# Lorentz scalar

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

### The length of a position vector

World lines for two particles at different speeds.

In special relativity the location of a particle in 4-dimensional spacetime is given by

${\displaystyle x^{\mu }=(ct,\mathbf {x} )}$

where ${\displaystyle \mathbf {x} =\mathbf {v} t}$ is the position in 3-dimensional space of the particle, ${\displaystyle \mathbf {v} }$ is the velocity in 3-dimensional space and ${\displaystyle c}$ is the speed of light.

The "length" of the vector is a Lorentz scalar and is given by

${\displaystyle x_{\mu }x^{\mu }=\eta _{\mu \nu }x^{\mu }x^{\nu }=(ct)^{2}-\mathbf {x} \cdot \mathbf {x} \ {\stackrel {\mathrm {def} }{=}}\ (c\tau )^{2}}$

where ${\displaystyle \tau }$ is the proper time as measured by a clock in the rest frame of the particle and the Minkowski metric is given by

${\displaystyle \eta ^{\mu \nu }=\eta _{\mu \nu }={\begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}}}$.

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.

${\displaystyle \eta ^{\mu \nu }=\eta _{\mu \nu }={\begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}}$.

This is a space-like metric.

In the Minkowski metric the space-like interval ${\displaystyle s}$ is defined as

${\displaystyle x_{\mu }x^{\mu }=\eta _{\mu \nu }x^{\mu }x^{\nu }=\mathbf {x} \cdot \mathbf {x} -(ct)^{2}\ {\stackrel {\mathrm {def} }{=}}\ s^{2}}$.

We use the space-like Minkowski metric in the rest of this article.

### The length of a velocity vector

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

${\displaystyle v^{\mu }\ {\stackrel {\mathrm {def} }{=}}\ {dx^{\mu } \over d\tau }=\left(c{dt \over d\tau },{dt \over d\tau }{d\mathbf {x} \over dt}\right)=\left(\gamma c,\gamma {\mathbf {v} }\right)=\gamma \left(c,{\mathbf {v} }\right)}$

where

${\displaystyle \gamma \ {\stackrel {\mathrm {def} }{=}}\ {1 \over {\sqrt {1-{{\mathbf {v} \cdot \mathbf {v} } \over c^{2}}}}}}$.

The magnitude of the 4-velocity is a Lorentz scalar,

${\displaystyle v_{\mu }v^{\mu }=-c^{2}\,}$.

Hence, c is a Lorentz scalar.

### The inner product of acceleration and velocity

The 4-acceleration is given by

${\displaystyle a^{\mu }\ {\stackrel {\mathrm {def} }{=}}\ {dv^{\mu } \over d\tau }}$.

The 4-acceleration is always perpendicular to the 4-velocity

${\displaystyle 0={1 \over 2}{d \over d\tau }\left(v_{\mu }v^{\mu }\right)={dv_{\mu } \over d\tau }v^{\mu }=a_{\mu }v^{\mu }}$.

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:

${\displaystyle {dE \over d\tau }=\mathbf {F} \cdot {\mathbf {v} }}$

where ${\displaystyle E}$ is the energy of a particle and ${\displaystyle \mathbf {F} }$ is the 3-force on the particle.

## Energy, rest mass, 3-momentum, and 3-speed from 4-momentum

The 4-momentum of a particle is

${\displaystyle p^{\mu }=mv^{\mu }=\left(\gamma mc,\gamma {m\mathbf {v} }\right)=\left(\gamma mc,{\mathbf {p} }\right)=\left({E \over c},{\mathbf {p} }\right)}$

where ${\displaystyle m}$ is the particle rest mass, ${\displaystyle \mathbf {p} }$ is the momentum in 3-space, and

${\displaystyle E=\gamma mc^{2}\,}$

is the energy of the particle.

### Measurement of the energy of a particle

Consider a second particle with 4-velocity ${\displaystyle u}$ and a 3-velocity ${\displaystyle \mathbf {u} _{2}}$. In the rest frame of the second particle the inner product of ${\displaystyle u}$ with ${\displaystyle p}$ is proportional to the energy of the first particle

${\displaystyle p_{\mu }u^{\mu }=-{E_{1}}}$

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. ${\displaystyle E_{1}}$, the energy of the first particle in the frame of the second particle, is a Lorentz scalar. Therefore,

${\displaystyle {E_{1}}=\gamma _{1}\gamma _{2}m_{1}c^{2}-\gamma _{2}\mathbf {p} _{1}\cdot \mathbf {u} _{2}}$

in any inertial reference frame, where ${\displaystyle E_{1}}$ 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

${\displaystyle p_{\mu }p^{\mu }=-(mc)^{2}\,}$.

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 ${\displaystyle m_{0}}$ to avoid confusion with the relativistic mass, which is ${\displaystyle \gamma m_{0}}$

### Measurement of the 3-momentum of the particle

Note that

${\displaystyle \left(p_{\mu }u^{\mu }/c\right)^{2}+p_{\mu }p^{\mu }={E_{1}^{2} \over c^{2}}-(mc)^{2}=\left(\gamma _{1}^{2}-1\right)(mc)^{2}=\gamma _{1}^{2}{\mathbf {v} _{1}\cdot \mathbf {v} _{1}}m^{2}=\mathbf {p} _{1}\cdot \mathbf {p} _{1}}$.

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

${\displaystyle v_{1}^{2}=\mathbf {v} _{1}\cdot \mathbf {v} _{1}={{\mathbf {p} _{1}\cdot \mathbf {p} _{1}c^{4}} \over {E_{1}^{2}}}}$.

## 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.

## References

• 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.