being much more massive than electrons, produce less degeneracy pressure at the same density than electrons do, as it takes more energy to cause them to move at the same speeds. Therefore, a neutron star is much denser, and therefore smaller, than a white dwarf. These remarkable bodies are less than 10 miles in radius!

Neutron stars have many interesting properties, since they are among the densest forms of matter. First, they are in many ways similar to a giant atomic nucleus, but containing trillions and trillions instead of merely hundreds of nucleon; the two entities are quite comparable in density, however-each packs approximately 300,000,000,000,000,000 (3*10¹⁷) kilograms into each cubic meter. If the entire mass of the Earth were compressed to this density, it would occupy a volume roughly equivalent to that of the Great Pyramid of Giza! An object on the surface of a neutron star would have to achieve a velocity one-third of the speed of light to escape its gravity, and any matter that falls onto a neutron star impacts with such force that the atoms themselves are broken apart.

Another interesting phenomenon arising from the strong gravity of neutron stars is gravitational lensing. The light emitted from the surface of a neutron star, though having sufficient velocity to escape its gravitational pull, is distorted and curved back towards the surface. As a result, when looking at a neutron star from any given side, one can see more than half of the surface (illustrated below).

A diagram indicating the portion of the surface visible looking at a neutron star with the equator head-on. 

Each patch covers 30° by 30° of surface. The poles are visible as the points of convergence of the longitude lines. Note that, without distortion, one could only see up to the poles, but on a neutron star, one can see more than 40° beyond each pole and around the far side.

In addition, neutron stars rotate. The force of rotation of a parent star is conserved when it becomes a neutron star, but since a neutron star is many times smaller, the momentum causes it to spin extremely rapidly: usually completing each rotation in less than a second! This causes many neutron stars to have a "bulge" near the equatorial regions.

Another important property of a neutron star is its magnetic field. The magnetic field of a neutron star is on the order of a billion times stronger than Earth's, and in some cases it is over a trillion times stronger. Young neutron stars, at high temperatures, emit large amounts of electromagnetic radiation. The structure of the magnetic field causes these beams of energy to be released at the poles. Neutron stars with these electromagnetic beams are called pulsars. Pulsars can be easily detected when one of the poles faces Earth at some time during the neutron star's rotation.

A combined optical, and X-ray image of the Vela pulsar. A distinct beam of radiation in evident from its north pole.

The composition of neutron stars is not definitively known, and various types of matter appear between the outer crust and the center of a neutron star, as the density of the object increases significantly as one journeys inward.

A hypothetical cross-section of a neutron star, based on many simulations and models.

The densities are in terms of the constant ρ₀, the density at which isolated protons and neutrons actually touch one another.  The crust generally consists of atomic nuclei, whose valence electrons have been pushed out by the extreme pressures. The electrons themselves float freely, creating an "electron sea" around the nuclei. This makes the matter of the crust extremely conductive. A similar phenomenon actually occurs in regular metals, with free-flowing electrons surrounding ions. Metals are therefore by some definitions electron degenerate! However, they are not degenerate for the same reason as stellar remnants, as the metals obviously are not (usually) at extraordinarily high densities.

Proceeding inward, one reaches the outer core. Here, the high density causes electrons and protons to become neutrons through a reverse of beta decay, called electron capture. Therefore, the nuclei of the outer core are very neutron rich, to the extent that they would not be stable under normal conditions. Only the density of the neutron star keeps such neutron-rich nuclei together. As one penetrates farther into the outer core, however, neutrons begin to "drip" out of nuclei and become free-flowing, as even the tremendous pressure is not enough to hold together the neutron-rich nuclei.

Near the core, as the pressure exceeds ρ₀, neutrons come into physical contact. Being fermions, the Pauli Exclusion Principle (which states that no two fermions can have the same quantum state, that is, occupy the same location simultaneously) causes pressure counteracting the force of gravity. In a manner similar to the situation in white dwarfs, this counterpressure is known as neutron degeneracy pressure.

It is not known what type of matter exists at the very center of a neutron star, and the answer may vary with the neutron star's mass. However, matter consisting completely of neutrons by definition "should" compose the inner core of a neutron star (by its name), and if, as displayed above, other exotic types of matter exist there, the object would more appropriately be called a "quark star". Such hypothetical objects, their composition, and observational evidence for them, are discussed in the next post.

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Sources: http://en.wikipedia.org/wiki/Neutron_star, http://en.wikipedia.org/wiki/Degenerate_matter#Neutron_degeneracy, http://www.astro.umd.edu/~miller/nstar.html