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Causal structure, the Glossary

In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold.^[1]

64 relations: Absolute horizon, Alfred Robb, Anti-de Sitter space, Arrow of time, Big Bang, Black hole, Cartesian coordinate system, Cauchy horizon, Cauchy surface, Causal dynamical triangulation, Causal sets, Causal structure, Causality (physics), Causality conditions, Closed timelike curve, Closure (mathematics), Conformal map, Cosmic censorship hypothesis, Curvature, Curve, David Malament, De Sitter space, Determinism, Diffeomorphism, Disjoint sets, Equivalence class, Equivalence relation, Gary Gibbons, General relativity, Globally hyperbolic manifold, Gravitational singularity, Holonomy, Homeomorphism, Interior (topology), Light cone, Lorentz transformation, Malament–Hogarth spacetime, Manifold, Mathematical physics, Metric signature, Metric tensor, Minkowski space, Modern physics, Monotonic function, Null infinity, Penrose diagram, Penrose–Hawking singularity theorems, Poincaré group, Pseudo-Riemannian manifold, Rafael Sorkin, ... Expand index (14 more) »
Lorentzian manifolds

Absolute horizon

In general relativity, an absolute horizon is a boundary in spacetime, defined with respect to the external universe, inside which events cannot affect an external observer. Causal structure and absolute horizon are general relativity.

See Causal structure and Absolute horizon

Alfred Robb

Alfred Arthur Robb FRS (18 January 1873 in Belfast – 14 December 1936 in Castlereagh) was a Northern Irish physicist.

See Causal structure and Alfred Robb

Anti-de Sitter space

In mathematics and physics, n-dimensional anti-de Sitter space (AdSn) is a maximally symmetric Lorentzian manifold with constant negative scalar curvature.

See Causal structure and Anti-de Sitter space

Arrow of time

The arrow of time, also called time's arrow, is the concept positing the "one-way direction" or "asymmetry" of time.

See Causal structure and Arrow of time

Big Bang

The Big Bang is a physical theory that describes how the universe expanded from an initial state of high density and temperature.

See Causal structure and Big Bang

Black hole

A black hole is a region of spacetime where gravity is so strong that nothing, not even light and other electromagnetic waves, is capable of possessing enough energy to escape it. Causal structure and black hole are theory of relativity.

See Causal structure and Black hole

Cartesian coordinate system

In geometry, a Cartesian coordinate system in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, called coordinate lines, coordinate axes or just axes (plural of axis) of the system.

See Causal structure and Cartesian coordinate system

Cauchy horizon

In physics, a Cauchy horizon is a light-like boundary of the domain of validity of a Cauchy problem (a particular boundary value problem of the theory of partial differential equations). Causal structure and Cauchy horizon are general relativity.

See Causal structure and Cauchy horizon

Cauchy surface

In the mathematical field of Lorentzian geometry, a Cauchy surface is a certain kind of submanifold of a Lorentzian manifold. Causal structure and Cauchy surface are Lorentzian manifolds.

See Causal structure and Cauchy surface

Causal dynamical triangulation

Causal dynamical triangulation (CDT), theorized by Renate Loll, Jan Ambjørn and Jerzy Jurkiewicz, is an approach to quantum gravity that, like loop quantum gravity, is background independent.

See Causal structure and Causal dynamical triangulation

Causal sets

The causal sets program is an approach to quantum gravity. Causal structure and causal sets are theoretical physics.

See Causal structure and Causal sets

Causal structure

In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold. Causal structure and causal structure are general relativity, Lorentzian manifolds, theoretical physics and theory of relativity.

See Causal structure and Causal structure

Causality (physics)

Physical causality is a physical relationship between causes and effects.

See Causal structure and Causality (physics)

Causality conditions

In the study of Lorentzian manifold spacetimes there exists a hierarchy of causality conditions which are important in proving mathematical theorems about the global structure of such manifolds. Causal structure and causality conditions are general relativity, Lorentzian manifolds, theoretical physics and theory of relativity.

See Causal structure and Causality conditions

Closed timelike curve

In mathematical physics, a closed timelike curve (CTC) is a world line in a Lorentzian manifold, of a material particle in spacetime, that is "closed", returning to its starting point. Causal structure and closed timelike curve are Lorentzian manifolds.

See Causal structure and Closed timelike curve

Closure (mathematics)

In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset.

See Causal structure and Closure (mathematics)

Conformal map

In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths.

See Causal structure and Conformal map

Cosmic censorship hypothesis

The weak and the strong cosmic censorship hypotheses are two mathematical conjectures about the structure of gravitational singularities arising in general relativity. Causal structure and cosmic censorship hypothesis are general relativity.

See Causal structure and Cosmic censorship hypothesis

Curvature

In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane.

See Causal structure and Curvature

Curve

In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.

See Causal structure and Curve

David Malament

David B. Malament (born 21 December 1947) is an American philosopher of science, specializing in the philosophy of physics.

See Causal structure and David Malament

De Sitter space

In mathematical physics, n-dimensional de Sitter space (often denoted dSn) is a maximally symmetric Lorentzian manifold with constant positive scalar curvature.

See Causal structure and De Sitter space

Determinism

Determinism is the philosophical view that all events in the universe, including human decisions and actions, are causally inevitable.

See Causal structure and Determinism

Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds.

See Causal structure and Diffeomorphism

Disjoint sets

In set theory in mathematics and formal logic, two sets are said to be disjoint sets if they have no element in common.

See Causal structure and Disjoint sets

Equivalence class

In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes.

See Causal structure and Equivalence class

Equivalence relation

In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.

See Causal structure and Equivalence relation

Gary Gibbons

Gary William Gibbons (born 1 July 1946) is a British theoretical physicist.

See Causal structure and Gary Gibbons

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. Causal structure and general relativity are theory of relativity.

See Causal structure and General relativity

Globally hyperbolic manifold

In mathematical physics, global hyperbolicity is a certain condition on the causal structure of a spacetime manifold (that is, a Lorentzian manifold). Causal structure and Globally hyperbolic manifold are general relativity.

See Causal structure and Globally hyperbolic manifold

Gravitational singularity

A gravitational singularity, spacetime singularity or simply singularity is a condition in which gravity is predicted to be so intense that spacetime itself would break down catastrophically. Causal structure and gravitational singularity are general relativity and Lorentzian manifolds.

See Causal structure and Gravitational singularity

Holonomy

In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported.

See Causal structure and Holonomy

Homeomorphism

In mathematics and more specifically in topology, a homeomorphism (from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function.

See Causal structure and Homeomorphism

Interior (topology)

In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in.

See Causal structure and Interior (topology)

Light cone

In special and general relativity, a light cone (or "null cone") is the path that a flash of light, emanating from a single event (localized to a single point in space and a single moment in time) and traveling in all directions, would take through spacetime. Causal structure and light cone are Lorentzian manifolds and theory of relativity.

See Causal structure and Light cone

Lorentz transformation

In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former.

See Causal structure and Lorentz transformation

Malament–Hogarth spacetime

A Malament–Hogarth (M-H) spacetime, named after David B. Malament and Mark Hogarth, is a relativistic spacetime that possesses the following property: there exists a worldline \lambda and an event p such that all events along \lambda are a finite interval in the past of p, but the proper time along \lambda is infinite. Causal structure and Malament–Hogarth spacetime are general relativity.

See Causal structure and Malament–Hogarth spacetime

Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

See Causal structure and Manifold

Mathematical physics

Mathematical physics refers to the development of mathematical methods for application to problems in physics. Causal structure and mathematical physics are theoretical physics.

See Causal structure and Mathematical physics

Metric signature

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.

See Causal structure and Metric signature

Metric tensor

In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there.

See Causal structure and Metric tensor

Minkowski space

In physics, Minkowski space (or Minkowski spacetime) is the main mathematical description of spacetime in the absence of gravitation. Causal structure and Minkowski space are Lorentzian manifolds.

See Causal structure and Minkowski space

Modern physics

Modern physics is a branch of physics that developed in the early 20th century and onward or branches greatly influenced by early 20th century physics.

See Causal structure and Modern physics

Monotonic function

In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order.

See Causal structure and Monotonic function

Null infinity

In theoretical physics, null infinity is a region at the boundary of asymptotically flat spacetimes. Causal structure and null infinity are general relativity, Lorentzian manifolds and theoretical physics.

See Causal structure and Null infinity

Penrose diagram

In theoretical physics, a Penrose diagram (named after mathematical physicist Roger Penrose) is a two-dimensional diagram capturing the causal relations between different points in spacetime through a conformal treatment of infinity. Causal structure and Penrose diagram are Lorentzian manifolds.

See Causal structure and Penrose diagram

Penrose–Hawking singularity theorems

The Penrose–Hawking singularity theorems (after Roger Penrose and Stephen Hawking) are a set of results in general relativity that attempt to answer the question of when gravitation produces singularities. Causal structure and Penrose–Hawking singularity theorems are general relativity.

See Causal structure and Penrose–Hawking singularity theorems

Poincaré group

The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the isometry group of Minkowski spacetime. Causal structure and Poincaré group are theory of relativity.

See Causal structure and Poincaré group

Pseudo-Riemannian manifold

In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. Causal structure and pseudo-Riemannian manifold are Lorentzian manifolds.

See Causal structure and Pseudo-Riemannian manifold

Rafael Sorkin

Rafael Dolnick Sorkin (born c. 1945) is an American physicist.

See Causal structure and Rafael Sorkin

Raychaudhuri equation

In general relativity, the Raychaudhuri equation, or Landau–Raychaudhuri equation, is a fundamental result describing the motion of nearby bits of matter. Causal structure and Raychaudhuri equation are general relativity.

See Causal structure and Raychaudhuri equation

Reflexive relation

In mathematics, a binary relation R on a set X is reflexive if it relates every element of X to itself.

See Causal structure and Reflexive relation

Relation (mathematics)

In mathematics, a relation on a set may, or may not, hold between two given members of the set.

See Causal structure and Relation (mathematics)

Smoothness

In mathematical analysis, the smoothness of a function is a property measured by the number, called differentiability class, of continuous derivatives it has over its domain.

See Causal structure and Smoothness

Spacetime

In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Causal structure and spacetime are theoretical physics and theory of relativity.

See Causal structure and Spacetime

Stephen Hawking

Stephen William Hawking, (8 January 1942 – 14 March 2018) was an English theoretical physicist, cosmologist, and author who was director of research at the Centre for Theoretical Cosmology at the University of Cambridge.

See Causal structure and Stephen Hawking

Subset

In mathematics, a set A is a subset of a set B if all elements of A are also elements of B; B is then a superset of A. It is possible for A and B to be equal; if they are unequal, then A is a proper subset of B. The relationship of one set being a subset of another is called inclusion (or sometimes containment).

See Causal structure and Subset

Tangent space

In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions.

See Causal structure and Tangent space

Tangent vector

In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point.

See Causal structure and Tangent vector

The Large Scale Structure of Space–Time

The Large Scale Structure of Space–Time is a 1973 treatise on the theoretical physics of spacetime by the physicist Stephen Hawking and the mathematician George Ellis.

See Causal structure and The Large Scale Structure of Space–Time

Topology

Topology (from the Greek words, and) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself.

See Causal structure and Topology

Transitive relation

In mathematics, a binary relation on a set is transitive if, for all elements,, in, whenever relates to and to, then also relates to.

See Causal structure and Transitive relation

Wolfram Demonstrations Project

The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-sized) interactive programmes called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields.

See Causal structure and Wolfram Demonstrations Project

World line

The world line (or worldline) of an object is the path that an object traces in 4-dimensional spacetime. Causal structure and world line are theory of relativity.

See Causal structure and World line

References

[1] https://en.wikipedia.org/wiki/Causal_structure

Also known as Absolute future, Absolute past, Achronal, Causal curve, Causal diamond, Causal future, Causal past, Causal relation, Causal spacetime structure, Causality relation, Causality violating set, Chronological future, Chronological past, Chronological relation, Chronology violating set, Future null infinity, Future set, Horismos, Horismos relation, Indecomposable past set, Non-spacelike curve, Null curve, Past set, Proper indecomposable past set, Spacelike curve, Terminal indecomposable past set, Timelike curve.

, Raychaudhuri equation, Reflexive relation, Relation (mathematics), Smoothness, Spacetime, Stephen Hawking, Subset, Tangent space, Tangent vector, The Large Scale Structure of Space–Time, Topology, Transitive relation, Wolfram Demonstrations Project, World line.

Causal structure, the Glossary

Table of Contents

See also

Lorentzian manifolds

References