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Hartle–Hawking State
The Hartle–Hawking state is a proposal in theoretical physics concerning the state of the Universe prior to the Planck epoch. It is named after James Hartle and Stephen Hawking. Hartle and Hawking suggest that if we could travel backwards in time towards the beginning of the Universe, we would note that quite near what might otherwise have been the beginning, time gives way to space such that at first there is only space and no time. According to the Hartle–Hawking proposal, the Universe has no origin as we would understand it: the Universe was a singularity in both space and time, pre-Big Bang. However, Hawking does state "...the universe has not existed forever. Rather, the universe, and time itself, had a beginning in the Big Bang, about 15 billion years ago.", but that the Hartle-Hawking model is not the steady state Universe of Hoyle; it simply has no initial boundaries in time or space. Technical explanation The Hartle–Hawking state is the wave function of the U ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Physics
Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain and predict natural phenomena. This is in contrast to experimental physics, which uses experimental tools to probe these phenomena. The advancement of science generally 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.There is some debate as to whether or not theoretical physics uses mathematics to build intuition and illustrativeness to extract physical insight (especially when normal experience fails), rather than as a tool in formalizing theories. This links to the question of it using mathematics in a less formally rigorous, and more intuitive or heuristic way than, say, mathematical physics. For example, while developing special relativity, Albert Einstein was concerned with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise naturally and frequently in mathematics and physics, typically as function spaces. Formally, a Hilbert space is a vector space equipped with an inner product that defines a distance function for which the space is a complete metric space. The earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the term ''Hilbert space'' for the abstract concept that u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signature Change
In mathematics, the signature of a metric tensor ''g'' (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive, negative and zero eigenvalues of the real symmetric matrix of the metric tensor with respect to a basis. In relativistic physics, the ''v'' represents the time or virtual dimension, and the ''p'' for the space and physical dimension. Alternatively, it can be defined as the dimensions of a maximal positive and null subspace. By Sylvester's law of inertia these numbers do not depend on the choice of basis. The signature thus classifies the metric up to a choice of basis. The signature is often denoted by a pair of integers implying ''r''= 0, or as an explicit list of signs of eigenvalues such as or for the signatures and , respectively. The signature is said to be indefinite or mixed if both ''v'' and ''p'' are nonzero, and degenerate if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiple Histories
In theoretical physics, top-down cosmology is a proposal to regard the many possible past histories of a given event as having real existence. This idea of multiple histories has been applied to cosmology, in a theoretical interpretation in which the universe has multiple possible cosmologies, and in which reasoning ''backwards'' from the current state of the universe to a quantum superposition of possible cosmic histories makes sense. Stephen Hawking has argued that the principles of quantum mechanics forbid a single cosmic history, and has proposed cosmological theories in which the lack of a past boundary condition naturally leads to multiple histories, called the 'no-boundary proposal', the proposed Hartle–Hawking state. According to Hawking and Thomas Hertog, "The top-down approach we have described leads to a profoundly different view of cosmology, and the relation between cause and effect. Top down cosmology is a framework in which one essentially traces the histories ba ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Imaginary Time
Imaginary time is a mathematical representation of time which appears in some approaches to special relativity and quantum mechanics. It finds uses in connecting quantum mechanics with statistical mechanics and in certain cosmological theories. Mathematically, imaginary time is real time which has undergone a Wick rotation so that its coordinates are multiplied by the imaginary unit ''i''. Imaginary time is ''not'' imaginary in the sense that it is unreal or made-up (any more than, say, irrational numbers defy logic), it is simply expressed in terms of what mathematicians call imaginary numbers. Origins In mathematics, the imaginary unit i is the square root of -1, such that i^2 is defined to be -1. A number which is a direct multiple of i is known as an imaginary number. In certain physical theories, periods of time are multiplied by i in this way. Mathematically, an imaginary time period \tau may be obtained from real time t via a Wick rotation by \pi/2 in the complex plane: \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wheeler–DeWitt Equation
The Wheeler–DeWitt equation for theoretical physics and applied mathematics, is a field equation attributed to John Archibald Wheeler and Bryce DeWitt. The equation attempts to mathematically combine the ideas of quantum mechanics and general relativity, a step towards a theory of quantum gravity. In this approach, time plays a role different from what it does in non-relativistic quantum mechanics, leading to the so-called ' problem of time'. More specifically, the equation describes the quantum version of the Hamiltonian constraint using metric variables. Its commutation relations with the diffeomorphism constraints generate the Bergman–Komar "group" (which ''is'' the diffeomorphism group on-shell). Quantum gravity All defined and understood descriptions of string/M-theory deal with fixed asymptotic conditions on the background spacetime. At infinity, the "right" choice of the time coordinate "t" is determined (because the space-time is asymptotic to some fixed space-ti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John D
John is a common English name and surname: * John (given name) * John (surname) John may also refer to: New Testament Works * Gospel of John, a title often shortened to John * First Epistle of John, often shortened to 1 John * Second Epistle of John, often shortened to 2 John * Third Epistle of John, often shortened to 3 John People * John the Baptist (died c. AD 30), regarded as a prophet and the forerunner of Jesus Christ * John the Apostle (lived c. AD 30), one of the twelve apostles of Jesus * John the Evangelist, assigned author of the Fourth Gospel, once identified with the Apostle * John of Patmos, also known as John the Divine or John the Revelator, the author of the Book of Revelation, once identified with the Apostle * John the Presbyter, a figure either identified with or distinguished from the Apostle, the Evangelist and John of Patmos Other people with the given name Religious figures * John, father of Andrew the Apostle and Saint Peter * Pope ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Time
Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to compare the duration of events or the intervals between them, and to quantify rates of change of quantities in material reality or in the conscious experience. Time is often referred to as a fourth dimension, along with three spatial dimensions. Time has long been an important subject of study in religion, philosophy, and science, but defining it in a manner applicable to all fields without circularity has consistently eluded scholars. Nevertheless, diverse fields such as business, industry, sports, the sciences, and the performing arts all incorporate some notion of time into their respective measuring systems. 108 pages. Time in physics is operationally defined as "what a clock reads". The physical nature of time is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Induced Metric
In mathematics and theoretical physics, the induced metric is the metric tensor defined on a submanifold that is induced from the metric tensor on a manifold into which the submanifold is embedded, through the pullback. It may be determined using the following formula (using the Einstein summation convention In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...), which is the component form of the pullback operation: :g_ = \partial_a X^\mu \partial_b X^\nu g_\ Here a, b describe the indices of coordinates \xi^a of the submanifold while the functions X^\mu(\xi^a) encode the embedding into the higher-dimensional manifold whose tangent indices are denoted \mu, \nu. Example – curve on a torus Let : \Pi\colon \mathcal \to \mathbb^3,\ \tau \mapsto \begin\beginx^1&= (a+b\cos(n\cdot \t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why different observers perceive differently where and when events occur. Until the 20th century, it was assumed that the three-dimensional geometry of the universe (its spatial expression in terms of coordinates, distances, and directions) was independent of one-dimensional time. The physicist Albert Einstein helped develop the idea of spacetime as part of his theory of relativity. Prior to his pioneering work, scientists had two separate theories to explain physical phenomena: Isaac Newton's laws of physics described the motion of massive objects, while James Clerk Maxwell's electromagnetic models explained the properties of light. However, in 1905, Einstein based a work on special relativity on two postulates: * The laws of physics are invari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Surface
In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only ''non-compact'' components. Examples The only connected one-dimensional example is a circle. The sphere, torus, and the Klein bottle are all closed two-dimensional manifolds. A line is not closed because it is not compact. A closed disk is a compact two-dimensional manifold, but it is not closed because it has a boundary. Open manifolds For a connected manifold, "open" is equivalent to "without boundary and non-compact", but for a disconnected manifold, open is stronger. For instance, the disjoint union of a circle and a line is non-compact since a line is non-compact, but this is not an open manifold since the circle (one of its components) is compact. Abuse of language Most books generally define a manifold as a space that is, locally, homeomorphic to Euclidean space (along with some other technical con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
