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Type of exact solution in general relativity of Einstein's field equations
relativity, a scalar field solution is an exact solution of the Einstein field equation in which the gravitational field is due entirely to the field energy
Scalar_field_solution
Assignment of numbers to points in space
physics, a scalar field is a function associating a single number to each point in a region of space – possibly physical space. The scalar may either
Scalar_field
Field theory of scalar fields
theoretical physics, scalar field theory can refer to a relativistically invariant classical or quantum theory of scalar fields. A scalar field is invariant under
Scalar_field_theory
Field-equations in general relativity
theory of a scalar field, Φ {\displaystyle \Phi } , which is the gravitational potential in joules per kilogram of the gravitational field g = − ∇ Φ {\displaystyle
Einstein_field_equations
Algebraic structure in linear algebra
based on different kinds of scalars: real numbers and complex numbers. Scalars can also be, more generally, elements of any field. Vector spaces generalize
Vector_space
Proposed theories of gravity
scalar theories because the gravitational field is a scalar. Other proposed alternatives include scalar–tensor theories that contain a scalar field in
Alternatives to general relativity
Alternatives_to_general_relativity
Line integral of the electric field
also known as the electrostatic potential or electric field potential, is a field of scalar quantities through space, often denoted with the letter
Electric_potential
Relativistic generalization of Mordehai Milgrom's MOND paradigm
on the following ingredients: A unit vector field; A dynamical scalar field; A nondynamical scalar field; A matter Lagrangian constructed using an alternate
Tensor–vector–scalar_gravity
particle physics. It emerged as one potential solution to the symmetron field, a hypothesized scalar field. List of hypothetical particles "Space Has Invisible
Symmetron
Relativistic wave equation in quantum mechanics
relativistic scalar fields play a prominent role in modern physics, the Klein–Gordon is likewise indispensable in many areas. For example, such scalar fields are
Klein–Gordon_equation
Application of Lagrangian mechanics to field theories
fields, which include scalar and vector fields as special cases. For example, if there are m {\displaystyle m} real-valued scalar fields, φ 1 , … , φ m {\displaystyle
Lagrangian_(field_theory)
In general relativity, an exact solution is a (typically closed form) solution of the Einstein field equations whose derivation does not invoke simplifying
Exact solutions in general relativity
Exact_solutions_in_general_relativity
Measure of curvature in differential geometry
In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To
Scalar_curvature
When potential energy difference depends only on displacement
traveling from one position to the other. It is a scalar field in three-space: a directionless value (scalar) that depends only on its location. A familiar
Scalar_potential
Physical quantities taking values at each point in space and time
In science, a field or field quantity is a physical quantity – represented by a scalar, vector, spinor, or tensor – that has a value for each point in
Field_(physics)
Visual representation of solutions to a differential equation
slope field (also called a direction field) is a graphical representation of the solutions to a first-order differential equation of a scalar function
Slope_field
Assignment of a vector to each point in a subset of Euclidean space
} Given vector fields V, W defined on S and a smooth function f defined on S, the operations of scalar multiplication and vector addition
Vector_field
Theoretical framework in physics
In the case of the real scalar field, the existence of these operators was a consequence of the decomposition of solutions of the classical equations
Quantum_field_theory
Conjectured dark matter in cosmology
In astrophysics and cosmology scalar field dark matter is a classical, minimally coupled, scalar field postulated to account for the inferred dark matter
Scalar_field_dark_matter
Elementary particle involved with rest mass
excitation of the Higgs field, one of the fields in particle physics theory. In the Standard Model, the Higgs particle is a massive scalar boson that couples
Higgs_boson
Magnetic analog of electric potential valid outside materials
surfaces, piecemeal solutions can be stitched together to provide a description of the magnetic field at all points in space. The scalar potential is a useful
Magnetic_scalar_potential
Integration over a non-flat region in 3D space
integrate over this surface a scalar field (that is, a function of position which returns a scalar as a value), or a vector field (that is, a function which
Surface_integral
Field theory in physics that aims to unify the fundamental forces and particles
fields. Gauge boson fields also have quanta, such as photons for the electromagnetic field. The Standard Model has a unique fundamental scalar field,
Unified_field_theory
Quadratic scalar invariant
context of applications to general relativity, the Kretschmann scalar is a quadratic scalar invariant. It was introduced by Erich Kretschmann. The Kretschmann
Kretschmann_scalar
Branch of mathematics
e., nonzero if the scalars belong to a field). Cramer's rule is a closed-form expression, in terms of determinants, of the solution of a system of n linear
Linear_algebra
Equations describing classical electromagnetism
represent scalar quantities, unless otherwise indicated. The equations introduce the electric field, E, a vector field, and the magnetic field, B, a pseudovector
Maxwell's_equations
Algebraic structure with addition, multiplication, and division
purposes, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Number fields, the siblings
Field_(mathematics)
Quantum field theory with four-point interactions
In quantum field theory, a quartic interaction or φ4 theory is a type of self-interaction of a scalar field. Other types of quartic interactions may be
Quartic_interaction
Physical theory describing classical fields
{Q}{r^{2}}}{\hat {\mathbf {r} }}\,.} The electric field is conservative, and hence is given by the gradient of a scalar potential, V(r) E ( r ) = − ∇ V ( r ) .
Classical_field_theory
Proposed theory of gravitation
of a scalar–tensor theory, a gravitational theory in which the gravitational interaction is mediated by a scalar field as well as the tensor field of general
Brans–Dicke_theory
Theory of gravitation as curved spacetime
Ricci scalar, take on infinite values. Well-known examples of spacetimes with future singularities—where worldlines end—are the Schwarzschild solution, which
General_relativity
Differential equation for the description of waves or standing wave
an electrical field, magnetic field, and magnetic vector potential and elastic waves. By comparison with vector wave equations, the scalar wave equation
Wave_equation
Broad concept generalizing scalars in mathematics and physics
single scalar quantity. The term may also be used to refer to elements of vector spaces, that can be added together and multiplied ("scaled") by scalars. In
Vector (mathematics and physics)
Vector_(mathematics_and_physics)
Electric and magnetic fields produced by moving charged objects
An electromagnetic field (also EM field) is a physical field, varying in space and time, that represents the electric and magnetic influences generated
Electromagnetic_field
Electrodynamics of spin 0 particles
In theoretical physics, scalar electrodynamics is a theory of a U(1) gauge field coupled to a charged spin 0 scalar field that takes the place of the
Scalar_electrodynamics
brought together. A quadrupole field is shown in the lower figure on the right. Spherical harmonics can represent any scalar field (function of position) that
Earth's_magnetic_field
Theorem in geometric topology
Sometimes, an otherwise complicated operation reduces to multiplication by a scalar (a number). Such numbers are called eigenvalues of that operation. Eigenvalues
Poincaré_conjecture
Field with an imaginary mass
a field—usually a scalar field—whose squared mass is negative, and is used to describe spontaneous symmetry breaking: The existence of such a field implies
Tachyonic_field
Features that do not change if length or energy scales are multiplied by a common factor
scale-invariant classical field theory is the massless scalar field (note that the name scalar is unrelated to scale invariance). The scalar field, φ(x, t) is a function
Scale_invariance
Millennium Prize Problem
dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined
Navier–Stokes existence and smoothness
Navier–Stokes_existence_and_smoothness
Hypothetical topological feature of spacetime
The drainhole is a solution manifold of Einstein's field equations for a vacuum spacetime, modified by inclusion of a scalar field minimally coupled to
Wormhole
Scalar theories of gravitation are field theories of gravitation in which the gravitational field is described using a scalar field, which is required
Scalar theories of gravitation
Scalar_theories_of_gravitation
Relativistic vector field
vector function from which the electromagnetic field can be derived. It combines both an electric scalar potential and a magnetic vector potential into
Electromagnetic four-potential
Electromagnetic_four-potential
Internal magnetic field generated by a magnet
M is the magnetisation. The general solution of the first equation can be expressed as the gradient of a scalar potential U(r): Inside the magnetic body
Demagnetizing_field
Certain vector fields are the sum of an irrotational and a solenoidal vector field
a scalar field is a scalar field, and the Fourier transform of a vector field is a vector field of same dimension. Now consider the following scalar and
Helmholtz_decomposition
Process in quantum mechanical theories
motion for a free scalar field, but also the quantum equation for a scalar particle wave-function. This meant that quantizing a field appeared to be similar
Canonical_quantization
Generating function for quantum correlation functions
Feynman diagrams. In a d {\displaystyle d} -dimensional field theory with a real scalar field ϕ {\displaystyle \phi } and action S [ ϕ ] {\displaystyle
Partition function (quantum field theory)
Partition_function_(quantum_field_theory)
Approximation in plasma physics
parallel to the magnetic field. If the current density is identically zero, then the magnetic field is the gradient of a magnetic scalar potential ϕ {\displaystyle
Force-free_magnetic_field
Center of gravity of a material body
is the (scalar) weight of the ith particle and W is the (scalar) total weight of all the particles. This equation always has a unique solution, and in
Centers of gravity in non-uniform fields
Centers_of_gravity_in_non-uniform_fields
Four-dimensional number system
.} Looking at the scalar and vector parts in this equation separately yields two equations, which when solved gives the solutions q = ( r , v → ) = ±
Quaternion
Linear perturbations to solutions of nonlinear Einstein field equations
represents the solutions of the equation. Although succinct when written out using Einstein notation, hidden within the Ricci tensor and Ricci scalar are exceptionally
Linearized_gravity
Modified theory of gravity developed by John Moffat
postulates the existence of a vector field, while elevating the three constants of the theory to scalar fields. In the weak-field approximation, STVG produces
Scalar–tensor–vector_gravity
Predecessor to the theory of relativity
scalar theories of gravitation (while Einstein explored tensor theories). Nordström's first attempt to propose a suitable relativistic scalar field equation
Nordström's theory of gravitation
Nordström's_theory_of_gravitation
Modification of general relativity
theories that add tensor fields, under the name Bimetric gravity or both scalar and vector fields can be added, as in Tensor–vector–scalar gravity. The name
Einstein-aether_theory
Russian theoretical physicist
"Inflationary Attractor". This is the unique solution of the gravitational equations with massive scalar field which solution has maximal degree of inflation and
Vladimir_Belinski
Gravity theories that are invariant under Weyl transformations
is the scalar 4-derivative wave equation: ◻ 2 Φ = 0 {\displaystyle \operatorname {\Box } ^{2}\Phi =0} The solution for this in a central field of force
Conformal_gravity
Second-order partial differential equation
real-valued function. The Laplace operator therefore maps a scalar function to another scalar function. If the right-hand side is specified as a given function
Laplace's_equation
Transport of a substance by bulk motion
{\displaystyle \mathbf {a} } is a vector field instead of the scalar field ψ {\displaystyle \psi } . Solutions to the advection equation can be approximated
Advection
Mathematical concept
several scalarizations. The solution to each scalarization yields a Pareto optimal solution, whether locally or globally. The scalarizations of the NBI
Multi-objective_optimization
of a scalar field, this monopole is an example of an 't Hooft–Polyakov monopole and should not be confused with the singular monopole solutions to Maxwell's
Monopole_(mathematics)
Unified field theory
English-language Kaluza field equations, including the scalar field, were provided by Williams. To obtain the 5D Kaluza–Klein–Einstein field equations, the 5D
Kaluza–Klein_theory
Mathematical function, in linear algebra
vector spaces, which respects the basic operations of vector addition and scalar multiplication. A standard example of a linear map is an m × n {\displaystyle
Linear_map
Black holes are characterized only by mass, charge, and spin
carry a finite scalar charge which might be a result of interaction with cosmological scalar fields such as the inflaton. The solution is stable and does
No-hair_theorem
Hypothetical gravitational particle
is one of the possible solutions to the present non-observation of various signals coming from massless or light scalar fields that are generically predicted
Pressuron
Concept in mathematical physics
tensor is null. Such a spacetime can be interpreted as an exact solution of Einstein's field equation, in which the only mass–energy present in the spacetime
Null_dust_solution
Chinese-American mathematician (born 1949)
have constant scalar curvature. The key tool in their analysis was an extension of Hermann Weyl's differential identity used in the solution of the Weyl
Shing-Tung_Yau
Theory of gravity
permeability of flat spacetime, εo and μo respectively by multiplying them by a scalar function, K: ε 0 → ε = K ε 0 ; μ 0 → μ = K μ 0 {\displaystyle \varepsilon
Polarizable_vacuum
Type of artificial neural network
spatial coordinates, time) to continuous outputs (i.e., scalars, vectors, etc.). This makes neural fields not only discretization independent, but also easily
Neural_field
Ternary operation on vectors
different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product. The scalar triple product (also called
Triple_product
Configurations of a system that do or do not satisfy classical equations of motion
opposing flows of positive energy. An example comes from considering a scalar field in D-dimensional Minkowski space. Consider a Lagrangian density given
On_shell_and_off_shell
Equation in physics
vector Laplacian, not Laplacian applied on scalar functions.) gives the wave equation for the electric field E: 1 c 2 ∂ 2 E ∂ t 2 − ∇ 2 E = − ( 1 ε 0 ∇
Inhomogeneous electromagnetic wave equation
Inhomogeneous_electromagnetic_wave_equation
Method to solve scalar wave equation
value of the solution of the homogeneous scalar wave equation at an arbitrary point P in terms of the values of the solution and the solution's first-order
Kirchhoff_integral_theorem
Theory of gravity
different function, f, of the Ricci scalar, R. The simplest case is just the function being equal to the scalar; this is general relativity. As a consequence
F(R)_gravity
Theory proposed by Roger Penrose
Yang–Mills field equations. Witten showed that a further extension, within the framework of super Yang–Mills theory, including fermionic and scalar fields, gave
Twistor_theory
Physics problem related to laws of motion and gravity
second-order vector differential equations are equivalent to 18 first order scalar differential equations."[better source needed] As June Barrow-Green notes
Three-body_problem
Concepts from linear algebra
often solved using finite element analysis, but neatly generalize the solution to scalar-valued vibration problems. In mechanics, the eigenvectors of the moment
Eigenvalues_and_eigenvectors
Approximation of physical behavior
statistical inference. In mean field theory, the mean field appearing in the single-site problem is a time-independent scalar or vector quantity. However
Mean-field_theory
Class of exact solutions to Einstein's field equations
relativity, a dust solution is a fluid solution, a type of exact solution of the Einstein field equation, in which the gravitational field is produced entirely
Dust_solution
Property of space that quantifies the magnetic influence at a given location
Magnetic scalar potential SI electromagnetism units – common units used in electromagnetism Orders of magnitude (magnetic field) – list of magnetic field sources
Magnetic_field
Solution of Einstein field equations
exact solution of Einstein's field equations that describes a black hole or massive object immersed in an expanding cosmological spacetime. The solution was
McVittie_metric
Method for numerically solving time-dependent incompressible fluid-flow problems
_{\text{sol}}=0)} This is a Poisson equation for the scalar function ϕ {\displaystyle \,\phi } . If the vector field u {\displaystyle \mathbf {u} } is known, the
Projection method (fluid dynamics)
Projection_method_(fluid_dynamics)
Theory extending Einstein gravity
an additional scalar boson and the massless graviton, ensuring that general relativity is recovered at low energies. The additional scalar particle appears
Quadratic_gravity
Formulations of electromagnetism
potential, A, for the magnetic field. The electric potential is a scalar field, while the magnetic potential is a vector field. This is why sometimes the
Mathematical descriptions of the electromagnetic field
Mathematical_descriptions_of_the_electromagnetic_field
Theories of higher-dimensional general relativity
conformal field theory on its boundary. In this context, black hole solutions in higher dimensions correspond to thermal states in the dual quantum field theory
Higher-dimensional Einstein gravity
Higher-dimensional_Einstein_gravity
Type of unphysical field in quantum field theory which provides mathematical consistency
for regular complex scalar fields, and the second term describes the interaction with the gauge fields as well as the Higgs field. Note that in abelian
Faddeev–Popov_ghost
Exact gravitational-wave solution to Einstein's field equations
A.R.; Tiwari, R.N. (1972). "A class of exact solutions for coupled electromagnetic and scalar fields for einstein-rosen metric. I". Annals of Physics
Einstein–Rosen_metric
variable for their measurements, i.e., if they have a scalar time series. If one only has a scalar time series, they need to use the method of time delay
Vector_field_reconstruction
Calculation of electric field generated by current distribution
electric scalar potential and the Lorenz gauge, we can now write what is called the electric field integral equation (EFIE), relating the electric field E to
Electric-field integral equation
Electric-field_integral_equation
Mechanism that explains the generation of mass for gauge bosons
is scalar, but its phase is capable of defining a gauge in gauge based field theories. To do this, the field must be charged. A charged scalar field must
Higgs_mechanism
Solution to the Einstein field equations
known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass
Schwarzschild_metric
Branch of physics about magnetism in systems with steady electric currents
has the general solution H = − ∇ Φ M , {\displaystyle \mathbf {H} =-\nabla \Phi _{M},} where Φ M {\displaystyle \Phi _{M}} is a scalar potential. Substituting
Magnetostatics
Hypothetical elementary particle that mediates gravity
is a quantum of gravitational wave energy. There is no complete quantum field theory of gravitons due to the unsolved mathematical problem of renormalization
Graviton
Dirac equation as a true field equation. The quantized "Dirac field" or "electron field" was introduced, with the "solutions of negative energy" pointing
History of quantum field theory
History_of_quantum_field_theory
Black hole solution in 3+1 spacetime with a scalar field
black hole is a black hole solution for (3+1)-dimensional gravity with a minimally coupled self-interacting scalar field. It is named after Cristián
MTZ_black_hole
Now-discarded theory in particle physics
strong CP problem. Motivated as a solution to this problem, Peccei–Quinn (PQ) theory introduces a new complex scalar field φ {\displaystyle \varphi } in addition
Peccei–Quinn_theory
Extension of the scalar spherical harmonics for use with vector fields
spherical harmonics (VSH) are an extension of the scalar spherical harmonics for use with vector fields. The components of the VSH are complex-valued functions
Vector_spherical_harmonics
Elliptic partial differential equation
the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with the potential field known
Poisson's_equation
Wave equations respecting special and general relativity
operator into the relativistic energy–momentum relation: The solutions to (1) are scalar fields. The KG equation is undesirable due to its prediction of negative
Relativistic_wave_equations
Mathematical function that preserves angles
transformation is conformal whenever the Jacobian at each point is a positive scalar times a rotation matrix (orthogonal with determinant one). Some authors
Conformal_map
Scattering of an electromagnetic plane wave by a sphere
electromagnetism, the Mie solution to Maxwell's equations (also known as the Lorenz–Mie solution, the Lorenz–Mie–Debye solution or Mie scattering) describes
Mie_scattering
SCALAR FIELD-SOLUTION
SCALAR FIELD-SOLUTION
Male
English
Variant spelling of English unisex Schuyler, SKYLAR means "protection, shelter."Â
Girl/Female
African, American, Australian, British, Chinese, Dutch, English, Scandinavian
Sheltering; Scholar; Eternal Life; Strength; Love and Beauty; Learned One; Protection; Shelter
Boy/Male
Shakespearean
Henry VI, Part 2' Lord Scales.
Boy/Male
English
Gathering field; meeting field.
Male
Hebrew
(ש×ָרָר) Hebrew name SHARAR means "enemy" or "to be firm, hard." In the bible, this is the name of the father of Ahiam.
Boy/Male
Muslim
Leader
Boy/Male
Australian, British, English
A Field
Biblical
(or Aijeleth Shahar) the land of the morning
Boy/Male
English
In the field.
Surname or Lastname
English
English : topographic name for someone who lived on land which had been cleared of forest, but not brought into cultivation, from Old English feld ‘pasture’, ‘open country’, as opposed on the one hand to æcer ‘cultivated soil’, ‘enclosed land’ (see Acker) and on the other to weald ‘wooded land’, ‘forest’ (see Wald).Possibly also Scottish or Irish : reduced form of McField (see McPhail).Jewish (American) : Americanized and shortened form of any of the many Jewish surnames containing Feld.
Boy/Male
Dutch
Scholar.
Female
Hebrew
(ש×ַחַר) Variant spelling of Hebrew unisex Shachar, SHAHAR means "dawn" or "morning."
Male
English
 Anglicized form of Hebrew Shelach, SALAH means "a missile, weapon." In the bible, this is the name of a son of Arphaxad. Compare with another form of Salah.
Male
Irish
Irish name SCANLAN means "scandal."
Surname or Lastname
English
English : variant of Field.
Male
Norse
Variant form of Old Norse Ãsvaldr, ÓSVALDR means "divine power" or "divine ruler."
Female
Hebrew
(סְמָדַר) Variant form of Hebrew Semadar, SMADAR means "bud" or "blossom."
Boy/Male
American, British, Chinese, Dutch, English, Scandinavian
Scholar; Shield; Protection; Learned One; Shelter
Surname or Lastname
English
English : topographic name from Middle English feldes, plural or possessive of feld ‘open country’. This name is also found as a translation of equivalent names in other languages, in particular French Deschamps, Duchamp.
Boy/Male
Muslim/Islamic
Leader
SCALAR FIELD-SOLUTION
SCALAR FIELD-SOLUTION
SCALAR FIELD-SOLUTION
SCALAR FIELD-SOLUTION
SCALAR FIELD-SOLUTION
SCALAR FIELD-SOLUTION
SCALAR FIELD-SOLUTION