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
.
4vector

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 4vector 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 4vectors. The velocity 4vector
(γv, γbfv)
and the energymomentum 4vector
(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 4vectors 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 3space. When
defined in this way, the inner product of any two 4vectors 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
3×10^{8}
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 energymomentum
4vector 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 ydirection.

F
_{x} = F
_{x}' 



F
_{y} =




