Martin David Kruskal
Martin David Kruskal was an American mathematician and physicist. He made fundamental contributions in many areas of mathematics and science, ranging from plasma physics to general relativity and from nonlinear analysis to asymptotic analysis, his single most celebrated contribution was the theory of solitons. He was a student at the University of Chicago and at New York University, where he completed his Ph. D. under Richard Courant in 1952. He spent much of his career at Princeton University, as a research scientist at the Plasma Physics Laboratory starting in 1951, as a professor of astronomy and chair of the Program in Applied and Computational Mathematics, professor of mathematics, he retired from Princeton University in 1989 and joined the mathematics department of Rutgers University, holding the David Hilbert Chair of Mathematics. Apart from his research, Kruskal was known as a mentor of younger scientists, he always aimed not just to prove a result but to understand it thoroughly. And he was notable for his playfulness.
He invented the Kruskal Count, a magical effect, known to perplex professional magicians because – as he liked to say – it was based not on sleight of hand but on a mathematical phenomenon. Martin David Kruskal grew up in New Rochelle, he was known as Martin to the world and David to his family. His father, Joseph B. Kruskal, Sr. was a successful fur wholesaler. His mother, Lillian Rose Vorhaus Kruskal Oppenheimer, became a noted promoter of the art of origami during the early era of television and founded the Origami Center of America in New York City, which became OrigamiUSA, he was one of five children. His two brothers, both eminent mathematicians, were William Kruskal. Martin Kruskal was married to his wife of 56 years. Laura is well known as a writer about origami and originator of many new models. Martin, who had a great love of games and word play of all kinds invented several quite unusual origami models including an envelope for sending secret messages. Martin and Laura traveled extensively to scientific meetings and to visit Martin’s many scientific collaborators.
Laura used to call Martin "my ticket to the world." Wherever they went, Martin would be hard at work and Laura would keep busy teaching origami workshops in schools and institutions for elderly people and people with disabilities. Martin and Laura had a great love of traveling and hiking, their three children are Karen and Clyde, who are known as an attorney, an author of children’s books, a mathematician. Martin Kruskal's scientific interests covered a wide range of topics in pure mathematics and applications of mathematics to the sciences, he had lifelong interests in many topics in partial differential equations and nonlinear analysis and developed fundamental ideas about asymptotic expansions, adiabatic invariants, numerous related topics. His Ph. D. dissertation, written under the direction of Richard Courant and Bernard Friedman at New York University, was on the topic "The Bridge Theorem For Minimal Surfaces." He received his Ph. D. in 1952. In the 1950s and early 1960s, he worked on plasma physics, developing many ideas that are now fundamental in the field.
His theory of adiabatic invariants was important in fusion research. Important concepts of plasma physics that bear his name include the Kruskal–Shafranov instability and the Bernstein–Greene–Kruskal modes. With I. B. Bernstein, E. A. Frieman, R. M. Kulsrud, he developed the MHD Energy Principle, his interests extended to plasma astrophysics as well as laboratory plasmas. Martin Kruskal's work in plasma physics is considered by some to be his most outstanding. In 1960, Kruskal discovered the full classical spacetime structure of the simplest type of black hole in General Relativity. A spherically symmetric black hole can be described by the Schwarzschild solution, discovered in the early days of General Relativity. However, in its original form, this solution only describes the region exterior to the horizon of the black hole. Kruskal discovered the maximal analytic continuation of the Schwarzschild solution, which he exhibited elegantly using what are now called Kruskal–Szekeres coordinates; this led Kruskal to the astonishing discovery that the interior of the black hole looks like a "wormhole" connecting two identical, asymptotically flat universes.
This was the first real example of a wormhole solution in General Relativity. The wormhole collapses to a singularity before any observer or signal can travel from one universe to the other; this is now believed to be the general fate of wormholes in General Relativity. In the 1970s, when the thermal nature of black hole physics was discovered, the wormhole property of the Schwarzschild solution turned out to be an important ingredient. Nowadays, it is considered a fundamental clue in attempts to understand quantum gravity. Kruskal's most known work was the discovery in the 1960s of the integrability of certain nonlinear partial differential equations involving functions of one spatial variable as well as time; these developments began with a pioneering computer simulation by Kruskal and Norman Zabusky of a nonlinear equation known as the Korteweg–de Vries equation (
A blueshift is any decrease in wavelength, with a corresponding increase in frequency, of an electromagnetic wave. In visible light, this shifts the color from the red end of the spectrum to the blue end. Doppler blueshift is caused by movement of a source towards the observer; the term applies to any decrease in wavelength and increase in frequency caused by relative motion outside the visible spectrum. Only objects moving at near-relativistic speeds toward the observer are noticeably bluer to the naked eye, but the wavelength of any reflected or emitted photon or other particle is shortened in the direction of travel. Doppler blueshift is used in astronomy to determine relative motion: The Andromeda Galaxy is moving toward our own Milky Way galaxy within the Local Group. Components of a binary star system will be blueshifted when moving towards Earth When observing spiral galaxies, the side spinning toward us will have a slight blueshift relative to the side spinning away from us. Blazars are known to propel relativistic jets toward us, emitting synchrotron radiation and bremsstrahlung that appears blueshifted.
Nearby stars such as Barnard's Star are moving toward us, resulting in a small blueshift. Doppler blueshift of distant objects with a high z can be subtracted from the much larger cosmological redshift to determine relative motion in the expanding universe. Unlike the relative Doppler blueshift, caused by movement of a source towards the observer and thus dependent on the received angle of the photon, gravitational blueshift is absolute and does not depend on the received angle of the photon: It is a natural consequence of conservation of energy and mass–energy equivalence, was confirmed experimentally in 1959 with the Pound–Rebka experiment. Gravitational blueshift contributes to cosmic microwave background anisotropy via the Sachs–Wolfe effect: when a gravitational well evolves while a photon is passing, the amount of blueshift on approach will differ from the amount of gravitational redshift as it leaves the region. There are faraway active galaxies. One of the largest blueshifts is found in the narrow-line quasar, PG 1543+489, which has a relative velocity of -1150 km/s.
These types of galaxies are called "blue outliers". In a hypothetical universe undergoing a runaway big crunch contraction, a cosmological blueshift would be observed, with galaxies further away being blueshifted—the exact opposite of the observed cosmological redshift in the present expanding universe. Gravitational potential Redshift Relativistic Doppler effect
In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rotations and translations in space and time form the inhomogeneous Galilean group. Without the translations in space and time the group is the homogeneous Galilean group; the Galilean group is the group of motions of Galilean relativity acting on the four dimensions of space and time, forming the Galilean geometry. This is the passive transformation point of view; the equations below, although obvious, are valid only at speeds much less than the speed of light. In special relativity the Galilean transformations are replaced by Poincaré transformations. Galileo formulated these concepts in his description of uniform motion; the topic was motivated by his description of the motion of a ball rolling down a ramp, by which he measured the numerical value for the acceleration of gravity near the surface of the Earth.
Though the transformations are named for Galileo, it is absolute time and space as conceived by Isaac Newton that provides their domain of definition. In essence, the Galilean transformations embody the intuitive notion of addition and subtraction of velocities as vectors; this assumption is abandoned in the Poincaré transformations. These relativistic transformations are applicable to all velocities, while the Galilean transformation can be regarded as a low-velocity approximation to the Poincaré transformation; the notation below describes the relationship under the Galilean transformation between the coordinates and of a single arbitrary event, as measured in two coordinate systems S and S', in uniform relative motion in their common x and x′ directions, with their spatial origins coinciding at time t = t′ = 0: x ′ = x − v t y ′ = y z ′ = z t ′ = t. Note that the last equation expresses the assumption of a universal time independent of the relative motion of different observers. In the language of linear algebra, this transformation is considered a shear mapping, is described with a matrix acting on a vector.
With motion parallel to the x-axis, the transformation acts on only two components: = Though matrix representations are not necessary for Galilean transformation, they provide the means for direct comparison to transformation methods in special relativity. The Galilean symmetries can be uniquely written as the composition of a rotation, a translation and a uniform motion of spacetime. Let x represent a point in three-dimensional space, t a point in one-dimensional time. A general point in spacetime is given by an ordered pair. A uniform motion, with velocity v, is given by ↦, where v ∈ ℝ3. A translation is given by ↦, where a ∈ ℝ3 and s ∈ ℝ. A rotation is given by ↦; as a Lie group, the Galilean transformations span 10 dimensions, i.e. comprise 10 generators. Two Galilean transformations G compose to form a third Galilean transformation, G G = G; the set of all Galilean transformations Gal on space forms a group with composition as the group operation. The group is sometimes represented as a matrix group with spacetime events as vectors where t is real and x ∈ ℝ3 is a position in space.
The action is given by =, where s is real and v, x, a ∈ ℝ3 and R is a rotation matrix. The composition of transformations is accomplished through matrix multiplication. Gal has named sub
In physics, spacetime is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams can be used to visualize relativistic effects such as why different observers perceive where and when events occur; until the turn of the 20th century, the assumption had been that the three-dimensional geometry of the universe was independent of one-dimensional time. However, in 1905, Albert Einstein based his seminal work on special relativity on two postulates: The laws of physics are invariant in all inertial systems; the logical consequence of taking these postulates together is the inseparable joining together of the four dimensions, hitherto assumed as independent, of space and time. Many counterintuitive consequences emerge: in addition to being independent of the motion of the light source, the speed of light has the same speed regardless of the frame of reference in which it is measured. Einstein framed his theory in terms of kinematics.
His theory was a breakthrough advance over Lorentz's 1904 theory of electromagnetic phenomena and Poincaré's electrodynamic theory. Although these theories included equations identical to those that Einstein introduced, they were ad hoc models proposed to explain the results of various experiments—including the famous Michelson–Morley interferometer experiment—that were difficult to fit into existing paradigms. In 1908, Hermann Minkowski—once one of the math professors of a young Einstein in Zürich—presented a geometric interpretation of special relativity that fused time and the three spatial dimensions of space into a single four-dimensional continuum now known as Minkowski space. A key feature of this interpretation is the formal definition of the spacetime interval. Although measurements of distance and time between events differ for measurements made in different reference frames, the spacetime interval is independent of the inertial frame of reference in which they are recorded. Minkowski's geometric interpretation of relativity was to prove vital to Einstein's development of his 1915 general theory of relativity, wherein he showed how mass and energy curve this flat spacetime to a Pseudo Riemannian manifold.
Non-relativistic classical mechanics treats time as a universal quantity of measurement, uniform throughout space and, separate from space. Classical mechanics assumes that time has a constant rate of passage, independent of the state of motion of an observer, or indeed of anything external. Furthermore, it assumes that space is Euclidean, to say, it assumes that space follows the geometry of common sense. In the context of special relativity, time cannot be separated from the three dimensions of space, because the observed rate at which time passes for an object depends on the object's velocity relative to the observer. General relativity, in addition, provides an explanation of how gravitational fields can slow the passage of time for an object as seen by an observer outside the field. In ordinary space, a position is specified by three numbers, known as dimensions. In the Cartesian coordinate system, these are called x, y, z. A position in spacetime is called an event, requires four numbers to be specified: the three-dimensional location in space, plus the position in time.
Spacetime is thus four dimensional. An event is something that happens instantaneously at a single point in spacetime, represented by a set of coordinates x, y, z and t; the word "event" used in relativity should not be confused with the use of the word "event" in normal conversation, where it might refer to an "event" as something such as a concert, sporting event, or a battle. These are not mathematical "events" in the way the word is used in relativity, because they have finite durations and extents. Unlike the analogies used to explain events, such as firecrackers or lightning bolts, mathematical events have zero duration and represent a single point in spacetime; the path of a particle through spacetime can be considered to be a succession of events. The series of events can be linked together to form a line which represents a particle's progress through spacetime; that line is called the particle's world line. Mathematically, spacetime is a manifold, to say, it appears locally "flat" near each point in the same way that, at small enough scales, a globe appears flat.
An large scale factor, c relates distances measured in space with distances measured in time. The magnitude of this scale factor, along with the fact that spacetime is a manifold, implies that at ordinary, non-relativistic speeds and at ordinary, human-scale distances, there is little that humans might observe, noticeably different from what they might observe if the world were Euclidean, it was only with the advent of sensitive scientific measurements in the mid-1800s, such as the Fizeau experiment and the Michelson–Morley experiment, that puzzling discrepancies began to be noted between observation versus predictions based on the implicit assumption of Euclidean space. In special relativity, an observer will, in most
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize and predict natural phenomena. This is in contrast to experimental physics; the advancement of science depends on the interplay between experimental studies and theory. In some cases, theoretical physics adheres to standards of mathematical rigour while giving little weight to experiments and observations. For example, while developing special relativity, Albert Einstein was concerned with the Lorentz transformation which left Maxwell's equations invariant, but was uninterested in the Michelson–Morley experiment on Earth's drift through a luminiferous aether. Conversely, Einstein was awarded the Nobel Prize for explaining the photoelectric effect an experimental result lacking a theoretical formulation. A physical theory is a model of physical events, it is judged by the extent. The quality of a physical theory is judged on its ability to make new predictions which can be verified by new observations.
A physical theory differs from a mathematical theorem in that while both are based on some form of axioms, judgment of mathematical applicability is not based on agreement with any experimental results. A physical theory differs from a mathematical theory, in the sense that the word "theory" has a different meaning in mathematical terms. A physical theory involves one or more relationships between various measurable quantities. Archimedes realized that a ship floats by displacing its mass of water, Pythagoras understood the relation between the length of a vibrating string and the musical tone it produces. Other examples include entropy as a measure of the uncertainty regarding the positions and motions of unseen particles and the quantum mechanical idea that energy are not continuously variable. Theoretical physics consists of several different approaches. In this regard, theoretical particle physics forms a good example. For instance: "phenomenologists" might employ empirical formulas to agree with experimental results without deep physical understanding.
"Modelers" appear much like phenomenologists, but try to model speculative theories that have certain desirable features, or apply the techniques of mathematical modeling to physics problems. Some attempt to create approximate theories, called effective theories, because developed theories may be regarded as unsolvable or too complicated. Other theorists may try to unify, reinterpret or generalise extant theories, or create new ones altogether. Sometimes the vision provided by pure mathematical systems can provide clues to how a physical system might be modeled. Theoretical problems that need computational investigation are the concern of computational physics. Theoretical advances may consist in setting aside old, incorrect paradigms or may be an alternative model that provides answers that are more accurate or that can be more applied. In the latter case, a correspondence principle will be required to recover the known result. Sometimes though, advances may proceed along different paths. For example, an correct theory may need some conceptual or factual revisions.
However, an exception to all the above is the wave–particle duality, a theory combining aspects of different, opposing models via the Bohr complementarity principle. Physical theories become accepted if they are able to make correct predictions and no incorrect ones; the theory should have, at least as a secondary objective, a certain economy and elegance, a notion sometimes called "Occam's razor" after the 13th-century English philosopher William of Occam, in which the simpler of two theories that describe the same matter just as adequately is preferred. They are more to be accepted if they connect a wide range of phenomena. Testing the consequences of a theory is part of the scientific method. Physical theories can be grouped into three categories: mainstream theories, proposed theories and fringe theories. Theoretical physics began at least 2,300 years ago, under the Pre-socratic philosophy, continued by Plato and Aristotle, whose views held sway for a millennium. During the rise of medieval universities, the only acknowledged intellectual disciplines were the seven liberal arts of the Trivium like grammar and rhetoric and of the Quadrivium like arithmetic, geometry and astronomy.
During the Middle Ages and Renaissance, the concept of experimental science, the counterpoint to theory, began with scholars such as Ibn al-Haytham and Francis Bacon. As the Scientific Revolution gathered pace, the concepts of matter, space and causality began to acquire the form we know today, other sciences spun off from the rubric of natural philosophy, thus began the modern era of theory with the Copernican paradigm shift in astronomy, soon followed by Johannes Kepler's expressions for planetary orbits, which summarized the meticulous observations of Tycho Brahe.
A black hole is a region of spacetime exhibiting such strong gravitational effects that nothing—not particles and electromagnetic radiation such as light—can escape from inside it. The theory of general relativity predicts that a sufficiently compact mass can deform spacetime to form a black hole; the boundary of the region from which no escape is possible is called the event horizon. Although the event horizon has an enormous effect on the fate and circumstances of an object crossing it, no locally detectable features appear to be observed. In many ways, a black hole acts like an ideal black body. Moreover, quantum field theory in curved spacetime predicts that event horizons emit Hawking radiation, with the same spectrum as a black body of a temperature inversely proportional to its mass; this temperature is on the order of billionths of a kelvin for black holes of stellar mass, making it impossible to observe. Objects whose gravitational fields are too strong for light to escape were first considered in the 18th century by John Michell and Pierre-Simon Laplace.
The first modern solution of general relativity that would characterize a black hole was found by Karl Schwarzschild in 1916, although its interpretation as a region of space from which nothing can escape was first published by David Finkelstein in 1958. Black holes were long considered a mathematical curiosity; the discovery of neutron stars by Jocelyn Bell Burnell in 1967 sparked interest in gravitationally collapsed compact objects as a possible astrophysical reality. Black holes of stellar mass are expected to form when massive stars collapse at the end of their life cycle. After a black hole has formed, it can continue to grow by absorbing mass from its surroundings. By absorbing other stars and merging with other black holes, supermassive black holes of millions of solar masses may form. There is general consensus. Despite its invisible interior, the presence of a black hole can be inferred through its interaction with other matter and with electromagnetic radiation such as visible light.
Matter that falls onto a black hole can form an external accretion disk heated by friction, forming some of the brightest objects in the universe. If there are other stars orbiting a black hole, their orbits can be used to determine the black hole's mass and location; such observations can be used to exclude possible alternatives such as neutron stars. In this way, astronomers have identified numerous stellar black hole candidates in binary systems, established that the radio source known as Sagittarius A*, at the core of the Milky Way galaxy, contains a supermassive black hole of about 4.3 million solar masses. On 11 February 2016, the LIGO collaboration announced the first direct detection of gravitational waves, which represented the first observation of a black hole merger; as of December 2018, eleven gravitational wave events have been observed that originated from ten merging black holes. On 10 April 2019, the first direct image of a black hole and its vicinity was published, following observations made by the Event Horizon Telescope in 2017 of the supermassive black hole in Messier 87's galactic centre.
Larry Kimura, a Hawaiian language professor at the University of Hawaii at Hilo, named the hole Pōwehi—a Hawaiian phrase referring to an "embellished dark source of unending creation." The idea of a body so massive that light could not escape was proposed by astronomical pioneer and English clergyman John Michell in a letter published in November 1784. Michell's simplistic calculations assumed that such a body might have the same density as the Sun, concluded that such a body would form when a star's diameter exceeds the Sun's by a factor of 500, the surface escape velocity exceeds the usual speed of light. Michell noted that such supermassive but non-radiating bodies might be detectable through their gravitational effects on nearby visible bodies. Scholars of the time were excited by the proposal that giant but invisible stars might be hiding in plain view, but enthusiasm dampened when the wavelike nature of light became apparent in the early nineteenth century. If light were a wave rather than a "corpuscle", it became unclear what, if any, influence gravity would have on escaping light waves.
Modern relativity discredits Michell's notion of a light ray shooting directly from the surface of a supermassive star, being slowed down by the star's gravity and free-falling back to the star's surface. In 1915, Albert Einstein developed his theory of general relativity, having earlier shown that gravity does influence light's motion. Only a few months Karl Schwarzschild found a solution to the Einstein field equations, which describes the gravitational field of a point mass and a spherical mass. A few months after Schwarzschild, Johannes Droste, a student of Hendrik Lorentz, independently gave the same solution for the point mass and wrote more extensively about its properties; this solution had a peculiar behaviour at what is now called the Schwarzschild radius, where it became singular, meaning that some of the terms in the Einstein equations became infinite. The nature of this surface was not quite understood at the time. In 1924, Arthur Eddington showed that the singularity disappeared after a change of coordinates, although it took until 1933 for Georges Lemaître to realize that this meant the singularity at the Schwarzschild radius was a non-physical coordinate singularity.
Arthur Eddington did however comment on the possibility of a star with mass c
A wormhole is a speculative structure linking disparate points in spacetime, is based on a special solution of the Einstein field equations solved using a Jacobian matrix and determinant. A wormhole can be visualized as a tunnel with two ends, each at separate points in spacetime. More it is a transcendental bijection of the spacetime continuum, an asymptotic projection of the Calabi–Yau manifold manifesting itself in Anti-de Sitter space. Wormholes are consistent with the general theory of relativity, but whether wormholes exist remains to be seen. A wormhole could connect long distances such as a billion light years or more, short distances such as a few meters, different universes, or different points in time. For a simplified notion of a wormhole, space can be visualized as a two-dimensional surface. In this case, a wormhole would appear as a hole in that surface, lead into a 3D tube re-emerge at another location on the 2D surface with a hole similar to the entrance. An actual wormhole would be analogous with the spatial dimensions raised by one.
For example, instead of circular holes on a 2D plane, the entry and exit points could be visualized as spheres in 3D space. Another way to imagine wormholes is to take a sheet of paper and draw two somewhat distant points on one side of the paper; the sheet of paper represents a plane in the spacetime continuum, the two points represent a distance to be traveled, however theoretically a wormhole could connect these two points by folding that plane so the points are touching. In this way it would be much easier to traverse the distance. In 1928, Hermann Weyl proposed a wormhole hypothesis of matter in connection with mass analysis of electromagnetic field energy. American theoretical physicist John Archibald Wheeler coined the term "wormhole" in a 1957 paper co-authored by Charles Misner: This analysis forces one to consider situations... where there is a net flux of lines of force, through what topologists would call "a handle" of the multiply-connected space, what physicists might be excused for more vividly terming a "wormhole".
Wormholes have been defined both topologically. From a topological point of view, an intra-universe wormhole is a compact region of spacetime whose boundary is topologically trivial, but whose interior is not connected. Formalizing this idea leads to definitions such as the following, taken from Matt Visser's Lorentzian Wormholes. If a Minkowski spacetime contains a compact region Ω, if the topology of Ω is of the form Ω ~ R × Σ, where Σ is a three-manifold of the nontrivial topology, whose boundary has topology of the form ∂Σ ~ S2, if, the hypersurfaces Σ are all spacelike the region Ω contains a quasipermanent intrauniverse wormhole. Geometrically, wormholes can be described as regions of spacetime that constrain the incremental deformation of closed surfaces. For example, in Enrico Rodrigo's The Physics of Stargates, a wormhole is defined informally as: a region of spacetime containing a "world tube" that cannot be continuously deformed to a world line; the equations of the theory of general relativity have valid solutions.
The first type of wormhole solution discovered was the Schwarzschild wormhole, which would be present in the Schwarzschild metric describing an eternal black hole, but it was found that it would collapse too for anything to cross from one end to the other. Wormholes that could be crossed in both directions, known as traversable wormholes, would only be possible if exotic matter with negative energy density could be used to stabilize them. Schwarzschild wormholes known as Einstein–Rosen bridges, are connections between areas of space that can be modeled as vacuum solutions to the Einstein field equations, that are now understood to be intrinsic parts of the maximally extended version of the Schwarzschild metric describing an eternal black hole with no charge and no rotation. Here, "maximally extended" refers to the idea that the spacetime should not have any "edges": it should be possible to continue this path arbitrarily far into the particle's future or past for any possible trajectory of a free-falling particle.
In order to satisfy this requirement, it turns out that in addition to the black hole interior region that particles enter when they fall through the event horizon from the outside, there must be a separate white hole interior region that allows us to extrapolate the trajectories of particles that an outside observer sees rising up away from the event horizon. And just as there are two separate interior regions of the maximally extended spacetime, there are two separate exterior regions, sometimes called two different "universes", with the second universe allowing us to extrapolate some possible particle trajectories in the two interior regions; this means that the interior black hole region can contain a mix of particles that fell in from either universe, particles from the interior white hole region can escape into either universe. All four regions can be seen in a spacetime diagram that uses Kruskal–Szekeres coordinates. In this spacetime, it is possible to come up with coordinate systems such that