Contents

Lorentz scalar

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.

Inspecial 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 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, ⁠${\displaystyle c}$⁠ is a Lorentz scalar.

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.

The 4-momentum of a particle is $${\displaystyle p^{\mu }=mv^{\mu }=\left(\gamma mc,\gamma m\mathbf {v} \right)=\left(\gamma mc,\mathbf {p} \right)=\left({\frac {E}{c}},\mathbf {p} \right)}$$ 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.

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.

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 ${\displaystyle \gamma m_{0}}$.

Note that $${\displaystyle \left({\frac {p_{\mu }u^{\mu }}{c}}\right)^{2}+p_{\mu }p^{\mu }={E_{1}^{2} \over c^{2}}-($$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.

The 3-speed, in the frame of the second particle, can be constructed from two Lorentz scalars $${\displaystyle v_{1}^{2}=\mathbf {v} _{1}\cdot \mathbf {v} _{1}={\frac {\mathbf {p} _{1}\cdot \mathbf {p} _{1}}{E_{1}^{2}}}c^{4}.}$$

Scalars may also be constructed from the tensors and vectors, from the contraction of tensors (such as ${\displaystyle F_{\mu \nu }F^{\mu \nu }}$), or combinations of contractions of tensors and vectors (such as ${\displaystyle g_{\mu \nu }x^{\mu }x^{\nu }}$).

• 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 ed.). Oxford: Pergamon. ISBN 0-08-018176-7.

• Media related to Lorentz scalar at Wikimedia Commons