Gravitational wave for Solo mass system

every equation i found in internet for gravitational wave radiation is for binary system which is need 2 mass in order to find strain (h).

my question is: if we have one mass which is oscillate , how about this one for calculating strain(h) ?

for example: for a single mass 1.000.000 kg & frequency 100 times per second

if there is formula or reference link , let us know.

• en.wikipedia.org/wiki/Quadrupole_formula has the relevant equations. But the configuration of your oscillator is unclear. Is it just moving back & forth on a line? How far does it travel on each oscillation? The calculations for binary systems assume that the centre of mass doesn't move, but that's not the case for your oscillator. Commented Sep 24, 2023 at 12:50
• Define what you mean by a single source and how its mass distribution oscillates. Commented Sep 24, 2023 at 14:00
• @PM2Ring 1-what is 'r' in that formula? 2-i can put kg mass in 'I' in the formula? 3-frequency is not important? 4-this formula give us strain? Commented Sep 24, 2023 at 14:59
• The formula is given in en.wikipedia.org/wiki/Quadrupole_formula . If you don't understand that formula then I suggest you reformulate your question to ask about that. Commented Sep 24, 2023 at 15:41
• Note that you have to conserve momentum, so if you're pushing a mass back and forth, you also have to account for the motion of yourself pushing it. This is basically why gravitational waves are sourced by time-varying mass quadrupoles. Time-varying mass dipoles are not even allowed.
– Sten
Commented Sep 24, 2023 at 16:11

Suppose you have a mass $$m$$ moving up and down along the z-axis according to:

$$z(t) = A \cos\omega t$$

The traceless part of the quadrople moment tensor then has components

\begin{align} I_{xx}^T &= -\frac{1}{3} m A^2 \cos^2\omega t,\\ I_{yy}^T &= -\frac{1}{3} m A^2 \cos^2\omega t,\\ I_{zz}^T &= \frac{2}{3} m A^2 \cos^2\omega t,\\ \end{align}

with all other components zero.

This can be plugged into quadrupole formula:

$$\bar{h}_{ij}(t,r) = \frac{2 G}{c^4 r} \ddot{I}_{ij}(t-r/c)$$

to find the spatial part of the trace reversed perturbation of the metric, $$\bar{h}_{ij}$$.

The average power emitted in gravitational waves is $$\frac{16G}{15c^5} A^4m^2\omega^6.$$

Note that the quadrupole formula is an approximation only valid in the non-relativistic approximation, i.e. $$A\omega\ll c.$$

Suppose I am waving my hand up and down, and want to know the gravitational waves produced. For sake of simplicity we put $$m = 1 kg$$, $$A=1 m$$, and $$\omega = 1Hz$$, the emitted power is then about $$1.5\times10^{-52} W$$. That is very small. To see how small, lets suppose that gravity is in fact quantized. The energy of a single quantum (i.e. graviton) should then be $$\hbar\omega\approx 1\times10^{-34} J$$. So, I'd expect to emitted less then one graviton per Hubble time. So while you might sometimes see a gravitational wave physicist on stage claim that wave their hands about would generate gravitational waves, this might not quite be true due to quantum effects!

• thanks a zillion !!! really good answer. can you calculate strain (h) also so that i can see how formula works for that? for your aboved example Commented Sep 25, 2023 at 16:39
• also what is A ? you meant in 1 square meter deposit??? Commented Sep 25, 2023 at 16:42
• A is the amplitude of the oscillation. Commented Sep 25, 2023 at 18:17
• can also caculate strain (h) for your example ? thanks Commented Sep 25, 2023 at 18:22
• I can, but so can you. Everything you need is here. Commented Sep 25, 2023 at 18:25