Relativistic lifetime, energy and Doppler shift
| English | 中文 | Pinyin |
|---|---|---|
| proper time/ˈprɒpə taɪm/ | 固有时 | gù yǒu shí |
| rest energy | 静能 | jìng néng |
A decision before an answer
- A fast unstable particle can travel farther in the laboratory than its rest-frame lifetime multiplied by its speed would suggest.
- Your goal: Relate proper lifetime to laboratory time and distance.
Read the relationship
- For relative speed v, define β=v/c and γ=1/sqrt(1−β²). Proper time 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.
- Use invariant energy–momentum and relativistic kinetic energy.
A particle at 0.8c has proper mean lifetime 6 μs. Its laboratory mean lifetime is:
γ=5/3, so γτ0=10 μs.
Use the defining rule
- Total energy 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 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.
- Infer longitudinal recession speed from wavelength ratio.
For E=10 GeV and pc=6 GeV, rest energy is:
sqrt(100−36)=8 GeV.
Check the conditions
- 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.
- Infer longitudinal recession speed from wavelength ratio.
A particle has proper mean lifetime 3 μs and moves at 0.8c. Its lab lifetime is 5 μs and mean travel distance is 0.8·3×10⁸·5×10⁻⁶=1200 m. A longitudinal wavelength ratio 2 implies recession speed 0.6c. A particle with total energy 13 GeV and pc=12 GeV has rest energy 5 GeV.
At speed 0.6c, K/(mc²)=____ (decimal).
γ=1.25, so γ−1=0.25.
Apply the task format
- 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.
- Infer longitudinal recession speed from wavelength ratio.
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.
Which answer fits this case?
Relate proper lifetime to laboratory time and distance
A wavelength redshift ratio of 2 implies superluminal recession under the stipulated longitudinal special-relativity model.
β=(4−1)/(4+1)=0.6.
Keep the distinctions
- proper time 固有时 — Time recorded by a clock moving with the object along its worldline.
- rest energy 静能 — Energy mc² associated with an object’s rest mass.
- Relate proper lifetime to laboratory time and distance.
- Use invariant energy–momentum and relativistic kinetic energy.
- Infer longitudinal recession speed from wavelength ratio.
Match each term with its precise meaning in this lesson.
Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
The order follows the stated process; check each stage before the next.