There has been a stunning revolution in astronomy over the past ~6 years with the detection of gravitational waves by the LIGO and VIRGO consortium. Now that we are detecting "naturally" made gravitational waves, I've been wondering could we generate them artificially - and detect them?
Would be possible to generate artificial gravitational waves that could be detected with the current detectors (LIGO, VIRGO)? Short version - seems like no. Sources that have currently been detected are mergers of black holes and neutron stars that are 10's to 100's of megaparsecs away, so first I consider reducing the mass and having the source be closer - and this still requires incredibly large masses moving very fast (now in close dangerously close proximity!)
I also tried a quick idea about using motion of electrons within atoms to generate gravitational waves, but electron motion is not that much faster, and the mass of the electrons is much too small. Motion of atoms within molecules would not achieve the required speeds and thus would require even more mass.
Here is a helpful chart gravitational of wave events that have been detected by LIGO & VIRGO as of this writing April 3 2021:
I'm going to focus on the neutron star merger in the lower right corner of the above - GW170817 -
here is a link to the wikipedia article,
here is a link to the PRL paper.
How to calculate the "intensity" (strain) of gravitational waves
His equation 8 (p. 4) specifies the "amplitude of the oscillation" - h - the strain:
$$ h \sim 2 \epsilon \frac{GM}{rc^2} $$
$\epsilon$ is defined later, roughly speaking related to the internal velocity of the source
$G$ is the gravitational constant
$M$ is the mass of "oscillating" system
$r$ is the distance to the oscillating system
$c$ is the speed of light
He provides a helpful equation to calculate this (partly anyway):
$$ \frac{GM}{rc^2} = 2.4 \times 10^{-21} \left( \frac{M}{M_\odot} \right) \left( \frac{20 Mpc}{r} \right)$$
$20 Mpc$ is the unit of distance for comparison: 20 mega
parsecs
As a sanity check, plug in the the values for the neutron star merger GW170817 from above:
$$ \frac{GM}{rc^2} = 2.4 \times 10^{-21} \left( \frac{2.74 M_\odot}{M_\odot} \right) \left( \frac{20 Mpc}{40 Mpc} \right)$$
$$ = 3.29 \times 10^{-21} $$
From Figure 2 in the PRL paper, the strain observed is roughly $1 \times 10^{-21}$ to $7.5 \times 10^{-20}$, which is in rough agreement with this calculation
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| Figure 2 from PRL paper referenced / linked to above |
Attempting to create gravitational waves macroscopically by trading mass for distance
As a thought experiment, what if we moved the system much closer - as close as possible - and reduced the mass so that proportionally the same intensity was observed? To do this, we keep the previously calculated strain, set a new distance of ~50 km (1.62 \times 10^{12} Mpc) and solve for the new mass:
$$ 3.29 \times 10^{-21} = 2.4 \times 10^{-21} \left( \frac{M_{new}}{M_\odot} \right) \left( \frac{20 Mpc}{1.62 \times 10^{-12} Mpc} \right) $$
$$ 1.37 = \left( \frac{M_{new}}{M_\odot} \right) \left( 1.23 \times 10^{13} \right) $$
$$ M_{new} = 1.1 \times 10^{-13} M_\odot $$
or in $kg$:
$$ M_{new} = 2.21 \times 10^{17} kg $$
$M_{new}$ is a lot less than previously, and it's a lot less than the mass of the sun or the earth or even the moon ($7 \times 10^{22} kg$), but it's still a lot - not something that can be built practically to generate gravitational waves.
The relationship between distance and and mass in this equation is inversely proportional, so even if you could move the system 0.5 km from the detector, it would still have to have a mass of $2.21 \times 10^{15} kg$ - still a lot!!
And we haven't even checked in on how fast it has to be moving ...
Effect of rotational speed on strain / intensity
Schutz's equation 13 provides the relationship between rotational speed and the observed strain (reminder, strain is the intensity of gravitational radiation):
$$ h \sim \epsilon \frac{GM }{rc^2} $$
$$ = \frac{v^{2}_{N,S}}{c^2} \frac{GM }{rc^2} $$
(Note this equation defines the previously-vaguely-defined $\epsilon$ parameter)
Based on the data in Figure 2 from the PRL paper, very roughly speaking when the neutron stars were near each other, they had an orbital frequency of between 30-500 Hz. Let's define "near each" other as ~3 radii separation = 30 km. Let's use 100 Hz as the orbital frequency*, then the angular frequency is 628 radians/s. The velocity of the stars are then:
$$ v = \omega r = 628 \times 30 km $$
$$ \sim 19,000 \frac{km}{s} $$
$$ = 0.063 c $$
About 6.3% the speed of light. That's pretty fast for somethings that are so heavy. For the above calculation trading mass for distance, we would need this speed for our $2.21 \times 10^{15} kg$ object - Not going to happen any time soon!
* I think there may be a factor of two I'm neglecting here, but this a rough calculation so ¯\_(ツ)_/¯
Can we make further tradeoffs on distance and velocity for mass by using an atomic or molecular system?
Short version: probably not.
Slightly longer version: imagine we could have a chunk of material that we could cause coherent excitation of some atomic or molecular motion throughout all constituents (atoms or molecules in the chunk). Then in theory we could move this chunk very close to the detector - let's say 1 meter away. That gives us a factor of 500 reduction in the mass required. Now let's say we're able create motion with velocity 0.6 c (see appendix 1) this gives us an $\epsilon$ value of:
$$ \epsilon = \frac{v^2}{c^2} = 0.6^2 = 0.36 $$
For the merging neutron stars we estimated their speed as 0.063 c, leading to $\epsilon = 0.004$. Therefore the change in epsilon is a factor of ~100 for the atomic/molecular system. Combining the proximity and speed factors, we have an increase of ~50,000. Applying that to the previously estimate mass leads to a new mass for the system of $44.2 \times 10^9 kg$. ~44 billion kilograms, and at this point we've made several unlikely assumptions to get there - we're not going to be able to put 44 billion kg of material within 1 meter of our detector, and we're definitely not going to be able to excite even a tiny fraction of it into a coherent motion of the nuclei*. Game over friend, game over.
* that's right: in the appendix I'm discussing motion of electron in a highly idealized system, but in this case we would need to have motion of the nuclei, since that's where the vast majority of the mass in the atom is located.
Appendices
An atomic system - electron orbiting an ion core
Rydberg atoms follow approximately Newtonian equations of motion:
$$F = ma = \frac{ke^2}{r^2} = \frac{mv^2}{r}$$
$e$ is the electric charge
$m$ is the mass of the electron
$v$ is the velocity of the electron
$r$ is the radius of the circular orbit of the electron
$k = \frac{1}{4 \pi \epsilon_0}$
$\epsilon_0$ is the vacuum permittivity
Rearranging to solve for $v$:
$$v^2 = \frac{ke^2}{mr}$$
Following the referenced wikipedia article, the Bohr radius follows this equation:
$$mvr = n \hbar$$
$n$ is the principal quantum number of the electron
$\hbar$ is the reduced Planck's constant
rearranging to solve for r:
$$r = \frac{n \hbar}{m v}$$
combining these two equations:
$$v^2 = \frac{ke^2}{m \frac{n \hbar}{m v}}$$
$$v = \frac{ke^2}{n \hbar}$$
$$ = \frac{e^2}{n \hbar} \frac{1}{4 \pi \epsilon_0} $$
$$ = \frac{e^2}{2 n h \epsilon_0}$$
plugging in values for the constants:
$$v = \frac{e^2}{n (4.14 \times 10^{-15} eV \cdot s) \cdot 2 \cdot (55.3 \frac{e^2}{GeV \cdot fm})}$$
$$ = \frac{1}{n (4.14 \times 10^{-15} eV \cdot s) \cdot 2 \cdot (55.3 \times 10^{-9} \frac{1}{eV \cdot fm})}$$
$$ = \frac{1}{n (4.58 \times 10^{-22} \frac{s}{fm})}$$
$$ = \frac{1}{n} (2.18 \times 10^{21} \frac{fm}{s})$$
in some alternate units:
$$v = \frac{1}{n} (2.18 \times 10^{6} \frac{m}{s})$$
$$ v = \frac{1}{n} (0.00727 c) $$
$c$ is the speed of light in vacuum
So the fastest speed will be in the "ground state" n=1. We can in fact boost this by increasing the charge on the nucleus / ion core - for example, if we used lead (atomic number 82) we could very crudely estimating that this core electron would be 82x faster, leading to a relationship of $ v = \frac{1}{n} (0.596 c) $
A couple strong caveats / outright mistakes in the above logic:
- having an effective charge of +82 for lead is just wrong, but it's certainly an upper limit
- An electron within a state is not actually moving with this velocity in the classical sense, but it's probably fair to say this is a safe upper limit
- In theory we would need to create a superposition between states to have something remotely like classical motion ...
$$ 50 km \times \frac{1 Mpc}{30.9 \times 10^{12} km} = 1.62 \times 10^{-12} Mpc $$
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