Terms and Formulae for Relativistic Dynamics
Terms
Relativistic energy
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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.
Relativistic momentum
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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
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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') |
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| A1 = γ(A1' + (v/c)A0') |
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| A2 = A2' |
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| A3 = A3' |
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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
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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
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The inner product of two 4-vectors is defined as:
| A⋅B = A0B0 - A1B1 - A2B2 - A3B3 |
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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
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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
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Lorentz Transformations for Energy and Momentum
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| E = γ(E' + vp') |
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| | p = γ(p' + vE'/c2) |
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Formula for Velocity in terms of Energy and Momentum
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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.
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Lorentz transformations for force undergoing a boost in the y-direction.
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1 and p 
are the most common.
