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General Covariance, Diffeomorphism Invariance, and Background

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General Covariance, Diffeomorphism Invariance, and Background General Covariance, Diffeomorphism Invariance, and Background Independence in 5 Dimensions . Antonio Vassallo [email protected] Section de Philosophie Université de Lausanne March, 19th 2015 Antonio Vassallo (UNIL) GC, DI, and BI in 5 Dimensions March, 19th 2015 1 / 13 General Relativity General Relativity is nowadays considered one of the pillars of modern physics. Main reasons for this success: Empirical adequacy. Richness in applications. Mathematical beauty. Antonio Vassallo (UNIL) GC, DI, and BI in 5 Dimensions March, 19th 2015 2 / 13 General Relativity Paul Dirac in 1978: Let us now face the question, suppose a discrepancy had appeared, well confirmed and substantiated, between the theory and observations. [...] Should one then consider the theory to be basically wrong? [...]I would say that the answer to the [q]uestion is emphatically NO.The Einstein theory of gravitation has a character of excellence of its own. [...]If a discrepancy should appear in some applications of the theory, it must be caused by some secondary feature relating to this application [...], and not by a failure of the general principles of the theory. Antonio Vassallo (UNIL) GC, DI, and BI in 5 Dimensions March, 19th 2015 3 / 13 General Relativity What is it that gives general relativity this “character of excellence”? Tentative answer: . 1 Diff (M4)-invariance (with no a priori restriction on the topology of M4). 2. Spacetime has no more geometrical structure than that encoded in a dynamically selected semi-Riemannian metric. ( 1. + 2. ) put a constraint on any spacetime theory that seek to supersede general relativity. This “GR-desideratum” is usually backed-up by metaphysical arguments (violation of reciprocity, indeterminism). What are the interpretational consequences of pursuing this desideratum beyond general relativity? Is it really a desideratum or just a (meta)physical prejudice? Antonio Vassallo (UNIL) GC, DI, and BI in 5 Dimensions March, 19th 2015 4 / 13 Kaluza-Klein Theory R [g ] = 0; A; B = 0;:::; 4. AB AB( ) 2 2 p gij + κ Ai Aj κ Ai gAB = 2 ; i; j = 0;:::; 3; κ = 4 π. κ Aj 1 The fifth dimension has a closed short-scale topology (compactification condition). Under these conditions, the “Kaluza-Klein miracle” happens: . kl − k Gij = 2πgij Fkl F Fi Fjk ; F ;i = 0: . ij Antonio Vassallo (UNIL) GC, DI, and BI in 5 Dimensions March, 19th 2015 5 / 13 Kaluza-Klein Theory Standard Kaluza-Klein theory does not satisfy the GR-desideratum: It is a Diff (M4) × U(1)-invariant theory. 1 It is restricted to Ricci-flat manifolds of the form M4 × S . Should we dismiss the theory as mere mathematical trickery, or can we take it as a genuine (albeit empirically inadequate) 5-dimensional extension of general relativity? What’s wrong with accepting a “5-d cylinder world” as a primitive fact of the matter? Antonio Vassallo (UNIL) GC, DI, and BI in 5 Dimensions March, 19th 2015 6 / 13 Induced Matter Theory RAB [gAB ] = 0. No particular condition placed on gAB . No compactification condition. Gij = 8πTij ; p ( p ) kl h i g44 1 g44 g g g g g ;i ;j − ;4 ij;4 − kl − kl ;4 ij;4 ij kl kl 2 Tij = p p gij;44 + g gik;4gjl ;4 + g ;4gkl ;4 + (g gkl ;4) : g44 2g44 g44 2 4 . 4 Whenever we consider a 4-hypersurface Σ4 by fixing x = const:, we obtain a 4-metric gij and a stress-energy tensor Tij both well-defined on Σ4. Antonio Vassallo (UNIL) GC, DI, and BI in 5 Dimensions March, 19th 2015 7 / 13 Induced Matter Theory The theory satisfies the GR-desideratum, but there is a price to pay. Consider a 5-d Minkowski spacetime: ds2 = dt2 − dr 2 − r 2dΩ2 − d 2. The sections = const: are 4-dimensional Minkowski spacetimes (Tij = 0). Let us change the coordinates to ( )− 1 ( ) 1 0 0 r r 2 2 0 r 2 2 t = t; r = 1 + 2 ; = 1 + 2 . The 5-d line element( becomes: ) 2 02 − 02 dr 02 02 2 − 02 ds = dt 1−r 02 + r dΩ d . 0 The sections = const: are FLRW spacetimes (Tij =6 0). Not surprisingly, this theory treats ordinary 4-d matter as “gauge fluff”. Antonio Vassallo (UNIL) GC, DI, and BI in 5 Dimensions March, 19th 2015 8 / 13 Induced Matter Theory In one sense, the theory is as empirically adequate as general relativity. In another sense, it seems to lead to empirical incoherence. Claiming the physical distinctness of 4-d states of affairs leads to hole-type problems. Adding 5-d matter is not a satisfactory solution. Not clear if and how distinguishing between partial and complete observables would help here. Why don’t we relax the GR-desideratum? Antonio Vassallo (UNIL) GC, DI, and BI in 5 Dimensions March, 19th 2015 9 / 13 Induced Matter Theory Let us add to the theory a dynamically selected distinguished vector A field ≡ (0; 0; 0; 0; ). A induces a privileged decomposition of the 5-manifold M5 in 4-hypersurfaces that are normal to the vector field and, hence, spacetime-like. In short, we are placing the condition that, for each 5-model of the theory, there is a distinguished way to foliate it in spacetime-like hypersurfaces. Antonio Vassallo (UNIL) GC, DI, and BI in 5 Dimensions March, 19th 2015 10 / 13 Induced Matter Theory This modified theory does not treat 4-matter as mere gauge fluff. Also in this theory 4-matter supervene on 5-d geometrical facts. In this case, the payoff of introducing more geometrical structure than that encoded in the metric tensor is enough to justify the move. Antonio Vassallo (UNIL) GC, DI, and BI in 5 Dimensions March, 19th 2015 11 / 13 Conclusions A too strict pursue of the GR-desideratum might lead to conceptual problems. This does not necessarily mean that general covariance, diffeomorphism invariance, and background independence should be given up. Relaxing the GR-desideratum might be a viable strategy when facing problems such as the embedding of quantum non-locality into general relativistic spacetimes. Antonio Vassallo (UNIL) GC, DI, and BI in 5 Dimensions March, 19th 2015 12 / 13 References Dirac, P. (1978). The excellence of Einstein’s theory of gravitation. Paper given at the symposium on the impact of modern scientific ideas on society held in Munich, 18-20 September 1978. Available at http://unesdoc.unesco.org/images/0003/000311/031102eb.pdf. Overduin, J. and P. Wesson (1997). Kaluza-Klein gravity. Physics Reports 283, 303–378. http://arxiv.org/abs/grqc/9805018. Wesson, P. (1995). Consequences of covariance in Kaluza-Klein theory. Modern Physics Letters A 10(1), 15–24. Antonio Vassallo (UNIL) GC, DI, and BI in 5 Dimensions March, 19th 2015 13 / 13.
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