Test particle

Classical gravity

The easiest case for the application of a test particle arises in Newton's law of universal gravitation. The general expression for the gravitational force between any two point masses ${\displaystyle m_1}$ and ${\displaystyle m_2}$ is:

${\displaystyle F=-G\frac{m_1{m}_2}{|{\mathbf{r}}_1-{\mathbf{r}}_2{|}^2}}$,

where ${\displaystyle {\mathbf{r}}_1}$ and ${\displaystyle {\mathbf{r}}_2}$ represent the position of each particle in space. In the general solution for this equation, both masses rotate around their center of mass R, in this specific case:

${\displaystyle \mathbf{R}=\frac{m_1{\mathbf{r}}_1+m_2{\mathbf{r}}_2}{m_1+m_2}}$.

In the case where one of the masses is much larger than the other (${\displaystyle m_1\gg m_2}$), one can assume that the smaller mass moves as a test particle in a gravitational field generated by the larger mass, which does not accelerate. We can define the gravitational field as

${\displaystyle \mathbf{g}(r)=-\frac{G{m}_1}{r^2}\hat{\mathbf{r}}}$,

with ${\displaystyle r}$ as the distance between the massive object and the test particle, and ${\displaystyle \hat{r}}$ is the unit vector in the direction going from the massive object to the test mass. Newton's second law of motion of the smaller mass reduces to

${\displaystyle \mathbf{a}(r)=\frac{F}{m_2}\hat{\mathbf{r}}=\mathbf{g}(r)}$,

and thus only contains one variable, for which the solution can be calculated more easily. This approach gives very good approximations for many practical problems, e.g. the orbits of satellites, whose mass is relatively small compared to that of the Earth.

General relativity

In metric theories of gravitation, particularly general relativity, a test particle is an idealized model of a small object whose mass is so small that it does not appreciably disturb the ambient gravitational field.

According to the Einstein field equations, the gravitational field is locally coupled not only to the distribution of non-gravitational mass–energy, but also to the distribution of momentum and stress (e.g. pressure, viscous stresses in a perfect fluid).

In the case of test particles in a vacuum solution or electrovacuum solution, this turns out to imply that in addition to the tidal acceleration experienced by small clouds of test particles (with spin or not), test particles with spin may experience additional accelerations due to spin–spin forces.