Signs in relative vector addition

So I know the formula for relative vector addition is $$w = \frac{v-u}{1-\frac{uv}{c^2}}.$$

1. How do I chose when to use this formula or its inverse transformation for solving vector addition problems in special relativity.

2. In which direction/scenario is the sign of u negative?

• Gallie $v_{3}=v_{1}+v_{2}$ SRT sign with Lorentz transformation $\Lambda _{3}=\Lambda _{2}\Lambda _{1}$ – Eli Mar 13 at 8:09

When I was studying for the PGRE I came up with my own way of writing the relativistic velocity addition law. I'm sure I'm not the only/first person to think of it this way, but I found it to be quite useful.

The way I do it is to first not assume any sign dependence. In which case it looks like this

$$v_{ac} = \frac{v_{ab} + v_{bc}}{1+ \frac{v_{ab}v_{bc}}{c^2}}.$$

where $$v_{ij}$$ is "the velocity of $$i$$ with respect to observer $$j$$". Then draw your coordinate system and account for signs accordingly.

When writing it this way it is purely just regular vector addition, and is no different than Galilean velocity addition.

• I thing the equation is also correct if v is a vector – Eli Mar 13 at 12:24