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The | In [[physics]], a particle is called '''ultrarelativistic''' when its speed is very close to the speed of light <math>c</math>. | ||
[[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]]). | |||
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): | |||
* 1-v ≈ 1/(2γ<sup>2</sup>) | |||
* E-p = E*(1-v) ≈ m<sup>2</sup>/(2E) = m/(2γ) | |||
* [[rapidity]] φ ≈ ln(2γ) | |||
* 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). | |||
* 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. | |||
== Accuracy of the approximation == | |||
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. | |||
== Other limits == | |||
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>. | |||
== See also == | |||
* [[Classical mechanics]] | |||
* [[Special relativity]] | |||
* [[Aichelburg-Sexl ultraboost]] | |||
== References == | |||
<references/> | |||
[[Category:Special relativity]] | |||
Revision as of 20:29, 18 October 2013
In physics, a particle is called ultrarelativistic when its speed is very close to the speed of light .
Max Planck showed that the relativistic expression for the energy of a particle whose rest mass is and momentum is is given by . The energy of an ultrarelativistic particle is almost completely due to its momentum (), and thus can be approximated by . 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).
In general, the ultrarelativistic limit of an expression is the resulting simplified expression when is assumed. Or, similarly, in the limit where the Lorentz factor is very large ().[1] Here are some ultrarelativistic approximations (in units with c=1):
- 1-v ≈ 1/(2γ2)
- E-p = E*(1-v) ≈ m2/(2E) = m/(2γ)
- rapidity φ ≈ ln(2γ)
- Motion with constant proper acceleration: d ≈ eaτ/(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: ECM ≈ where E1 and E2 are energies of the particle and the target respectively (so E1≫E2), and ECM 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 is about 10%, and for it is just 2%. For particles such as neutrinos, whose γ (Lorentz factor) are usually above 106 ( very close to c), the approximation is essentially exact.
Other limits
The opposite case is a so-called classical particle, where its speed is much smaller than and so its energy can be approximated by .
See also
References
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