In a relativistic theory of physics, a Lorentz scalar is an expression, formed from items of the theory, which evaluates to a scalar, invariant under any Lorentz transformation. A Lorentz scalar may be generated from e.g., the scalar product of vectors, or from contracting tensors of the theory. While the components of vectors and tensors are in general altered under Lorentz transformations, Lorentz scalars remain unchanged.
A Lorentz scalar is not always immediately seen to be an invariant scalar in the mathematical sense, but the resulting scalar value is invariant under any basis transformation applied to the vector space, on which the considered theory is based. A simple Lorentz scalar in Minkowski spacetime is the spacetime distance ("length" of their difference) of two fixed events in spacetime. While the "position"-4-vectors of the events change between different inertial frames, their spacetime distance remains invariant under the corresponding Lorentz transformation. Other examples of Lorentz scalars are the "length" of 4-velocities (see below), or the Ricci curvature in a point in spacetime from General relativity, which is a contraction of the Riemann curvature tensor there.
Often the alternate signature of the Minkowski metric is used in which the signs of the ones are reversed.
Therefore, we can regard acceleration in spacetime as simply a rotation of the 4-velocity. The inner product of the acceleration and the velocity is a Lorentz scalar and is zero. This rotation is simply an expression of energy conservation:
In the rest frame of the particle the inner product of the momentum is
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.
The 3-speed, in the frame of the second particle, can be constructed from two Lorentz scalars