Lorentz scalar

 ( Concepts In Physics ) 

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, e.g., the scalar product of vectors, or by contracting tensors. 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 ("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.

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Simple scalars in special relativity

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Length of a position vector In special relativity the location of a particle in 4-dimensional spacetime is given by $$x^\backslash mu\; =\; (ct,\; \backslash mathbf\{x\})$$ where $\backslash mathbf\{x\}\; =\; \backslash mathbf\{v\}\; t$ is the position in 3-dimensional space of the particle, $\backslash mathbf\{v\}$ is the velocity in 3-dimensional space and $c$ is the speed of light.

The "length" of the vector is a Lorentz scalar and is given by $$x\_\{\backslash mu\}\; x^\{\backslash mu\}\; =\; \backslash eta\_\{\backslash mu\; \backslash nu\}\; x^\{\backslash mu\}\; x^\{\backslash nu\}\; =\; (ct)^2\; -\; \backslash mathbf\{x\}\; \backslash cdot\; \backslash mathbf\{x\}\; \backslash \; \backslash stackrel\{\backslash mathrm\{def\}\}\{=\}\backslash \; (c\backslash tau)^2$$ where $\backslash tau$ is the proper time as measured by a clock in the rest frame of the particle and the Minkowski metric is given by $$\backslash eta^\{\backslash mu\backslash nu\}\; =\; \backslash eta\_\{\backslash mu\backslash nu\}\; =\; \backslash begin\{pmatrix\}$$

 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. $$\backslash eta^\{\backslash mu\backslash nu\}\; =\; \backslash eta\_\{\backslash mu\backslash nu\}\; =\; \backslash begin\{pmatrix\}$$

 -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 $s$ is defined as $$x\_\{\backslash mu\}\; x^\{\backslash mu\}\; =\; \backslash eta\_\{\backslash mu\; \backslash nu\}\; x^\{\backslash mu\}\; x^\{\backslash nu\}\; =\; \backslash mathbf\{x\}\; \backslash cdot\; \backslash mathbf\{x\}\; -\; (ct)^2\; \backslash \; \backslash stackrel\{\backslash mathrm\{def\}\}\{=\}\backslash \; s^2.$$

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

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Length of a velocity vector The velocity in spacetime is defined as $$v^\{\backslash mu\}\; \backslash \; \backslash stackrel\{\backslash mathrm\{def\}\}\{=\}\backslash \; \{dx^\{\backslash mu\}\; \backslash over\; d\backslash tau\}\; =\; \backslash left(\; c\; \{dt\; \backslash over\; d\backslash tau\},\; \{dt\; \backslash over\; d\backslash tau\}\{d\backslash mathbf\{x\}\; \backslash over\; dt\}\; \backslash right)\; =\; \backslash left(\; \backslash gamma\; c,\; \backslash gamma\; \{\; \backslash mathbf\{v\}\; \}\; \backslash right)\; =\; \backslash gamma\; \backslash left(\; c,\; \{\; \backslash mathbf\{v\}\; \}\; \backslash right)$$ where $$\backslash gamma\; \backslash \; \backslash stackrel\{\backslash mathrm\{def\}\}\{=\}\backslash \; \{\; 1\; \backslash over\; \{\backslash sqrt\; \{1\; -\; \{\; \{\backslash mathbf\{v\}\; \backslash cdot\; \backslash mathbf\{v\}\}\; \backslash over\; c^2\}\; \}\; \}\; \}\; .$$

The magnitude of the 4-velocity is a Lorentz scalar, $$v\_\backslash mu\; v^\backslash mu\; =\; -c^2\backslash ,.$$

Hence, is a Lorentz scalar.

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Inner product of acceleration and velocity The 4-acceleration is given by $$a^\{\backslash mu\}\; \backslash \; \backslash stackrel\{\backslash mathrm\{def\}\}\{=\}\backslash \; \{dv^\{\backslash mu\}\; \backslash over\; d\backslash tau\}.$$

The 4-acceleration is always perpendicular to the 4-velocity $$0\; =\; \{1\; \backslash over\; 2\}\; \{d\; \backslash over\; d\backslash tau\}\; \backslash left(\; v\_\backslash mu\; v^\backslash mu\; \backslash right)\; =\; \{d\; v\_\backslash mu\; \backslash over\; d\backslash tau\}\; v^\backslash mu\; =\; a\_\backslash mu\; v^\backslash mu.$$

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: $$\{d\; E\; \backslash over\; d\backslash tau\}\; =\; \backslash mathbf\{F\}\; \backslash cdot\; \backslash mathbf\{v\}$$ where $E$ is the energy of a particle and $\backslash mathbf\{F\}$ is the 3-force on the particle.

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Energy, rest mass, 3-momentum, and 3-speed from 4-momentum The 4-momentum of a particle is $$p^\backslash mu\; =\; m\; v^\backslash mu\; =\; \backslash left(\; \backslash gamma\; m\; c,\; \backslash gamma\; m\; \backslash mathbf\{v\}\; \backslash right)\; =\; \backslash left(\; \backslash gamma\; m\; c,\; \backslash mathbf\{p\}\; \backslash right)\; =\; \backslash left(\; \backslash frac\; E\; c\; ,\; \backslash mathbf\{p\}\; \backslash right)$$ where $m$ is the particle rest mass, $\backslash mathbf\{p\}$ is the momentum in 3-space, and $$E\; =\; \backslash gamma\; m\; c^2$$ is the energy of the particle.

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Energy of a particle Consider a second particle with 4-velocity $u$ and a 3-velocity $\backslash mathbf\{u\}\_2$. In the rest frame of the second particle the inner product of $u$ with $p$ is proportional to the energy of the first particle $$p\_\backslash mu\; u^\backslash mu\; =\; -\; E\_1$$ 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. $E\_1$, the energy of the first particle in the frame of the second particle, is a Lorentz scalar. Therefore, $$E\_1\; =\; \backslash gamma\_1\; \backslash gamma\_2\; m\_1\; c^2\; -\; \backslash gamma\_2\; \backslash mathbf\{p\}\_1\; \backslash cdot\; \backslash mathbf\{u\}\_2$$ in any inertial reference frame, where $E\_1$ is still the energy of the first particle in the frame of the second particle.

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Rest mass of the particle In the rest frame of the particle the inner product of the momentum is $$p\_\backslash mu\; p^\backslash mu\; =\; -(mc)^2\; \backslash ,.$$

Therefore, 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 $m\_0$ to avoid confusion with the relativistic mass, which is $\backslash gamma\; m\_0$.

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3-momentum of a particle Note that $$\backslash left(\; \backslash frac\{p\_\{\backslash mu\}\; u^\backslash mu\}\{c\}\; \backslash right)^2\; +\; p\_\{\backslash mu\}\; p^\{\backslash mu\}\; =\; \{E\_1^2\; \backslash over\; c^2\}\; -\; (mc)^2\; =\; \backslash left(\; \backslash gamma\_1^2\; -\; 1\; \backslash right)\; (mc)^2\; =\; \backslash gamma\_1^2\; \{\backslash mathbf\{v\}\_1\; \backslash cdot\; \backslash mathbf\{v\}\_1\}\; m^2\; =\; \backslash mathbf\{p\}\_1\; \backslash cdot\; \backslash mathbf\{p\}\_1.$$

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.

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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 $$v\_1^2\; =\; \backslash mathbf\{v\}\_1\; \backslash cdot\; \backslash mathbf\{v\}\_1\; =\; \backslash frac\; \{\; \backslash mathbf\{p\}\_1\; \backslash cdot\; \backslash mathbf\{p\}\_1\; \}\; \{\; E\_1^2\; \}\; c^4.$$

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More complicated scalars Scalars may also be constructed from the tensors and vectors, from the contraction of tensors (such as $F\_\{\backslash mu\backslash nu\}F^\{\backslash mu\backslash nu\}$), or combinations of contractions of tensors and vectors (such as $g\_\{\backslash mu\backslash nu\}x^\{\backslash mu\}x^\{\backslash nu\}$).

• (1973). 9780716703440, W. H. Freeman. ISBN 9780716703440
• (1975). 9780080181769, Pergamon. ISBN 9780080181769

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