In general, the (transformational) nature of a Lorentz tensor 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 is used throughout the article.
Scalars ;
Spacetime interval:\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 ;
Proper time (for
timelike intervals):\Delta \tau = \sqrt{\frac{\Delta s^2}{c^2}},\, \Delta s^2 > 0 ;
Proper distance (for
spacelike intervals):L = \sqrt{-\Delta s^2},\, \Delta s^2 ;
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:\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} ;
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}
Four-vectors ;
4-displacement: \Delta X^a = \left(c\Delta t, \Delta\vec{x}\right) = (c\Delta t, \Delta x, \Delta y, \Delta z) ;
4-position: X^a = \left(ct, \vec{x}\right) = (ct, x, y, z) ;
4-gradient: which is the 4D
partial derivative: \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: U^a = \gamma\left(c, \vec{u}\right) = \gamma \left(c, \frac{dx}{dt}, \frac{dy}{dt}, \frac{dz}{dt}\right) where U^a = \frac{dX^a}{d\tau} ;
4-momentum: 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 P^a = m U^a and m is the
rest mass. ;
4-current: J^a = \left(c\rho, \vec{j}\right) = \left(c\rho, j_x, j_y, j_z\right) where J^a = \rho_o U^a ;
4-potential: A^a = \left(\frac{\phi}{c}, \vec{A}\right)= \left(\frac{\phi}{c}, A_x, A_y, A_z\right)
Four-tensors ;
Kronecker delta:\delta^a_b = \begin{cases} 1 & \mbox{if } a = b, \\ 0 & \mbox{if } a \ne b. \end{cases} ;
Minkowski metric (the metric of flat 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} ;
Electromagnetic field tensor (using a
metric signature of + − − −):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: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} ==Lorentz violating models==