Fay Dowker at 1:04:06 I side with Leibniz The great debate between defining notions of space and time as real objects themselves (absolute), or mere orderings upon actual objects (relational), began between physicists Isaac Newton (via his spokesman, Samuel Clarke) and Gottfried Leibniz in the papers of the Leibniz–Clarke correspondence. it took physics a long time to catch up to Leibniz’s thinking. Even if philosophers were convinced that Leibniz had the better argument, Newton’s view was easier to develop, and took off, whereby Leibniz’s remained philosophy. This is easy to understand: 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. Leibniz’s criticisms of Newton’s physics were sharpened by several thinkers, the most influential of which was Mach, who in the late 19th century gave an influential critique of Newtonian physics on the basis of its treatment of acceleration as absolute. Einstein was among those whose thinking was changed by Mach. There is a certain historical complication, because what Einstein called ”Mach’s principle” was not exactly what Mach wrote. But that need not concern us here. The key idea that Einstein got from,or read into, Mach, was that acceleration should be defined relative to a frame of reference that is dynamically determined by the configuration of the whole universe, rather than being fixed absolutely, as in Newton’s theory. In Newton’s mechanics, the distinction between who is accelerating and who is moving uniformly is a property of an absolute background space-time geometry, that is fixed independently of the history or configuration of matter. Mach proposed, in essence eliminating absolute space as a cause of the distinction between accelerated and non-accelerated motion, and replacing it with a dynamically determined distinction. This resolves the problem of under-determination, by replacing an a priori background with a dynamical mechanism. By doing this Mach showed us that a physics that respects Leibniz’s principle of sufficient reason is more predictive, because it replaces an arbitrary fact with a dynamically caused and observationally falsifiable relationship between the local inertial frames and the distribution of matter in the universe. This for the first time made it possible to see how, in a theory without a fixed background, properties of local physics, thought previously to be absolute, might be genuinely explained, self consistently, in terms of the whole universe. There is a debate about whether general relativity is ”Machian”, which is partly due to confusion over exactly how the term is to be applied. But there is no doubt that general relativity can be characterized as a partly relational theory, in a precise sense that I will explain below. Any theory postulates that the world is made up of a very large collection of elementary entities (whether particles, fields, or events or processes.) Indeed, the fact that the world has many things in it is essential for these considerations-it means that the theory of the world may be expected to differ in important aspects from models that describe the motion of a single particle, or a few particles in interaction with each other. The basic formof a physical theory is framed by howthesemany entities acquire properties. In an absolute framework the properties of any entity are defined with respect to a single entity-which is presumed to be unchanging. An example is the absolute space and time of Newton, according to which positions and motions are defined with respect to this unchanging entity. Thus, in Newtonian physics the background is three dimensional space, and the fundamental properties are a list of the positions of particles in absolute space as a function of absolute time: xai (t). Another example of an absolute background is a regular lattice, which is often used in the formulation of quantum field theories. Particles and fields have the property of being at different nodes in the lattice, but the lattice does not change in time. The entities that plays this role may be called the background for the description of physics. The background consists of presumed entities that do not change in time, but which are necessary for the definition of the kinematical quantities and dynamical laws. The most basic statement of the relational view is that R1 There is no background. How then do we understand the properties of elementary particles and fields? The relational view presumes that R2 The fundamental properties of the elementary entities consist entirely in relationships between those elementary entities. Dynamics is then concerned with how these relationships change in time. An example of a purely relational kinematics is a graph. The entities are the nodes. The properties are the connections between the nodes. The state of the system is just which nodes are connected and which are not. The dynamics is given by a rule which changes the connectivity of the graph. We may summarize this as R3 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. Thus, we often take background independent and relational as synonymous. The debate between philosophers that used to be phrased in terms of absolute vs. relational theories of space and time is continued in a debate between physicists who argue about background dependent vrs background independent theories. It should also be said that for physicists relationalism is a strategy. As we shall see, theories may be partly relational, i.e.. they can have varying amounts of background structure. 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. Mach’s principle is the paradigm for this strategic view of relationalism. As discussed above, 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. General relativity is a partly relational theory. one cannot define the physical observables of the theory without solving the dynamics. In other words, as Stachel emphasizes, there is no kinematics without dynamics. This is because all observables are relational, in that they describe relations between physical degrees of freedom. You cannot just ask what is happening at a manifold point, or an event, labeled by some coordinate, and assume you are asking a physically meaningful question. The problem is that because of diffeomorphism invariance, points are not physically meaningful without a specification of how a point or event is to be identified by the values of some physical degrees of freedom. As a result, even observables that refer to local points or regions of physical spacetime are non-local in the sense that as functions of initial data they depend on data in the whole initial slice. As a result, the physical interpretation of classical general relativity is more subtle than is usually appreciated. In fact, most of what we think we understand naively about how to interpret classical GR applies only to special solutions with symmetries, where we use the symmetries to define special coordinates. These methods do not apply to generic solutions, which have no symmetries. It is possible to give a physical interpretation to the generic solutions of the theory, but only by taking into account the issues raised by the facts that all physical observables must be diffeomorphism invariant, and the related fact that the hamiltonian is a sum of constraints. We see here a reflection of Leibniz’s principles, in that the interpretation that must be given to generic solutions, without symmetries, is completely different from that given to the measure zero of solutions with symmetries. even at the classical level, there is a distinction between background independent and background dependent approaches to the physical interpretation. If one is interested only in observables for particles moving within a given spacetime, one can use a construction that regards that spacetime as fixed. But if one wants to discuss observables of the gravitational field itself, one cannot use background dependent methods, for those depend on fixing the gravitational degrees of freedom to one solution. To discuss how observables vary as we vary the solution to the Einstein equations we need functions of the phase space variables that make sense for all solutions. Then one must work on the full space of solutions, either in configuration space or phase space. The problem of time. One can see this with the issue of time. If by time you mean time experienced by observers following worldlines in a given spacetime, then we can work within that spacetime. For example, in a given spacetime time can be defined in terms of the causal structure. But if one wants to discuss time in the context in which the gravitational degrees of freedom are evolving, then one cannot work within a given spacetime. One constructs instead a notion of time on the infinite dimensional phase or configuration space of the gravitational field itself. Thus, at the classical level, there are clear solutions to the problems of what is time and what is an observable in general relativity. - LEE SMOLIN
Posted on: Mon, 22 Dec 2014 15:50:47 +0000
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