The Large Hadron Collider can accelerate protons to 99.9999991% the speed of light. Why can’t we reach 100%?
In: Physics
Using special relativity the formula for the energy of a proton can be found as E = γmc^2. Where m is the mass of the proton and c is the speed of light. γ is an interesting number called the ‘Lorentz factor’ and depends on the speed of the proton.
The Lorentz factor with a speed of zero is equal to 1 and for slow, everyday speeds is basically still equal to 1. This is the familiar E = mc^2 equation and tells you how much energy a particle (like our proton) has when it is stationary, due its mass alone.
As you might expect, as you accelerate this proton faster its energy increases. It now has kinetic energy too. If you’ve done any physics you might be familiar with the equation KE = 1/2 mv^2 for kinetic energy, but really this is an approximation that only works for speeds much slower than c. Instead we stick to γmc^2 to describe the total mass and kinetic energy of the proton.
The problem now is that rather than simply increasing with the square of speed like our old formula for kinetic energy, the gamma factor, and therefore the energy, actually starts to increase much more rapidly as you get close to the speed of light. So to go from 99.991% to 99.992% is a lot more energy than you’d otherwise expect. As you get closer and closer to c, smaller increments in speed require much larger increments in energy in such a way that reaching c would require an infinite amount of energy.
Picture it this way: accelerating the proton is like rolling a ball along a path that represents the speed of our proton. We can roll the ball to any point along our path, except that it has an end, the speed of light. As you get toward the end the hill gets steeper and steeper until it may as well be a vertical wall that goes on forever. You can roll the ball as high up as you like but you can never reach the top
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