NationMaster - Encyclopedia: Lorentz covariance

In physics, Lorentz covariance is a key property of spacetime that follows from the special theory of relativity, where it applies globally. Local Lorentz covariance refers to Lorentz covariance applying only locally in an infinitesimal region of spacetime at every point, which follows from general relativity. Lorentz covariance has two distinct, but closely related meanings. This is a discussion of a present category of science. ... For other uses of this term, see Spacetime (disambiguation). ... Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein. ... For a less technical and generally accessible introduction to the topic, see Introduction to general relativity. ...

A physical quantity is said to be Lorentz covariant if it transforms under a given representation of the Lorentz group. According to the representation theory of the Lorentz group, these quantities are built out of scalars, four-vectors, four-tensors, and spinors. In particular, a scalar (e.g. the space-time interval) remains the same under Lorentz transformations 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 are that if they hold in one inertial frame, then they hold in any inertial frame (this is a result of the fact 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.

Note: this usage of the term covariant should not be confused with the related concept of a covariant vector. On manifolds, 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. A physical quantity is either a quantity within physics that can be measured (e. ... Group representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... The Lorentz group is the group of all Lorentz transformations of Minkowski spacetime. ... In physics, a scalar is a simple physical quantity that does not depend on direction, and therefore does not depend on the choice of a coordinate system. ... In relativity, a four-vector is a vector in a four-dimensional real vector space, whose components transform like the space and time coordinates (ct, x, y, z) under spatial rotations and boosts (a change by a constant velocity to another inertial reference frame). ... Four-tensor is a frequent abbreviation for a tensor in a four-dimensional spacetime. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... In special relativity and general relativity, time and three-dimensional space are treated together as a single four-dimensional pseudo-Riemannian manifold called spacetime. ... A Lorentz transformation (LT) is a linear transformation that preserves the spacetime interval between any two events in Minkowski space, while leaving the origin fixed (=rotation of Minkowski space). ... In mathematics, in particular group representation theory, a group representation of the group G is called a trivial representation if (i) it is defined on a one-dimensional vector space V over a field K and (ii) all elements g of G act on V as the identity mapping. ... An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). ... Wikisource has original text related to this article: Relativity: The Special and General Theory A principle of relativity is a criterion for judging physical theories, stating that they are inadequate if they do not prescribe the exact same laws of physics in certain similar situations. ... â€œGravityâ€ redirects here. ... All frames of reference that move with constant velocity with respect to any other inertial frame of reference are members of the group of inertial reference frames. ... In category theory, see covariant functor. ... On a sphere, the sum of the angles of a triangle is not equal to 180Â° (see spherical trigonometry). ... In category theory, see covariant functor. ... Contravariant is a mathematical term with a precise definition in tensor analysis. ... In category theory, see covariant functor. ... Contravariant is a mathematical term with a precise definition in tensor analysis. ...

There is a generalization of this concept to cover Poincaré covariance and Poincaré invariance. Lorentz covariance is a term in physics for the property of space time, that in two different frames of reference, located at the same event in spacetime but moving relative to each other, all non-gravitational laws must make the same predictions for identical experiments. ... Poincare symmetry is the full symmetry of special relativity and includes translations (ie, displacements) in time and space (these form the Abelian Lie group of translations on space-time) rotations in space (this forms the non-Abelian Lie group of 3-dimensional rotations) boosts, ie, transformations connecting two uniformly moving...

1 Examples
2 Lorentz violation
- 2.1 Constraints
3 See also
4 References
5 External links

Examples

In general, the nature of a Lorentz tensor can be identified by 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, that we use the metric sign convention such that η = diag (1, -1, -1, -1) throughout the article.

Lorentz scalars

Spacetime interval: A scalar may be: Look up scalar in Wiktionary, the free dictionary. ... For other uses of this term, see Spacetime (disambiguation). ...

Δ s 2 = x a x b η a b = c 2 Δ t 2 - Δ x 2 - Δ y 2 - Δ z 2

Proper time (for timelike intervals): In relativity, proper time is time measured by a single clock between events that occur at the same place as the clock. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...

$Delta tau = sqrt{frac{Delta s^2}{c^2}},, Delta s^2 > 0$

Rest mass: The term mass in special relativity is used in a couple of different ways, occasionally leading to a great deal of confusion. ...

$m_0^2 c^2 = p^a p^b eta_{ab}= frac{E^2}{c^2} - p_x^2 - p_y^2 - p_z^2$

Electromagnetism invariants:

$F_{ab} F^{ab} = 2 left( B^2 - frac{E^2}{c^2} right)$

$G_{cd}F^{cd}=epsilon_{abcd}F^{ab} F^{cd} = frac{2}{c} left( vec B cdot vec E right)$

D'Alembertian/wave operator: In special relativity, electromagnetism and wave theory, the dAlembert operator, also called dAlembertian, is the Laplace operator of Minkowski space. ...

$Box = partial_a partial_b eta^{ab} = frac{1}{c^2}frac{partial^2}{partial t^2} - frac{partial^2}{partial x^2} - frac{partial^2}{partial y^2} - frac{partial^2}{partial z^2}$

Lorentz 4-vectors

4-Displacement: In relativity, a four-vector is a vector in a four-dimensional real vector space, called Minkowski space, whose components transform like the space and time coordinates (t, x, y, z) under spatial rotations and boosts (a change by a constant velocity to another inertial reference frame). ... In Newtonian mechanics, displacement is the vector that specifies the position of a point or a particle in reference to an origin or to a previous position. ...

x a = [c t, x, y, z]

Partial derivative:

$partial_a = left[ frac{1}{c}frac{partial}{partial t}, frac{partial}{partial x}, frac{partial}{partial y}, frac{partial}{partial z} right]$

4-velocity: In physics, in particular in special relativity and general relativity, the four-velocity of an object is a four-vector (vector in four-dimensional spacetime) that replaces classical velocity (a three-dimensional vector). ...

$U^a = frac{dx^a}{dt} = gamma left[c, frac{dx}{dt}, frac{dy}{dt}, frac{dz}{dt}right]$

4-momentum: In special relativity, four-momentum is a four-vector that replaces classical momentum; the four-momentum of a particle is defined as the particles mass times the particles four-velocity. ...

$p^a = m_0 U^a = left[frac{E}{c}, p_x, p_y, p_zright]$

4-current: In special and general relativity, the four-current is the Lorentz covariant four-vector that replaces the electromagnetic current density where c is the speed of light, ρ the charge density, and j the conventional current density. ...

j a = [c ρ, j x, j y, j z]

Lorentz 4-tensors

The Kronecker delta: Four-tensor is a frequent abbreviation for a tensor in a four-dimensional spacetime. ... In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...

$delta^a_b = begin{cases} 1 & mbox{if } a = b, 0 & mbox{if } a ne b. end{cases}$

The Minkowski metric: In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...

$eta_{ab} = eta^{ab} = begin{cases} -1 & mbox{if } a = b = 0, 1 & mbox{if }a = b = 1, 2, 3, 0 & mbox{if } a ne b. end{cases}$

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

$epsilon_{abcd} = -epsilon^{abcd} = begin{cases} +1 & mbox{if } {abcd} mbox{ is an even permutation of } {0123}, -1 & mbox{if } {abcd} mbox{ is an odd permutation of } {0123}, 0 & mbox{otherwise.} end{cases}$

Electromagnetic field tensor: In electromagnetism, the electromagnetic tensor, or electromagnetic field tensor, F, is defined as: where Ai is the vector potential. ...

$F_{ab} = begin{bmatrix} 0 & E_x/c & E_y/c & E_z/c -E_x/c & 0 & -B_z & B_y -E_y/c & B_z & 0 & -B_x -E_z/c & -B_y & B_x & 0 end{bmatrix}$

Dual electromagnetic field tensor:

$G_{cd} = frac{1}{2}epsilon_{abcd}F^{ab} = begin{bmatrix} 0 & B_x & B_y & B_z -B_x & 0 & -E_z/c & E_y/c -B_y & E_z/c & 0 & -E_x/c -B_z & -E_y/c & E_x/c & 0 end{bmatrix}$

Lorentz violation

Lorentz violation refers to theories which are approximately relativistic 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. Albert Einsteins theory of relativity is a set of two theories in physics: special relativity and general relativity. ...

Such models typically fall into four classes:

The laws of physics are exactly Lorentz covariant but this symmetry is spontaneously broken. In special relativistic theories, this leads to phonons, which are the Goldstone bosons. The phonons travel at LESS than the speed of light. In general relativistic theories, this leads to a massive graviton (note that this is different from massive gravity, which is Lorentz covariant) which travels at less than the speed of light (because the graviton devours the phonon).
The laws of physics are NOT Lorentz covariant but Lorentz covariance emerges as an approximate symmetry (at least in the so-called "visible sector"). Models of these sort are typically ether theories.
The laws of physics are symmetric under a deformation of the Lorentz or more generally, the Poincaré group, and this deformed symmetry is exact and unbroken. This deformed symmetry is also typically a quantum group 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.
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.

Lorentz covariance is a term in physics for the property of space time, that in two different frames of reference, located at the same event in spacetime but moving relative to each other, all non-gravitational laws must make the same predictions for identical experiments. ... Spontaneous symmetry breaking in physics takes place when a system that is symmetric with respect to some symmetry group goes into a vacuum state that is not symmetric. ... For a less technical and generally accessible introduction to the topic, see Introduction to special relativity. ... Normal modes of vibration progression through a crystal. ... In particle and condensed matter physics, Goldstone bosons (also known as Nambu-Goldstone bosons) are bosons that appear in models with spontaneously broken symmetry. ... A line showing the speed of light on a scale model of Earth and the Moon, about 1. ... In theoretical physics, massive gravity is a particular generalization of general relativity studied by van Dam and Veltman; and by Zakharov. ... A termite cathedral mound produced by a termite colony: a classic example of emergence in nature. ... Chinese Wood (æœ¨) | Fire (ç«) | Earth (åœŸ) | Metal (é‡‘) | Water (æ°´) Hinduism and Buddhism The Pancha Mahabhuta (The Five Great Elements) Vayu/Pavan (Air/Wind) Agni/Tejas (Fire) Akasha (Aether) Prithvi/Bhumi (Earth) Ap/Jala (Water) Aether (also spelled ether) is a concept used in ancient and medieval science as a substance. ... In engineering mechanics, deformation is a change in shape due to an applied force. ... In physics and mathematics, the PoincarÃ© group is the group of isometries of Minkowski spacetime. ... In abstract algebra, a Hopf algebra is a bialgebra H over a field K together with a K-linear map such that the following diagram commutes . (Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. ... Doubly-special relativity -- also called deformed special relativity or, by some, extra-special relativity -- is a new theory of special relativity. ... Ignoring gravity, experimental bounds seem to suggest that special relativity with its Lorentz symmetry and Poincare symmetry describes spacetime. ...

Constraints

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. In physics, the term renormalization refers to a variety of theoretical concepts and computational techniques revolving either around the idea of rescaling transformations, or around the process of removing infinities from the calculated quantities (see also regularization). ... Look up relevant in Wiktionary, the free dictionary. ... Quantum electrodynamics (QED) is a relativistic quantum field theory of electrodynamics. ... The Standard Model of Fundamental Particles and Interactions For the Standard Model in Cryptography, see Standard Model (cryptography). ... This article needs cleanup. ... In theoretical physics, cutoff usually represents a particular energy scale or length scale. ...

Models belonging to the first two classes have a problem in explaining just why the low energy physics "conspires" in such a way as to look extremely relativistic. This is especially true of emergent Lorentz symmetry models. Most models of this sort will predict that photons and gravitons and the maximum speed of various particles will travel at different speeds. DSR gives us 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 and is still protected from radiative corrections as we do have an exact (quantum) symmetry.

References

http://www.physics.indiana.edu/~kostelec/faq.html
http://relativity.livingreviews.org/Articles/lrr-2005-5/
http://www.nature.com/nature/journal/v393/n6687/full/393763a0_fs.html
http://www.nature.com/nature/journal/v424/n6952/full/nature01882.html
http://www.nature.com/nature/journal/v424/n6952/full/4241007a.html
http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=PRVDAQ000067000012124011000001

External links

Categories: Special relativity | Symmetry

Results from FactBites:

Corresponding States (4032 words)
	On the basis of this hypothesis, Lorentz showed that the description of the equilibrium configuration of a uniformly moving material object in terms of its “local coordinates” is identical to the description of the same object at absolute rest in terms of the ether rest frame coordinates.
	The most significant point is that Poincare had recognized that Lorentz had reached the limit of his constructive approach, and instead he (Poincare) was proceeding not to deduce the necessity of relativity from the phenomena of electromagnetism or gravity, but rather to deduce the necessary attributes of electromagnetism and gravity from the principle of relativity.
	To place Lorentz’s achievement in context, recall that toward the end of the 19th century it appeared electromagnetism was not relativistic, because the property of being relativistic was equated with being invariant under Galilean transformations, and it was known that Maxwell’s equations (unlike Newton’s laws of mechanics) do not possess this invariance.

STR: A Brief History of Einstein's Special Theory of Relativity (8622 words)
	Lorentz's theory was not, however, an adequate explanation of the principle of relativity, for there is still something puzzling about the empirical equivalence entailed by the symmetry of the Lorentz transformation equations.
	Lorentz meant his transformation equations to be a way of describing the length contraction and time dilation in material objects with absolute motion, for that would explain the Michelson-Morley experiment, that is, why absolute motion cannot be detected by measuring the velocity of light in different directions.
	Lorentz’s explanation of length contraction assumed that the ether is totally unaffected by the motion of material objects through it, and he had no explanation of such first order effects except to state transformation equations by which one could obtain the coordinates used on the moving object from those used at absolute rest.

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