Special Relativity: Dynamics: Terms and Formulae for Relativistic Dynamics

Terms

Relativistic energy

In Special Relativity the concept of total energy in the absence of a potential E = 1/2mv² is replaced with another conserved quantity E = γmc², where m is the mass or rest mass of the object. This quantity is conserved in all collisions and decays. Where there is a potential involved it is the total energy γmc² + V which is conserved. Notice that an object at rest still has an amount of energy proportional to its mass E_v=0 = mc².
Relativistic momentum

The quantity that is conserved in all collisions in relativity is not p = mv but p = γmv. This is called the relativistic momentum. When v < < c then γ 1 and pmv.
4-vector

A vector with four components that, under a Lorentz transformation, transforms as (cdt, dx, dy, dz) does. That is, for A = (A₀, A₁, A₂, A₃) the 4-vector in another frame must be:

A₀ = γ(A₀' + (v/c)A₁')

A₁ = γ(A₁' + (v/c)A₀')

A₂ = A₂'

A₃ = A₃'

Only those vectors for which the result of the above transformation is equal to the transformation of the individual coordinates under the Lorentz transformations are 4-vectors. The velocity 4-vector (γv, γbfv) and the energy-momentum 4-vector (E/c, are the most common.
Proper time

The proper time interval between any two events is defined as:

Δτ =

This is a particularly useful quantity because it is in independent of the frame in which it is measured.
Inner product invariance

The inner product of two 4-vectors is defined as:

AƒB = A₀B₀ - A₁B₁ - A₂B₂ - A₃B₃

Note that the minus signs make this inner product different from the usual dot product in 3-space. When defined in this way, the inner product of any two 4-vectors is a constant, independent of frame (that is, it is independent of the frame in which the vectors are written).
Relativistic units

Are units in which c, the speed of light is given the value 1. This can be done in any number of ways; setting the unit of distance equal to 3×10⁸ meters is one way. Setting the unit of distance as approximately 1 foot and the unit of time to 1 nanosecond also does the trick since the speed of light is approximately 1 foot/nanosecond. This simplifies calculations immensely. If you need to find an exact answer it is always possible to put the right number of factors of c back in at the end of a calculation by looking at the units and working out where factors of m/s are missing.

Formulae

Lorentz Transformations for Energy and Momentum

E = γ(E' + vp')
p = γ(p' + vE'/c²)

Formula for Velocity in terms of Energy and Momentum

Relativistic relationship between mass, energy, and momentum. (Specifically, this equation states that the square of the energy-momentum 4-vector is equal to m²c⁴.) The formula reduces to the familiar E = mc² when the momentum p is zero.