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

Famous quotes containing the words special and/or relativity:

    “People generally will soon understand that writers should be judged, not according to rules and species, which are contrary to nature and art, but according to the immutable principles of the art of composition, and the special laws of their individual temperaments.”
    —Victor Hugo (1802–1885)

    “By an application of the theory of relativity to the taste of readers, to-day in Germany I am called a German man of science, and in England I am represented as a Swiss Jew. If I come to be regarded as a bête noire the descriptions will be reversed, and I shall become a Swiss Jew for the Germans and a German man of science for the English!”
    —Albert Einstein (1879–1955)