Is there an approximation for the Lorentz factor for very large velocities? I am aware of the approximation generally used for low speeds to calculate the Lorentz factor, that being, 
$$\gamma \approx 1 + \frac{1}{2} \left(\frac{v}{c} \right)^2$$
But I need the exact opposite thing -- is there any suitable approximation for when v is extremely close to c?
 A: This is roughly the simplest you can get it:
$$\gamma = \frac{1}{\sqrt{1-v^2/c^2}} = \frac{1}{\sqrt{1-v/c} \sqrt{1+v/c}} \approx \frac{1}{\sqrt{2}} \frac{1}{\sqrt{1-v/c}}.$$
In other words, if $\Delta v$ is how far the speed is below $c$, then 
$$\gamma \approx \sqrt{\frac{c}{2 \Delta v}}.$$
A: For ultra-relativistic particles, $c-v$ basically stops being an experimentally accessible observable. Unless you are extremely careful about timing, you assume that the beam is traveling at $c$ and measure the Lorenz factor by comparing the kinetic energy per particle to the particle mass, $\gamma = E/mc^2$. (If you care about the difference between total energy $E=\gamma m c^2$ and the kinetic energy $K=(\gamma-1)mc^2$, you're not ultra-relativistic yet.)
I first understood this when I got to Jefferson Lab, which has two antiparallel 1 GeV electron linacs connected like a racetrack.  The electrons are injected at 50 MeV. After a lap they're at 2 GeV. After five laps, they're at 10 GeV. The accelerator feeds beam to four different halls at once, each of which can request a different energy by accepting the beam after a different number of passes around the track. So at any given instant while the beam is on, the linac might have beam bunches with five or six different energies interleaved. And --- here's what's interesting to you --- the fast ones don't have different timing than the slow ones. Even the electrons from the 50 MeV injector already have $\gamma = 100$, and there are a bunch of 1% issues that make it hard to distinguish between their speed and exactly $c$.
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