= Solution
<Diffusive shock acceleration> is viable in principle at $511\ \mathrm{GeV}$: repeated scattering off magnetic irregularities on opposite sides of a strong nonrelativistic <shock wave> can reach such <energies>. Actual viability requires a sufficiently large and long-lived accelerator with an acceleration time shorter than its escape and loss times; $\gamma$ alone does not establish those conditions.
Take positive <velocity> toward the upstream material, and let the front move at $+v_s$ in the upstream rest frame. For a strong adiabatic <shock wave> in a gas with <adiabatic index> $5/3$, the <Rankine-Hugoniot conditions> give $\rho_d/\rho_u=4$ and the shock-frame inflow <speeds> satisfy $u_1=v_s$, $u_2=v_s/4$. The signed <velocities> in the three frames are
$$
\begin{array}{c|ccc}
\text{rest frame}&v_u&v_d&v_{\rm front}\\ \hline
\text{upstream}&0&3v_s/4&v_s\\
\text{downstream}&-3v_s/4&0&v_s/4\\
\text{front}&-v_s&-v_s/4&0
\end{array}
$$
Thus the two scattering media converge at $V=u_1-u_2=3v_s/4$. For a crossing particle with incident <energy> $E$ in its \i[local plasma rest frame], a <Lorentz transformation> to the approaching other plasma gives
$$
E'=\Gamma_V(E+Vp\mu)=E\left[1+\frac{V}{c}\mu+O(V^2/c^2)\right],
$$
where $\mu>0$ is the incident direction cosine measured toward the approaching medium and $pc\simeq E$. For an almost <isotropic> crossing population, flux weighting gives $\langle\mu\rangle=\int_0^1\mu^2d\mu/\int_0^1\mu d\mu=2/3$. Scattering randomizes directions without changing <energy> in that plasma frame, and the reverse crossing has the same mean gain. Therefore
$$
\boxed{\left\langle\frac{\Delta E}{E}\right\rangle_{\rm crossing}=\frac{2V}{3c},\qquad
\left\langle\frac{\Delta E}{E}\right\rangle_{\rm cycle}=\frac{4V}{3c}=\frac{v_s}{c}.}
$$
These are first-order mean gains. They are not an increase of <energy> in one fixed frame caused by an instantaneous boundary crossing. Literally choosing the <electron>'s instantaneous rest frame gives incident $E=m_ec^2$, and the other gas approaches at nearly $c$, so the expansion in small $V/c$ is invalid. This concrete frame check explains why the printed description of $V$ must instead refer to the upstream/downstream scattering frames.
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