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.
Simple scalars in special relativity
The length of a position vector
where is the position in 3-dimensional space of the particle, is the velocity in 3-dimensional space and is the speed of light.
The "length" of the vector is a Lorentz scalar and is given by
where is the proper time as measured by a clock in the rest frame of the particle and the Minkowski metric is given by
This is a time-like metric.
Often the alternate signature of the Minkowski metric is used in which the signs of the ones are reversed.
This is a space-like metric.
In the Minkowski metric the space-like interval is defined as
We use the space-like Minkowski metric in the rest of this article.
The length of a velocity vector
The velocity in spacetime is defined as
The magnitude of the 4-velocity is a Lorentz scalar,
Hence, c is a Lorentz scalar.
The inner product of acceleration and velocity
The 4-acceleration is given by
The 4-acceleration is always perpendicular to the 4-velocity
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:
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 (m) 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
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 ).