12.3.8 Relativistic Mass

• Einstein showed in a1905 paper that matter can be considered a form of energy and hence, he proposed:
  • Mass can be converted into energy
  • Energy can be converted into mass
• This is known as mass-energy equivalence, and can be summarised by the equation:

• Where:
  • E = energy (J)
  • m = mass (kg)
  • c = the speed of light (m s^-1)
• Scenarios where mass is converted to energy or energy to mass include:
  • The fusion of hydrogen into helium in the centre of the Sun
  • The fission of uranium in nuclear power plants
  • Nuclear weapons
  • High-energy particle collisions in particle accelerators

• An object's proper mass (that is, its mass measured by an observer at rest relative to the object) is also called its rest mass, m₀
• An observer in an inertial reference frame moving at speed v  relative to the object measures the object's mass as m, which is given by:

• This can also be written as:

• Because γ > 1, when an object is moving relative to your frame of reference, you measure its mass as larger than it's rest mass
  • The derivation for this relativistic mass is a good deal more complex than the derivation for relativistic time and length and is not examinable so it is omitted from this revision note
• One interesting outcome of this relativistic mass is the speed limit imposed on objects
  • As an object's speed nears the speed of light, it's mass gets larger and larger
  • Very close to the speed of light, an object's mass tends towards an infinite mass

Graph showing how relativistic mass changes with velocity

The effect of speed, as a fraction of c, on an object's mass, in multiples of rest mass. Mass increases rapidly as the object's speed approaches c.

Relativistic Energy

• Let's reconsider Einstein's famous equation, taking relativistic effects into account:

• The mass of a stationary object is its rest mass, m₀, so it has a rest energy of E = m₀c²