Gravitational Waves as Fiction

We recall the following Newtonian model of the Universe from this recent post:

• $\rho=\Delta\phi$                        (N)             (conservation of gravitational force)
• $\dot\rho +\nabla\cdot m =0$                        (conservation of mass)
• $\dot m +\nabla\cdot (um) +\rho\nabla\phi =0$    (conservation of momentum)

describing a (zero pressure for simplicity) distribution of matter subject to gravitation, where $\rho$ is mass density, $\phi$ gravitational potential, $\nabla\phi$ gravitational force per unit mass, $m$ is momentum, and $u=\frac{m}{\rho}$ is material velocity, all depending on a Euclidean spatial coordinate $x$ plus time $t$ with the dot representing differentiation with respect to time.

We focus on the connection between gravitational potential $\phi$ and mass distribution $\rho$ expressed through Laplace/Poisson's equations (N), which formally involves infinite speed of propagation into $\phi$ from a local change of $\rho$. Let us compare with the following wave equation as a Neo-Newtonian variant of (N) with finite speed of propagation $c$

• $\frac{1}{c^2}\ddot\phi -\Delta\phi = -\rho$.            (NN)

Let us now compare (N) and (NN) in a situation where the mass distribution changes/moves with velocity $v$ which is small compared to $c$, which is the typical situation within a planetary system and collection of stars or even galaxy. This means that $\nabla\cdot m$ is small of size $v$, which means that $\dot\rho$ is small of size $v$. We conclude from (NN) that $\dot\phi$ is small of size $v$ and so also $\ddot\phi$ assuming $\nabla\cdot\dot m$ is small of size $v$. This means that the difference between (N) and (NN) is of size $\frac{v}{c^2}$ thus very small.

We conclude that even if we extend (N) (without gravitational waves) to (NN) (with gravitational waves), the difference is very small. This is in line with the LIGO experiment supposedly identifying a very very small gravitational wave from a very very large source. Indeed, very very small. More precisely from LIGO documentation:

• For physicists, a strong gravitational wave will produce displacements on the order of $10^{-18}$ meters - this is 1000 times smaller than the diameter of a proton. Waves of this strength will be produced by very massive systems undergoing large accelerations, like two orbiting black holes that are about to merge into one. Since systems like these are rare, these sources will be light-years away. Therefore, the search for gravitational waves is seeking the minute effects of some of the most energetic astrophysical systems from the depths of the universe.

We thus have theoretical and observational support of an idea that we can view gravitational waves to be fiction,  which we do not have to worry about. This makes theory simpler and also computational simulation, since (N) is much simpler to solve computationally than (NN), and so makes cosmology simpler. This is a gift from Newton.

Napoleon criticised Laplace, expert in infinitesimal Calculus, for work on infinitely small issues in his administration. Napoleon would probably similarly criticise Einstein for working with infinitely small deviations from Newton's mechanics.