RA.1 · Relativistic lifetime, energy and Doppler shift

A fast unstable particle can travel farther in the laboratory than its rest-frame lifetime multiplied by its speed would suggest.

Prerequisites: 6, 42.

• Relate proper lifetime to laboratory time and distance
• Use invariant energy–momentum and relativistic kinetic energy
• Infer longitudinal recession speed from wavelength ratio

For relative speed v, define β=v/c and γ=1/sqrt(1−β²). Proper time 固有时 is measured along the particle’s worldline by a clock at rest with it. If its proper mean lifetime is τ0, its laboratory mean lifetime is γτ0 and mean travel distance is vγτ0 at constant speed. The decay is statistical: mean lifetime is not a guaranteed decay time for each particle. At β=0.8, γ=5/3. Compute spacetime events consistently in one frame; proper time and coordinate time are different quantities.

Total energy 总能量 is E=γmc², momentum p=γmv and invariant E²−p²c²=m²c⁴. When E and pc are expressed in the same energy units, find mc²=sqrt(E²−(pc)²), not E−pc. The nonnegative square root is required for positive rest mass. For E=13 GeV and pc=12 GeV, rest energy 静能 is 5 GeV and mass is 5 GeV/c². A massless particle has E=pc but need not have zero energy or momentum. These relations concern isolated-particle four-momentum, not classical mv at relativistic speed.

Work accelerating a particle from rest equals kinetic energy K=E−mc²=(γ−1)mc². At β=0.6, γ=1.25 and K=0.25mc², while the classical ½mv² gives 0.18mc². Classical kinetic energy is the low-speed expansion and becomes inaccurate near c. Finite acceleration work increases γ rather than allowing a massive particle to reach or exceed c. Keep total energy, rest energy and kinetic energy distinct when interpreting answer units.

For purely longitudinal relative recession in special relativity, wavelength ratio r=λ_observed/λ_emitted=sqrt((1+β)/(1−β)). Rearranging gives β=(r²−1)/(r²+1). A ratio r=2 gives β=3/5, not c times r−1 from a low-speed approximation. Blueshift uses r<1 and a negative recession parameter under this convention. The formula assumes the shift is entirely kinematic; cosmological expansion or gravitational redshift requires a different model. State the question’s stipulated model before interpreting a spectral ratio.

Total energy and momentum determine invariant rest energy.

For E = 17 GeV and pc = 15 GeV, Rest mass is 8 GeV/c squared. Do not subtract momentum directly from energy.

Do not use the proper lifetime as laboratory time, subtract pc from E to find rest mass, or apply the classical Doppler approximation to a large wavelength ratio.

proper time: Time recorded by a clock moving with the object along its worldline.

rest energy: Energy mc² associated with an object’s rest mass.