söndag 14 februari 2016

Gravitational Waves in Newtonian Mechanics?

We consider a cosmological model in the form of a compressible gas under interaction by gravitation: Find $(\phi ,m,e)$ depending on a space-time coordinate $(x,t)$, such that for all $(x,t)$
  • $\Delta\dot\phi + \nabla\cdot m =0$
  • $\dot m +\nabla\cdot (um) +\nabla p + \nabla\phi =0$
  • $\dot e +\nabla\cdot (ue) +p\nabla\cdot u =0$,       
where $\rho = \Delta \phi$ is matter density with $\phi$ gravitational potential, $u=\frac{m}{\rho}$ is matter velocity with $m$ momentum, $p=(\gamma -1)e$ is pressure with $\gamma > 1$ a gas constant and $e$ is internal (heat) energy, and the dot denotes differentiation with respect to time, see Many-Minds Relativity 20.3.

In the case $u=0$ and $\nabla p=0$, this model reduces to
  • $\Delta\dot\phi + \nabla\cdot m =0$,                      (1a)
  • $\dot m +\nabla\phi =0$,                                (1b)     
from which follows by differentiating the first equation with respect to time:
  • $\Delta\ddot\phi + \nabla\cdot\dot m =0$,
  • $\dot m +\nabla\phi =0$,      
that is
  • $\Delta\ddot\phi + \Delta\phi =0$,
and thus
  • $\ddot\phi + \phi =0$,
which expresses a periodic oscillation of $\phi (x,t)$ in time.

Alternatively, eliminating instead $\phi$ in (1), gives the following wave equation in $m$:
  • $\ddot m + \nabla\Delta^{-1}\nabla\cdot m=0$. 
A universe in the form of a compressible gas thus allows a periodically varying gravitational potential $\phi (x,t)$ with corresponding periodic mass density $\rho (x,t)=\Delta\phi (x,t)$, if we extend the model to include mass density varying from positive to negative mass, along with periodic momentum $m$.  The effect of such a locally in space oscillating gravitational potential may at distance be perceived as a varying gravitational field and and thus as a gravitational wave. In this context, recall the recent post on dark energy and dark matter where you can follow the expansion a gravitational wave originating from a local oscillation of gravitational potential:

Newtonian mechanics extended to include both positive and negative mass,  thus appears to allow gravitational waves.  The conclusion that gravitational waves somehow require Einstein's general theory of relativity, may be drawn too quickly. Maybe Newton's theory extended to negative mass is enough.

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