Does gravity have electromagnetic properties

I'd like to add mainly to Frederic Brünner's and Anna V's answers.

Let's begin with, as Frederic does:

Sunlight does not point back to the sun’s true center of gravity, whereas gravity always points back to the sun’s true center of gravity.

and

And if it propagated at the speed of light, gravity (like sunlight) would not point back to the sun’s true location; and as a result the planets would drift away from the sun and leave the solar system.

Precisely these arguments that your paper proposes have a long, long history of having been thoroughly studied, beginning with the great Laplace. See the Wiki discussion of the speed of gravity, in particular its summary of Laplace's thoughts on the matter and also this contemporary "review" from the original Usenet physics FAQ. Before general relativity, you could indeed argue that the planet orbits would not be stable with a finite lightspeed.

After General relativity is accounted for, guess what? The orbits are STILL unstable!! And this is exactly what is observed!. I'm being slightly mischievous here, because the effect on the Earth's orbit is fantastically small: Earth radiates about 200 watts of gravitational radiation. See the "power radiated by by orbiting bodies" section in the Gravitational Wave Wiki Page. So the instability is not going to show a perceptible difference in orbit any time soon! But there is an astronomical system which allows us to experimentally check the instability and that is the Hulse-Taylor binary system: this is a binary star system which has been carefully observed and measured since its discovery in 1974 and the observed spin down carefully compared with the spindown foretold by General Relativity (one calculates, by GTR, the gravitational wave power emitted). GTR exactly matches observation here. Moreover, early this year, direct observation of gravitational waves in the early cosmos is thought to have been made by the BICEP2 experiment as frozen ripples in the CBR.

So there is a great deal of evidence directly amassing for finite speed propagtion of gravitation. And that's before one looks at the theoretical argument against infinite gravitational speed propagation made by special relativity and the thoroughly experimentally tested notion of Lorentz invariance.

Lastly, let me copy Aaron Dufour's excellent comment here lest it should be deleted:

[It's] Worth noting that falling off like $1/r^2$ is a generic property of things that propagate in 3 spatial dimensions; anything else would imply energy being regularly gained/lost along the way.

and let me add to it as follows. If we go back to Laplace's simple model, where he assumes Newton's inverse square law (which, as Aaron says can be construed as a property arising in 3 spatial dimensions) and simply adds a retardation, but if we do it in a way that is Lorentz invariant in freespace, we find again that the orbit instability is much smaller. Interestingly, what you now have is the theory of Gravitoelectromagnetism, which is precisely analogous to Maxwell's equations. So here you have the full "magnetic" and "electric" laws arising simply from the $1/r^2$ property of three spatial dimensions and then requiring the laws to be Lorentz invariant. So you would expect electric/magnetic like equations that at least roughly describe utterly unrelated phenomenonse, which is an even stronger version of Aaron's argument. Incidentally, if we note that the universal gravitation constant corresponds to $1/(4\pi\epsilon_0)$ in Maxwell's equation, then the Gravitoelectromagnetism version of the orbital instability, i.e. of the Larmor formula is:

$$P = \frac{2}{3} G^3 \frac{m_e^2\,m_s^2}{r_e^4\,c^3}$$

with $m_s$ = Sun's mass = $2\times10^30{\rm kg}$, $m_e$ = Earth's mass = $6\times10^24{\rm kg}$ and $r_e=1.5\times10^{11}{\rm m}$ I get about $3{\rm GW}$ radiation. THis sounds much more significant than the GTR loss but it would still take of the order of $10^8$ times the age of the universe for the Earth to spiral into the Sun. Gravitoelectromagnetism is falsified by the Hulse-Taylor binary. The difference is essentially that GTR only allows quadrupole and higher order radiation sources, not the much more energetic dipole radiation that Gravitoelectromagnetism (and Maxwell's equations) allows.

Footnote: Actually, we don't quite quite get Lorentz invariance with Gravitoelectromagnetism even though the equations in freespace are Lorentz invariant. It turns out that $(\rho_g,\,\vec{J}_g)$, the analogue of the current density four-vector from Maxwell's equations, is not a four-vector in GTR but merely an incomplete representation of the stress energy tensor $T$,

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