Lorentz scalar
In a relativistic theory of physics, a Lorentz scalar is a scalar expression whose value is invariant under any Lorentz transformation.
A Lorentz scalar may be generated from, e.g., the scalar product of vectors, or by contracting tensors.
While the components of the contracted quantities may change under Lorentz transformations, the Lorentz scalars remain unchanged.
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.
In special relativity the location of a particle in 4-dimensional spacetime is given by
is the position in 3-dimensional space of the particle,
The "length" of the vector is a Lorentz scalar and is given by
is the proper time as measured by a clock in the rest frame of the particle and the Minkowski metric is given by
Often the alternate signature of the Minkowski metric is used in which the signs of the ones are reversed.
In the Minkowski metric the space-like interval
We use the space-like Minkowski metric in the rest of this article.
The velocity in spacetime is defined as
The magnitude of the 4-velocity is a Lorentz scalar,
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 second particle the inner product of
is proportional to the energy of the first particle
Since the relationship is true in the rest frame of the second particle, it is true in any reference frame.
in any inertial reference frame, where
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
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
), or combinations of contractions of tensors and vectors (such as
