**3.1.1 Lorentz transformation of vectors**
Correct. But you need to pay attention

We both need to pay careful attention to exactly what vector transformations we're describing.

the transformation of velocity vectors is

**based** on the

**general** transformation of displacement vectors, see formulas (1.1)-(1.2)

here.

Equation (1.1) comes from Moller:

$$\begin{align}

\vec{r}' &= \vec{r} + \vec{V}(\frac{\gamma-1}{V^2}\vec{r}.\vec{V} - \gamma t) \\

t' &= \gamma(t - \vec{V}.\vec{x}/c^2)

\end{align}$$

This is the general transform of an arbitrary

**four-vector**.

In order to get the transformation for velocity, one needs to perform a couple of divisions, one by $$dt' \ne 0$$ followed by a second one, by $$dt \ne 0$$, see formula (1.3). Therefore, one **cannot** turn around and plug in $$dt=0$$ **or** $$dt' =0$$ into the **general** formula for transformation (1.1) as Moller does.

You certainly can, if you're transforming space-like four-vectors, as Moller does.

While Moller formulas are correct for the **particular** case $$dt=0$$ , you are not allowed to apply it for the problem we are discussing because the derivation of the velocity transformation already **precludes** EITHER $$dt=0$$ OR $$dt'=0$$. You need to proceed in your calculations by using the **general** formula (1.2) for displacement vector transformation, you cannot use the particular formula derived by setting $$dt=0$$.

We already agreed on the velocity transformation.

You don't need differentiation to derive it, since a timelike four-vector easily translates to a velocity 3-vector, but that's irrelevant - we agree on the transformation.

Differentiation is irrelevant when transforming a

*spatial displacement* vector.

On page 47 of Moller, (the specific citation you gave), Moller is explicitly talking about transforming the spatial displacement vector between two points at rest in

**S'**.

To do so, he considers the four-vector between simultaneous events in

**S**, one at each point. Obviously, this four vector has a t value of zero.

Transforming that four-vector to

**S'** and considering the spatial component produces Moller's result:

$$\vec{r}' = \vec{r} + \vec{v}(\gamma-1)\frac{\vec{r}.\vec{v}}{v^2}$$

Note that the t' component of the transformed four-vector is non-zero, but is of no interest because the two points in question are at rest in

**S'**.

So there's no problem with Moller, unless it's misapplied.

You used the inverse of this transformation yourself in your document, so we clearly agree on how to transform the spatial displacement between points which are both at rest in some frame.

We agree on the general transformation of a four-vector.

We agree on the transformation of a velocity vector.

We agree on the transformation of spatial vectors between points stationary in one frame.

So what's the issue?

It seems that the specific issue we're having is about how $$\hat{P_t}(t)$$ should be transformed.

You want to differentiate it and transform the resulting velocity vector.

I want to transform it as a simple spatial vector (actually not quite, but the result turns out to be the same).

I think a new subheading is in order:

**3.1.1.1 Lorentz transformation of vectors**