Energy & Time Dilation in Large Hadron Collider (LHC)

[stevebd1]

Due to the blogs being removed, I thought it might be worthwhile posting a few in the forums-The Large Hadron Collider has produced collisions at 7 TeV. For collisions at 7 TeV, protons need to be ‘ramped’ to 3.5 TeV, the proton has a mass of 1.6726e−27 kg which, according to mass–energy equivalence (E=mc²), is 938.272 MeV where 1 eV= 1.6022e−19 Joules. The proton will be accelerated to 0.999999964c (11,103.4 revolutions of the LHC per second) which means the following relativistic equation can be used-

E_T= γmc²

where E_T is energy total and γ=1/√(1-(v²/c²)). γ is the Lorentz factor and tells us how much the energy of an object increases due to kinetic energy.

which produces a total of E_T=3.4967 TeV

This is also supported by Einstein’s more complete mass-energy equation-

E_T=√(m²c⁴+p²c²)

where p is momentum and is expressed p=γmv (or p=h/λ in the case of light where h is Planck’s constant and λ is wavelength).

which produces a total of E_T=3.4959 TeV

CERN hope to conduct collisions at 14 TeV which would require speeds of up to 0.999999991c (1-8.98e−9c)

Another interesting aspect is the effect on time for the proton. The relativistic equation for time dilation is-

τ=t/γ

where τ is proper time relative to the proton and t is coordinate time, or time according to a (relatively) static frame. In this case, at energys of 3.5 Tev, the time dilation for the proton is 2.6833e−4 which means for every hour that passes outside the accelerator, only 1 second passes for the proton (0.966 seconds), at 7 TeV, only ½ a second would pass (0.483 seconds). The application of the Lorentz factor to time dilation can be supported by looking at a basic spacetime metric derived from Minkowski space time, accordingly-

c²dτ²=c²dt²-dx²

again, where τ is the proper time of the moving object, t is coordinate time and x is the distance covered. x can be rewritten as-

dx²=v²dt²

where v=dx/dt (i.e. velocity is m/s), v²=dx²/dt² which can be rewritten as above. The spacetime metric can now be written as-

c²dτ²=c²dt²-v²dt²

dτ²=(c²dt²-v²dt²)/c²

dτ=√((c²dt²-v²dt²)/c²)

=√(dt²(1-v²/c²))

= dt√(1-v²/c²)

which is equivalent to τ=t/γ.

Time dilation for relativistic sub-atomic particles is also supported by muons (high energy leptons) which enter the atmosphere from space, according to our clocks, the muon should decay at 660 m into the atmosphere based on a life span of 2.2e−6 seconds and a velocity of 0.9996678c but due to time dilation (τ=0.02577), muons survive the flight to Earth's surface and can penetrate tens of meters of rock before decaying.

 

[Ambitwistor]

Thank you for sharing this information about the Large Hadron Collider and its collisions at 7 TeV. It is fascinating to see the application of Einstein's theories of relativity in the study of sub-atomic particles. The time dilation effect on the protons is particularly interesting, and it is amazing to think that only 1 second passes for the proton for every hour that passes outside the accelerator. I am also intrigued by the potential for even higher energy collisions at 14 TeV and the implications for time dilation and other effects. It is clear that the study of particle physics is constantly pushing the boundaries of our understanding of the universe. Thank you for bringing this topic to our attention in the forum.