Special Relativity: Dynamics
Terms and Formulae for Relativistic Dynamics
|A 0 = γ(A 0' + (v/c)A 1')|
|A 1 = γ(A 1' + (v/c)A 0')|
|A 2 = A 2'|
|A 3 = A 3'|
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.
This is a particularly useful quantity because it is in independent of the frame in which it is measured.
|AƒB = A 0 B 0 - A 1 B 1 - A 2 B 2 - A 3 B 3|
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).
|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 m 2 c 4 .) The formula reduces to the familiar E = mc 2 when the momentum p is zero.||
|Lorentz transformations for force undergoing a boost in the y-direction.||
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