LHC: How fast do these protons go?


Context

The Large Hadron Collider has been successfully tested as a particle accelerator on Wednesday 10th September. A beam of protons was accelerated and completed several loops through the whole structure (26,659 m), clockwise and counter-clockwise. (This first tests did not involve particle collisions, and the LHC will not be used as a collider before Spring 2009 because of a serious incident.)
When the power of this machine is discussed, the energy of each proton is often mentioned: The protons each have an energy of 7 TeV. What does that mean?

Einstein to the rescue

At low velocities (conceivable for human beings), the energy of a point object is measured by:

e=1-2mv2.png

However, this formula can not be applied at speeds close to the speed of light. We must use Einstein's theory of Special relativity, which gives:

e=gamma-mc2.png

where m is the mass at rest and γ is the Lorentz factor, defined as

gamma-def.png

When the particle is at rest (v=0), this yields the famous equivalence between mass and energy:

e=mc2.png

Speed of the protons in the LHC

The original version of this article contained a mistake, now fixed; thanks to Ian Bryce for spotting it!

The energy reported by the LHC is only the kinetic energy of the particles, it doesn’t include the rest energy. Indeed, the rest energy of a proton is around 938 MeV, whereas accelerators such as Linac 2 at CERN accelerate particles at 50 MeV.

With E the total energy, KE the kinetic energy, m0 the mass at rest, m the relativistic mass (equal to γ×m0) and E0 the rest energy, we have the following:

v-over-c.png

With E = 7 TeV:

Time frame consequences

The time implications are always fascinating. At such a speed, the particle experiences time differently. The difference is given by:

t-t0-gamma.png

This means that time passes 7460 times more slowly for the particles than it does for us observers. A clock traveling at that speed from Earth to Proxima Centauri would measure a journey time of under 5 hours, while an observer who would remain on Earth would have aged over 4 years (Proxima Centauri is about 4.243 light-years away from us).

Getting even faster is expensive

When getting very close to c, the energy difference between two particles going at almost the same speed can be very large. A famous example of a very high-energy particle detected on Earth is the so-called “Oh-My-God particle”, probably a proton detected at a speed close to 0.9999999999999999999999951 c:

The energy of the Oh My God particle seen by the Fly's Eye is equivalent to 51 joules—enough to light a 40 watt light bulb for more than a second—equivalent, in the words of Utah physicist Pierre Sokolsky,  to “a brick falling on your toe.” The particle's energy is equivalent to an American baseball travelling fifty-five miles an hour.
[...]
After traveling one light year, the particle would be only 0.15 femtoseconds—46 nanometres—behind a photon that left at the same time.
 

This small difference in velocity is responsible for a massive difference in energy. Plotting the Lorentz factor as a function of v shows how fast the energy needs grow when approaching c:

Lorentz factor
γ as a function of v
(Image courtesy of Wikipedia)

I read Stephen Hawking's A Brief History Of Time recently, and it was a fascinating book; It inspired me to look up these theories and learn about it more. I haven't posted here in a long time, and I thought that would be an interesting subject to write on.