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Lorentz scalar
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In physics a Lorentz scalar is a scalar which is invariant under a Lorentz transformation. A Lorentz scalar is generated from vectors and tensors. While the vectors and tensors are altered by Lorentz transformations, scalars are unchanged.
e 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 c times the proper time as measured by a clock in the rest frame of the particle and the metric is given by
.
This is a time-like metric.
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Encyclopedia
In physics a Lorentz scalar is a scalar which is invariant under a Lorentz transformation. A Lorentz scalar is generated from vectors and tensors. While the vectors and tensors are altered by Lorentz transformations, scalars are unchanged.
Simple scalars in special relativity
The length of a position vector
In Special relativity the location of a particle in 4-dimensional spacetime is given by its world line
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 c times the proper time as measured by a clock in the rest frame of the particle and the metric is given by
.
This is a time-like metric. Often 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 Minkowski metric in the rest of this article.
The length of a velocity vector
The velocity in spacetime is defined as
where
.
The magnitude of the 4-velocity is a Lorentz scalar and is minus one,
.
The 4-velocity is therefore, not only a representation of the velocity in spacetime, is also a unit vector in the direction of the position of the particle in spacetime.
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
See [Ref. 2, P. 65]. A space-like metric is used.
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 intertial 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 is a Lorentz scalar. The relationship remains true independent of the frame in which the inner product is calculated.
Measurement of the 3-momentum of the particle
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
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, or combinations of contractions of tensors and vectors.
See also
- Albert Einstein
- Fermi-Walker transport
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