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.

Simple scalars in special relativity

Length of a position vector

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 $${\displaystyle x_{\mu }{x}^{\mu }={\eta }_{\mu \nu }{x}^{\mu }{x}^{\nu }=(ct{)}^2-\mathbf{x}\cdot \mathbf{x}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}(c\tau {)}^2}$$ 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 $${\displaystyle {\eta }^{\mu \nu }={\eta }_{\mu \nu }=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right).}$$ 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. $${\displaystyle {\eta }^{\mu \nu }={\eta }_{\mu \nu }=\left(\begin{array}{cccc}-1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right).}$$ This is a space-like metric.

In the Minkowski metric the space-like interval ${\displaystyle s}$ is defined as $${\displaystyle x_{\mu }{x}^{\mu }={\eta }_{\mu \nu }{x}^{\mu }{x}^{\nu }=\mathbf{x}\cdot \mathbf{x}-(ct{)}^2\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}{s}^2.}$$

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

Length of a velocity vector

The velocity in spacetime is defined as $${\displaystyle v^{\mu }\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{d{x}^{\mu }}{d\tau }=\left(c\frac{dt}{d\tau },\frac{dt}{d\tau }\frac{d\mathbf{x}}{dt}\right)=\left(\gamma c,\gamma \mathbf{v}\right)=\gamma \left(c,\mathbf{v}\right)}$$ where $${\displaystyle \gamma \stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{1}{\sqrt{1-\frac{\mathbf{v}\cdot \mathbf{v}}{c^2}}}.}$$

The magnitude of the 4-velocity is a Lorentz scalar, $${\displaystyle v_{\mu }{v}^{\mu }=-c^2.}$$

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

Inner product of acceleration and velocity

The 4-acceleration is given by $${\displaystyle a^{\mu }\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{d{v}^{\mu }}{d\tau }.}$$

The 4-acceleration is always perpendicular to the 4-velocity $${\displaystyle 0=\frac{1}{2}\frac{d}{d\tau }\left(v_{\mu }{v}^{\mu }\right)=\frac{d{v}_{\mu }}{d\tau }{v}^{\mu }=a_{\mu }{v}^{\mu }.}$$

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