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Concept
Lorentz scalar
Summary
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.
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, 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 velocity in spacetime is defined as
where
The magnitude of the 4-velocity is a Lorentz scalar,
Hence, c is a Lorentz scalar.
Official source
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