Lorentz scalar

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 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 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.

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

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