In previous posts I have presented a non-standard view on the connection between mass density $\rho (x)$ and gravitational potential $\phi (x)$ connected by Poisson's equation
- $\Delta \phi = \rho$
where $\Delta $ is the Laplacian differential operator with respect to the spatial coordinate $x$ in 3d Euclidean space and the gravitational force is given by $\nabla\phi (x)$ as the gradient of $\phi$.
The standard view is to consider the mass density $\rho (x)$ somehow creating the gravitational potential and so the gravitational force, through the formula
- $\phi (x) = \frac{1}{4\pi}\int \frac{\rho (y)}{\vert x-y\vert}dy$,
that is, the gravitational potential/force is created by global summation over the global presence of mass as an instant action at distance. The appearance is that the gravitational potential/force at the position $x$ is the instant effect of the total presence of mass at all points $y$ different from $x$.
The trouble with this standard view is that generates questions which have never been satisfactorily answered:
- How can action be instantly transmitted over arbitrarily large distances?
- Is there self-interaction between a point mass and the gravitational potential/force it creates?
In the Standard Model of fundamental/particle physics, forces are transmitted by force carrying particles but no such particle (named graviton) for gravitational force has been found. It seems that the standard view has reached a dead end.
In the non-standard view the roles of mass density and potential are switched: The potential $\phi (x)$ is then viewed to endow matter with mass density $\rho (x)$ through the formula
- $\rho (x) = \Delta\phi (x)$ (*)
that is, $\rho (x)$ is created by differentiation of $\phi (x)$. This is a local instant operation which does not require instant action at distance and thus eliminates questions 1. Further, there is no self-interaction because the flow of information is one-way from $\phi (x)$ to $\rho (x)$.
The dynamics of a Newtonian Universe can thus be described by (*) combined with Newton's 2nd Law, more precisely in the form of the Euler equations for a compressible gas in terms of gravitational potential, momentum and internal energy as shown in detail here.
To see the role of a potential, let us we compare with what we see on weather maps as rotational flow around high and low pressure zones with the pressure acting like a potential (click on arrow to start simulation)
We see the trace of air particles moving around high and low pressure zones driven by pressure/Coriolis and inertial forces, in a way similar to planets moving around Suns subject to gravitational and inertial forces, with pressure and gravitational potential playing the same role.
The power of the gravitational potential is to supply matter with the quality of mass as the capability to react to gravitational force according to Newton's 2nd Law, and then in a next step react the same way subject to inertial forces as an expression of equality of gravitational and inertial mass.
In this non-standard view, the gravitational potential thus supplies matter with mass and so opens to motion under gravitational force from the potential. The gravitational potential is thus primordial, while it changes according to the dynamics it creates, which can be seen as a form of feed-back. The non-standard view avoids the unresolved problems of the standard view with matter/mass as primordial, with further rationale here.
The fact that we can see celestial bodies move subject to gravitational force, while we cannot see the gravitational potential, only feel its gravitational force, can give us the impression of the standard view that it is the celestial bodies, which creates the gravitational force by Newton's Law of gravitation. If we could see the gravitational potential we would be led to the non-standard view with the gravitational potential somehow giving mass to matter.
Ultimately motion is created by the gravitational potential giving gravitational mass to matter to react to gravitational force and then to inertial force with inertial mass equal to gravitational mass. The origin of dynamics as motion in space is thus the gravitational potential.
PS The gravitational potential vs mass is a hen vs egg problem, with the gravitational potential playing the role of the hen laying an egg by local action, which is in a way understandable. On the other hand the creation of a hen out of an egg is more mysterious as a creation seemingly out of nothing, and so is the original creation of the gravitational potential.
However it is possible to think of the source as a local oscillation $\bar\phi$ of an original null state $\phi_0 =0$ satisfying $\Delta\phi_0 =0$ with $\Delta\bar\phi =\bar\rho$ where $\bar\rho (x)$ has variable sign representing positive and negative mass, which by gravitational dynamics repel each other and so separate into two Universa with no further contact, where we happen to live in the Universe with positive mass. The creation of the gravitational potential would thus result from a perturbation in an original null state $\phi_0$ satisfying the equation $\Delta\phi_0 =0$ serving as a laboratory ready for perturbation, like a violin string before the stroke by a bow. The creation miracle is then reduced to the stretching of a string.
Notice that by the nature of the differentiation action of the Laplacian $\Delta$ a small localised perturbation $\bar\phi$ will give an amplified mass density $\bar\rho =\Delta\bar\phi$ output as an apparent creation out of nothing.
Inlägg
What is then creating the primordial gravitational potential in the first place? A hen and egg problem?
SvaraRaderaIf this is correct would then kinetic energy in a collision between two objects having a speed x be less if it takes place, say further from the sun, since the gravitationnal field has less intensity?
SvaraRadera