<p>Suppose you have a mass <span class="math-container">$m$</span> moving up and down along the z-axis according to:</p>
<p><span class="math-container">$$ z(t) = A \cos\omega t $$</span></p>
<p>The traceless part of the quadrople moment tensor then has components</p>
<p><span class="math-container">\begin{align}
I_{xx}^T &amp;= -\frac{1}{3} m A^2 \cos^2\omega t,\\
I_{yy}^T &amp;= -\frac{1}{3} m A^2 \cos^2\omega t,\\
I_{zz}^T &amp;= \frac{2}{3} m A^2 \cos^2\omega t,\\
\end{align}</span></p>
<p>with all other components zero.</p>
<p>This can be plugged into <a href="https://en.wikipedia.org/wiki/Quadrupole_formula" rel="nofollow noreferrer">quadrupole formula</a>:</p>
<p><span class="math-container">$$ \bar{h}_{ij}(t,r) = \frac{2 G}{c^4 r} \ddot{I}_{ij}(t-r/c)$$</span></p>
<p>to find the spatial part of the trace reversed perturbation of the metric, <span class="math-container">$\bar{h}_{ij}$</span>.</p>
<p>The average power emitted in gravitational waves is
<span class="math-container">$$\frac{16G}{15c^5} A^4m^2\omega^6.$$</span></p>
<p>Note that the quadrupole formula is an approximation only valid in the non-relativistic approximation, i.e. <span class="math-container">$A\omega\ll c.$</span></p>
<p>Suppose I am waving my hand up and down, and want to know the gravitational waves produced. For sake of simplicity we put <span class="math-container">$m = 1 kg$</span>, <span class="math-container">$A=1 m$</span>, and <span class="math-container">$\omega = 1Hz$</span>, the emitted power is then about <span class="math-container">$1.5\times10^{-52} W$</span>. 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 <span class="math-container">$\hbar\omega\approx 1\times10^{-34} J$</span>. So, I'd expect to emitted less then one graviton per <a href="https://en.wikipedia.org/wiki/Hubble%27s_law#Hubble_time" rel="nofollow noreferrer">Hubble time</a>. 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!</p>

CC BY-SA 4.0
2023-09-25T16:39:39.670Z
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
Hamidreza Abdollahi
CC BY-SA 4.0
2023-09-25T16:42:29.980Z
also what is A ? you meant in 1 square meter deposit???
Hamidreza Abdollahi
CC BY-SA 4.0
2023-09-25T18:17:47.260Z
A is the amplitude of the oscillation.
TimRias
CC BY-SA 4.0
2023-09-25T18:22:07.360Z
can also caculate strain (h) for your example ? thanks
Hamidreza Abdollahi
CC BY-SA 4.0
2023-09-25T18:25:28.440Z
I can, but so can you. Everything you need is here.
TimRias
CC BY-SA 4.0
2023-09-25T18:48:41.620Z
i do not know what is 'r'  and what is 't' in that formula
Hamidreza Abdollahi
CC BY-SA 4.0
2023-09-27T08:13:02.380Z
after calculate each component, i must sum ix,iy, iz together in order to place in strain fromula.?
Hamidreza Abdollahi
CC BY-SA 4.0
2023-09-25T13:03:29.920Z
2023-09-25T13:16:43.063Z
2023-09-25T13:16:43.063Z
TimRias
<p>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).</p>
<p>my question is:
if we have one mass which is oscillate , how about this one for calculating strain(h) ?</p>
<p>for example: for a single mass 1.000.000 kg &amp; frequency 100 times per second</p>
<p>if there is formula or reference link , let us know.</p>

CC BY-SA 4.0
2023-09-24T12:50:36.053Z
https://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.
PM 2Ring
CC BY-SA 4.0
2023-09-24T14:00:40.590Z
Define what you mean by a single source and how its mass distribution oscillates.
ProfRob
CC BY-SA 4.0
2023-09-24T14:59:39.290Z
@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?
Hamidreza Abdollahi
CC BY-SA 4.0
2023-09-24T15:03:35.997Z
@ProfRob suppose that i have a ball in my hand shake it in different frequency. the single mass is constant(for example 1.000.000 kg) but frequency may differ in any moment. how can i solve and calculate Strain in any moment.
Hamidreza Abdollahi
CC BY-SA 4.0
2023-09-24T15:41:55.883Z
The formula is given in https://en.wikipedia.org/wiki/Quadrupole_formula . If you don't understand that formula then I suggest you reformulate your question to ask about that.
ProfRob
CC BY-SA 4.0
2023-09-24T16:11:31.870Z
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
CC BY-SA 4.0
2023-09-24T16:42:10.740Z
@ProfRob no i do not meant move through a line , shake it with momentum. in your link above , i do not see frequency of that mass?  there must be mass and frequency as an input in formula in order to find strain (h).
Hamidreza Abdollahi
CC BY-SA 4.0
2023-09-24T16:44:36.940Z
@Sten yes , with momentum , mass and frequency as an input in formula in order to find strain (h). how can i solve it? or if you can send it as an example pf solving with virtual mass and frequency as a thought experiment
Hamidreza Abdollahi
CC BY-SA 4.0
2023-11-21T11:10:16.560Z
If you want an object to swing around a point of equilibrium, you need a restoring force. Who or what does this when there is no second body anywhere?
9herbert9
<gravitational-waves>
Gravitational wave for Solo mass system
Hamidreza Abdollahi
# Gravitational wave for Solo mass system

(Asked by: username_0 on 2023-09-24)

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.

**Tags:** gravitational-waves

> **Comments:**
> **username_1**: https://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.  
> **username_2**: Define what you mean by a single source and how its mass distribution oscillates.  
> **username_0**: @username_1 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?  
> **username_0**: @username_2 suppose that i have a ball in my hand shake it in different frequency. the single mass is constant(for example 1.000.000 kg) but frequency may differ in any moment. how can i solve and calculate Strain in any moment.  
> **username_2**: The formula is given in https://en.wikipedia.org/wiki/Quadrupole_formula . If you don't understand that formula then I suggest you reformulate your question to ask about that.  
> **username_3**: 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.  
> **username_0**: @username_2 no i do not meant move through a line , shake it with momentum. in your link above , i do not see frequency of that mass?  there must be mass and frequency as an input in formula in order to find strain (h).  
> **username_0**: @username_3 yes , with momentum , mass and frequency as an input in formula in order to find strain (h). how can i solve it? or if you can send it as an example pf solving with virtual mass and frequency as a thought experiment  
> **username_4**: If you want an object to swing around a point of equilibrium, you need a restoring force. Who or what does this when there is no second body anywhere?  

---

## 1 answers

### Answer by username_5 on 2023-09-25. Score: 6 | Accepted answer

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](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/Hubble%27s_law#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!

> **Comments:**
> **username_0**: 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  
> **username_0**: also what is A ? you meant in 1 square meter deposit???  
> **username_5**: A is the amplitude of the oscillation.  
> **username_0**: can also caculate strain (h) for your example ? thanks  
> **username_5**: I can, but so can you. Everything you need is here.  
> **username_0**: i do not know what is 'r'  and what is 't' in that formula  
> **username_0**: after calculate each component, i must sum ix,iy, iz together in order to place in strain fromula.?  

---

