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| In [[physics]], a particle is called '''ultrarelativistic''' when its speed is very close to the speed of light <math>c</math>.
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| [[Max Planck]] showed that the relativistic expression for the energy of a particle whose rest mass is <math> m </math> and momentum is <math> p </math> is given by <math>E^2 = m^2 c^4 + p^2 c^2</math>. The energy of an ultrarelativistic particle is almost completely due to its momentum (<math>p c \gg m c^2</math>), and thus can be approximated by <math>E = p c</math>. This can result from holding the mass fixed and increasing ''p'' to very large values (the usual case); or by holding the energy ''E'' fixed and shrinking the mass ''m'' to negligible values. 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]]).
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| In general, the '''ultrarelativistic limit''' of an expression is the resulting simplified expression when <math>p c \gg m c^2</math> is assumed. Or, similarly, in the limit where the [[Lorentz factor]] is very large (<math>\gamma \gg 1</math>).<ref>{{cite journal | last=Dieckmann | first=ME | title=Particle simulation of an ultrarelativistic two-stream instability | journal=Physical Review Letters | volume=94 | issue=15 | pages=155001 |date=April 2005 | pmid=15904153 | doi=10.1103/PhysRevLett.94.155001 | bibcode=2005PhRvL..94o5001D}}</ref> Here are some ultrarelativistic approximations (in units with c=1):
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| * 1-v ≈ 1/(2γ<sup>2</sup>)
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| * E-p = E*(1-v) ≈ m<sup>2</sup>/(2E) = m/(2γ)
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| * [[rapidity]] φ ≈ ln(2γ)
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| * Motion with constant proper acceleration: d ≈ e<sup>aτ</sup>/(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).
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| * Fixed target collision with ultrarelativistic motion of the center of mass: E<sub>CM</sub> ≈ <math>\sqrt{2E_{1}E_{2}}</math> where E<sub>1</sub> and E<sub>2</sub> are energies of the particle and the target respectively (so E<sub>1</sub>≫E<sub>2</sub>), and E<sub>CM</sub> is energy in the center of mass frame.
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| == Accuracy of the approximation ==
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| For calculations of the energy of a particle, the [[relative error]] of the ultrarelativistic limit for a speed <math>v = 0.95 c</math> is about 10%, and for <math>v = 0.99 c</math> it is just 2%. For particles such as [[neutrinos]], whose γ ([[Lorentz factor]]) are usually above 10<sup>6</sup> (<math>v</math> very close to c), the approximation is essentially exact.
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| == Other limits ==
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| The opposite case is a so-called '''classical particle''', where its speed is much smaller than <math>c</math> and so its energy can be approximated by <math>E = m c^2 + \frac{p^2}{2m}</math>.
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| == See also ==
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| * [[Classical mechanics]]
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| * [[Special relativity]]
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| * [[Aichelburg-Sexl ultraboost]]
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| == References ==
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| <references/>
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| [[Category:Special relativity]]
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