Composition of Velocities

In classical mechanics, a velocity ${\text{u}}^{\prime }$ in the moving frame will be measured in the stationary frame as $\text{u}=\text{v}+{\text{u}}^{\prime }$, which is the well-known law of the parallelograms. In relativity, the parallelogram formula holds only for velocities well below the speed of light, and the composition of velocities needs to account for the fact that no speed can be measured, which is faster than the speed of light. We will use the same reference systems as defined in the section of The Transformation of Time and Space. We consider the general motion of a body in $F^{\prime }$ described by the four-vectorwhere we assume that ${\text{x}}^{\prime }={\text{x}}^{\prime }(t^{\prime })$. The motion of the body observed in $F$ is given by the Lorentz transformationwhere $\mathbf{\Lambda }(\text{v})$ is the Loading....and $\text{x}=\text{x}(t)$. The velocity of the body in $F$ is given by the derivative of the position vector with respect to timeFrom the transformation of time and space,we can compute the relation between the velocities $\text{u}$ and ${\text{u}}^{\prime }$ in $F$ and $F^{\prime }$ respectively,which using the Lorentz transformation in velocity, readsand simplifies to,The equation above can be written in a more compact form as,or,from which we readily obtain the velocity ${|\text{u}⟩}^{\prime }$ as,The velocity $|\text{u}⟩$ can be easliy expressed in terms of the velocity $|{\text{u}}^{\prime }⟩$ by exchanging the roles of $|\text{u}⟩$ and $|{\text{u}}^{\prime }⟩$ in the equation above, and changing the sign of the velocity.
As an exercise, we will derive the same result in a more general way by inverting the equation above. We start by factoring the velocity $|\text{u}⟩$ in Loading...and express the equation as an operator on the velocity $|\text{u}⟩$,Further expanding the equation above in the components of $|{\text{u}}^{\prime }⟩$ , we getThe operator,can be easily inverted as,We can express the velocity $|\text{u}⟩$ as,

which finally simplifies to,

By dropping the explicit time dependence, we can write the above equation in a more elengant form as,

In this section we are going to derive the limit formulas for the composition of velocities when one of the velocities approache the speed of light. We start by considering the limit of the velocity composition formula Loading... when the velocity $|{\text{u}}^{\prime }⟩$ approaches the speed of light.

$|\text{u}⟩$ Approaching Speed of Light

We start by first considering the case where $|\text{u}⟩$ approaches the speed of light $c$ and we will determine the value that $|{\text{u}}^{\prime }⟩$ reaches. To achieve this, we need to compute the module of $|{\text{u}}^{\prime }⟩$ from Loading..., which reads

$$\begin{array}{c}{u}^{\prime 2}=\frac{1}{{\left(1-\frac{⟨\text{v}|\text{u}⟩}{c^2}\right)}^2}\left(\frac{{⟨\text{u}|\text{v}⟩}^2}{c^2}+v^2+\frac{u^2}{{\beta }^2}-2⟨\text{u}|\text{v}⟩\right)\end{array}$$

when $|\text{u}⟩=c|\text{n}⟩$, where $|\text{n}⟩$ is a unitary vector, the equation above simplifies to

$$\begin{array}{c}{u}^{\prime 2}=\frac{c^2}{{\left(1-\frac{⟨\text{v}|\text{n}⟩}{c}\right)}^2}{\left(1-\frac{⟨\text{v}|\text{n}⟩}{c}\right)}^2=c^2\end{array}$$

which implies that the velocity $|{\text{u}}^{\prime }⟩$ reaches the speed of light when $|\text{u}⟩$ approaches the speed of light.

$|{\text{u}}^{\prime }⟩$ Approaching Speed of Light

Similarly, we can determine the limit of the velocity $|\text{u}⟩$ when the velocity $|{\text{u}}^{\prime }⟩$ approaches the speed of light. From the velocity composition formula Loading..., the module of the velocity $|\text{u}⟩$ reads,

$$\begin{array}{c}{u}^2=\frac{1}{{\left(1+\frac{⟨\text{v}|{\text{u}}^{\prime }⟩}{c^2}\right)}^2}\left(\frac{{⟨{\text{u}}^{\prime }|\text{v}⟩}^2}{v^2}+v^2+\frac{u^{\prime 2}}{{\beta }^2}+2⟨{\text{u}}^{\prime }|\text{v}⟩\right)\end{array}$$

which is the same equation for the module of $|\text{u}⟩$ with the sign of the velocity $|\text{v}⟩$ reversed. When $|{\text{u}}^{\prime }⟩=c|\text{n}⟩$, where $|\text{n}⟩$ is a unitary vector, the equation above simplifies to

$$\begin{array}{c}{u}^2=\frac{c^2}{{\left(1+\frac{⟨\text{v}|\text{n}⟩}{c}\right)}^2}{\left(1+\frac{⟨\text{v}|\text{n}⟩}{c}\right)}^2=c^2\end{array}$$

which, also in this case, means that the velocity $|\text{u}⟩$ reaches the speed of light when $|{\text{u}}^{\prime }⟩$ approaches the speed of light.

$|\text{v}⟩$ Approaching Speed of Light

Following the same logic, when the velocity $|\text{v}⟩=c|\text{n}⟩$ , the module of the velocity $|\text{u}⟩$ reads,

$$\begin{array}{c}{u}^2=\frac{c^2}{{\left(1+\frac{⟨{\text{u}}^{\prime }|\text{n}⟩}{c}\right)}^2}{\left(1+\frac{⟨{\text{u}}^{\prime }|\text{n}⟩}{c}\right)}^2=c^2\end{array}$$

independently of the value of the velocity $|{\text{u}}^{\prime }⟩$.