Editing Lorentz covariance (section)

==Examples==

In general, the (transformational) nature of a Lorentz tensor{{clarify|this terminology should be introduced before use|date=March 2017}} can be identified by its [[tensor order]], which is the number of free indices it has. No indices implies it is a scalar, one implies that it is a vector, etc.  Some tensors with a physical interpretation are listed below.

The [[sign convention]] of the [[Minkowski metric]] {{nowrap|1=''η'' = [[diagonal matrix|diag]] (1, −1, −1, −1)}} is used throughout the article.

===Scalars===

;[[Spacetime interval]]:<math>\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</math>
;[[Proper time]] (for [[timelike]] intervals):<math>\Delta \tau = \sqrt{\frac{\Delta s^2}{c^2}},\, \Delta s^2 > 0</math>
;[[Proper distance]] (for [[spacelike]] intervals):<math>L = \sqrt{-\Delta s^2},\, \Delta s^2 < 0</math>
;[[Mass]]:<math>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</math>
;Electromagnetism invariants:<math>\begin{align}
  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{align}</math>
;[[D'Alembertian]]/wave operator:<math>\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}</math>

===Four-vectors===
;[[Displacement (vector)|4-displacement]]: <math>\Delta X^a = \left(c\Delta t, \Delta\vec{x}\right) = (c\Delta t, \Delta x, \Delta y, \Delta z)</math>
;[[Four-position|4-position]]: <math>X^a = \left(ct, \vec{x}\right) = (ct, x, y, z)</math>
;[[Four-gradient|4-gradient]]: which is the 4D [[partial derivative]]:{{paragraph}} <math>\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)</math>
;[[Four-velocity|4-velocity]]: <math>U^a = \gamma\left(c, \vec{u}\right) = \gamma \left(c, \frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt}\right)</math>{{paragraph}} where <math>U^a = \frac{dX^a}{d\tau}</math>
;[[Four-momentum|4-momentum]]: <math>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)</math>{{paragraph}} where <math>P^a = m U^a</math> and <math>m</math> is the [[Mass_in_special_relativity|rest mass]].
;[[Four-current|4-current]]: <math>J^a = \left(c\rho, \vec{j}\right) = \left(c\rho, j_x, j_y, j_z\right)</math>{{paragraph}} where <math>J^a = \rho_o U^a</math>
;[[Electromagnetic four-potential|4-potential]]: <math>A^a = \left(\frac{\phi}{c}, \vec{A}\right)= \left(\frac{\phi}{c}, A_x, A_y, A_z\right)</math>

===Four-tensors===

;[[Kronecker delta]]:<math>\delta^a_b = \begin{cases} 1 & \mbox{if } a = b, \\ 0 & \mbox{if } a \ne b. \end{cases}</math>
;[[Minkowski metric]] (the metric of flat space according to [[general relativity]]):<math>\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}</math>
;[[Electromagnetic field tensor]] (using a [[sign convention#Metric signature|metric signature]] of + − − −):<math>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}</math>
;[[Hodge dual|Dual]] electromagnetic field tensor:<math>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}</math>