Inphysics, a particle is calledultrarelativistic when its speed is very close to the speed of lightc. Notations commonly used are${\displaystyle v\approx c}$ or${\displaystyle \beta \approx 1}$ or${\displaystyle \gamma \gg 1}$ where${\displaystyle \gamma }$ is theLorentz factor,${\displaystyle \beta =v/c}$ and${\displaystyle c}$ is the speed of light.

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

This can result from holding the mass fixed and increasing the kinetic energy to very large values or by holding the energyE fixed and shrinking the massm 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${\displaystyle c}$. The latter is used to derive orbits of massless particles such as thephoton from those of massive particles (cf.Kepler problem in general relativity).^{[citation needed]}

Below are few ultrarelativistic approximations when${\displaystyle \beta \approx 1}$. Therapidity 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), whered is the distance traveled,a =dφ/dτ is proper acceleration (withaτ ≫ 1),τ is proper time, and travel starts at rest and without changing direction of acceleration (seeproper acceleration for more details).
• Fixed target collision with ultrarelativistic motion of the center of mass:E_CM ≈√2E₁E₂ whereE₁ andE₂ are energies of the particle and the target respectively (soE₁ ≫E₂), andE_CM is energy in the center of mass frame.

For calculations of the energy of a particle, therelative error of the ultrarelativistic limit for a speedv = 0.95c is about10%, and forv = 0.99c it is just2%. For particles such asneutrinos, whoseγ (Lorentz factor) are usually above10⁶ (v practically indistinguishable fromc), the approximation is essentially exact.

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

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

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

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