Is there a way to detect gravitational waves in subatomic particles?

Question

Consider the hypothetical situation of two electrons orbiting each other with a certain radius between them, going at extremely high speeds. Would this create gravitational waves strong enough to be detected? If not, would there be another method in doing so? Also consider that we are in classical physics; no weird quantum stuff.

Answer 1

The dimensionless amplitude of a gravitational wave (known as "the strain") is given approximately by $$ h \sim \frac{2G}{rc^4}\ddot{Q}, $$ where $\ddot{Q}$ is the second derivative of the mass quadrupole moment and $r$ is the distance to the potential source of gravitational waves. It is this strain amplitude that is detected by a gravitational wave detector.

Leaving aside that two electrons wouldn't "orbit each other" gravitationally (however, they could be forced to circle a common centre of mass by applying an appropriate magnetic field), a pair of masses in orbit have a mass quadrupole moment of (just considering one of the possible two polarisations) $$ Q = \mu a^2 (\cos^2 \omega t - 1/3),$$ where $a$ is the separation of the masses and $\mu$ is the "reduced mass" $m_1 m_2/(m_1 + m_2)$, which for equal masses $m_e$, we have $\mu = m_e/2$. The angular frequency $\omega$ is equal to $2\pi$ divided by the orbital period.

Differentiating this twice we get $$ \ddot{Q} = \omega^2 m_e a^2 \cos (2\omega t),$$ and an amplitude of $$ h \sim \frac{2G}{rc^4} \omega^2 m_e a^2.$$

A more useable expression comes from noting that $\omega a/2$ would be the speed of the orbit, leaving $$ h \sim 10^{-57} \left(\frac{r}{1\ {\rm m}}\right)^{-1} \left(\frac{v}{c}\right)^2$$

The best gravitational wave detectors in the world are capable of detecting strains of about $10^{-23}$ at frequencies of $\omega \sim 1000$ rad/s. To get close to $v \sim c$ at this frequency, the electrons would need to circle their centre of mass at a separation of $600$ km (hard to see how you would arrange that) and in order to see them as some coherent source you would need to observe them from considerably more distance than that, i.e. at $r > 10^{6}$ m. The strain produced would then be $\sim 40$ orders of magnitude smaller than could be detected.

You could of course get closer to a smaller source, but to keep the orbital frequency in the detectable band of instruments like LIGO, you would have to reduce the speed to well below that of light.