Lorentz covariance

The

Spacetime interval

${\displaystyle \Delta s^{2}=\Delta x^{a}\Delta x^{b}\eta _{ab}=c^{2}\Delta t^{2}-\Delta x^{2}-\Delta y^{2}-\Delta z^{2}}$

timelike

intervals)

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

spacelike

intervals)

${\displaystyle L={\sqrt {-\Delta s^{2}}},\,\Delta s^{2}<0}$

Mass

${\displaystyle m_{0}^{2}c^{2}=P^{a}P^{b}\eta _{ab}={\frac {E^{2}}{c^{2}}}-p_{x}^{2}-p_{y}^{2}-p_{z}^{2}}$

Electromagnetism invariants

${\displaystyle {\begin{aligned}F_{ab}F^{ab}&=\ 2\left(B^{2}-{\frac {E^{2}}{c^{2}}}\right)\\G_{cd}F^{cd}&={\frac {1}{2}}\epsilon _{abcd}F^{ab}F^{cd}=-{\frac {4}{c}}\left({\vec {B}}\cdot {\vec {E}}\right)\end{aligned}}}$

D'Alembertian

/wave operator

${\displaystyle \Box =\eta ^{\mu \nu }\partial _{\mu }\partial _{\nu }={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-{\frac {\partial ^{2}}{\partial x^{2}}}-{\frac {\partial ^{2}}{\partial y^{2}}}-{\frac {\partial ^{2}}{\partial z^{2}}}}$

Four-vectors

4-displacement

${\displaystyle \Delta X^{a}=\left(c\Delta t,\Delta {\vec {x}}\right)=(c\Delta t,\Delta x,\Delta y,\Delta z)}$

4-position

${\displaystyle X^{a}=\left(ct,{\vec {x}}\right)=(ct,x,y,z)}$

4-gradient

which is the 4D partial derivative:

${\displaystyle \partial ^{a}=\left({\frac {\partial _{t}}{c}},-{\vec {\nabla }}\right)=\left({\frac {1}{c}}{\frac {\partial }{\partial t}},-{\frac {\partial }{\partial x}},-{\frac {\partial }{\partial y}},-{\frac {\partial }{\partial z}}\right)}$

4-velocity

${\displaystyle U^{a}=\gamma \left(c,{\vec {u}}\right)=\gamma \left(c,{\frac {dx}{dt}},{\frac {dy}{dt}},{\frac {dz}{dt}}\right)}$

where ${\displaystyle U^{a}={\frac {dX^{a}}{d\tau }}}$

4-momentum

${\displaystyle P^{a}=\left(\gamma mc,\gamma m{\vec {v}}\right)=\left({\frac {E}{c}},{\vec {p}}\right)=\left({\frac {E}{c}},p_{x},p_{y},p_{z}\right)}$

where ${\displaystyle P^{a}=mU^{a}}$ and ${\displaystyle m}$ is the rest mass.

4-current

${\displaystyle J^{a}=\left(c\rho ,{\vec {j}}\right)=\left(c\rho ,j_{x},j_{y},j_{z}\right)}$

where ${\displaystyle J^{a}=\rho _{o}U^{a}}$

4-potential

${\displaystyle A^{a}=\left({\frac {\phi }{c}},{\vec {A}}\right)=\left({\frac {\phi }{c}},A_{x},A_{y},A_{z}\right)}$

Four-tensors

Kronecker delta

${\displaystyle \delta _{b}^{a}={\begin{cases}1&{\mbox{if }}a=b,\\0&{\mbox{if }}a\neq b.\end{cases}}}$

Minkowski metric (the metric of flat space according to general relativity

)

${\displaystyle \eta _{ab}=\eta ^{ab}={\begin{cases}1&{\mbox{if }}a=b=0,\\-1&{\mbox{if }}a=b=1,2,3,\\0&{\mbox{if }}a\neq b.\end{cases}}}$

Electromagnetic field tensor (using a metric signature

of + − − −)

${\displaystyle F_{ab}={\begin{bmatrix}0&{\frac {1}{c}}E_{x}&{\frac {1}{c}}E_{y}&{\frac {1}{c}}E_{z}\\-{\frac {1}{c}}E_{x}&0&-B_{z}&B_{y}\\-{\frac {1}{c}}E_{y}&B_{z}&0&-B_{x}\\-{\frac {1}{c}}E_{z}&-B_{y}&B_{x}&0\end{bmatrix}}}$

Dual

electromagnetic field tensor

${\displaystyle G_{cd}={\frac {1}{2}}\epsilon _{abcd}F^{ab}={\begin{bmatrix}0&B_{x}&B_{y}&B_{z}\\-B_{x}&0&{\frac {1}{c}}E_{z}&-{\frac {1}{c}}E_{y}\\-B_{y}&-{\frac {1}{c}}E_{z}&0&{\frac {1}{c}}E_{x}\\-B_{z}&{\frac {1}{c}}E_{y}&-{\frac {1}{c}}E_{x}&0\end{bmatrix}}}$

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 lead to violations of Lorentz invariance,^[2] these studies are part of phenomenological quantum gravity. Lorentz violations are allowed in string theory, supersymmetry and Hořava–Lifshitz gravity.^[3]

Lorentz violating models typically fall into four classes:^{[citation needed]}

• The laws of physics are exactly
  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
  .
• 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.[4] 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.^[5]
• The laws of physics are symmetric under a
  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.
• 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 preonic models,^[6] 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.^[7]

Lorentz invariance is also violated in QFT assuming non-zero temperature.^[8][9][10]

There is also growing evidence of Lorentz violation in

Notes