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# 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 Scalar s, Four-vector s, Four-tensor s, and Spinor s. In particular, a scalar (e.g. the Space-time Interval ) remains the same under Lorentz Transformation 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 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- Gravitation 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 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.

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 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 Scalar s


Spacetime Interval :
:\Delta s^2=x^a x^b \eta_{ab}=c^2 \Delta t^2 - \Delta x^2 - \Delta y^2 - \Delta z^2

Proper Time (for Timelike intervals):
:\Delta au = \sqrt{ rac{\Delta s^2}{c^2}},\, \Delta s^2 > 0

Rest Mass :
:m_0^2 c^2 = p^a p^b \eta_{ab}= rac{E^2}{c^2} - p_x^2 - p_y^2 - p_z^2

Electromagnetism invariants:
:F_{ab} F^{ab} = \ 2 \left( B^2 - rac{E^2}{c^2} ight)
:G_{cd}F^{cd}=\epsilon_{abcd}F^{ab} F^{cd} = rac{2}{c} \left( ec B \cdot ec E ight)

D'Alembertian /wave operator:
:\Box = \partial_a \partial_b \eta^{ab} = rac{1}{c^2} rac{\partial^2}{\partial t^2} - rac{\partial^2}{\partial x^2} - rac{\partial^2}{\partial y^2} - rac{\partial^2}{\partial z^2}


Lorentz 4-vector s


4- Displacement :
:x^a = x, y, z

Partial derivative:
:\partial_a = \left[ rac{1}{c} rac{\partial}{\partial t}, rac{\partial}{\partial x}, rac{\partial}{\partial y}, rac{\partial}{\partial z} ight]

4-velocity :
:U^a = rac{dx^a}{dt} = \gamma \left[c, rac{dx}{dt}, rac{dy}{dt}, rac{dz}{dt} ight]

4-momentum :
:p^a = m_0 U^a = \left[ rac{E}{c}, p_x, p_y, p_z ight]

4-current :
:j^a = [c ho, j_x, j_y, j_z]


Lorentz 4-tensor s


The Kronecker Delta :
:\delta^a_b = \begin{cases} 1 & \mbox{if } a = b, \ 0 & \mbox{if } a
e b. \end{cases}

The Minkowski Metric :
:\eta_{ab} = \eta^{ab} = \begin{cases} -1 & \mbox{if } a = b = 0, \ 1 & \mbox{if }a = b = 1, 2, 3, \ 0 & \mbox{if } a
e b. \end{cases}

The Levi-Civita Symbol :
:\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 :
: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} = rac{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.

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 Phonon s, which are the Goldstone Boson s. 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 .



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 Correction s. So, we also have very strict and severe constraints on irrelevant Lorentz violating operators.

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.


SEE ALSO




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



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