In the initial post of this thread, I introduced Dewey B. Larson's Reciprocal System of Physical Theory (RST) as a new system of physical theory that goes beyond Newton's system and thus provides the basis for a new program of research that, while it has the same grand goal of Newton's program, is based on a new definition of motion.
Larson's new definition of motion is based, in turn, on his novel definition of space and time. The nature of space and time has been at the center of the physics philosophical debate for centuries. Discussing this in a recent paper, Lee Smolin, of the Perimeter Institute for Theoretical Physics, summarizes the, as yet unanswered, fundamental challenges facing the mainstream physics community for the last three decades: [1]
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During the last three decades research in theoretical physics has focused on four key problems, which, however, remain unsolved. These are
1. The problem of quantum gravity.
2. The problem of further unifying the different forces and particles, beyond the partial unification of the standard model.
3. The problem of explaining how the parameters of the standard models of particles physics and cosmology, including the cosmological constant, were chosen by nature.
4. The problem of what constitutes the dark matter and energy, or whether the evidence for them are to be explained by modifications in the laws of physics at very large scales.
One can also mention a fifth unsolved problem, that of resolving the controversies concerning the foundations of quantum mechanics.
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In his paper, Smolin shows how, in spite of some tantalyzing clues, the vigorous pursuit of a solution to these challenges by a cast of thousands of the most bright and talented people in the world, using the string theory approach, has only led to some perplexing and recalcitrant difficulties. This situation drives him to hypothesize that "some wrong assumption was made somewhere in the course of the development of [string] theory," and he further surmises that the false assumption has to do with the nature of space and time. Specifically, he refers to the age old debate between relational and absolute theories of space and time, "which, he notes, has been central to the thinking about the nature of space and time going back to the beginning of physics."
Thus, Smolin asserts that, if string theory is to succeed in meeting the key issues facing modern physics, as many hope it will, it must be reformulated in a way that does not depend on the current assumption regarding the nature of space and time as a background for physical phenomena. He writes:
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The reason that we do not have a fundamental formulation of string theory, from which it might be possible to resolve the challenge posed by the landscape, is that it has been so far developed as a background dependent theory. This is despite there being compelling arguments that a fundamental theory must be background independent. Whether string theory turns out to describe nature or not, there are now few alternatives but to approach the problems of unification and quantum gravity from a background independent perspective.
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The idea of background independence is the modern articulation of the famous debate between Newton and Leibniz over whether space and time are properly regarded as something that exists substantively and absolutely, or whether they have no meaning other than that given to them by the relative positions of objects in different spatial locations at different moments in time. Newton argued that space and time are to be regarded as absolute, and he eventually won the argument after presenting his famous water bucket thought experiment.
However, Smolin points out that Leibniz's argument for the "principle of sufficient reason," eliminates the philosophical problem that Newton's position raises, namely that "a theory that begins with the choice of a background geometry, among many equally consistent choices," must provide the justification for that choice. But, since no theory can justify the position or orientation of the universe as a whole, relative to a given background, the theoretical requirement for a fixed background of space and time becomes a philosophical liability. Smolin writes:
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This is sometimes called the problem of under determination: nothing in the laws of physics answers the question of why the whole universe is where it is, rather than translated or rotated.
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This is not the only philosophical argument Leibniz raised against a background dependent theory, there are others having to do with global symmetries and conserved charges that modern physics eventually has come to recognize in the context of general relativity. Nevertheless, Smolin notes, "a physics where space and time are absolute can be developed one particle at a time, while a relational view requires that the properties of any one particle are determined self-consistently by the whole universe." Of course, eventually, the dues have to be paid, and it appears that the time has come for modern theoretical physics to pay up.
Change comes slowly, however, and the recognition of the need for a background independent theory is not as universally acknowledged as Smolin would like. He and his colleagues, however, have tried to come to grips with the problem, and in so doing have arrived at a "rough consensus" as to what a relational view of space and time actually is. They refer to it as "the physicists' relational conception of space and time." There are three elements to this concept that Smolin discusses:
1) There is no background.
2) The fundamental properties of the elementary entities consist entirely in relationships between those elementary entities.
3) The relationships are not fixed, but evolve according to law. Time is nothing but changes in the relationships, and consists of nothing but their ordering.
Smolin characterizes the dynamics of such a concept as consisting of the changes of the relationship of its entities over time, which he summarizes in statement 3 above. "Thus," he continues, "we often take background independent and relational as synonymous. The debate between philosophers that used to be phrased in terms of absolute vrs relational theories of space and time is continued in a debate between physicists who argue about background dependent vrs background independent theories."
In this debate, Smolin articulates a strategy for those seeking background-independent theories:
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Relational strategy: Seek to make progress by identifying the background structure in our theories and removing it, replacing it with relations which evolve subject to dynamical law.
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Smolin cites Mach's ideas, and Einstein's successful exploitation of them, as an encouraging indication that the correct paradigm for the relational strategy is Mach's principle:
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Mach’s principle is the paradigm for this strategic view of relationalism. ...Mach’s suggestion was that replacing absolute space as the basis for distinguishing acceleration from uniform motion with the actual distribution of matter would result in a theory that is more explanatory, and more falsifiable. Einstein took up Mach’s challenge, and the resulting success of general relativity can be taken to vindicate both Mach’s principle and the general strategy of making theories more relational.
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However, this is obviously a compromise, since Mach's principle provides a relational background, which, while, in the final analysis, it clearly is better than an absolute background of space and time, it is a background nevertheless, and, though it addresses the global symmetry problem, it does not affect the description of motion, which still must be defined as a function of x(t).
(see continuation in following post)