Lorentz scalar

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In a relativistic theory of physics, a Lorentz scalar is a scalar which is invariant under a Lorentz transformation. A Lorentz scalar may be generated from multiplication of vectors or tensors.[further explanation needed] While the components of vectors and tensors are in general altered by Lorentz transformations, scalars remain unchanged.

A Lorentz scalar is not a scalar in the mathematical sense, that is, invariant under any basis transformation. For example, the determinant of the matrix of basis vectors[clarification needed] is a number that is invariant under Lorentz transformations, but it is not invariant under any basis transformation.[clarification needed]

Simple scalars in special relativity[edit]

The length of a position vector[edit]

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, 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[edit]

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, c is a Lorentz scalar.

The inner product of acceleration and velocity[edit]

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[edit]

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[edit]

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[edit]

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[edit]

Note that

.

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[edit]

The 3-speed, in the frame of the second particle, can be constructed from two Lorentz scalars

.

More complicated scalars[edit]

Scalars may also be constructed from the tensors and vectors, from the contraction of tensors, or combinations of contractions of tensors and vectors.

References[edit]

  • Misner, Charles; Thorne, Kip S. & Wheeler, John Archibald (1973). Gravitation. San Francisco: W. H. Freeman. ISBN 0-7167-0344-0. 
  • Landau, L. D. & Lifshitz, E. M. (1975). Classical Theory of Fields (Fourth Revised English Edition). Oxford: Pergamon. ISBN 0-08-018176-7.