Local Coexistence: Leibniz vs Einstein

Local coexistence.

A basic idea of modern physics is that the conceptions of space and time of classical physics formed by Leibniz/Newton have to be replaced by a concept of curved space-time of Einstein's theory of relativity. Einstein struggled through his entire scientific life to give meaning to this new concept, but today very few modern physicists claim to have a proper understanding of what Einstein wanted to tell humanity, while it is commonly agreed that Einstein was right and so that Leibniz/Newton was wrong.   

But the physics of Leibniz/Newton is the classical physics of the scientific revolution reaching full power with Maxwell's equations at the end of the 19th century thus capturing all of macroscopic mechanics and electromagnetics in differential equations in terms of functions $f(x,t)$ of classical Euclidean space coordinates $x$ and time $t$. It was a formidable success, which met the young patent clerk Einstein preparing for a scientific career at the turn to the 20th century with the demanding question: How to impress the World?

At the patent office Einstein tested his abilities on some main open problems of classical physics including black body radiation, photo-electricity and the (non-)existence of an ether as medium for the propagation of electromagnetic waves. Einstein's efforts resulted in 5 articles on these topics in 1905 named Annus Mirabilis. 

In one of the papers Einstein picked up an observation by Lorentz that wave equations take the same mathematical form under a simple linear transformation of coordinates mixing space and time, named Lorentz transformation. Lorentz cautioned that coordinates transformed from physical coordinates were absolutely not to be viewed as physical coordinates. But here young Einstein found a thread to follow, by opposing Lorentz and claiming that all coordinates subject to Lorentz transformation have a physical meaning. So was Einstein's special theory of relativity formed as the physics of Lorentz transformations, which showed a very new form of physics with space contraction and time dilation unheard of by Leibniz/Newton and classical physics. But this did not impress the physics community at all, which made Einstein double down by extending his special theory to a General Theory of Relativity GR during 10 years of hard struggle to make sense of his curved space-time. Again the physics community was not impressed and GR was not hailed as a trade mark of modern physics until Einstein had passed away and no longer could be asked to clarify his ideas, and so the physics community was free to exploit Einstein's $E=mc^2$ and nuclear power to attract funding to Big Physics, which succeeded very well until lack of results resulted in a backlash as the crisis of modern physics of today...

The mantra of modern physics is that the classical concepts of space and time by Leibniz/Newton have to be replaced by Einstein's space-time coming to expression in the Lorentz transformation mixing space and time into strange new physics. The mantra was based on a perception that Newton's physics required absolute space and time, which was hard to rationalise. 

But Leibniz viewed both space and time to be relational (and so relative) and thus not absolute in Newton's sense: 

• space is order of coexistence
• time is order of succession/change    

which can pull the rug under the necessity to give up classical concepts. It is natural to qualify Leibniz view to 

• space is measure of local coexistence
• time is measure of change of local coexistence.    

The central concept is local coexistence or simultaneity as something appearing with nearby space coordinates at the same time, like an extended body existing at a specified time. The novelty as compared with Newton's absolute space as global space is the aspect of locality. 

This aspect is expressed in explicit time stepping of the heat equation $\dot T(x,t) = \Delta T(x,t)=0$ with $T(x,t)$ temperature depending on a (1d for simplicity) Euclidean space coordinate $x$ and a time coordinate $t$, which takes the form

• $T(x, t+dt) = T(x,t)+0.5(T(x+dx,t)-2T(x,t) + T(x-dx,t))$

where $dx$ is a small space step and $dt=0.5dx^2$ a time step. We see that $T(x,t+dt)$ at new time $t+dt$ is updated from $T(y,t)$ at previous time $t$ and $y=x$ and nearby $y=x\pm dx$ expressing local coexistence (with effectively finite speed of propagation). This connects to the previous post on Newton's law of gravitation and Universe Model.

The Lorentz transformation connects space-time events without extension in space and so has no room for local coexistence/simultaneity. But without local coexistence/simultaneity the World falls apart and so must a physical theory based on the Lorentz transformation. 

Locality of space and time comes to expression by the clock of the local village church telling the villagers when to quit work and go home to eat, or a local Sunset. Global coordination is not needed to make physics without human intervention to go around. Explicit time stepping under local coexistence/simultaneity is enough. 

It is important to make a distinction between man-made physics and natural physics without human intervention. Man-made physics like the GPS system uses global time shared by satellites realised by man-made synchronisation, while natural physics only requires local time as local coexistence. 

Sum up: With a Leibnizian concept of space as local coexistence and time as change of local coexistence there is no compelling reason to replace classical concepts (which make sense) by Einstein's concept of curved space-time (which does not make sense). Leibniz thus offers a theory of relativity without the confusing aspects Einstein's theories.