## fredag 23 februari 2024

### Motion vs Appearance or Emergence

This is a continuation of earlier posts on Zeno's paradox as an unresolved mystery of the physics of motion:

• How can an arrow move, when at each time instant it is still, that is, not moving?
• Is the arrow jumping from one position to the next in a discrete series of events in increasing time?
No convincing resolution is offered by either classical or modern physics, and so the question is dismissed as a no-question so obvious that it does not need any explanation: Just look and see how things are moving  or shifting positions in space $x(t)$ with time $t$, with velocity $v(t)=\dot x(t)$ and the dot signifies differentiation with respect to time.

Given a velocity $v(t)$, the corresponding motion/trajectory $x(t)$ is created by exactly solving the differential equation $\dot x(t)=v(t)$ (as if the arrow is smoothly changing position in time without jumps), or by time-stepping from one discrete time instant to a next (as if the arrow de facto is jumping).

But a child eager to understand the World may not be satisfied with such an empty explanation, but maybe by the following argument:

Let us compare the concept of motion with that of appearance or emergence. If a certain person appears at a party, invited or not, the question may come up how the person got there, more precisely what trajectory of motion the person had followed? Today the path would be stored in the cloud, but then as a discrete sequence of still-positions just like the arrow, and the basic question would remain: How is motion possible at all? But fact is that the person did appear and so let us shift focus from motion to appearance.

We then take Newtonian mechanics to our help which describes the World by the following conservation laws in Eulerian form:
• $\dot\rho +\nabla\cdot m=0$                                (conservation of mass)                          (1)
• $\rho =\Delta\phi$                                          (conservation of gravitational force)     (2)
• $\dot m +\nabla\cdot (vm)-\rho\nabla\phi=0$        (conservation of momentum)                (3)
where $\rho (x,t)$ is mass density, $\phi (x,t)$ gravitational potential, $m$ is momentum, $v= \frac{m}{\rho}$ is velocity and $x$ a Euclidean space coordinate.

The standard way of interpreting (1)-(3) is to say that presence of mass at $(x,t)$ creates the gravitational potential $\phi (y,t)$ for all points $y$ different from $x$ by instant action at distance at time $t$, which however lacks physics explanation. Further, trajectories of motion $x(t)$ appear as solutions to $\dot x=v(x,t)$.

I have suggested a different possibility, which is to view instead the potential $\phi (x,t)$ as primary from which mass $\rho (x,t)=\Delta\phi (x,t)$ is created by differentiation as an instant local action expressed by the Laplacian $\Delta$, which possibly is not inexplicable. The potential $\phi (x,t)$ then changes or evolves in time according to (1) with connection (2), without any need of particle trajectories of motion,

In this view mass emerges or appears at different locations in space following the evolution of the gravitational potential, and we do not have to speak about particle/mass motion and explain exactly how the motion is realised. It connects to time-stepping corresponding to jumping from one discrete time event to the next.

So it may be fruitful to think of appearance evolving in time rather than motion. In this perspective motion is illusionary, like a water wave appearing to move in space without corresponding motion of water.