Test particle

In physical theories, a test particle is an idealized model of an object whose physical properties (usually mass, charge, or size) are assumed to be negligible except for the property being studied, which is considered to be insufficient to alter the behavior of the rest of the system. The concept of a test particle often simplifies problems, and can provide a good approximation for physical phenomena. In addition to its uses in the simplification of the dynamics of a system in particular limits, it is also used as a diagnostic in computer simulations of physical processes.

Classical gravity

The easiest case for the application of a test particle arises in Newtonian gravity. The general expression for the gravitational force between two masses $m_{1}$ and $m_{2}$ is:

$F(r)=-G{\frac {m_{1}m_{2}}{(r_{1}-r_{2})^{2}}}$

where $r_{1}$ and $r_{2}$ represent the position of each particle in space. In the general solution for this equation, both masses rotate around their center of mass, in this specific case:

$R={\frac {m_{1}r_{1}+m_{2}r_{2}}{m_{1}+m_{2}}}$^[1]

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. By defining the gravitational field as

$g(r)={\frac {Gm_{1}}{r^{2}}}$

with $r$ as the distance between the two objects, the equation for the motion of the smaller mass reduces to

$a(r)={\frac {F(r)}{m_{2}}}=-g(r)$

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.

Test particles in 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 equation, 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 (spinning or not), spinning test particles may experience additional accelerations due to spin-spin forces.^[2]

Test particles in plasma physics or electrodynamics

In simulations with electromagnetic fields the most important characteristics of a test particle is its electric charge and its mass. In this situation it is often referred to as a test charge.

An electric field is defined by ${\textbf {E}}=k{\frac {q}{r^{2}}}{\hat {r}}$. Multiplying the field by a test charge $q_{{\textrm {test}}}$ gives an electric force exerted by the field on a test charge. Note that both the force and the electric field are vector quantities, so a positive test charge will experience a force in the direction of the electric field.

In a magnetic field, the behavior of a test charge is determined by effects of special relativity described by the Lorentz force. In this case, a positive test charge will be deflected clockwise if moving perpendicular to a magnetic field pointing toward you, and counterclockwise if moving perpendicular to a magnetic field directed away from you.

See also

• Magnetogravitic tensor and the Bel decomposition of the Riemann tensor
• Papapetrou-Dixon equations
• Point mass
• Point charge

References

1. Herbert Goldstein (1980). Classical Mechanics, 2nd Ed. Addison-Wesley. p. 5.
2. Poisson, Eric. "The Motion of Point Particles in Curved Spacetime". Living Reviews in Relativity. Retrieved March 26, 2004.