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Ultrarelativistic limit

The energy of an ultrarelativistic particle is almost completely due to its kinetic energy ${\displaystyle E_{k}=(\gamma -1)mc^{2}}$. The total energy can also be approximated as ${\displaystyle E=\gamma mc^{2}\approx pc}$ where ${\displaystyle p=\gamma mv}$ is the Lorentz invariant momentum.

This can result from holding the mass fixed and increasing the kinetic energy to very large values or by holding the energy E fixed and shrinking the mass m to very small values which also imply a very large ${\displaystyle \gamma }$. Particles with a very small mass do not need much energy to travel at a speed close to c. The latter is used to derive orbits of massless particles such as the photon from those of massive particles (cf. Kepler problem in general relativity). ^{[citation needed]}

Ultrarelativistic approximations[edit]

Below are few ultrarelativistic approximations when ${\displaystyle \beta \approx 1}$. The rapidity is denoted ${\displaystyle w}$:

${\displaystyle 1-\beta \approx {\frac {1}{2\gamma ^{2}}}}$

${\displaystyle w\approx \ln(2\gamma )}$

• Motion with constant proper acceleration: d ≈ e^aτ/(2a), where d is the distance traveled, a = dφ/dτ is proper acceleration (with aτ ≫ 1), τ is proper time, and travel starts at rest and without changing direction of acceleration (see proper acceleration for more details).
• Fixed target collision with ultrarelativistic motion of the center of mass: E_CM ≈ √2E₁E₂ where E₁ and E₂ are energies of the particle and the target respectively (soE₁ ≫ E₂), and E_CM is energy in the center of mass frame.

Accuracy of the approximation[edit]

For calculations of the energy of a particle, the relative error of the ultrarelativistic limit for a speed v = 0.95c is about 10%, and for v = 0.99c it is just 2%. For particles such as neutrinos, whose γ (Lorentz factor) are usually above 10⁶ (v practically indistinguishable from c), the approximation is essentially exact.

Other limits[edit]

The opposite case (v ≪ c) is a so-called classical particle, where its speed is much smaller than c. Its kinetic energy can be approximated by first term of the ${\displaystyle \gamma }$ binomial series:

${\displaystyle E_{k}=(\gamma -1)mc^{2}={\frac {1}{2}}mv^{2}+\left[{\frac {3}{8}}m{\frac {v^{4}}{c^{2}}}+...+mc^{2}{\frac {(2n)!}{2^{2n}(n!)^{2}}}{\frac {v^{2n}}{c^{2n}}}+...\right]}$

See also[edit]

• Relativistic particle
• Classical mechanics
• Special relativity
• Aichelburg–Sexl ultraboost

References[edit]

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