Note 18 Relativistic Momentum and Energy
Lastly we look at the conservation laws of momentum and energy. The total relativistic
momentum of a mass is this. These are derived from the spacial components of spacetime.
p = γmv
The relativistic total energy of a mass is this. This is derived from the temporal components of
spacetime.
E = γmc 2
The mass in both of these equations is the rest mass of the object, the mass when the object is
not moving. You can look at the rest energy as the potential energy from creating the object in the
first place. When energy is put into an object to make it move, you can think of it as increasing the
relativistic mass (γ times m with gamma greater than 1) or increasing the kinetic energy.
E = ( γm )c 2 = mc 2 + K
The first term mc2 is the rest energy of the object. This is the energy stored in the mass of the
object itself. The relativistic kinetic energy is thus
K = ( γm )c 2 − mc 2 = ( γ − 1 )mc 2
If the velocity is small, then we can expand like before to get this approximation.
E = ( γm )c =
2
⎛
β 2 ⎞⎟
1
⎜
≅ mc ⎜ 1 + ⎟⎟ = mc 2 + mv 2 = mc 2 + K
⎜
2
⎟
2 ⎠
2
⎝
1− β
mc 2
2
1
⇒ K = mv 2
2
page 1
Example
A proton has a rest mass of 1.6726 x 10-27 kg. Its relativistic total energy is
2
E = γmc 2 = mc 2 = ( 1.672622 ×10−27 kg )( 299,792, 458 m/s ) = 1.503278 ×10−10 J
This is usually expressed in term of the equivalent energy in eV or MeV.
⎛
⎞⎟
1 eV
E = ( 1.503278 ×10−10 J )⎜⎜
⎟ = 9.38272 ×108 eV = 938.272 MeV
⎜⎝ 1.602177 ×10−9 J ⎟⎟⎠
A proton is traveling at one quarter the speed of light. The value of gamma is this.
γ=
1
1 − β2
=
1
1 − 0.252
= 1.032796
Its relativistic total energy is then equal to this.
2
E = γmc 2 = ( 1.032796 )( 1.672622 ×10−27 kg )( 299,792, 458 m/s ) = 1.552579 ×10−10 J
⎛
⎞⎟
1 eV
E = ( 1.552579 ×10−10 J )⎜⎜
⎟ = 9.690431×108 eV = 969, 043 MeV
⎜⎝ 1.602177 ×10−19 J ⎟⎟⎠
The kinetic energy is this.
K = γmc 2 − mc 2 = 969.043 MeV − 938.272 MeV = 30.770 MeV
Here is a table of the kinetic energy for various speeds.
speed
gamma
kinetic
energy
(MeV)
Increase
factor
0
1
0
–
0.005c
1.0000125
0.011728
0.010c
1.000050
0.046917
0.05c
1.001252
1.175
0.10c
1.005038
4.727
0.20c
1.020621
19.35
4.09
0.40c
1.091089
85.47
4.42
0.80c
1.666667
625.6
7.32
4.0002
4.02
Note that at slow speeds, the doubling of the speed produces a 4 times increase in the kinetic
energy. However, at higher speeds, more than 4 times the kinetic energy is required to double the
speed. Near the speed of light, the energy required goes to infinity. Again, at 0.1c, the error is
about half of a percent.
page 2