Editing Virial theorem (section)

== In special relativity ==
{{Unreferenced section|date=April 2020}}For a single particle in special relativity, it is not the case that {{math|1=''T'' = {{sfrac|1|2}}'''p''' · '''v'''}}. Instead, it is true that {{math|1=''T'' = (''γ'' − 1) ''mc''<sup>2</sup>}}, where {{mvar|γ}} is the [[Lorentz factor]]
<math display="block">
 \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}},
</math>
and {{math|1='''β''' = {{sfrac|'''v'''|''c''}}}}. We have
<math display="block">\begin{align}
 \frac 12 \mathbf{p} \cdot \mathbf{v}
  &= \frac 12 \boldsymbol{\beta} \gamma mc \cdot \boldsymbol{\beta} c \\
  &= \frac 12 \gamma \beta^2 mc^2 \\[5pt]
  &= \left(\frac{\gamma \beta^2}{2(\gamma - 1)}\right) T.
\end{align}</math>
The last expression can be simplified to
<math display="block">
 \left(\frac{1 + \sqrt{1 - \beta^2}}{2}\right) T = \left(\frac{\gamma + 1}{2 \gamma}\right) T.
</math>
Thus, under the conditions described in earlier sections (including [[Newton's third law of motion]], {{math|1='''F'''<sub>''jk''</sub> = −'''F'''<sub>''kj''</sub>}}, despite relativity), the time average for {{mvar|N}} particles with a power law potential is
<math display="block">
 \frac{n}{2} \left\langle V_\text{TOT} \right\rangle_\tau =
 \left\langle \sum_{k=1}^N \left(\tfrac{1 + \sqrt{1 - \beta_k^2}}{2}\right) T_k \right\rangle_\tau =
 \left\langle \sum_{k=1}^N \left(\frac{\gamma_k + 1}{2 \gamma_k}\right) T_k \right\rangle_\tau.
</math>
In particular, the ratio of kinetic energy to potential energy is no longer fixed, but necessarily falls into an interval:
<math display="block">
 \frac{2 \langle T_\text{TOT} \rangle}{n \langle V_\text{TOT} \rangle} \in [1, 2],</math>
where the more relativistic systems exhibit the larger ratios.