From gamma decay to photon rockets
Suppose an atomic nucleus with mass m₁ emits a gamma ray and ends up with mass m₂. How fast does the nucleus recoil?
In units where the speed of light is 1, the time and space components of the energy-momentum vectors are:
(𝑚₁, 0) for the initial nucleus
(γ, –γ) for the gamma ray
𝑚₂ (1, 𝑣)/√(1–𝑣²) for the recoiling nucleus
where γ, 𝑣 are unknowns.
Conservation of energy-momentum means:
(𝑚₁, 0) = (γ, –γ) + 𝑚₂ (1, 𝑣)/√(1–𝑣²)
This is solved by:
𝑣 = (𝑚₁² – 𝑚₂²) / (𝑚₁² + 𝑚₂²)
γ = (𝑚₁² – 𝑚₂²) / (2 𝑚₁)
We can also solve for the mass ratio:
𝑚₂/𝑚₁ = √[(1–𝑣)/(1+𝑣)]
But this result also gives us the “rocket equation” for an idealised relativistic rocket, that emits only photons as exhaust and works with perfect efficiency.
For example, it tells us that to accelerate such a rocket to half lightspeed, 𝑣=1/2, it will end up with a payload fraction of √[1/3].