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In
relativistic physics In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR). It provides a non-quantum mechanical description of a system of particles, or of a fluid, in cases where the velocities of ...
, Lorentz symmetry or Lorentz invariance, named after the Dutch physicist Hendrik Lorentz, is an equivalence of observation or observational symmetry due to special relativity implying that the laws of physics stay the same for all observers that are moving with respect to one another within an inertial frame. It has also been described as "the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space". Lorentz covariance, a related concept, is a property of the underlying spacetime manifold. Lorentz covariance has two distinct, but closely related meanings: # A
physical quantity A physical quantity is a physical property of a material or system that can be quantified by measurement. A physical quantity can be expressed as a ''value'', which is the algebraic multiplication of a ' Numerical value ' and a ' Unit '. For examp ...
is said to be Lorentz covariant if it transforms under a given representation of the
Lorentz group In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physic ...
. According to the representation theory of the Lorentz group, these quantities are built out of
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such a ...
s, four-vectors,
four-tensor In physics, specifically for special relativity and general relativity, a four-tensor is an abbreviation for a tensor in a four-dimensional spacetime.Lambourne, Robert J A. Relativity, Gravitation and Cosmology. Cambridge University Press. 2010. ...
s, and
spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
s. In particular, a Lorentz covariant scalar (e.g., the space-time interval) remains the same under
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. The respective inverse transformation i ...
s and is said to be a ''Lorentz invariant'' (i.e., they transform under the trivial representation). # An equation is said to be Lorentz covariant if it can be written in terms of Lorentz covariant quantities (confusingly, some use the term ''invariant'' here). The key property of such equations is that if they hold in one inertial frame, then they hold in any inertial frame; this follows from the result that if all the components of a tensor vanish in one frame, they vanish in every frame. This condition is a requirement according to the principle of relativity; i.e., all non- gravitational laws must make the same predictions for identical experiments taking place at the same spacetime event in two different
inertial frames of reference In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleratio ...
. On
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
s, the words ''covariant'' and ''contravariant'' refer to how objects transform under general coordinate transformations. Both covariant and contravariant four-vectors can be Lorentz covariant quantities. Local Lorentz covariance, which follows from general relativity, refers to Lorentz covariance applying only ''locally'' in an infinitesimal region of spacetime at every point. There is a generalization of this concept to cover Poincaré covariance and Poincaré invariance.


Examples

In general, the (transformational) nature of a Lorentz tensor can be identified by its
tensor order In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
, which is the number of free indices it has. No indices implies it is a scalar, one implies that it is a vector, etc. Some tensors with a physical interpretation are listed below. The
sign convention In physics, a sign convention is a choice of the physical significance of signs (plus or minus) for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly describ ...
of the Minkowski metric is used throughout the article.


Scalars

; Spacetime interval:\Delta s^2=\Delta x^a \Delta x^b \eta_=c^2 \Delta t^2 - \Delta x^2 - \Delta y^2 - \Delta z^2 ;
Proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. It is thus independent of coordinates, and is a Lorentz scalar. The proper time interval b ...
(for
timelike 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 differen ...
intervals):\Delta \tau = \sqrt,\, \Delta s^2 > 0 ;
Proper distance Proper length or rest length is the length of an object in the object's rest frame. The measurement of lengths is more complicated in the theory of relativity than in classical mechanics. In classical mechanics, lengths are measured based on t ...
(for
spacelike 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 differen ...
intervals):L = \sqrt,\, \Delta s^2 < 0 ; Mass:m_0^2 c^2 = P^a P^b \eta_= \frac - p_x^2 - p_y^2 - p_z^2 ;Electromagnetism invariants:\begin F_ F^ &= \ 2 \left( B^2 - \frac \right) \\ G_ F^ &= \frac\epsilon_F^ F^ = - \frac \left( \vec \cdot \vec \right) \end ;
D'Alembertian In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box: \Box), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator (''cf''. nabla symbol) is the Laplace operator of ...
/wave operator:\Box = \eta^\partial_\mu \partial_\nu = \frac\frac - \frac - \frac - \frac


Four-vectors

; 4-displacement: \Delta X^a = \left(c\Delta t, \Delta\vec\right) = (c\Delta t, \Delta x, \Delta y, \Delta z) ;
4-position In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
: X^a = \left(ct, \vec\right) = (ct, x, y, z) ;
4-gradient In differential geometry, the four-gradient (or 4-gradient) \boldsymbol is the four-vector analogue of the gradient \vec from vector calculus. In special relativity and in quantum mechanics, the four-gradient is used to define the properties and re ...
: which is the 4D partial derivative: \partial^a = \left(\frac, -\vec\right) = \left(\frac\frac, -\frac, -\frac, -\frac \right) ;
4-velocity In physics, in particular in special relativity and general relativity, a four-velocity is a four-vector in four-dimensional spacetimeTechnically, the four-vector should be thought of as residing in the tangent space of a point in spacetime, ...
: U^a = \gamma\left(c, \vec\right) = \gamma \left(c, \frac, \frac, \frac\right) where U^a = \frac ;
4-momentum In special relativity, four-momentum (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is ...
: P^a = \left(\gamma mc, \gamma m\vec\right) = \left(\frac, \vec\right) = \left(\frac, p_x, p_y, p_z\right) where P^a = m U^a and m is the rest mass. ;
4-current In special and general relativity, the four-current (technically the four-current density) is the four-dimensional analogue of the electric current density. Also known as vector current, it is used in the geometric context of ''four-dimensional spa ...
: J^a = \left(c\rho, \vec\right) = \left(c\rho, j_x, j_y, j_z\right) where J^a = \rho_o U^a ; 4-potential: A^a = \left(\frac, \vec\right)= \left(\frac, A_x, A_y, A_z\right)


Four-tensors

;
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 & ...
:\delta^a_b = \begin 1 & \mbox a = b, \\ 0 & \mbox a \ne b. \end ; Minkowski metric (the metric of flat space according to general relativity):\eta_ = \eta^ = \begin 1 & \mbox a = b = 0, \\ -1 & \mboxa = b = 1, 2, 3, \\ 0 & \mbox a \ne b. \end ; Electromagnetic field tensor (using a metric signature of + − − −):F_ = \begin 0 & \fracE_x & \fracE_y & \fracE_z \\ -\fracE_x & 0 & -B_z & B_y \\ -\fracE_y & B_z & 0 & -B_x \\ -\fracE_z & -B_y & B_x & 0 \end ;
Dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...
electromagnetic field tensor:G_ = \frac\epsilon_F^ = \begin 0 & B_x & B_y & B_z \\ -B_x & 0 & \fracE_z & -\fracE_y \\ -B_y & -\fracE_z & 0 & \fracE_x \\ -B_z & \fracE_y & -\fracE_x & 0 \end


Lorentz violating models

In standard field theory, there are very strict and severe constraints on marginal and relevant Lorentz violating operators within both QED and the Standard Model. Irrelevant Lorentz violating operators may be suppressed by a high cutoff scale, but they typically induce marginal and relevant Lorentz violating operators via radiative corrections. So, we also have very strict and severe constraints on irrelevant Lorentz violating operators. Since some approaches to
quantum gravity Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...
lead to violations of Lorentz invariance, these studies are part of phenomenological quantum gravity. Lorentz violations are allowed in string theory, supersymmetry and Hořava–Lifshitz gravity. Lorentz violating models typically fall into four classes: * The laws of physics are exactly Lorentz covariant but this symmetry is
spontaneously broken Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or the ...
. In special relativistic theories, this leads to
phonon In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phonon is an excited state in the quantum mechanical ...
s, which are the Goldstone bosons. The phonons travel at ''less'' than the speed of light. * Similar to the approximate Lorentz symmetry of phonons in a lattice (where the speed of sound plays the role of the critical speed), the Lorentz symmetry of special relativity (with the speed of light as the critical speed in vacuum) is only a low-energy limit of the laws of physics, which involve new phenomena at some fundamental scale. Bare conventional "elementary" particles are not point-like field-theoretical objects at very small distance scales, and a nonzero fundamental length must be taken into account. Lorentz symmetry violation is governed by an energy-dependent parameter which tends to zero as momentum decreases. Such patterns require the existence of a privileged local inertial frame (the "vacuum rest frame"). They can be tested, at least partially, by ultra-high energy cosmic ray experiments like the
Pierre Auger Observatory The Pierre Auger Observatory is an international cosmic ray observatory in Argentina designed to detect ultra-high-energy cosmic rays: sub-atomic particles traveling nearly at the speed of light and each with energies beyond 1018  eV. In Ear ...
. * The laws of physics are symmetric under a
deformation Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies. * Defo ...
of the Lorentz or more generally, the
Poincaré group The Poincaré group, named after Henri Poincaré (1906), was first defined by Hermann Minkowski (1908) as the group of Minkowski spacetime isometries. It is a ten-dimensional non-abelian Lie group that is of importance as a model in our und ...
, and this deformed symmetry is exact and unbroken. This deformed symmetry is also typically a
quantum group In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebr ...
symmetry, which is a generalization of a group symmetry. Deformed special relativity is an example of this class of models. The deformation is scale dependent, meaning that at length scales much larger than the Planck scale, the symmetry looks pretty much like the Poincaré group. Ultra-high energy cosmic ray experiments cannot test such models. * Very special relativity forms a class of its own; if charge-parity (CP) is an exact symmetry, a subgroup of the Lorentz group is sufficient to give us all the standard predictions. This is, however, not the case. Models belonging to the first two classes can be consistent with experiment if Lorentz breaking happens at Planck scale or beyond it, or even before it in suitable
preon In particle physics, preons are point particles, conceived of as sub-components of quarks and leptons. The word was coined by Jogesh Pati and Abdus Salam, in 1974. Interest in preon models peaked in the 1980s but has slowed, as the Standard Mo ...
ic models, and if Lorentz symmetry violation is governed by a suitable energy-dependent parameter. One then has a class of models which deviate from Poincaré symmetry near the Planck scale but still flows towards an exact Poincaré group at very large length scales. This is also true for the third class, which is furthermore protected from radiative corrections as one still has an exact (quantum) symmetry. Even though there is no evidence of the violation of Lorentz invariance, several experimental searches for such violations have been performed during recent years. A detailed summary of the results of these searches is given in the Data Tables for Lorentz and CPT Violation. Lorentz invariance is also violated in QFT assuming non-zero temperature. There is also growing evidence of Lorentz violation in
Weyl semimetal Weyl fermions are massless chiral fermions embodying the mathematical concept of a Weyl spinor. Weyl spinors in turn play an important role in quantum field theory and the Standard Model, where they are a building block for fermions in quantum f ...
s and Dirac semimetals.


See also

*
4-vector In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
* Antimatter tests of Lorentz violation * Fock–Lorentz symmetry *
General covariance In theoretical physics, general covariance, also known as diffeomorphism covariance or general invariance, consists of the invariance of the ''form'' of physical laws under arbitrary differentiable coordinate transformations. The essential idea ...
* Lorentz invariance in loop quantum gravity *
Lorentz-violating electrodynamics Searches for Lorentz violation involving photons provide one possible test of relativity. Examples range from modern versions of the classic Michelson–Morley experiment that utilize highly stable electromagnetic resonant cavities to searches for ...
*
Lorentz-violating neutrino oscillations Lorentz-violating neutrino oscillation refers to the quantum phenomenon of neutrino oscillations described in a framework that allows the breakdown of Lorentz invariance. Today, neutrino oscillation or change of one type of neutrino into another i ...
* Planck length * Symmetry in physics


Notes


References

* Background information on Lorentz and CPT violation: http://www.physics.indiana.edu/~kostelec/faq.html * * * * * {{cite journal, doi=10.1103/PhysRevD.67.124011, title=Threshold effects and Planck scale Lorentz violation: Combined constraints from high energy astrophysics, year=2003, last1=Jacobson, first1=T., last2=Liberati, first2=S., last3=Mattingly, first3=D., journal=Physical Review D, volume=67, issue=12, pages=124011, arxiv = hep-ph/0209264 , bibcode = 2003PhRvD..67l4011J , s2cid=119452240 Special relativity Symmetry Hendrik Lorentz