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Shape Dynamics Shape Dynamics Tim A. Koslowski Abstract Barbour’s formulation of Mach’s principle requires a theory of gravity to implement local relativity of clocks, local relativity of rods and spatial covariance. It turns out that relativity of clocks and rods are mutually exclusive. General Relativity implements local relativity of clocks and spatial covariance, but not local relativity of rods. It is the purpose of this contribution to show how Shape Dynamics, a theory that is locally equivalent to General Relativity, implements local relativity of rods and spatial covariance and how a BRST formulation, which I call Doubly General Relativity, implements all of Barbour’s principles. 1 Introduction A reflection on Mach’s principle lead Barbour to postulate that rods and spatial frames of reference should be locally determined by a procedure that he calls “best matching,” while clocks should be locally determined by what he calls “objective change.” (For more, see [1].) More concretely, Barbour’s principles postulate lo- cal time reparametrization invariance, local spatial conformal invariance and spatial covariance. The best matching algorithm for spatial covariance and local spatial conformal invariance turns out to be equivalent to the imposition of linear diffeo- morphism and conformal constraints Z Z 3 ab 3 H(x) = d xp (Lx g)ab; C(r) = d xr p; (1) S S Tim A. Koslowski Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, Canada New address: Department of Mathematics and Statistics, University of New Brunswick Fredericton, New Brunswick E3B 5A3, Canada e-mail: [email protected] 1 2 Tim A. Koslowski where we use a compact Cauchy surface S without boundary with Riemannian met- ab ric gab and metric momentum density p with trace p. The vector field x and the scalar field r are Lagrange multipliers. A more involved procedure, which I will not explain here, leads to the implementation of local time reparametrization invariance through quadratic Hamilton constraints Z ( ) 3 ab cd Sˆ(N) = d xN p Fabcdp −V ; (2) S where Fabcd(x) and V(x) are constructed from gab(x) and its derivatives at x and N denotes a Lagrange multiplier. There is no reason for Sˆ to have homogeneous con- formal weight, so a system containing the constraints Sˆ(N) and C(r) will not be first class except for very special choices of Fabcd;V. This means that time reparametriza- tion symmetry and spatial conformal symmetry generically exclude one another. An interesting situation occurs when we choose the Fabcd;V to reproduce the Hamilton constraints of General Relativity ! Z ab 1 cd p p (gacgbd − gabgcd)p S(N) = d3xN p 2 − (R − 2L) jgj ; (3) S jgj where the constraint system S(N);Q(r) is completely second class, while the con- straint system S(N);H(x) is first class as is the constraint system Q(r);H(x). We will shortly see that this is the reason, why a Shape Dynamics and Doubly General Relativity can be constructed [2]. 2 Symmetry Trading Gauge theories describe a physical system using redundant degrees of freedom. The physical degrees of freedom are identified with orbits of the action of the gauge group. This redundant description is very useful in field theory, because it is often the only local description of a given system. The canonical description of gauge theories (see e.g. [3]) is provided by a regular irreducible set of first class constraints fca ga2A , whose elements ca are smooth functions on a phase space G with Poisson bracket f:;:g. First class means that the constraint surface C = fx 2 G : ca (x) = 0; 8a 2 A g is foliated into gauge orbits, whose infinitesimal generators are the Hamilton vector fields va : f 7! fca ; f g. For simplicity we will assume that the system is generally covariant, so it has vanishing on-shell Hamil- tonian. Observables [O] of the system are equivalence classes of smooth gauge- invariant functions O on G , where two functions are equivalent if they coincide on C and where gauge invariance means that O is constant along gauge orbits. This means that an observable is completely determined by determining its dependence on a reduced phase space Gred, which contains one and only one point out of each gauge orbit. There is no unique choice of Gred and the simplest description is through Shape Dynamics 3 a a regular irreducible set of gauge fixing conditions fs ga2A , such that a proper re- a duced phase is defined through Gred = fx 2 G : ca (x) = 0 = s (x); 8a 2 A g. The observable algebra can then be identified with the Dirac algebra on reduced phase space, where the Dirac bracket takes the form ( ) a b b a f f ;ggD := f f ;gg − f f ; ca gMb fs ;gg − f f ;s gMb fca ;gg ; (4) where M denotes the inverse of the linear operator fc;sg. The condition that Gred contains one and only one point out of each gauge orbit poses important restrictions on the gauge fixing conditions, but the set of gauge fixing conditions is not required to be first class. A very interesting situa- tion arises when the set of gauge fixing condition is itself first class: In this case one can switch the role of gauge fixing conditions and constraints and describe the same observable algebra, and thus the same physical system, with the gauge theory a A = (G ;f:;:g;fca ga2A ) or with the gauge theory B = (G ;f:;:g;fs ga2A ). The manifest equivalence of the two theories is established by gauge fixing theory A a with the gauge fixing set fs ga2A and gauge theory B with the gauge fixing set fca ga2A . One thus trades one gauge symmetry for another. A very useful tool for the construction of equivalent gauge theories is a linking theory, see [4] and figure 1. Let us start with a phase space extension and denote the configuration variables of the extension by fa and their canonically conjugate ! "#$%&'() !"#!$!$ %&'(! (%')! *+#& !"#!$!$ ,'-,! (.//!#0. %'0#+'1 ,'-,! 2"+, %'0#+'1 ,'-,! 2"+, $%&'()#* $%&'()#+ ('/! %&'(! (%')! '( 3 !0. 5 ('/! %&'(! (%')! '( 3 !0. 4 ,'-,! (.//!#0. 4 ,'-,! (.//!#0. 5 6-0# !0 ,'-,! 2"+, 6-0# !0 ,'-,! 2"+, ,-.' /()#$%&'() 0!$-)!$ %'&(! (%')! 0!$-)!$ ,'-,! (.//!#0. Fig. 1 Construction of equivalent gauge theories from a linking theory. 4 Tim A. Koslowski momenta by pb and local Darboux coordinates on the original phase space G by j (qi; p ). A linking theory on extended phase space is a set of regular irreducible 1 first class constraints that can be split into three subsets: The set fca ga2A can be 1 f fca g pa weakly solved for a , the set 2 a2A can be weakly solved for and the set 3 fcm gm2M is weakly independent of the phase space extension. In this case, we can simplify the discussion by noticing that the three constraint sets are equivalent to the sets ff − f a g fpa − po g fc3 g a o (q; p) a2A ; a (q; p) a2A and ˜m (q; p) m2M : (5) a There are two sets of natural gauge fixing conditions ffa ga2A and fp ga2A . Im- a o posing fa = 0 gauge fixes the constraints p − pa (q; p) and leads to the phase b o space reduction (fa ;p ) ! (0;pb (q; p)), so the reduced phase space is G . More- over, the Dirac bracket associated with this phase space reduction coincides with the Poisson bracket on G . The result of the phase space reduction is the gauge theory o 3 B = (G ;f:;:g;fpa ga2A [ fc˜m gm2M ). pa f pb ! f a Similarly, imposing = 0 yields a phase space reduction ( a ; ) ( o (q; p);0) o 3 and the resulting gauge theory is A = (G ;f:;:g;ffa ga2A [ fc˜m gm2M ). The gauge theories A and B describe obviously the same physical system. It turns out that we would have obtained the same result even if had we not solved the first two subsets of constraints for the phase space extension. 3 Shape Dynamics Let us now extend the phase space of General Relativity by a conformal factor f and its conjugate momentum density pf . The linking theory between General Relativity on a compact manifold S without boundary and Shape Dynamics can be obtained by canonical best matching General Relativity in the ADM formulation with respect to conformal transformations that do not change the total spatial volume. This yields the following set of constraints ! Z a b p p s s − fˆ 1 fˆ fˆ fˆ TS(N) = d3xN pb a e 6 + (2L − hpi2) jgje6 − R(e6 g) jgje2 + a S jgj 6 Z ( p ) 3 Q(r) = d xr pf − 4p + 4hpi jgj (6) ZS ( ) 3 ab H(x) = d x p (Lx g)ab + pf Lx f ; S fˆ f − 1 h 6f i s a pac − 1 pd a where := 6 ln ep , b = gcb 3 b and where triangle bracketsp de- note the mean w.r.t. jgj and where the term a vanishes when p = hpi jgj. Imposing the gauge fixing condition f = 0 results in a phase space reduction 1 “Weakly” means on the constraint surface. Shape Dynamics 5 (f;pf ) ! (0;4(p − hpi)) and reduces the system back to the ADM formulation of General Relativity. Imposing the gauge fixing condition pf = 0 gauge fixes all TS(N), except for one. The simplest way to see this is to observe that TS = 0 is equivalent to imposing that f solves the Lichnerowicz-York equation ( ) a b 1 s s − 8D W = hpi2 − 2L W 5 + RW − b a W 7 (7) g 6 jgj W f for = e R, but withp the( reducibility) condition that the conformal factor is volume 3 6f preserving S d x jgj 1 − e = 0.
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