Would there be any consequences that we could measure?
For instance, there is an upper bound of the amount of EM energy that can be poured through a square mm of area – not predicted by Maxwell’s Eqn’s of course, as they are linear, but by quantum field effects. If we instead look at how gravitational energy we can pass through that same square mm, is it the same number of joules/sec? http://en.wikipedia.org/wiki/Schwinger_limit
Well there are a few problems with the Schwinger limit too:
"A single plane wave is insufficient to cause nonlinear effects, even in QED. The basic reason for this is that a single plane wave of a given energy may always be viewed in a different reference frame, where it has less energy (the same is the case for a single photon)."
So according to QED, we can actually make a laser of any power – and as long as its in a vacuum, there are no non linear effects. Can that really be true?
The Schwinger limit is about 2.3 E33 Watts/metre^2.
I have calculated the limit of gravitational wave energy (which depends on frequency) to be
P (max gravity waves) = 3/(5pi)*c^3/G*w^2,
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 http://en.wikipedia.org/wiki/Gravitational_wave:
There is a formula there for the total power radiated by a two body system:
(1) P = 32/5*G^4/c^5*m^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 = 1/2*G^2/c^4*2m^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 = 64/5*c^3/4*m/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 (http://voyager.egglescliffe.org.uk/physics/gravitation/binary/binary.html)
(4) Pmax = 16/5 c^3/G*w^2*r^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) Pmax/per sq meter = 3/(5*pi)*c^3/G*w^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 watts/m^2 !
That’s a lot of power, dwarfing the Schwinger limit.
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.
w = frequency
So for gravitation, we have linear behavoir holds up until some fantastic power level:
1e65 watts per sq metre at visible light frequencies is about the linear limit for gravitational waves at a frequency of 10^14 .
This means that gravity has ‘lots of headroom’ to create the phenomena of electromagnetism.
Perhaps one could dream up a super efficient way to generate ‘normal’ quadrupole gravitational radiation using some radio sources arranged in some way. Or a way to generate anti-gravity, etc.
GR certainly has a large enough range of linearity to power all of the EM we know today. Its also possible to generate monopole and spin 1 radiation from gravity, look up Brady’s papers on EM generation from simple compressible fluids, for instance.
Also do the joules/sec per square mm or whatever calc.
Also look at some other consequences in the dark recesses of the proton and electron (my models of them, or effects just based on size and field levels). Would we start to get non-linear EM effects at what distance from the centre of an electron? Same for quarks?