|
GHP Formalism
The GHP formalism (or Geroch–Held–Penrose formalism) is a technique used in the mathematics of general relativity that involves singling out a pair of null directions at each point of spacetime. It is a rewriting of the Newman–Penrose formalism which respects the covariance of Lorentz transformations preserving two null directions. This is desirable for Petrov Type D spacetimes, where the pair is made up of degenerate principal null directions, and spatial surfaces, where the null vectors are the natural null orthogonal vectors to the surface. The New Covariance The GHP formalism notices that given a spin-frame (o^A,\iota^A) with o_A \iota^A = 1, the complex rescaling (o^A,\iota^A)\rightarrow (\lambda o^A, \lambda^ \iota^A ) does not change normalization. The magnitude of this transformation is a boost, and the phase tells one how much to rotate. A quantity of weight (p,q) is one that transforms like \eta \rightarrow \lambda^p \bar^q \eta. One then defines derivative operator ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Of General Relativity
When studying and formulating Albert Einstein's theory of general relativity, various mathematical structures and techniques are utilized. The main tools used in this geometrical theory of gravitation are tensor fields defined on a Lorentzian manifold representing spacetime. This article is a general description of the mathematics of general relativity. ''Note: General relativity articles using tensors will use the abstract index notation''. Tensors The principle of general covariance was one of the central principles in the development of general relativity. It states that the laws of physics should take the same mathematical form in all reference frames. The term 'general covariance' was used in the early formulation of general relativity, but the principle is now often referred to as ' diffeomorphism covariance'. Diffeomorphism covariance is not the defining feature of general relativity, .html" ;"title="/sup>">/sup> and controversies remain regarding its present status in ... [...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 invariant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newman–Penrose Formalism
The Newman–Penrose (NP) formalism The original paper by Newman and Penrose, which introduces the formalism, and uses it to derive example results.Ezra T Newman, Roger Penrose. ''Errata: An Approach to Gravitational Radiation by a Method of Spin Coefficients''. Journal of Mathematical Physics, 1963, 4(7): 998. is a set of notation developed by Ezra T. Newman and Roger Penrose for general relativity (GR). Their notation is an effort to treat general relativity in terms of spinor notation, which introduces complex forms of the usual variables used in GR. The NP formalism is itself a special case of the tetrad formalism, where the tensors of the theory are projected onto a complete vector basis at each point in spacetime. Usually this vector basis is chosen to reflect some symmetry of the spacetime, leading to simplified expressions for physical observables. In the case of the NP formalism, the vector basis chosen is a null tetrad: a set of four null vectors—two real, and a com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General relativity generalizes special relativity and refines Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time or four-dimensional spacetime. In particular, the ' is directly related to the energy and momentum of whatever matter and radiation are present. The relation is specified by the Einstein field equations, a system of second order partial differential equations. Newton's law of universal gravitation, which describes classical gravity, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass distributions. Some predictions of general relativity, however, are beyond Newton's law of universal gravitat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NP Formalism
NP may refer to: Arts and entertainment * ''NP'' (novel), by Japanese author Banana Yoshimoto Organizations * Nashua-Plainfield Community School District, Iowa, United States * National Party (other), various political parties * Ngee Ann Polytechnic, Singapore * Nigeria Police Force * Northern Pacific Railway (AAR reporting mark NP) * November Project, free, open-to-the-public exercise group Places * NP postcode area, Newport, Wales, UK * Nepal (ISO 3166-1 alpha-2 country code NP) ** .np, the country code top level domain (ccTLD) for Nepal Science, technology and mathematics Biology and medicine * Nucleoside phosphorylase, an enzyme * Nurse practitioner * Kallikrein 8, an enzyme * Neptunium, symbol Np, a chemical element * Nosocomial pneumonia * Natriuretic peptide Mathematics and computer science * NP (complexity), Nondeterministic Polynomial, a computational complexity class ** NP-complete, a class of decision problems ** NP-hard, a class of problems in computation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Geroch
Robert Geroch (born 1 June 1942 in Akron, Ohio) is an American theoretical physicist and professor at the University of Chicago. He has worked prominently on general relativity and mathematical physics and has promoted the use of category theory in mathematics and physics. He was the Ph.D. supervisor for Abhay Ashtekar, Basilis Xanthopoulos and Gary Horowitz. He also proved an important theorem in spin geometry. Education Geroch obtained his Ph.D. degree from Princeton University in 1967 under the supervision of John Archibald Wheeler, with a thesis on ''Singularities in the spacetime of general relativity: their definition, existence, and local characterization''. Writings Chapters *Geroch R.P. (1977) "Asymptotic Structure of Space-Time", p.1--105 in: Esposito F.P., Witten L. (eds) Asymptotic Structure of Space-Time. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-2343-3_1 *Horowitz, G.T and Geroch, R.P. (1979) "Global structure of spacetimes", p.212--293. In Hawki ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roger Penrose
Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, philosopher of science and Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics in the University of Oxford, an emeritus fellow of Wadham College, Oxford, and an honorary fellow of St John's College, Cambridge and University College London. Penrose has contributed to the mathematical physics of general relativity and cosmology. He has received several prizes and awards, including the 1988 Wolf Prize in Physics, which he shared with Stephen Hawking for the Penrose–Hawking singularity theorems, and one half of the 2020 Nobel Prize in Physics "for the discovery that black hole formation is a robust prediction of the general theory of relativity". He is regarded as one of the greatest living physicists, mathematicians and scientists, and is particularly noted for the breadth and depth of his work in both natural and formal sciences. Early life and education Bor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.svg "Click for more on -> Roger Penrose")