Conservation of components of relativistic momentum transverse to applied force

Question

Suppose a particle, say a positron is moving with initial velocity ${\bf v} = (c/3,c/3,c/3)$, where $c$ is the speed of light. Then the relativistic momentum is ${\bf p} = \gamma m{\bf v}$. Now suppose we apply a force, perhaps due to an electric field E $=(0,0,E)$, so the force is F$= (0,0,eE)$. In Newtonian mechanics this force can only change the z component the momentum and the x and y components are conserved as no force acts in their direction.
In the relativistic case I don't understand how the components of the momentum will change as they cannot be independent of each other. Here is my frame of thought. An increase in the $z$ component of the velocity will increase the $\gamma$ factor. But if the momentum in the $x$ and $y$ direction is conserved then the $x$ and $y$ components of the velocity must decrease to compensate for an increasing $\gamma$ factor.
Is this reasoning the physical reality?

Answer 1

Note that 4-velocity can be written:

$$ u^{\mu} =\gamma c (1, \vec{\beta}) $$

so you can read off $\gamma$ and the 3-velocity as long as there is a "1" in time-slot (that is, $\gamma c$ is factored out).

So you have

$$ u^{\mu} = c\sqrt{\frac 3 2}(1, \frac 1 3, \frac 1 3, \frac 1 3) $$

and

$$F^{\mu\nu}=\frac E c\left(\begin{array}{cccc}0&0&0&-1\\0&0&0&0\\0&0&0&0\\1&0&0&0\end{array}\right) $$

The Lorentz force law says:

$$ m\frac{du^{\mu}}{d\tau}=\frac{dp^{\mu}}{d\tau}=qF^{\mu\nu}u_{\nu}$$

$$ a^{\mu}\equiv\frac{du^{\mu}}{d\tau} = \frac{eE}m\sqrt{\frac 1 6}(1, 0, 0, 3)$$

First, note that 4-acceleration is orthogonal to 4-velocity (which is always the case for every force and every particle):

$$ a^{\mu}u_{\mu}=[\frac{eE}m\sqrt{\frac 1 6}(1,0,0,3)][\sqrt{\frac 1 6} c\gamma(3,-1,-1,-1)]=0$$

so that $||u^{\mu}|| = c$ and this $||p^{\mu}|| = mc^2$ after the acceleration (as it must).

Moreover, the transverse component of the 4-velocity and momentum don't change:

$$\frac{d u_1}{d\tau} = \frac{d u_2}{d\tau}=0$$

$$\frac{d p_1}{d\tau} = \frac{d p_2}{d\tau}=0$$

But....

$$u_i = \gamma v_i, \ \ i\in(1,2)$$

and if $\gamma$ changes, the $v_i$ must change.

Let's look at a small change in 4-velocity:

$$ u^{\mu} \rightarrow u'^{\mu} = u^{\mu} + a^{\mu} \delta\tau $$

Then:

$$u'^{\mu} = c\sqrt{\frac 3 2}(1+x, \frac 1 3, \frac 1 3, \frac 1 3+3x)$$

where $ x = (\delta\tau)\frac{eE}{cm}\frac 1{\sqrt 6}\sqrt{\frac 2 3}$.

Doing some rearranging:

$$u'^{\mu} = c\sqrt{\frac 3 2}(1+x)(1, \frac 1 {3(1+x)}, \frac 1 {3(1+x)}, (\frac 1 3+3x)/(1+x))$$

What we see here is that:

$$ \gamma \rightarrow \gamma \cdot (1+x) $$

and the transverse components of 3 velocity:

$$ v_i \rightarrow v_i / (1+x), \ \ \ i\in (1,2)$$

which shows exactly what the OP suspected: the change in Lorentz factor accounts for a change in transverse 3 velocity while conserving transverse momentum.