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Special Relativity: Dynamics

Terms and Formulae for Relativistic Dynamics

Introduction to Relativistic Dynamics

Energy and Momentum

Terms

Relativistic energy  -  In Special Relativity the concept of total energy in the absence of a potential E = 1/2mv 2 is replaced with another conserved quantity E = γmc 2 , 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 γmc 2 + V which is conserved. Notice that an object at rest still has an amount of energy proportional to its mass E v=0 = mc 2 .
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 p mv .
4-vector  -  A vector with four components that, under a Lorentz transformation, transforms as (cdt, dx, dy, dz) does. That is, for A = (A 0, A 1, A 2, A 3) the 4-vector in another frame must be:


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.
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 = 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).
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'/c 2)      

 
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.

E 2 - |    

 
Lorentz transformations for force undergoing a boost in the y-direction.


F x = F x'      
F y =      

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