Gravitational Field of Light and MOND

One of the candidates of solving the problem of flat galaxy rotation curves is MOND. The theory introduces a universal acceleration scale signified by $$a_0\sim cH_0.$$

The question should now be asked that whose physical entity is this acceleration?

For the first time I show that this is the acceleration of light: \(H_0\) suggests we begin with the FLRW metric $$ds^2 = -c^2 dt^2 + a^2(t)\left(dr^2 + r^2 d\Omega^2 \right)$$

Consider the null radial curves (light) \(ds=0\). We find $$ \frac{d^2 r}{dt^2}=\pm \frac{cH(t)}{a(t)},$$

for the acceleration of null paths (light) which in the present epoch of the Universe is the same as the acceleration scale given by \(a_0\). Although I have not yet been able to develop this and derive a modification of gravity, this result is quite striking in itself: In flat spacetime, in absence of gravity (that is produced by material sources), light accelerates due to the expansion of the Universe. But by the equivalence principle acceleration is gravity, so by the sole reason that the Universe is expanding, light gravitates.

In this light it is no wonder that MOND is related to `Dark matter'. If we assume that a new authentic entity called `Dark matter' does not really exist, according to the Principle of Identity of Indiscernibles there is but one `Dark matter' that does not interact with light: light itself!

The current theory of Electrodynamics (including QED) does not allow light to accelerate. Light is a wave and waves (by the current definition) propagate at constant speed. I am currently trying to prove that Maxwell equations change in an expanding Universe, with the result $$\frac{1}{c+a_0t}\frac{\partial}{\partial t}\left(\frac{1}{c+a_0t}\frac{\partial u}{\partial t}\right)=\nabla^2 u$$

for Electric and Magnetic fields. This is an equation for an accelerating `wave'. 

An immediately-testable consequence of Electric and Magnetic fields (light) accelerating is that there exists a critical value for electric field $$E_0=\frac{a_0}{\sqrt{4\pi \epsilon_0 G}}\approx 1.39 \text{ Volts/meter},$$

below which it is expected that the inverse-square Coulomb's law fails and gives place to

$$\varphi=\sqrt{\frac{E_0}{4\pi\epsilon_0}}\sqrt{|Q|}\log r.$$

This possible modification could have not been tested so far in principle, because all the tests of Coulomb's law that I see, only test the distance dependence, but as we see here, the dependence on electric charge is also different and not linear.