Gravitational Waves Affecting EM Fields: Evidence & Implications

[vibe3]

Suppose a gravitational wave propagating through space encounters a strong magnetic field (for example the wave might pass through a magnetar with a B field strength of $10^{11}$ Tesla). Would there be any observable perturbation in the magnetic field itself? In other words would the gravitational wave create an oscillation in the magnetic field which could potentially be observed?

I found one paper here:
http://www.sciencedirect.com/science/article/pii/S0370269303004155
which indicates that the answer is yes. But I haven't been able to find much more on this in terms of calculations or even just a simple explanation of what would happen.

The obvious implication is that if GWs perturb EM fields, this could be another avenue to observe them, provided we put a magnetic sensor in the right place, etc.

 

[Paul Colby]

I've looked at questions of this kind for some time now. It's easy to expand in the weak field limit in which case one gets an effective EM current and charge source term for Maxwells equations in flat space-time. This isn't a view point I would recommend but it can make some calculations simpler. The issues I've found with this effective current approach is in determining the effect of material boundaries which are subject to the same GW as the EM fields. For example, is the voltage drop across a charged capacitor modulated or changes by a passing GW? The answer yes but it depends on the reaction of the capacitor structure to the GW which itself is a complex question.

BTW, there are a number of researchers that look at microwave cavities as potential GW detectors. Yo might try to dig up some of their papers.

 

[vibe3]

I've looked at questions of this kind for some time now. It's easy to expand in the weak field limit in which case one gets an effective EM current and charge source term for Maxwells equations in flat space-time. This isn't a view point I would recommend but it can make some calculations simpler. The issues I've found with this effective current approach is in determining the effect of material boundaries which are subject to the same GW as the EM fields. For example, is the voltage drop across a charged capacitor modulated or changes by a passing GW? The answer yes but it depends on the reaction of the capacitor structure to the GW which itself is a complex question.

BTW, there are a number of researchers that look at microwave cavities as potential GW detectors. Yo might try to dig up some of their papers.

Thanks for your response. Do you know of a paper which works out an equation for the predicted effect on a B field? For example if a GW with a given amplitude and frequency passes through a B field, what is the amplitude/frequency of the expected perturbation of the B field? The weak (B field) limit would be fine with me for now since it could be interesting to understand the effect a GW would have on Earth's magnetic field, which probably satisfies the weak limit.

 

[Paul Colby]

Do you know of a paper which works out an equation for the predicted effect on a B field?

No, sorry. There is a footnote in Landau "Classical Theory of Fields" but I recall not finding it very helpful for what I was doing. I would be willing to summarize my notes for what that's worth. As with any calculation one may check it independently given the assumptions and starting point. From a classical EM viewpoint the leading gravitational effect looks like an effective current linear in the metric stress and the unperturbed electromagnetic field. For static electric or magnetic fields, the effective currents radiate at the frequency of the illuminating GW. It will take some time to put this together. Sounds like a job for tomorrow.

 

[vibe3]

No, sorry. There is a footnote in Landau "Classical Theory of Fields" but I recall not finding it very helpful for what I was doing. I would be willing to summarize my notes for what that's worth. As with any calculation one may check it independently given the assumptions and starting point. From a classical EM viewpoint the leading gravitational effect looks like an effective current linear in the metric stress and the unperturbed electromagnetic field. For static electric or magnetic fields, the effective currents radiate at the frequency of the illuminating GW. It will take some time to put this together. Sounds like a job for tomorrow.

Your notes, or even an outline of the derivation would be most welcome! If indeed this result isn't in the literature, I think it would be worth publishing, have you thought about that?

 

[Paul Colby]

Okay, some background. This was a project I started in 2011. About the time I retired I said to myself, self, wouldn't it be good for me to understand gravitational wave detection in terms of received power. In radio communication (and measurements in general) there is no free lunch. Signal detection reduces to received power and how this power compares to system and background noise power. I reasoned similar things must hold for the detection of gravitational waves since one ultimately detects electrical signals of some form. Along with this goal I wanted to look at possible high frequency detection schemes and needed a figure of merit with which to compare these (largely dumb) ideas. Basically, this figure of merit boils down to a power conversion area measure in $m^2$. Knowing the conversion area, $A$, for a detector the signal power received is then,

$P = A P_I$​

where, $P_I$, is the watts per square meter of gravitational wave illumination. What I soon learned, though, is femto-barns[1] are perhaps a more reasonable unit.

So the basic view I stated with is how does gravitational radiation give rise to electromagnetic radiation in a typical household environment for a given detector system. Household environment means we're working in the weak field limit where,

$g_{\mu\nu} = \eta_{mu\nu} + h_{\mu,\nu}$​

where, $\eta_{\mu\nu}$, is the usual Minkowski metric. I insisted upon keeping all constants like the speed of light and such because I want numbers in the end and I'm not the sharpest pencil in the box.

There are 4 fields in EM, $E$, $B$ and their hillbilly siblings, $H$ and $D$. It wasn't until I was faced with EM in an industrial setting that the truly awesome properties of $D$ and $H$ become apparent. So, Maxwells equations group into a set connecting $E$ and $B$ and a second independent set connecting $H$ and $D$. It is a well known fact that neither set depend at all on the metric, weakly curved or otherwise. This has to do with the symmetry of the covariant derivative. Clearly though, EM fields will be effected by curvature so what gives?

The answer is one must supply a constitutive relation between ($E$, $B$) and the other less popular fields, ($H$,$D$), in order for Maxwell's equations to form a dynamical system. For linear isotropic materials, people love to write the relations as,

$D = \epsilon E$
$B = \mu H$​

When written in tensor for ($E$,$B$) are the components of an antisymmetric tensor MTW call the Faraday, $F_{\mu\nu}$ while the ($H$,$D$) become the components of the other antisymmetric tensor, $M_{\mu\nu}$, call the Maxwell. As written above, the vacuum constitutive relations are not a tensor. The actual connection between fields in vacuum is,

(1) $M_{\mu\nu} = -\sqrt{\frac{\epsilon_o}{\mu_o}}\left(\frac{1}{2\sqrt{-g}}g_{\mu\alpha}g_{\nu\beta}\epsilon^{\alpha\beta\gamma\delta}F_{\gamma\delta}\right)$​

which does depend on the metric and is a tensor relationship.

So, what I did from here is expand (1) to first order in the metric strain, $h$. I'm tired of typing so I'll post the answer I got later.

[1] A Barn being $10^{-28}m^2$

 

[Paul Colby]

Okay, it's later. Now expanding (1) to linear terms in $h$ results in a modified free space constitutive relations,

$D = \epsilon_o E + \delta D$
$H = \mu_o^{-1}B - \delta H$​

where the small bits are

$\delta D = \epsilon_o \left[ h\cdot E - \Phi E - c (G \times B)\right]$
$\delta H = \mu_o^{-1}\left[ h\cdot B - \Phi B - c^{-1}(G \times E)\right]$​

where, $h$, are the space components of the metric strain, $G_i = h_{it}$, are time components and, $\Phi = h_{xx}+h_{yy}+h_{zz}$. For a transverse traceless wave, $G = 0$, as is $\Phi = 0$. Putting these into Maxwell's equations we may relegate them to the source terms yielding a gravitationally induces current and charge,

$\rho_G = -\epsilon_o \nabla\cdot\left[ h\cdot E - \Phi E -c(G \times B)\right]$
$J_G = \mu_o^{-1}\nabla\times\left[h\cdot B - \Phi B - c^{-1}(G\times E)\right] + \epsilon_o \partial_t\left[h\cdot E-\Phi E -c(G\times B)\right]$​

And, like all good currents, this one is conserved,

$\nabla \cdot J_G + \partial_t \rho_G = 0$​

So one may use this source term to compute the reaction of a detection system to an illuminating GW and from this compute the conversion area.

 

[vibe3]

Interesting result, thanks for typing it out. I'm not sure offhand what order of magnitude $h$ and $\Phi$ would be for the recent GW detection events, but assuming we know that we could estimate how large $B$ needs to be in order to have a chance of actually detecting something.

 

[Paul Colby]

Well, for gravitational waves in the transverse traceless gauge, $\Phi = 0$. For terrestrial observations $h\approx 10^{-21}$ but this could be larger in the neighborhood of the magnetic fields you're looking at. I've thought in passing that neutron stars might be a source of high frequency GW which would then convert to feeble EM radiation. Unfortunately, nuclear matter doesn't seem to support shear waves in any clear way. Most of the models it's more like a gas than a solid. One could wonder if the nucleon-nucleon tensor force might give rise to shear waves but I have insufficient background to tackle such a calculation.

 

[Paul Colby]

Interesting result, thanks for typing it out. I'm not sure offhand what order of magnitude $h$ and $\Phi$ would be for the recent GW detection events, but assuming we know that we could estimate how large $B$ needs to be in order to have a chance of actually detecting something.

For a simplified geometry where we have $B_x=10^{11}$ and $h_{xx} = 10^{-21}e^{ikz}$, I get $[J_G]_y$ at around $8.0\times 10^{-5} A/m^2$ which is definitely detectable in isolation. However, how does one distinguish radiation from this current from all other potential sources?

[edit] whoops, missed a factor of $k$ from the curl. The answer above needs to be multiplied by $2\pi/\lambda$ for the GW. Also, the induced current is along the y-axis for the geometry given.

 

[vibe3]

For a simplified geometry where we have $B_x=10^{11}$ and $h_{xx} = 10^{-21}e^{ikz}$, I get $[J_G]_y$ at around $8.0\times 10^{-5} A/m^2$ which is definitely detectable in isolation. However, how does one distinguish radiation from this current from all other potential sources?

[edit] whoops, missed a factor of $k$ from the curl. The answer above needs to be multiplied by $2\pi/\lambda$ for the GW. Also, the induced current is along the y-axis for the geometry given.

So it would seem we do indeed need the strong field of a magnetar or something similar to drive a reasonably strong current. The Sun's magnetic field (about $10^{-4} T$ would then yield a current of only $8 \times 10^{-20} A/m^2$ according to your numbers. This would be similar to the Earth's magnetic field also.

 

[Paul Colby]

So it would seem we do indeed need the strong field of a magnetar or something similar to drive a reasonably strong current. The Sun's magnetic field (about 10−4T10−4T10^{-4} T would then yield a current of only 8×10−20A/m28×10−20A/m28 \times 10^{-20} A/m^2 according to your numbers. This would be similar to the Earth's magnetic field also.

Yes, the conversion efficiency (cross section) is quite small and hence the 100 year delay between theory and observation. Strong electric fields are somewhat better IMO than magnetic fields. One detection system I considered was a fish tank filled with mineral oil and a high voltage electrode charged with a fluid based on a novel (IMO) mixed fluid Van De Graaff generator (an interesting discussion actually). Even at 10's of mega volts the sensitivity is still quite small. Another design I toyed with was a helical antenna based on a charged coaxial cable. The charm of this design is directivity. And then there was the scheme with a high voltage capacitor with a telescope to image optical frequency gravitons emitted by the sun. Even though the estimated graviton flux at the Earth's surface is about 1/4 $W/m^2$ the photon production rate is like one photon per age of the universe.