Background-Independence

Transcription

1 Gordon Belot University of Pittsburgh & CASBS March 14, 2007

2 Something for Everyone For Physicists:. For Philosophers: Counter-. For Everyone Else: and Counter-!

3 Einstein: The Usual Conception of Space and Time Ask an intelligent man who is not a scholar what space and time are, and he will perhaps answer as follows. If we imagine all physical things, all stars, all light taken out of the universe, what then remains is something like a giant vessel without walls called space. With respect to what is happening in the world, it plays the same role as the stage in a theater performance. In this space, in this vessel without walls, there is an eternally uniformly occurring tick-tock... that is time. Most natural scientists, up to the present, had this conception about the essence of space and time....

4 Einstein s Globes

5 Einstein on the Globes In classical mechanics, and no less in the special theory of relativity, there is an inherent epistemological defect.... What is the reason for this difference in the two bodies? No answer can be admitted as epistemologically satisfactory, unless the reason given is an observable fact of experience. Newtonian mechanics does not give a satisfactory answer to this question. It pronounces as follows: The laws of mechanics apply to the space R 1, in respect to which the body S 1 is at rest, but not to the space R 2, in respect to which the body S 2 is at rest. But the privileged space R 1 of Galileo, thus introduced, is a merely factitious cause [bloß fingierte Ursache], and not a thing that can be observed.

6 Factitious? factitious adj. 1. (Obsolete) Made by or resulting from art; artificial. Example: Beer, Ale, or other factitious drinks. 2. Got up, made up for a particular occasion or purpose; arising from custom, habit, or design; not natural or spontaneous; artificial, conventional. Example: The momentary and factitious joy which had greeted the day of William s crowning died utterly away. 3. (Medical) Of a disorder, symptom, or sign: feigned or self-induced by a patient. Example: Factitious purulent ophthalmia produced by the liquorice liana, or jequirity.

7 Upshot Einstein set out to create a theory in which space and time were among the actors rather than providing a fixed stage. These days, one says that general relativity is background-independent (i.e., the theory does not feature a spacetime geometry given a priori). This notion plays some role in polemics about the future of physics. How can one make the intuitive notion of background-independence precise?

8 Master-Builders and Philosophers The commonwealth of learning is not at this time without master-builders, whose mighty designs, in advancing the sciences, will leave lasting monuments to the admiration of posterity: but every one must not hope to be a Boyle or a Sydenham; and in an age that produces such masters as the great Huygenius and the incomparable Mr. Newton, with some others of that strain, it is ambition enough to be employed as an under-labourer in clearing the ground a little, and removing some of the rubbish that lies in the way to knowledge.... Locke, An Essay Concerning Human Understanding

9 Preliminary Topics A. Symmetries and Patterns. B. Relativistic Geometries. C. Relativistic Field Theories.

10 Example: Symmetry

11 Example: Pattern-Preserving Maps

12 More Formally Suppose we have configurations C 1, C 2,.... Each configuration involves an assignment of properties and relations to some fixed set of objects D. Consider d : x x, a means of matching up objects. d is a symmetry of configuration C i if for each x D, x and x play the same role in C i. d is a pattern-preserving map for C i and C j if for each x D, x plays the same role in C i that x plays in C j. We say that C i and C j instantiate the same pattern if there is such a pattern-preserving map. We write C i C j. (Sometimes D has internal structure; this will be preserved by any pattern-preserving map.)

13 Relativistic Geometries Ingredients: a manifold V and a metric g of Lorentz signature (special case: flat metrics Minkowski spacetime). V provides a very stretchy canvass (topology but no geometry). g provides V with a geometry including notions of straightness, distance, timelikeness, etc. by assigning geometrical properties to each point x V. Any given g has relatively few symmetries so few maps d : x x preserve the structure of (V, g). But many maps d : x x preserve the intrinsic structure of V. So for any g, there will be many g such that g g.

14 Relativistic Field Theories Ingredients: Spacetime V. A set Θ of fixed fields on V. A dynamical fields, φ 1,..., φ k. We denote a configuration of these by Φ. Technical conditions, determining the space K of kinematically possible Φ. Differential equations determining the space S K of solutions of the theory. A relativistic metric g must be among the fields of the theory.

15 Pattern-Preserving Maps for Field Theories A pattern-preserving map relating Φ, Φ K is a means d : x x of matching up points of V such that: (i) d leaves invariant the fixed fields of the theory and the intrinsic structure of V ; (ii) for each x V, Φ assigns x the same properties that Φ assigns x. NB. If Φ Φ then both are solutions or neither are our equations care only about structure.

16 Example: Theory with Fixed Fields : Pattern-Preserving Maps are Rigid and Scarce

17 Example: Theory without Fixed Fields : Pattern-Preserving Maps are Floppy and Common

18 Paragons Ordinary Wave Equation Let V = R 4 with a fixed Minkowski metric η. A real-valued dynamical field φ subject to η φ = 0. Fully background-dependent: the field propagates against Minkowski metric in each solution. Cosmological General Relativity V is spatially compact. No fixed fields. A single dynamical field a metric g subject to Ricci[g] = 0. Fully background-independent: maximal variation of geometry from solution to solution.

19 = No Fixed Fields? Any theory with fixed fields is background-dependent. An idea of Einstein: perhaps the converse is also true? No. Consider the following near-relative of our paragon of background-dependence: V = R 4. No fixed fields. Two dynamic fields a metric g subject to Riem[g] = 0, and a real-valued φ subject to g φ = 0. In this new theory, as in our paragon, each solution consists of a field obeying the wave equation living in Minkowski spacetime. Moral: A theory lacking fixed fields can be fully background-dependent.

20 = No Absolute Objects? Another idea of Einstein: in pre-1915 physics, geometry acts on matter but not vice versa; it is a virtue of general relativity to abandon this. Following Anderson, Friedman, et al. we say that a field theory has an absolute object if one of its fields instantiates the same pattern in every solution. A natural idea: a theory is background-independent iff it features no absolute objects. No. Consider general relativity with asymptotic boundary conditions: one takes V = R 4 and includes in K only those g that are asymptotically flat at spatial infinity. Consensus view: such a theory enjoys a large degree of background-independence but involves the introduction of geometrical background at infinity. Moral: Absoluteness test fails to detect some forms of background-dependence.

21 a Matter of Degree We have seen examples of full background-dependence and full background-independence. We have also seen a theory that falls just short of full background-independence. It is also possible to cook up theories that fall just short of full background-dependence. Example: V = R S 3 with two dynamic fields: a metric g subject to Weyl[g] = 0 and Ricci[g] 1 4gR[g] = 0; and a real-valued φ subject to g φ = 0. In each solution, g is de Sitter so spacetime has constant positive curvature k but the value of k can vary from solution to solution.

22 A Geometrically Ambiguous Example A variant on Nordström s scalar theory of gravity. V = R 4 ; no fixed fields. Dynamical fields: scalar field φ; metrics, η and g. Field Equations: Riem[η] = 0 Weyl[g] = 0 η φ = 4Gφ 3 T R[g] = 24πGT g = φ 2 η φ = ( detg) 1 8 η = g( detg) 1 4 Background dependent if η encodes the geometry; background-independent if g encodes the geometry. Moral: independence is not a formal feature.

23 Desiderata Theories lacking fixed fields can be background-dependent. (in)dependence is a matter of degree. (in)dependence is not a formal matter. Asymptotic boundary conditions can lead to a degree of background-dependence.

24 A Sketched: as Fine Dependence of on Fields Basic Idea: A theory is background-independent to the extent that the geometry of a solution depends on the fields of the theory. At one extreme we have theories in which this dependence is as fine as possible: two solutions have the same geometry iff they represent the same physical possibility. At the other extreme we have theories in which there is no such dependence: every solution has the same geometry. Required Ingredients: Appropriate notion of geometry; Schemes for counting possibilities.

25 and Content When do we think of a field theory as physics rather than mathematics? Plausible answer: when we understand its solutions are representing spatiotemporal processes. This is a substantive step: we assign a geometry to each solution; lay down a notion of sameness of geometry; require covariance. We restrict attention to cases where this is almost automatic: one of the fields of the theory encodes the Lorentzian geometry of spacetime; two solutions encode the same geometry iff their metrics are related by a pattern preserving map. But: In more general settings, things can become more interesting.

26 : Simple Cases When possible, the standard physicist assumes that the space of solutions of a theory parameterizes the space of possibilities allowed by a theory. In even the simplest cases, philosophers object: Haecceitist: Too few! Anti-Haecceitist: Too many! Claim: The disagreement between the standard physicist and the haecceitist is largely terminological. In what follows: focus on the strategy of the standard physicist; other strategies can be plugged into final proposal.

27 : Theories without Fixed Fields Rough and ready notion of determinism: if the instantaneous states are the same, then the global states are the same. In a well-behaved classical theory, indeterminism should be rare and illuminating (given appropriate boundary conditions). Consider a theory without fixed fields. It is easy to find distinct Φ, Φ S that induce the same initial data on some hypersurface Σ V look for solutions related by a pattern preserving map that is the identity on a neighbourhood of Σ. Indiscriminate and uninteresting indeterminism threatens unless we: (a) Deny that Φ Σ and Φ Σ correspond to the same instantaneous state. Bad idea. (b) Deny that Φ and Φ correspond to distinct physical possibilities. Good idea.

28 Determinism Threatened

29 Gauge Equivalence Star: We write Φ Φ if Φ and Φ instantiate the same pattern and induce the same initial data on some Σ V. Gauge Equivalence is the equivalence relation on S generated by : Φ and Φ are gauge equivalent iff there exist solutions Φ 1,..., Φ k such that Φ = Φ 1, Φ = Φ k, and Φ i Φ i+1 for each i = 1,..., k 1. Everyone agrees that gauge equivalent solutions always represent the same possibility. The standard physicist s approach is to take solutions to represent the same possibility iff they are gauge equivalent. Under this approach, the space of equivalence classes of gauge equivalence parameterizes the space of possibilities of a theory.

30 Features of Gauge Equivalence Gauge equivalent solutions instantiate the same pattern. In a theory with g a fixed field, each solution is gauge equivalent only to itself. In a theory without fixed fields: If there are no asymptotic boundary conditions then solutions are gauge equivalent iff they instantiate the same pattern. If there are asymptotic boundary conditions then (typically) solutions are gauge equivalent iff related by a pattern-preserving map asymptotic to the identity at infinity. NB some further technical conditions required...

31 Asymptotically Flat GR: Pattern-Preserving Maps

32 Asymptotically Flat GR: Gauge Equivalence

33 in Full A theory is fully background-dependent if every pair of solutions correspond to the same abstract geometry. A theory is nearly background-dependent if the family of abstract geometries instantiated is small (finite-dimensional, say). A theory is fully background-independent if two solutions correspond to the same abstract geometry iff they represent the same physical possibility. A theory is nearly background-independent if for each abstract geometry, the corresponding family of physical possibilities is small (finite-dimensional, say).

34 Asymptotically Flat General Relativity Under the standard scheme for counting possibilities the theory is nearly (but not fully) background-independent: the space of possibilities corresponding to any instantiated abstract geometry is ten-dimensional. There is an alternative scheme under which solutions corefer iff related by a pattern-preserving map. Under this scheme the theory is fully background-independent.

35 Desiderata Revisited Lack of fixed fields is consistent with full background-dependence. (Klein Gordon field on flat spacetime.) (in)dependence is non-formal, depending on choice of geometrization and on the choice of scheme for counting possibilities. (Nordström s theory; asymptotically flat general relativity.) (in)dependence is a matter of degree. (Klein Gordon field on de Sitter spacetime; asymptotically flat general relativity.) Full background-independence can be spoiled by asymptotic boundary conditions. (Asymptotically flat general relativity.)

36 What about trivial field theories? What about non-relativistic theories? Is string theory background-independent? What about non-spatiotemporal symmetries? What if some geometry is solution-independent? What about weirder asymptotic boundary conditions? What happens if matter is included in general relativity?