måndag 16 maj 2016

The Blind Space Traveler with Gravitational Potential Meter

              Hawking inside a space ship without windows with a Gravitational Potential Meter

Imagine you are a space traveler locked into a space ship without windows, or traveling through a  region of invisible dark matter. Imagine that in this difficult situation, you have access to an instrument capable of recording the gravitational potential around the space ship from near to far away, an instrument or sense which we may call a Gravitational Potential Meter. Below I discuss how such an instrument might be designed.

Would that allow you to create a normal picture of the distribution of celestial objects/matter around you including your own position, which would be the picture you could see if there were windows or dark matter somehow was made visible, a standard picture/map making it possible to navigate?

Yes, it would because the mass distribution $\rho (x)$ depending on a Euclidean space coordinate $x$ at any instant of time, is related to the gravitational potential $\phi (x)$ by Poisson's equation (in normalised form):
  • $\rho = \Delta\phi$,          (*)
where $\Delta$ is the Laplacian with respect to $x$. In this setting you would naturally view the gravitational potential $\phi (x)$ as primordial, because this is what you can record/sense, and you would view the mass distribution $\rho (x)$ as a derived quantity, because this is what you can compute knowing $\phi (x)$ by applying the Laplace operator, which is a differential operator acting locally in space. 

In this new setting you would not, as in the classical setting of viewing $\rho (x)$ as primordial and $\phi = \Delta^{-1}\rho$ as derived by the inverse of the Laplacian as a non-local operator, have to explain instant action at distance, only the local action of (*), and you would thus have eliminated the question of the physics of instant action at distance, which does not seem to have an answer, and as such may be the wrong question. 

We conclude that depending on what we can see through instruments or senses, we are led to questions, which may have answers or not.  It is natural to think that questions, which may have answers, are better questions than questions which do not have answers.

As to the design of a Gravitational Potential Meter or Gravitational Force Meter, imagine a system of little satellites in free fall distributed over the space of interest and connected to a GPS system allowing tracing of the satellites, thus giving information about the Gravitational Force and from that the Gravitational Potential. It is not unthinkable that such a system could cover any space accessible for space travel and beyond. 

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