Spinoza Principle

Continuing with my rationalist reconstruction of physics, I elaborate on a point mentioned here, that the Poisson equation $$\nabla^2 \phi=4\pi G\rho,$$ is possible to derive from Descartes' and Spinoza's metaphysical foundation for physics, which I name as Spinoza Principle. To state this principle, as a recap of Descartes' argument, and Spinoza's improvement, consider a single particle, isolated from all other objects in the universe. Descartes argues that if the state of this particle is to be altered from its initial state in which God put it in the creation of universe, God has to `change his mind'; because there is nothing else in the universe except God to change its course of motion, so as God is the cause of everything, any change in the particle's state of motion has to come from `God's decision', but for God to change his mind would be inglorious, therefore the particle continues its path `indifferently'. This is the same as Descartes' famous saying that the right question is not why a particle moves, but why it should not move! To remove vestiges of theology in Descartes' argument, Spinoza's Deus sive Natura implies that for the particle to change its course of motion, something other than Nature (=God) has to intervene, which is impossible for there is nothing other than God/Nature. 

You should probably now be thinking `it cannot be that simple', `why on earth the Scholastics idled so much on sophisticated Aristotelian arguments in favor of such a simple elegant principle ?' Indeed I have contemplated on this for a long time, and I can only answer, because people did not believe in the power of independent pure reflexion. The degree to which you have to take this argument seriously is enormous; to madness. You should take this as a word of God, a divine inviolable commandment, and assume it as a divine task. Without such confidence and even faith, you would not have the courage, motivation and perseverance to pursue this, which should guide you towards a construction of calculus for its purpose. I have done a similar reasoning for \(E=h\nu\) here, which again shows the necessity of this commitment. 

In modern language, this metaphysical consideration means that the path of a free particle \(x^\mu(s)\) is a geodesy (straight line in Euclidean space), $$\frac{d^2 x^\mu} {ds^2}=0\ \ *$$ But in the current understanding of theoretical physics, this principle is assumed to be applicable only in spacetime; for paths of particles on spacetime. Even in its most sophisticated statement, Hamilton Principle or Principle of Least Action, the integrals $$S=\int \mathcal{L}\sqrt{-g}\ d^4 x$$ and $$ S=\int L \ dt$$ are still taken on spacetime. 

Following some personal observations (and contributions) I promote this principle to one with universal validity:

Spinoza Principle

In all vector spaces which are in direct association with ontological entities, the geodesic equation holds. 

Roughly `the second derivative of something is zero'. I admit this statement is not without ambiguities and should be improved; sadly such improvement has not yet been achieved. 

 The three prominent examples are 

• Newton's Second Law, \(F=ma\), which can be written as $$\frac{d^2 F}{da^2}=0$$

• Laplace equation, \(\nabla^2 \phi=0\), `second ``derivative'' of gravitational potential is zero', a generalization of which becomes 
• Einstein Field Equations, `second ``derivative'' of the metric tensor is zero'
• Length contraction of Special Relativity \(L'=\gamma L\), similarly time dilation; which is my own contribution:

Length contraction can be written as $$\frac{d^2 L'}{dL^2}=0,$$ which is indeed a geodesic equation. I will show in a paper that using this one can derive Lorentz transformations. 

This brings me to a new equation. Note that it is easy to get from \(*\) to $${d^2 x^\mu \over ds^2}+\Gamma^\mu {}_{\alpha \beta}{d x^\alpha \over ds}{d x^\beta \over ds}=0.$$

Thanks to Spinoza Principle and by analogy, we can readily go from Laplace equation to $$\partial_\alpha \partial^\alpha \phi + A^{\mu\nu}(\partial_\mu \phi)(\partial_\nu \phi)=0,$$ 

where \(A^{\mu\nu}\) is the analogue of the Levi-Civita connection.

This brings in non-linearity to Newtonian mechanics, --before getting to General Relativity. This is something Born and Infeld tried to achieve but did not succeed, in my opinion. 

Elucidating the mathematical structure of this new geodesic equation is not my priority now but I can say it provides us with a natural connection for infinite-dimensional (probably Riemannian) manifolds.

Such is the power of Descartes' and Spinoza's metaphysical considerations. 

Two objectives now remain to complete a rationalist re-construction of theoretical physics,

1. Determination of constants,
2. Determination of which forces should exist, i.e. we must know about \(\phi\) before letting its second derivative equal to zero. Some relatively good arguments in this direction have been given by Kant and Schelling, but they are not satisfactory enough to me.

Both of these problems remain open to my future pure reflexions.