an expression of energy conservation: where is the energy of a particle and is the 3-force on the particle. Energy, rest mass, 3-momentum, and 3-speed from 4-momentum The 4-momentum of a particle is where is the particle rest mass, is the momentum in 3-space, and is the energy of the particle. Measurement of the energy of a particle Consider a second particle with 4-velocity and a 3-velocity . In the rest frame of the second particle the inner product of with is proportional to the energy of the first particle where the subscript 1 indicates the first particle. Since the relationship is true in the rest frame of the second particle, it is true in any reference frame. , the energy of the first particle in the frame of the second particle, is a Lorentz scalar. Therefore, in any inertial reference frame, where is still the energy of the first particle in the frame of the second particle. Measurement of the rest mass of the particle In the rest frame of the particle the inner product of the momentum is Therefore, the rest mass () is a Lorentz scalar. The relationship remains true independent of the frame in which the inner product is calculated. In many cases the rest mass is written as to avoid confusion with the relativistic mass, which is . Measurement of the 3-momentum of the particle Note that The square of the magnitude of the 3-momentum of the particle as measured in the frame of the second particle is a Lorentz scalar. Measurement of the 3-speed of the particle The 3-speed, in the frame of the second particle, can be constructed from two Lorentz scalars More complicated scalars Scalars may also be constructed from the tensors and vectors, from the contraction of tensors (such as ), or combinations of contractions of tensors and vectors (such as ). References Concepts in physics Minkowski spacetime Theory of relativity Hendrik Lorentz Scalars