Editing Special relativity (section)

==== Relativistic dynamics and invariance ====

The invariant magnitude of the [[four-momentum|momentum 4-vector]] generates the [[energy–momentum relation]]:
<math display="block">\mathbf{P}^2 = \eta^{\mu\nu}P_\mu P_\nu = -\left (\frac{E}{c} \right )^2 + p^2 .</math>

We can work out what this invariant is by first arguing that, since it is a scalar, it does not matter in which reference frame we calculate it, and then by transforming to a frame where the total momentum is zero.
<math display="block">\mathbf{P}^2 = - \left (\frac{E_\text{rest}}{c} \right )^2 = - (m c)^2 .</math>

We see that the rest energy is an independent invariant. A rest energy can be calculated even for particles and systems in motion, by translating to a frame in which momentum is zero.

The rest energy is related to the mass according to the celebrated equation discussed above:
<math display="block">E_\text{rest} = m c^2.</math>

The mass of systems measured in their center of momentum frame (where total momentum is zero) is given by the total energy of the system in this frame. It may not be equal to the sum of individual system masses measured in other frames.

To use [[Newton's third law of motion]], both forces must be defined as the rate of change of momentum with respect to the same time coordinate. That is, it requires the 3D force defined above. Unfortunately, there is no tensor in 4D that contains the components of the 3D force vector among its components.

If a particle is not traveling at ''c'', one can transform the 3D force from the particle's co-moving reference frame into the observer's reference frame. This yields a 4-vector called the [[four-force]]. It is the rate of change of the above energy momentum [[four-vector]] with respect to proper time. The covariant version of the four-force is:
<math display="block">F_\nu = \frac{d P_{\nu}}{d \tau} = m A_\nu </math>

In the rest frame of the object, the time component of the four-force is zero unless the "[[invariant mass]]" of the object is changing (this requires a non-closed system in which energy/mass is being directly added or removed from the object) in which case it is the negative of that rate of change of mass, times ''c''. In general, though, the components of the four-force are not equal to the components of the three-force, because the three force is defined by the rate of change of momentum with respect to coordinate time, that is, ''dp''/''dt'' while the four-force is defined by the rate of change of momentum with respect to proper time, that is, ''dp''/''dτ''.

In a continuous medium, the 3D ''density of force'' combines with the ''density of power'' to form a covariant 4-vector. The spatial part is the result of dividing the force on a small cell (in 3-space) by the volume of that cell. The time component is −1/''c'' times the power transferred to that cell divided by the volume of the cell. This will be used below in the section on electromagnetism.