In standard physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, Lorentz symmetry is "the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space". Lorentz covariance is a related concept, covariance being a measure of how much two variables change together.

Lorentz covariance (from Hendrik Lorentz

Hendrik Lorentz

Hendrik Antoon Lorentz was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect...

) is a key property of spacetime

Spacetime

In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...

 that follows from the special theory of relativity. Lorentz covariance has two distinct, but closely related meanings:

1. A physical quantity
   Physical quantity
   A physical quantity is a physical property of a phenomenon, body, or substance, that can be quantified by measurement.-Definition of a physical quantity:Formally, the International Vocabulary of Metrology, 3rd edition defines quantity as:...
   
    is said to be Lorentz covariant if it transforms under a given representation
   Group representation
   In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
   
    of the Lorentz group
   Lorentz group
   In physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting for all physical phenomena...
   
   . According to the representation theory of the Lorentz group, these quantities are built out of scalar
   Scalar (physics)
   In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or space-time translations . This is in contrast to a vector...
   
   s, four-vector
   Four-vector
   In the theory of relativity, a four-vector is a vector in a four-dimensional real vector space, called Minkowski space. It differs from a vector in that it can be transformed by Lorentz transformations. The usage of the four-vector name tacitly assumes that its components refer to a standard basis...
   
   s, four-tensors, and spinor
   Spinor
   In mathematics and physics, in particular in the theory of the orthogonal groups , spinors are elements of a complex vector space introduced to expand the notion of spatial vector. Unlike tensors, the space of spinors cannot be built up in a unique and natural way from spatial vectors...
   
   s. In particular, a scalar (e.g. the space-time interval) remains the same under Lorentz transformation
   Lorentz transformation
   In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...
   
   s and is said to be a Lorentz invariant (i.e. they transform under the trivial representation
   Trivial representation
   In the mathematical field of representation theory, a trivial representation is a representation of a group G on which all elements of G act as the identity mapping of V...
   
   ).
2. An equation
   Equation
   An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...
   
    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
   Principle of relativity
   In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference....
   
   , i.e. all non-gravitation
   Gravitation
   Gravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...
   
   al laws must make the same predictions for identical experiments taking place at the same spacetime event in two different inertial frames of reference.

Note: this usage of the term covariant should not be confused with the related concept of a covariant vector. On manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

s, the words covariant and contravariant refer to how objects transform under general coordinate transformations. Confusingly, both covariant and contravariant four-vectors can be Lorentz covariant quantities.

Local Lorentz covariance, which follows from general relativity

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

, refers to Lorentz covariance applying only locally

Local symmetry

In physics, a local symmetry is symmetry of some physical quantity, which smoothly depends on the point of the base manifold. Such quantities can be for example an observable, a tensor or the Lagrangian of a theory....

 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 nature of a Lorentz tensor can be identified by its tensor rank, which is the number of indices it has. No indices implies it is a scalar, one implies it is a vector etc. Furthermore, any number of new scalars, vectors etc. can be made by contracting any kinds of tensors together, but many of these may not have any real physical meaning. Some of those tensors that do have a physical interpretation are listed (by no means exhaustively) below.

Please note, the metric

Metric tensor

In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...

 sign convention such that η = diag

Diagonal matrix

In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero...

 (1, −1, −1, −1) is used throughout the article.

Lorentz scalars

Spacetime interval:

Proper time

Proper time

In relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two...

 (for timelike intervals):

Rest mass:

Electromagnetism invariants:

D'Alembertian/wave operator:

Lorentz 4-vectors

4-Displacement

Displacement (vector)

A displacement is the shortest distance from the initial to the final position of a point P. Thus, it is the length of an imaginary straight path, typically distinct from the path actually travelled by P...

:

Partial derivative:

4-velocity:

4-momentum:

4-current:

Lorentz 4-tensors

The Kronecker delta:

The Minkowski metric (the metric of flat space according to General Relativity

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

):

The Levi-Civita symbol

Levi-Civita symbol

The Levi-Civita symbol, also called the permutation symbol, antisymmetric symbol, or alternating symbol, is a mathematical symbol used in particular in tensor calculus...

:

Electromagnetic field tensor (using a metric signature of + − − − ):

Dual

Hodge dual

In mathematics, the Hodge star operator or Hodge dual is a significant linear map introduced in general by W. V. D. Hodge. It is defined on the exterior algebra of a finite-dimensional oriented inner product space.-Dimensions and algebra:...

 electromagnetic field tensor:

Lorentz violation

Lorentz violation refers to theories which are approximately relativistic

Theory of relativity

The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....

 when it comes to experiments that have actually been performed (and there are quite a number of such experimental tests) but yet contain tiny or hidden Lorentz violating corrections.

Such models typically fall into four classes:

• The laws of physics are exactly Lorentz covariant but this symmetry is spontaneously broken. In special relativistic
  Special relativity
  Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
  
   theories, this leads to phonon
  Phonon
  In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, such as solids and some liquids...
  
  s, which are the Goldstone boson
  Goldstone boson
  In particle and condensed matter physics, Goldstone bosons or Nambu–Goldstone bosons are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries...
  
  s. The phonons travel at less than the speed of light
  Speed of light
  The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
  
  .
• 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
  Preferred frame
  In theoretical physics, a preferred or privileged frame is usually a special hypothetical frame of reference in which the laws of physics might appear to be identifiably different from those in other frames....
  
   (the "vacuum rest frame"). They can be tested, at least partially, by ultra-high energy cosmic ray experiments like the Pierre Auger Observatory
  Pierre Auger Observatory
  The Pierre Auger Observatory is an international cosmic ray observatory designed to detect ultra-high-energy cosmic rays: single sub-atomic particles with energies beyond 1020 eV...
  
  .
• The laws of physics are symmetric under a deformation
  Deformation theory
  In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions Pε, where ε is a small number, or vector of small quantities. The infinitesimal conditions are therefore the result of applying the approach...
  
   of the Lorentz or more generally, the Poincaré group
  Poincaré group
  In physics and mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime.-Simple explanation:...
  
  , and this deformed symmetry is exact and unbroken. This deformed symmetry is also typically a quantum group
  Quantum group
  In mathematics and theoretical physics, the term quantum group denotes various kinds of noncommutative algebra with additional structure. In general, a quantum group is some kind of Hopf algebra...
  
   symmetry, which is a generalization of a group symmetry. Deformed special relativity is an example of this class of models. It is not accurate to call such models Lorentz-violating as much as Lorentz deformed any more than special relativity can be called a violation of Galilean symmetry rather than a deformation of it. 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.
• This is a class of its own; a subgroup of the Lorentz group is sufficient to give us all the standard predictions if CP is an exact symmetry. However, CP isn't exact. This is called Very Special Relativity
  Very special relativity
  Ignoring gravity, experimental bounds seem to suggest that special relativity with its Lorentz symmetry and Poincare symmetry describes spacetime. Surprisingly, Cohen and Glashow have demonstrated that a small subgroup of the Lorentz group is sufficient to explain all the current bounds.The minimal...
  
  .

Constraints

In standard field theory, there are very strict and severe constraints on marginal and relevant Lorentz violating operators within both QED

Quantum electrodynamics

Quantum electrodynamics is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and special relativity is achieved...

 and the Standard Model

Standard Model

The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...

. Irrelevant Lorentz violating operators may be suppressed by a high cutoff

Cutoff

In theoretical physics, cutoff is an arbitrary maximal or minimal value of energy, momentum, or length, used in order that objects with larger or smaller values than these physical quantities are ignored in some calculation...

 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.

Models belonging to the first two classes can be consistent with experiment if Lorentz breaking happens at Planck scale or beyond it, 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.

The Fermi telescope has not found any modification in the dispersion relation for gamma rays up to the level given by the Planck scale

Planck scale

In particle physics and physical cosmology, the Planck scale is an energy scale around 1.22 × 1019 GeV at which quantum effects of gravity become strong...

.

In September 2011, a paper from the OPERA

OPERA Experiment

The Oscillation Project with Emulsion-tRacking Apparatus is a scientific experiment for detecting tau neutrinos from muon neutrino oscillations. It is a collaboration between CERN in Geneva, Switzerland, and the Laboratori Nazionali del Gran Sasso in Gran Sasso, Italy and uses the CERN Neutrinos...

 collaboration at CERN indicated detection of muon neutrinos sent 730 kilometers (454 miles) from near Geneva, Switzerland to the Gran Sasso National Laboratory

Laboratori Nazionali del Gran Sasso

Laboratori Nazionali del Gran Sasso is a particle physics laboratory of the INFN, situated near the Gran Sasso mountain in Italy, between the towns of L'Aquila and Teramo, about 120 km from Rome. In addition to a surface portion of the laboratory, there are extensive underground facilities...

 in Italy

Italy

Italy , officially the Italian Republic languages]] under the European Charter for Regional or Minority Languages. In each of these, Italy's official name is as follows:;;;;;;;;), is a unitary parliamentary republic in South-Central Europe. To the north it borders France, Switzerland, Austria and...

 traveling faster than light by a factor of approximately 1 in 40,000, a statistic with 6.0-sigma significance. (see OPERA neutrino anomaly

OPERA neutrino anomaly

The OPERA neutrino anomaly is the detection of apparently faster-than-light neutrinos by the OPERA experiment as publicly announced in September 2011. The detection is anomalous because speeds exceeding that of light in a vacuum are generally thought to violate special relativity, a prevailing...

)

CPT violation

In 2002, Oscar Greenberg proved that CPT violation

CPT symmetry

CPT symmetry is a fundamental symmetry of physical laws under transformations that involve the inversions of charge, parity, and time simultaneously.-History:...

 implies the breaking of Lorentz 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.
Potential violation in high-speed-ion experiments has
been shown recently with many criticisms against the interpretation of the analysis done.

See also

• Antimatter tests of Lorentz violation
  Antimatter tests of Lorentz violation
  High-precision experiments could revealsmall previously unseen differences between the behaviorof matter and antimatter.This prospect is appealing to physicists because it mayshow that nature is not Lorentz symmetric.- Introduction :...
  
  
• Background independence
  Background independence
  Background independence, also called universality, is the concept or assumption, fundamental to all physical sciences, that the nature of reality is consistent throughout all of space and time...
  
  
• Deformed special relativity
• General covariance
  General covariance
  In theoretical physics, general covariance is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations...
  
  
• List of mathematical topics in relativity
• Loop quantum gravity
  Loop quantum gravity
  Loop quantum gravity , also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the theories of quantum mechanics and general relativity...
  
  
• Hendrik Lorentz
  Hendrik Lorentz
  Hendrik Antoon Lorentz was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect...
  
  
• Lorentz invariance in loop quantum gravity
  Lorentz invariance in loop quantum gravity
  Loop quantum gravity is a quantization of a classical Lagrangian field theory. It is equivalent to the usual Einstein-Cartan theory in that it leads to the same equations of motion describing general relativity with torsion. As such, it can be argued that LQG respects local Lorentz invariance...
  
  
• Lorentz transformation
  Lorentz transformation
  In physics, the Lorentz transformation or Lorentz-Fitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...
  
  
• Lorentz-violating neutrino oscillations
  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...
  
  
• Luminiferous aether
  Luminiferous aether
  In the late 19th century, luminiferous aether or ether, meaning light-bearing aether, was the term used to describe a medium for the propagation of light....
  
  
• Pierre Auger Observatory
  Pierre Auger Observatory
  The Pierre Auger Observatory is an international cosmic ray observatory designed to detect ultra-high-energy cosmic rays: single sub-atomic particles with energies beyond 1020 eV...
  
  
• Principle of locality
  Principle of locality
  In physics, the principle of locality states that an object is influenced directly only by its immediate surroundings. Experiments have shown that quantum mechanically entangled particles must violate either the principle of locality or the form of philosophical realism known as counterfactual...
  
  
• Relativistic mass
• Rotational symmetry
  Rotational symmetry
  Generally speaking, an object with rotational symmetry is an object that looks the same after a certain amount of rotation. An object may have more than one rotational symmetry; for instance, if reflections or turning it over are not counted, the triskelion appearing on the Isle of Man's flag has...
  
  
• Spacetime
  Spacetime
  In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
  
  
• Spin foam
  Spin foam
  In physics, a spin foam is a topological structure made out of two-dimensional faces that represents one of the configurations that must be summed to obtain a Feynman's path integral description of quantum gravity...
  
  
• Standard-Model Extension
  Standard-Model Extension
  Standard-Model Extension is an effective field theory that contains the Standard Model, General Relativity, and all possible operators that break Lorentz symmetry.Violations of this fundamental symmetry can be studied within this general framework...
  
  
• Symmetry in physics
  Symmetry in physics
  In physics, symmetry includes all features of a physical system that exhibit the property of symmetry—that is, under certain transformations, aspects of these systems are "unchanged", according to a particular observation...
  
  
• Test theories of special relativity
  Test theories of special relativity
  Test theories of special relativity give a mathematical framework for analyzing results of experiments to verify special relativity.An experiment to test the theory of relativity cannot assume the theory is true, and therefore needs some other framework of assumptions that are wider than those of...
  
  
• Translational symmetry
  Translational symmetry
  In geometry, a translation "slides" an object by a a: Ta = p + a.In physics and mathematics, continuous translational symmetry is the invariance of a system of equations under any translation...