Virial Theorem - in Special Relativity

In Special Relativity

For a single particle in special relativity, it is not the case that . Instead, it is true that and

$\begin{align} \frac 12 \mathbf{p} \cdot \mathbf{v} & = \frac 12 \vec{\beta} \gamma mc \cdot \vec{\beta} c = \frac 12 \gamma \beta^2 mc^2 = \left( \frac{\gamma \beta^2}{2(\gamma-1)}\right) T \,.\end{align}$

The last expression can be simplified to either or .

Thus, under the conditions described in earlier sections (including Newton's third law of motion, despite relativity), the time average for particles with a power law potential is

$\frac n2 \langle V_\mathrm{TOT} \rangle_\tau = \left\langle \sum_{k=1}^N \left(\frac{1 + \sqrt{1-\beta_k^2}}{2}\right) T_k \right\rangle_\tau = \left\langle \sum_{k=1}^N \left(\frac{\gamma_k + 1}{2 \gamma_k}\right) T_k \right\rangle_\tau \,.$

In particular, the ratio of kinetic energy to potential energy is no longer fixed, but necessarily falls into an interval:

where the more relativistic systems exhibit the larger ratios.

Read more about this topic: Virial Theorem

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