Lorentz covariance

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In standard 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.

  1. 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).
  2. 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.

There is a generalization of this concept to cover Poincaré covariance and Poincaré invariance.

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[edit] 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.

[edit] Lorentz scalars

Spacetime interval:

Δs2 = xaxbηab = c2Δt2 − Δx2 − Δy2 − Δz2

Proper time (for timelike intervals):

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

Rest mass:

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:

\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}

[edit] Lorentz 4-vectors

4-Displacement:

xa = [ct,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:

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

4-momentum:

p^a = m_0 U^a = \left[\frac{E}{c}, p_x, p_y, p_z\right]

4-current:

ja = [cρ,jx,jy,jz]

[edit] Lorentz 4-tensors

The Kronecker delta:

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

The Minkowski metric (the metric of empty space according to General Relativity):

\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:

\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} = \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}

[edit] 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 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).
  • 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 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. 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.

[edit] Constraints

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. However, if standard "elementary" particles are actually composite and made of superluminal constituents as postulated in the superbradyon hypothesis, such constraints do no longer hold.

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.

[edit] See also

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