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 Ev=0 = mc2.
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.
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.
The proper time interval between any two events is defined as:
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).
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.
|Lorentz Transformations for Energy and Momentum||
|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.||
|Lorentz transformations for force undergoing a boost in the y-direction.||