Outline General Relativity from Basic Principles General Relativity as an Extended Canonical Gauge Theory Jürgen Struckmeier GSI Helmholtzzentrum für Schwerionenforschung GmbH, Darmstadt, Germany j.struckmeier@gsi.de, web-docs.gsi.de/ struck Goethe Universität, Frankfurt am Main, Germany STARS 2015 Third Caribbean Symposium on Cosmology, Gravitation, Nuclear and Astroparticle Physics La Habana, Cuba, 10 13 May 2015 Outline 1 Relativistic field theory with variable space-time 2 Extended Lagrangians in field theory 3 Extended Hamiltonians in field theory 4 Extended canonical transformations 5 6 Conclusions 1 / 23 2 / 23 Rationale Outline General Relativity should obey the following principles: 1 Action Principle: The fundamental laws of nature should follow from action principles 2 General Principle of Relativity: The form of the action principle and hence the resulting field equations should be the same in any frame of reference extended canonical transformation 3 Einstein s conclusion:... the essential achievement of general relativity is only indirectly connected with the introduction of a Riemannian metric. The directly relevant conceptual element is the displacement field Γ α ν... 4 General Relativity must emerge from an extended canonical transformation of the Γ α ν. 3 / 23 Relativistic field theory with variable space-time Extended action principle, extended Lagrangian Generalized action functional for dynamical space-time: treat x ν and x ν /y as dynamical variables in the Lagrangian L Extended action principle S = R L ψ I, ψ I x ν, x det Λ d 4 y, δs =! 0, δψ R I = δx! R = 0 with y the new set of independent variables and x ν = x ν y x 0 x... 0 y 0 y 3 Λ =....., det Λ = x 0,..., x 3 y 0,..., y 3 0. x 3 x... 3 y 0 y 3 The integrand defines the extended Lagrangian L e L e ψ I y, ψ Iy y, x ν y, x ν y y = L ψ I y, y α ψ I y x y α, x ν y det Λ 4 / 23
Extended Lagrangians in field theory Extended Lagrangians in field theory Extended set of Euler-Lagrange equations The Euler-Lagrange equations adopt the usual form y α L e L e ψi y α ψ I = 0, y α L e L e x x = 0. y α The derivative of L e with respect to the space-time coefficients x /y ν yields the canonical energy-momentum tensor θ α x L e = L det Λ + L ψi det Λ x y x ν y ψi ν x x α y ν = δl α L ψ I y ν α ψi x det Λ = θ x α x y ν x α det Λ x α x α 5 / 23 The Einstein equations of General Relativity follow from the Lagrangian L e,eh = L R + L M det Λ, L R = R 2 = 1 2 g ν R ν, wherein R = g ν R ν denotes the Riemann curvature scalar, [Lenght] 2 a coupling constant, and L M the conventional Lagrangian of a given system. The Ricci tensor R ν = R ν is the contraction = β of the Riemann-Christoffel curvature tensor R βν = Γ ν y β Γ β y ν + Γ λ νγ λβ Γλ βγ λν. In the Palatini approach, the metric and the connection coefficients are a priori independent quantities, hence the Euler-Lagrange equations are L e,eh y β x y β L e,eh x = 0, y β L e,eh Γ y β L e,eh Γ = 0. 6 / 23 Extended Lagrangians in field theory Extended Hamiltonians in field theory Remarks regarding the Einstein GR The Einstein-Hilbert Lagrangian was postulated. Thus, the resulting theory is justified only inasmuch as it complies with experimental data. It perfectly describes the dynamics of our solar system. It is not compatible with the observed dynamics of remote galaxies. Possible solutions: 1 Introduce fictitious dark matter / dark energy to fit the observed dynamics to the dynamics following from Einstein s equations. So far, dark matter / dark energy have not been identified. 2 Consider an alternative GR that has the Einstein GR as the weak gravitational field limit. A reasonable candidate emerges from an extended gauge theory that provides a form-invariant Lagrangian of GR in analogy to the Yang-Mills theory. For option 2 we need to set up a particular canonical transformation in the extended Hamiltonian formalism of classical field theory. 7 / 23 Extended covariant Hamiltonian Corresponding to the π I, the tensor densities π I = π I det Λ are defined as the dual quantities of the derivatives of the fields π I x = L, π ψi I y = L e x ψi y. Similarly, the canonical variables t ν define the dual quantity to x ν /y t ν = L e = θ x ν α y y α x ν = θ ν α x y x α. y An extended Lagrangian L e = L det Λ is thus Legendre-transformed to the Extended Hamiltonian H e H e = H det Λ t α β x α y β H e = H θα α det Λ. 8 / 23
Extended canonical transformations Form-invariance for the extended action principle The extended action principle must be maintained for extended canonical transformations that also map x X, t ν T ν Condition for extended canonical transformations [ δ π α ψ I R I y α t β α x β ] y α H e d 4 y [ = δ Π α Ψ I I y α T β α X β ] y α H e d 4 y. R This condition again implies that the integrands may differ by the divergence of a vector field F 1 with δf 1 R = 0 π α I ψ I y α t β α x β y α H e = Π α I Ψ I y α T β α X β y α H e + F 1 α y α. F 1 may be defined to depend on ψ I, Ψ I, x ν, and X ν only. This defines the extended generating function of type F 1 F 1 = F 1 ψ I, Ψ I, x ν, X ν. 9 / 23 Extended canonical transformations Transformation rules for a generating function F 1 The divergence of a function F 1 ψ I, Ψ I, x ν, X ν is F α 1 y α = F α 1 ψ I ψ I y α + F α 1 Ψ I Ψ I y α + F α 1 x β x β y α + F α 1 X β X β y α. Comparing the coefficients with the integrand condition yields the Transformation rules for a generating function F 1 π I = F 1 ψ I, Π I = F 1 Ψ I, t ν = F 1 x ν, T ν = F 1 X ν, H e = H e. The second derivatives of the generating function F 1 yield the symmetry relations for canonical transformations from F 1 π I Ψ J = 2 F 1 ψ I Ψ J = Π J ψ I, t ν X α = 2 F 1 x ν X α = T α x ν. 10 / 23 Extended canonical transformations Extended generating function of type F 2 By means of a Legendre transformation F 2 ψ I, Π I, x ν, T ν = F 1 ψ I, Ψ I, x ν, X ν + Ψ I Π I X α T α, an equivalent set of transformation rules is encountered, hence the Rules for an extended generating function F 2 π I = F 2, Ψ I δ ν = F 2 ψ I Π ν, t ν = F 2 I x ν, X α δ ν = F 2 T α ν Symmetry relations for F 2 : π I Π ν J = 2 F 2 ψ I Π ν J = δ ν Ψ J ψ I, t β T ν α = 2 F 2 x β T ν α = δ ν, H e = H e. X α x β. Concept of extended gauge theory In Yang-Mills theories, gauge fields a KJ had to be introduced to convert a system that is form-invariant under a global transformation group Ψ I = u IJ ψ J into a locally form-invariant system when u IJ = u IJ x. Inhomogeneous transformation rule for gauge bosons A KJ = u KL a LI uij + 1 u KI ig x u IJ. We now set up the gauge formalism in order to convert a Lorentz-invariant system into a locally form-invariant system under a general metric. The connection coefficients Γ act as gauge fields that convert a global Lorentz form-invariance into a local one under a transition x X. Switching between general, non-inertial reference frames x X requires inhomogeneous transformation rule for the connection coefficients Γ X = γk ij x x i x j X X α X ξ x k + 2 x k X X α X ξ x k. 12 / 23 11 / 23
General space-time transformation General principle of relativity means: The description of physics must be form-invariant under the transition to another, possibly non-inertial frame of reference. For a physical theory derived from an action principle, the principle must be maintained in its form. The transition must be a canonical transformation, hence must be described by a generating function in the Hamiltonian formalism. The generating function is set up to yield the required transformation rule for the connection coefficients γ. The canonical transformation formalism yields simultaneously the rules for for their conjugates, r, and for the Hamiltonian. The transformation rule for the Hamiltonian then uniquely defines the particular Hamiltonian that is form-invariant under the given transformation rule γ x Γ X. 13 / 23 Generating function for the CT γ x Γ X F 2 γ, R ν + R λ, x α, T α ν = T α h α x y X λ γ k X ij x k The subsequent transformation rules are: Γ δ ν = F 2 X R y ν = δ ν γ k X ij x k r ij k = F 2 x γ k ij y = R λ t ν = F 2 x ν = T α [ R λ y X λ γ k ij h α x ν x ν X x k x i x j X α X ξ + X x k x i x j X α X ξ + X x k 2 x k X α X ξ 2 x k X α X ξ x i x j x X α X ξ X λ, X α δ ν = F 2 T =δ α ν ν h α X x k x i x j X α X ξ + X 2 x k ] x ν x k X α X ξ 14 / 23 Space-time transformation in classical vacuum We observe: The required transformation rule for the connection coefficients γ is reproduced. The formally introduced quantity r transforms as a tensor! The canonical formalism also defines the transformation rule for the Hamiltonians, namely H e = H e H det Λ H det Λ = T β α X β y α t β α x β y α. The transformation rule for the Hamiltonian is then [ H det Λ H det Λ = R γ k 2 X x ν x i x j ij x k x ν X X α X ξ + 2 x i X X α X x k + 2 X x k x ν x j X ξ + 2 x k x ν X α X ξ X + 2 x j X X ξ X x k x i X α 3 x k X X α X ξ X x k ]. 15 / 23 Transformation rule for the Hamiltonian Recipe to derive the physical Hamiltonian The task is now to express all derivatives of the X and x in terms of the connection coefficients γ and Γ, and their conjugates, r and according to the 1st and 2nd canonical transformation rules. R Remarkably, this works perfectly. The result is: H det Λ H det Λ = 1 Γ 2 R X + Γ α X ξ Γ i Γ i + Γi αγ iξ 1 γ 2 r x + γ α x ξ γ i γ i + γi αγ iξ The terms emerge in a symmetric form in the original and the transformed dynamical variables. We encounter unambiguously the particular Hamiltonian that is form-invariant under the transformation of the connection coefficients. 16 / 23
Final form-invariant Hamiltonian Similar to conventional gauge theory, the final form-invariant Hamiltonian must contain in addition a dynamics term H e,dyn = 1 4 R R, H 1 e,dyn = 4 r r. H e,dyn = H e,dyn must hold in order for the final extended Hamiltonians to maintain the required transformation rule H e = H e. This is ensured if det Λ = det Λ, hence if h x in F 2 satisfies X 0,..., X 3 x 0,..., x 3 = h 0 x,..., h 3 x x 0,..., x 3 = 1. The final form-invariant extended Hamiltonian that is compatible with the canonical transformation rules now writes for the x reference frame H e,gr r, γ, t = t α β x α 1 β 4 r r + y 1 2 r γ α x ξ + γ x + γk αγ kξ γk γ k First canonical equation The canonical equation for the connection coefficients follows as γ x = H e,gr γ r = 1 2 r + 1 α 2 x ξ + γ x + γk αγ kξ γk γ k Solved for r Riemann curvature tensor one finds exactly the representation of the r = γ α x ξ γ x + γk αγ kξ γk γ k It is manifestly skew-symmetric in the indices ξ and. We observe: The quantity r that was formally introduced setting up the generating function F 2 turns out to be the Riemann tensor. The CT requirement for r to constitute a tensor is satisfied. 18 / 23 17 / 23 Second canonical equation The second canonical equation follows as r τσα x α = H e,gr γ τσ = γ β α r τσα β This equation is actually a tensor equation with s σ αβ r τσα the torsion tensor. ;α = γ σ αβ r τβα = 1 2 γ τ αβ r βσα. γ σ αβ γ σ βα r τβα = s σ αβ r τβα For a torsion-free space-time, the canonical equation states that the covariant divergence of the Riemann tensor vanishes Form-invariant extended Lagrangian L e,gr With r expressed in terms of the γ α and their derivatives, the extended Lagrangian L e,gr can be set up as the Legendre transform Form-invariant extended Lagrangian in classical vacuum L e,gr = γ, γ x ν, x y ν = r γ x 1 4 r r 1 γ α 2 r x ξ = 1 4 r r det Λ = 1 4 g g βα g λξ g ζ r βλζ r t α β x α y β H e,gr γ x + γk αγ kξ γk γ k det Λ r τσα ;α = 0. 19 / 23 L e,gr is a particular quadratic function of the Riemann tensor. 20 / 23
Conclusions Conclusions Conclusions I Conclusions II The gauge principle can be applied as well to theories that are globally form-invariant under Lorentz transformations The theories can be rendered form-invariant under the corresponding local group by introducing connection coefficients γ as the respective gauge quantities. The canonical formalism yields unambiguously a Hamiltonian that describes the dynamics of the displacement fields γ. The resulting theories maintain the action principle general principle of relativity principle of scale invariance, hence the theories are form-invariant under transformations of the length scales of the space-time manifold. The theories are unambiguous in the sense that no other functions of the Riemann tensor emerge from the canonical transformation formalism. In contrast to standard GR that is based on the postulated Einstein- Hilbert action, the Lagrangian L e,gr is derived. For the source-free case L M 0, the theories are compatible with standard GR as they possess the Schwarzschild metric as a solution. For L M 0, the solutions differ from that of standard GR. A quantized theory that is based on the Lagrangian L e,gr is renormalizable as the coupling constant to L M is dimensionless. Remarkably, a Lagrangian of the derived form that is quadratic in the curvature tensor was already proposed by A. Einstein in a personal letter to H. Weyl, reasoning analogies with other classical field theories. 21 / 23 22 / 23 Conclusions References Struckmeier, J., Redelbach, A., Covariant Hamiltonian Field Theory, Int. J. Mod. Phys. E 17 2008, arxiv:0811.0508 [math-ph] Struckmeier, J., Reichau, H., General UN gauge transformations in the realm of covariant Hamiltonian field theory, in Exciting Interdisciplinary Physics, W. Greiner ed., Springer 2013, arxiv:1205.5754 [hep-th] Struckmeier, J., Generalized UN gauge transformations in the realm of the extended covariant Hamilton formalism of field theory, J. Phys. G: Nucl. Part. Phys. 40, 015007 2013, arxiv:1206.4452 [nucl-th] Struckmeier, J., General relativity as an extended canonical gauge theory, Phys. Rev. D 91, 085030 2015, arxiv:1411.1558 [gr-qc] Struckmeier, J., Greiner, W., Extended Lagrange and Hamilton Formalism for Point Mechanics and Covariant Hamilton Field Theory, to be published by World Scientific, ISBN: 9789814578417 23 / 23