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Unformatted text preview: 18. Experimental tests of gravitational theory 1 18. EXPERIMENTAL TESTS OF GRAVITATIONAL THEORY Revised November 2009 by T. Damour (IHES, Bures-sur-Yvette, France). Einstein’s General Relativity, the current “standard” theory of gravitation, describes gravity as a universal deformation of the Minkowski metric: g μν ( x λ ) = η μν + h μν ( x λ ) , where η μν = diag( − 1 , +1 , +1 , +1) . (18 . 1) Alternatively, it can be defined as the unique, consistent, local theory of a massless spin-2 field h μν , whose source must then be the total, conserved energy-momentum tensor . General Relativity is classically defined by two postulates. One postulate states that the Lagrangian density describing the propagation and self-interaction of the gravitational field is L Ein [ g μν ] = c 4 16 πG N √ gg μν R μν ( g ) , (18 . 2) R μν ( g ) = ∂ α Γ α μν − ∂ ν Γ α μα + Γ β αβ Γ α μν − Γ β αν Γ α μβ , (18 . 3) Γ λ μν = 1 2 g λσ ( ∂ μ g νσ + ∂ ν g μσ − ∂ σ g μν ) , (18 . 4) where G N is Newton’s constant, g = − det( g μν ), and g μν is the matrix inverse of g μν . A second postulate states that g μν couples universally, and minimally, to all the fields of the Standard Model by replacing everywhere the Minkowski metric η μν . Schematically (suppressing matrix indices and labels for the various gauge fields and fermions and for the Higgs doublet), L SM [ ψ, A μ , H, g μν ] = − 1 4 X √ gg μα g νβ F a μν F a αβ − X √ g ψ γ μ D μ ψ − 1 2 √ gg μν D μ HD ν H − √ g V ( H ) − X λ √ g ψ Hψ , (18 . 5) where γ μ γ ν + γ ν γ μ = 2 g μν , and where the covariant derivative D μ contains, besides the usual gauge field terms, a (spin-dependent) gravitational contribution Γ μ ( x ) . From the total action follow Einstein’s field equations, R μν − 1 2 Rg μν = 8 πG N c 4 T μν . (18 . 6) Here R = g μν R μν , T μν = g μα g νβ T αβ , and T μν = (2 / √ g ) δ L SM /δg μν is the (symmetric) energy-momentum tensor of the Standard Model matter. The theory is invariant K. Nakamura et al. , JPG 37 , 075021 (2010) (http://pdg.lbl.gov) July 30, 2010 14:36 2 18. Experimental tests of gravitational theory under arbitrary coordinate transformations: x μ = f μ ( x ν ). To solve the field equations Eq. (18 . 6), one needs to fix this coordinate gauge freedom. E.g. , the “harmonic gauge” (which is the analogue of the Lorenz gauge, ∂ μ A μ = 0, in electromagnetism) corresponds to imposing the condition ∂ ν ( √ gg μν ) = 0. In this Review , we only consider the classical limit of gravitation ( i.e. classical matter and classical gravity). Considering quantum matter in a classical gravitational background already poses interesting challenges, notably the possibility that the zero- point ﬂuctuations of the matter fields generate a nonvanishing vacuum energy density ρ vac , corresponding to a term...
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