fredag 14 februari 2014

The Equivalence Principle from Newton's 2nd Law

We now continue New View of Motion under Gravitation without Classical Mysteries with a study of the Equivalence Principle (EP) stating that inertial mass is equal to gravitational  (heavy) mass, or in other words, that all matter independent of mass and composition reacts the same way to gravitation. EP was identified by Galileo and by Newton made into a corner stone of his theory of gravitation.

The question is if EP is a deep mystery or a necessity which can be understood? Why does a stone and a feather fall the same way when dropped from the Tower of Pisa (assuming no air resistance)? Is it mystery or not?

We recall our Newtonian model of matter subject to gravitation expressed by the equations (to be satisfied for all $x$ and positive time $t$):
  • $\rho (x,t) =\Delta\phi (x,t)$                              (Newton's law of gravitation)
  • $\ddot x(t)= -\nabla\phi (x(t),t)$                          (Newton's 2nd law in math form)
  • $\Delta\dot\phi +\nabla\cdot (u\Delta\phi )=0$                        (Evolution equation for $\phi$),
where $\phi$ is a gravitational potential $\phi (x,t)$ depending on a Euclidean space coordinate $x$ and time $t$, and $\rho (x,t)$ is matter density and $x(t)$ denotes a trajectory with acceleration $\ddot x(t)$ and velocity $\dot x(t)=u(x(t),t)$ specified by initial conditions $x(0)$ and $\ddot x(0)$, with the dot denoting differentiation with respect to time. 

We thus consider the gravitational potential $\phi$ as the primordial object  and the matter density $\rho (x,t)$ as a derived physical quantity given by Newton's law of gravitation: $\rho (x,t) =\Delta\phi (x,t)$. 

We consider trajectories $x(t)$ as mathematical constructs given by Newton's 2nd law in its mathematical form: $\ddot x(t)= -\nabla\phi (x(t),t)$. Newton's 2nd law in math from states a connection between motion $\ddot x(t)$ and the gradient of the gravitational potential $\nabla\phi (x,t)$, which can be viewed to reflect a balance between two types of energy: 
  • KE(t) = $\frac{\vert\dot x(t)\vert^2}{2}$                                              (kinetic energy) 
  • GE(t) = $\phi (x(t),t)$,                                     (gravitational energy):
  • $\frac{d}{dt}(KE + GE)=0$. 
We view $\rho (x,t) =\Delta\phi (x,t)$ as expressing local "creation of matter" with density $\rho (x,t)$ by "action" of the Laplacian differential operator $\Delta$ operating on the gravitational potential $\phi (x,t)$ locally in space and time. We note that the "creation of matter" does not depend on the nature or composition of matter, and we can thus view the created matter to be a form of primordial matter with only quality being density $\rho$ from which all matter with different nature and composition is formed. 
Suppose now we multiply Newton's 2nd law by $\rho (x,t)$ to get
  • $\rho\ddot x(t)= -\rho (x,t)\nabla\phi (x(t),t)$,              (Newton's law in physics form)    
and make the following physical interpretation
  • $F =-\rho (x,t)\nabla\phi (x(t),t)$ is gravitational force
  • $\ddot x(t)$ is material acceleration
  • $\rho (x,t)$ is matter density.
We can then view Newton's 2nd law in physics form to express EP in the form:
  • Material motion under gravitation is independent of matter nature, composition and density.
We sum up:
  • EP is a consequence of Newton's 2nd law.
  • Newton's 2nd law serves to maintain a balance between kinetic energy depending on motion and gravitational energy depending on position.  
The essence given by the Creator is thus a Universe with a dynamics in time based on a balance between two forms of energy, kinetic energy of motion and gravitational energy of position, maintained by Newton's 2nd law. In this Universe, EP is valid as a consequence of Newton's 2nd law. 

Defining non-gravitational forces $F(x(t),t)$ by Newton's law in the form 
  • $F(x(t),t) \equiv \rho(x(t),t)\ddot x(t) + \rho (x(t),t)\nabla\phi (x(t),t)$,
extends this Universe to effects of forces other than gravitation. 

Einstein's general theory of relativity is supposedly based on EP. If now EP is a consequence of Newton's 2nd law, we are led to the conclusion that Einstein's mechanics is no different from Newton's. 

Comparison with Standard Newtonian Mechanics

If we now accept that EP is a consequence of Newton's 2nd law, it is natural to ask in what sense the New View may be more illuminating than the standard view?

In standard Newtonian mechanics, matter with density $\rho (x,t)$ is primordial with the gravitational  potential $\phi$ being generated from $\rho$ as a solution of $\Delta\phi =\rho$ (requiring instant action at distance). The gravitational force acting on a lump of matter of density $\rho$ will then be given by $\rho\nabla\phi$ and EP will then, as above, be equivalent to Newton's law in the form $\rho\ddot x + \rho\nabla\phi=0$. 

In this case matter of density $\rho$ appears in three forms: (i) as generator of gravitation in $\Delta\phi =\rho$, (ii) as gravitational mass in $\rho\nabla\phi$ and (iii) as inertial mass in $\rho\ddot x$,  all supposed to be the same, which may be viewed as coincidence or as a mystery. 

In the New View, $\rho =\Delta\phi$ appears in only one form as a product of $\phi$ combined with Newton's 2nd law in math form expressing a balance of kinetic and potential energies, and the equivalence of this form of mass with gravitational and inertial mass is simply a matter of definition and thus no mystery. 

The New View thus opens to understanding that EP is rather a matter of definition, which is true by the construction of language,  than a statement about physics, which may be true or false. In particular, it points to obvious difficulty of avoiding circular reasoning when seeking to test EP experimentally.

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