Background- Independence
G. Belot
Motivation
Preliminaries
Examples Background-Independence
Background- Independence Geometry Counting Gordon Belot Possibilities Proposal Worries University of Pittsburgh & CASBS
March 14, 2007 Something for Everyone
Background- Independence
G. Belot
Motivation
Preliminaries
Examples
Background- For Physicists: Examples. Independence Geometry For Philosophers: Counter-Examples. Counting Possibilities Proposal For Everyone Else: Examples and Counter-Examples! Worries Einstein: The Usual Conception of Space and Time
Background- Independence
G. Belot
Motivation “Ask an intelligent man who is not a scholar what space and
Preliminaries time are, and he will perhaps answer as follows. If we imagine Examples all physical things, all stars, all light taken out of the universe, Background- what then remains is something like a giant vessel without Independence Geometry walls called ‘space.’ With respect to what is happening in the Counting Possibilities world, it plays the same role as the stage in a theater Proposal Worries 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 . . . .” Einstein’s Globes
Background- Independence
G. Belot
Motivation
Preliminaries
Examples
Background- Independence Geometry Counting Possibilities Proposal Worries Einstein on the Globes
Background- Independence “In classical mechanics, and no less in the special theory of G. Belot relativity, there is an inherent epistemological defect . . . .” Motivation “What is the reason for this difference in the two bodies? Preliminaries No answer can be admitted as epistemologically Examples satisfactory, unless the reason given is an observable fact Background- Independence of experience.” Geometry Counting Possibilities “Newtonian mechanics does not give a satisfactory answer Proposal to this question. It pronounces as follows:—The laws of Worries mechanics apply to the space R1, in respect to which the body S1 is at rest, but not to the space R2, in respect to which the body S2 is at rest. But the privileged space R1 of Galileo, thus introduced, is a merely factitious cause [bloß fingierte Ursache], and not a thing that can be observed.” Factitious?
Background- Independence
G. Belot factitious adj.
Motivation 1. (Obsolete) Made by or resulting from art; artificial.
Preliminaries Example: Beer, Ale, or other factitious drinks. Examples 2. Got up, made up for a particular occasion or Background- Independence purpose; arising from custom, habit, or design; not Geometry Counting natural or spontaneous; artificial, conventional. Possibilities Proposal Example: The momentary and factitious joy which had Worries 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. Upshot
Background- Independence
G. Belot
Motivation Einstein set out to create a theory in which space and time Preliminaries were among the actors rather than providing a fixed stage. Examples These days, one says that general relativity is Background- Independence background-independent (i.e., the theory does not feature Geometry Counting a spacetime geometry given a priori). Possibilities Proposal This notion plays some role in polemics about the future Worries of physics. How can one make the intuitive notion of background-independence precise? Master-Builders and Philosophers
Background- Independence
G. Belot “The commonwealth of learning is not at this time without Motivation master-builders, whose mighty designs, in advancing the Preliminaries sciences, will leave lasting monuments to the admiration of Examples posterity: but every one must not hope to be a Boyle or a Background- Independence Sydenham; and in an age that produces such masters as the Geometry Counting Possibilities great Huygenius and the incomparable Mr. Newton, with some Proposal others of that strain, it is ambition enough to be employed as Worries 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 Preliminary Topics
Background- Independence
G. Belot
Motivation
Preliminaries
Examples
Background- A. Symmetries and Patterns. Independence Geometry B. Relativistic Geometries. Counting Possibilities Proposal C. Relativistic Field Theories. Worries Example: Symmetry
Background- Independence
G. Belot
Motivation
Preliminaries
Examples
Background- Independence Geometry Counting Possibilities Proposal Worries Example: Pattern-Preserving Maps
Background- Independence
G. Belot
Motivation
Preliminaries
Examples
Background- Independence Geometry Counting Possibilities Proposal Worries More Formally
Background- Independence
G. Belot Suppose we have configurations C1, C2,.... Each configuration
Motivation involves an assignment of properties and relations to some fixed 0 Preliminaries set of objects D. Consider d : x 7→ x , a means of matching up
Examples objects. Background- d is a symmetry of configuration C if for each x ∈ D, x Independence i 0 Geometry and x play the same role in Ci . Counting Possibilities Proposal d is a pattern-preserving map for Ci and Cj if for each 0 Worries x ∈ D, x plays the same role in Ci that x plays in Cj . We say that Ci and Cj instantiate the same pattern if there is such a pattern-preserving map. We write Ci ∼ Cj . (Sometimes D has internal structure; this will be preserved by any pattern-preserving map.) Relativistic Geometries
Background- Independence G. Belot Ingredients: a manifold V and a metric g of Lorentz signature Motivation (special case: flat metrics—Minkowski spacetime). Preliminaries V provides a very stretchy canvass (topology but no Examples geometry). Background- Independence g provides V with a geometry—including notions of Geometry Counting straightness, distance, timelikeness, etc.—by assigning Possibilities Proposal geometrical properties to each point x ∈ V . Worries Any given g has relatively few symmetries—so few maps d : x 7→ x0 preserve the structure of (V , g). But many maps d : x 7→ x0 preserve the intrinsic structure of V . So for any g, there will be many g 0 such that g ∼ g 0. Relativistic Field Theories
Background- Independence
G. Belot Ingredients: Motivation Spacetime V . Preliminaries
Examples A set Θ of fixed fields on V . Background- A dynamical fields, φ , . . . , φ . We denote a configuration Independence 1 k Geometry of these by Φ. Counting Possibilities Proposal Technical conditions, determining the space K of Worries 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. Pattern-Preserving Maps for Field Theories
Background- Independence
G. Belot
Motivation A pattern-preserving map relating Φ, Φ0 ∈ K is a means Preliminaries d : x 7→ x0 of matching up points of V such that: Examples
Background- (i) d leaves invariant the fixed fields of the theory and the Independence intrinsic structure of V ; Geometry Counting 0 Possibilities (ii) for each x ∈ V , Φ assigns x the same properties that Φ Proposal 0 Worries assigns x . NB. If Φ ∼ Φ0 then both are solutions or neither are—our equations care only about structure. Example: Theory with Fixed Fields
Background- Independence
G. Belot
Motivation
Preliminaries
Examples
Background- Independence Geometry Counting Possibilities Proposal Worries
: Pattern-Preserving Maps are Rigid and Scarce Example: Theory without Fixed Fields
Background- Independence
G. Belot
Motivation
Preliminaries
Examples
Background- Independence Geometry Counting Possibilities Proposal Worries
: Pattern-Preserving Maps are Floppy and Common Paragons
Background- Independence
G. Belot 4 Motivation Ordinary Wave Equation Let V = R with a fixed Minkowski Preliminaries metric η. A real-valued dynamical field φ subject Examples to ηφ = 0. Background- Fully background-dependent: the field propagates Independence Geometry against Minkowski metric in each solution. Counting Possibilities Proposal Cosmological General Relativity V is spatially compact. No Worries 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. Background-Independence = No Fixed Fields?
Background- Independence
G. Belot Any theory with fixed fields is background-dependent. An idea of Einstein: perhaps the converse is also true? Motivation
Preliminaries No. Consider the following near-relative of our paragon of
Examples background-dependence: 4 Background- V = R . Independence No fixed fields. Geometry Counting Two dynamic fields—a metric g subject to Riem[g] = 0, Possibilities Proposal and a real-valued φ subject to g φ = 0. Worries 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. Background-Independence = No Absolute Objects?
Background- Independence Another idea of Einstein: in pre-1915 physics, geometry
G. Belot acts on matter but not vice versa; it is a virtue of general relativity to abandon this. Motivation Following Anderson, Friedman, et al. we say that a field Preliminaries
Examples theory has an absolute object if one of its fields
Background- instantiates the same pattern in every solution. Independence A natural idea: a theory is background-independent iff it Geometry Counting Possibilities features no absolute objects. Proposal No. Consider general relativity with asymptotic boundary Worries 4 conditions: one takes V = R 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. Background-Independence a Matter of Degree
Background- Independence
G. Belot We have seen examples of full background-dependence and full background-independence. Motivation
Preliminaries We have also seen a theory that falls just short of full
Examples background-independence. Background- It is also possible to cook up theories that fall just short of Independence Geometry full background-dependence. Counting Possibilities 3 Proposal Example: V = R × S with two dynamic fields: a metric g Worries subject to Weyl[g] = 0 and 1 Ricci[g] − 4 gR[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. A Geometrically Ambiguous Example
Background- Independence
G. Belot A variant on Nordstr¨om’s scalar theory of gravity. 4 Motivation V = R ; no fixed fields. Preliminaries Dynamical fields: scalar field φ; metrics, η and g. Examples Field Equations: Background- Independence 3 Geometry Riem[η] = 0 φ = −4Gφ T Counting η Possibilities Proposal Weyl[g] = 0 R[g] = 24πGT Worries 2 1 −1 g = φ η φ = (−detg) 8 η = g(−detg) 4
Background dependent if η encodes the geometry; background-independent if g encodes the geometry. Moral: Background-independence is not a formal feature. Desiderata
Background- Independence
G. Belot
Motivation
Preliminaries
Examples Theories lacking fixed fields can be background-dependent. Background- Background-(in)dependence is a matter of degree. Independence Geometry Background-(in)dependence is not a formal matter. Counting Possibilities Proposal Asymptotic boundary conditions can lead to a degree of Worries background-dependence. A Proposal Sketched: Background-Independence as Fine Dependence of Geometry on Fields
Background- Independence G. Belot Basic Idea: A theory is background-independent to the extent
Motivation that the geometry of a solution depends on the
Preliminaries fields of the theory. Examples At one extreme we have theories in which this Background- Independence dependence is as fine as possible: two solutions Geometry have the same geometry iff they represent the Counting Possibilities Proposal same physical possibility. Worries 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. Geometry and Content
Background- Independence When do we think of a field theory as physics rather than G. Belot mathematics? Motivation Plausible answer: when we understand its solutions are Preliminaries representing spatiotemporal processes. Examples This is a substantive step: we assign a geometry to each Background- Independence solution; lay down a notion of sameness of geometry; Geometry Counting require covariance. Possibilities Proposal We restrict attention to cases where this is almost Worries 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. Counting Possibilities: Simple Cases
Background- Independence G. Belot When possible, the standard physicist assumes that the Motivation space of solutions of a theory parameterizes the space of Preliminaries possibilities allowed by a theory. Examples In even the simplest cases, philosophers object: Background- Independence Haecceitist: Too few! Geometry Counting Possibilities Anti-Haecceitist: Too many! Proposal Worries 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. Counting Possibilities: Theories without Fixed Fields
Background- Independence Rough and ready notion of determinism: if the
G. Belot instantaneous states are the same, then the global states are the same. In a well-behaved classical theory, Motivation
Preliminaries indeterminism should be rare and illuminating (given
Examples appropriate boundary conditions).
Background- Consider a theory without fixed fields. Independence 0 Geometry It is easy to find distinct Φ, Φ ∈ S that induce the same Counting Possibilities initial data on some hypersurface Σ ⊂ V —look for Proposal Worries solutions related by a pattern preserving map that is the identity on a neighbourhood of Σ. Indiscriminate and uninteresting indeterminism threatens unless we: 0 (a) Deny that Φ |Σ and Φ |Σ correspond to the same instantaneous state. Bad idea. (b) Deny that Φ and Φ0 correspond to distinct physical possibilities. Good idea. Determinism Threatened
Background- Independence
G. Belot
Motivation
Preliminaries
Examples
Background- Independence Geometry Counting Possibilities Proposal Worries Gauge Equivalence
Background- 0 0 Independence Star: We write Φ ∗ Φ if Φ and Φ instantiate the same G. Belot pattern and induce the same initial data on some
Motivation Σ ⊂ V . Preliminaries Gauge Equivalence is the equivalence relation on S generated Examples by ∗: Φ and Φ0 are gauge equivalent iff there Background- Independence exist solutions Φ1,..., Φk such that Φ = Φ1, 0 Geometry Φ = Φ , and Φ ∗ Φ for each i = 1,..., k − 1. Counting k i i+1 Possibilities Proposal Worries 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. Features of Gauge Equivalence
Background- Independence
G. Belot Gauge equivalent solutions instantiate the same pattern. Motivation In a theory with g a fixed field, each solution is gauge Preliminaries equivalent only to itself. Examples
Background- In a theory without fixed fields: Independence If there are no asymptotic boundary conditions then Geometry Counting Possibilities solutions are gauge equivalent iff they instantiate the same Proposal pattern. Worries 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 . . . Asymptotically Flat GR: Pattern-Preserving Maps
Background- Independence
G. Belot
Motivation
Preliminaries
Examples
Background- Independence Geometry Counting Possibilities Proposal Worries Asymptotically Flat GR: Gauge Equivalence
Background- Independence
G. Belot
Motivation
Preliminaries
Examples
Background- Independence Geometry Counting Possibilities Proposal Worries Proposal in Full
Background- Independence
G. Belot A theory is fully background-dependent if every pair of Motivation solutions correspond to the same abstract geometry. Preliminaries A theory is nearly background-dependent if the family of Examples abstract geometries instantiated is small Background- Independence (finite-dimensional, say). Geometry Counting Possibilities A theory is fully background-independent if two solutions Proposal correspond to the same abstract geometry iff they Worries 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). Asymptotically Flat General Relativity
Background- Independence
G. Belot
Motivation Preliminaries Under the standard scheme for counting possibilities the Examples theory is nearly (but not fully) background-independent: Background- Independence the space of possibilities corresponding to any instantiated Geometry abstract geometry is ten-dimensional. Counting Possibilities Proposal There is an alternative scheme under which solutions Worries corefer iff related by a pattern-preserving map. Under this scheme the theory is fully background-independent. Desiderata Revisited
Background- Independence
G. Belot Lack of fixed fields is consistent with full background-dependence. (Klein–Gordon field on flat Motivation spacetime.) Preliminaries
Examples Background-(in)dependence is non-formal, depending on
Background- choice of geometrization and on the choice of scheme for Independence Geometry counting possibilities. (Nordstr¨om’stheory; asymptotically Counting Possibilities flat general relativity.) Proposal Worries Background-(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.) Worries
Background- Independence
G. Belot
Motivation What about trivial field theories? Preliminaries
Examples What about non-relativistic theories?
Background- Independence Is string theory background-independent? Geometry Counting What about non-spatiotemporal symmetries? Possibilities Proposal What if some geometry is solution-independent? Worries What about weirder asymptotic boundary conditions? What happens if matter is included in general relativity?