# Maximum Power transmitted using General Relativity waves - cf Schwinger limit

In Electromagnetism, QED says that the linearity of Maxwell's equations comes to an end when field strengths approach the Schwinger limit. Its about 10^18 V/m.

What is the corresponding formula for gravitational waves? Since gravity is a non-linear theory, there should be a point where gravitational waves start to behave non linearly.

Here is my calculation, based on this:

There is a formula there for the total power radiated by a two body system: (1) $P = 32/5G^4/c^5m^5/r^5$ (for identical masses in orbit around each other)

Further down the same wiki page I see a formula for h, which has a max absolute value of (assuming h+ and standing at $R = 2r$ away from the system, $\theta = 0$):

(2) $h = 1/2G^2/c^42m^2/r^2$

Things will be highly non linear at $h = 1/2$ (which is the value of h used in the diagram on the wikipedia page!). So lets set $h = 1/2$, and then substitute (2) into (1) to get the power as radiated by the whole system when $h = 1/2$ (use a lower value like $h = 0.001$ perhaps to be more reasonable, if you like). I am not trying to calculate where the chirp stops in a binary spin-down here, I'm looking for the maximum field strength of a gravitational wave.

I get for the maximum power from a compact source

(3) $P = 64/5c^3/4m/r$

That's the total power radiated when h is well into the non linear region - you will never get more than this power out of a system using gravitational radiation.

The result depends on m/r , which makes sense as higher frequency waves with the same value of h carry more power.

Putting the result in terms of orbital frequency, w, we get (using newtonian orbit dynamics)

(4) $P_{max} = 16/5 c^3/Gw^2r^2$

That's the max coming out of a region r across, we want watts per sq metre, so divide by the surface area of a sphere:

(5) $P_{max}/sq.m = 3/(5\pi)c^3/Gw^2$

The maximum power that you can deliver at $10^{14} Hz$ (light wave frequencies, so as to compare to the E&M QED Schwinger limit) is $10^{65} W/m^2$ !

That's a lot of power, dwarfing the Schwinger limit, which is about $10^{31} W/m^2$ or so, I think.

Is that about right? The max power scales as the square of the frequency, and is truly huge, reflecting how close to linear GR is over large parameter spaces.

I think that there is a much more reliable way to estimate this GR wave power limit - there should be no need to work from the formula for a binary system, for instance.

• The Schwinger limit doesn't mean one can't ramp up power density beyond that point. There will be spontaneous pair creation, which limits the electromagnetic field density, but not the power density. The same is true for gravitational waves. At some power density they, too, will create significant electromagnetic waves and particle pairs, so the naive limits probably don't apply for a realistic coupled theory, which should converge towards one (in the low energy limit symmetry broken) field. – CuriousOne Apr 26 '15 at 19:28
• CuriousOne. Thanks. My question is not about some grand unified theory, its about General Relativity on its own. – Tom Andersen Apr 27 '15 at 0:17
• Jimmy360 - thanks for the prettification! – Tom Andersen Apr 27 '15 at 0:17
• My point was simply that what happens when you "ramp up the power" is that you go from electromagnetic waves to electron-positron pairs, eventually you will make quark gluon plasmas, then probably microscopic black holes, eventually you will be making macroscopic black holes, at which point you can't ramp up the density any longer (in a classical sense), because you will be making spacetime itself... or so many people think. – CuriousOne Apr 27 '15 at 0:25
• Either no one is trying, or I need to add a bounty – Jimmy360 Apr 29 '15 at 3:08