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, for example, the scalar product of vectors, or by contracting a tensor. 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 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 a 4-velocity, or the Ricci curvature at a point in spacetime in general relativity, which is a contraction of the Riemann curvature tensor.
Simple scalars in special relativity
Length of a position vector
[Image:Fermi walker 1.png|frame|left|World lines for two particles at different speeds.]In special relativity the location of a particle in 4-dimensional spacetime is given by
where is the position in 3-dimensional space of the particle with respect to a reference event, 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 Minkowski metric is given on a form in which the overall sign is 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.
Length of a velocity vector
[Image:Fermi walker 2.png|frame|left|The velocity vectors in spacetime for a particle at two different speeds. In relativity an acceleration is equivalent to a rotation in spacetime.]The velocity in spacetime is defined as
where
The magnitude of the 4-velocity is a Lorentz scalar,
Hence, is a Lorentz scalar.
Inner product of acceleration and velocity
The 4-acceleration is given byThe 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 iswhere is the particle rest mass, is the momentum in 3-space, and is the energy of the particle.
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 particlewhere 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.
Rest mass of the particle
In the rest frame of the particle the inner product of the momentum isTherefore, 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.
3-momentum of a particle
Note thatThe square of the magnitude of the 3-momentum of the particle as measured in the frame of the second particle is a Lorentz scalar.