Ultrarelativistic limit

Short description: Very close to the speed of light

In physics, a particle is called ultrarelativistic when its speed is very close to the speed of light c. Notation commonly used are [math]\displaystyle{ v \approx c }[/math] or [math]\displaystyle{ \beta \approx 1 }[/math] or [math]\displaystyle{ \gamma \gg 1 }[/math] where [math]\displaystyle{ \gamma }[/math] the Lorentz factor, [math]\displaystyle{ \beta = v/c }[/math] and c the speed of light.

The energy of an ultrarelativistic particle is almost completely due to its kinetic energy [math]\displaystyle{ E_k = (\gamma - 1) m c^2 }[/math]. The total energy can also be approximated as [math]\displaystyle{ E = \gamma m c^2 \approx pc }[/math] where [math]\displaystyle{ p = \gamma m v }[/math] 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 [math]\displaystyle{ \gamma }[/math]. Particles with a very small mass does 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).

Ultrarelativistic approximations

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

[math]\displaystyle{ 1 - \beta \approx \frac{1}{2\gamma^2} }[/math]

[math]\displaystyle{ w \approx ln(2 \gamma) }[/math]

• 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 (so E₁ ≫ E₂), and E_CM is energy in the center of mass frame.

Accuracy of the approximation

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

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 [math]\displaystyle{ \gamma }[/math] binomial series:

[math]\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] }[/math]

See also

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