and consistency with the earlier equations, Cartesian coordinates will be used. The simplest example of a four-vector is the position of an event in spacetime, which constitutes a timelike component ct and spacelike component , in a contravariant position four-vector with components: where we define so that the time coordinate has the same dimension of distance as the other spatial dimensions; so that space and time are treated equally. Now the transformation of the contravariant components of the position 4-vector can be compactly written as: where there is an implied summation on from 0 to 3, and is a matrix. More generally, all contravariant components of a four-vector transform from one frame to another frame by a Lorentz transformation: Examples of other 4-vectors include the four-velocity defined as the derivative of the position 4-vector with respect to proper time: where the Lorentz factor is: The relativistic energy and relativistic momentum of an object are respectively the timelike and spacelike components of a contravariant four-momentum vector: where m is the invariant mass. The four-acceleration is the proper time derivative of 4-velocity: The transformation rules for three-dimensional velocities and accelerations are very awkward; even above in standard configuration the velocity equations are quite complicated owing to their non-linearity. On the other hand, the transformation of four-velocity and four-acceleration are simpler by means of the Lorentz transformation matrix. The four-gradient of a scalar field φ transforms covariantly rather than contravariantly: which is the transpose of: only in Cartesian coordinates. It's the covariant derivative which transforms in manifest covariance, in Cartesian coordinates this happens to reduce to the partial derivatives, but not in other coordinates. More generally, the covariant components of a 4-vector transform according to the inverse Lorentz transformation: where is the reciprocal matrix of . The postulates of special relativity constrain the exact form the Lorentz transformation matrices take. More generally, most physical quantities are best described as (components of) tensors. So to transform from one frame to another, we use the well-known tensor transformation law where is the reciprocal matrix of . All tensors transform by this rule. An example of a four-dimensional second order antisymmetric tensor is the relativistic angular momentum, which has six components: three are the classical angular momentum, and the other three are related to the boost of the center of mass of the system. The derivative of the relativistic angular momentum with respect to proper time