LHC Relativistic Energies and Time Dilation at 3.5–14 TeV

Large Hadron Collider and proton energies

The Large Hadron Collider has produced collisions at 7 TeV. For collisions at 7 TeV, each proton must be ramped to 3.5 TeV. The proton has a mass of 1.6726e−27 kg, which corresponds to a rest energy of 938.272 MeV (1 eV = 1.6022e−19 J).

At 3.5 TeV each proton reaches a speed of about 0.999999964c (≈ 11,103.4 revolutions of the LHC per second). Given this ultra-relativistic speed, the relativistic energy expression applies:

E_T= γmc^2

where E_T is total energy and γ = 1/√(1-(v^2/c^2)). γ is the Lorentz factor and quantifies how much an object’s energy increases due to kinetic energy; using this for a 3.5 TeV proton produces a total of E_T = 3.4967 TeV.

Einstein’s full energy-momentum relation

Einstein’s more complete mass–energy relation is also applicable:

E_T=√(m^2c^4+p^2c^2)

Here p is the relativistic momentum, p = γmv (or p = h/λ in the case of light, where h is Planck’s constant and λ is wavelength). Using this form yields a total energy E_T ≈ 3.4959 TeV for the same conditions.

Planned 14 TeV operation and required speeds

CERN aims to conduct collisions at 14 TeV, which would require proton speeds up to about 0.999999991c (≈ (1 − 8.98e−9)c).

Time dilation and the Lorentz factor

Another important relativistic effect for these protons is time dilation. The time-dilation relation is:

τ = t/γ

where τ is the proper time experienced by the moving proton and t is the coordinate time measured in the laboratory (a relatively static frame). At 3.5 TeV the ratio τ/t ≈ 2.6833e−4. That means for every hour (3,600 seconds) that passes in the lab frame, about 0.966 seconds pass for the proton (3,600 × 2.6833e−4 ≈ 0.966 s). At 7 TeV the corresponding proper time for one hour in the lab is about 0.483 seconds.

Minkowski metric derivation (time dilation from spacetime)

The Lorentz-factor result for time dilation follows directly from a basic spacetime interval in Minkowski spacetime:

c^2dτ^2 = c^2dt^2 − dx^2

where τ is proper time of the moving object, t is coordinate time and x is the spatial distance covered. Writing dx^2 = v^2 dt^2 (with v = dx/dt) gives

c^2dτ^2 = c^2dt^2 − v^2dt^2

dτ^2 = (c^2dt^2 − v^2dt^2)/c^2

dτ = √((c^2dt^2 − v^2dt^2)/c^2)

= √(dt^2(1 − v^2/c^2))

= dt √(1 − v^2/c^2)

which is equivalent to τ = t/γ.

Experimental confirmation: atmospheric muons

Time dilation for relativistic subatomic particles is also observed in atmospheric muons (high-energy leptons) produced in cosmic-ray showers. Based on the muon rest lifetime (~2.2e−6 seconds) and a typical velocity of 0.9996678c, a non-relativistic estimate would predict muons to decay within ~660 m. However, time dilation increases their proper lifetime (τ ≈ 0.02577 s in the muon frame), allowing many muons to survive to the Earth’s surface and even penetrate tens of meters of rock before decaying—consistent with relativistic predictions.