Acquisition and understanding of the theoretical structures underlying General Relativity both in its geometric meaning and as a self-interacting theory for a massless spin-2 field.
Identification of analogies and differences between Einsteinian gravity and non-Abelian gauge theories for spin-1 fields, and discussion of the peculiarities associated with the localization of Poincaré and (anti-) de Sitter algebras.
Connection of the theory with current research aspects through the illustration of some notable solutions of the Einstein equations, both in the perturbative and in the non-perturbative regimes.
Identification of analogies and differences between Einsteinian gravity and non-Abelian gauge theories for spin-1 fields, and discussion of the peculiarities associated with the localization of Poincaré and (anti-) de Sitter algebras.
Connection of the theory with current research aspects through the illustration of some notable solutions of the Einstein equations, both in the perturbative and in the non-perturbative regimes.
Curriculum
teacher profile teaching materials
§ Gravity as a self-interacting spin-two field theory
sometries of flat spacetime. Lorentz transformations and the Poincar´e Group. Decomposition theorems and standard boost. Contractions: Galilei and Carroll limits. Symmetries: global vs local. Basics of Lie algebras. Noether’s theorem and conservation laws. The canonical stress-energy and angular momentum tensors. Non-canonical currents: improvements and their relevance for couplings. Belinfante’s symmetric energy-momentum tensor and angular momentum tensor. Scale invariance and traceless stress-energy tensor. Particles and fields in Special Relativity. Irreps of the Poincar´e group: Wigner’s induced representation method. Massless particles: ISO(D-2) little group and gauge invariance*. From relativistic massless spin-2 particles to full GR. Uniqueness of the quadratic Lagrangian. Noether method and non-linear completions. Noether’s construction of Yang-Mills Lagrangian*. The transverse-traceless gravitational cubic vertex. Weinberg’s Equivalence Principle from relativistic invariance of the S-matrix. Spin and sign of static forces*.
.
§ Isometries and maximally symmetric spaces
Lie derivative and isometries. Lie bracket. Conformal Killing vectors. Killing tensors. Integrability condition for Killing vectors. Maximal number of isometries. Homogeneous and isotropic spaces. Structure theorem for maximally symmetric spaces: identification of the curvature via the metric signature and a curvature constant. MSS as vacuum solutions to the EH equations with cosmological constant. Construction from embedding in (D+1) pseudo-Lorentzian spaces: metric and Christoffel coefficients. Conformally flat spaces. AdS in the Poincar´e patch. Weyl tensor. Riemann and Weyl in various dimensions: counting components for irreps of GL(D) and O(D). Conformal transformations of the metric tensor. Conformally coupled scalar fields*.
.
§ Elements of differential geometry
Topological spaces. Manifolds. Diffeomorphisms. Tangent spaces and vectors. Coordinate basis. Derivative operators on manifolds. Differential forms: definition, wedge product, interior and exterior derivatives, Cartan’s formula for the Lie derivative, Hodge dual. Integration. Maxwell and Yang-Mills theories in the language of forms. Stokes’s theorem.
§ Black holes
Spherically symmetric spaces. The Schwarzschild solution. Birkhoff’s theorem (partial proof). Singularities, definitions and criteria: curvature singularites and geodesic incompleteness. The tortoise coordinate. Extension of a space-time. Coordinates of Eddington-Finkelstein. Event horizons, black holes and white holes. Kruskal-Szekeres coordinates. Maximal extension of the Schwarzschild solution. Kruskal’s diagram and eternal black holes. The Reissner-Nordstr¨om solution. (A)dS-Schwarzschild space-time*. Killing horizons. The Kerr solution.
§ Conformal diagrams
Conformal compactifications and causal structure. Penrose diagrams of Minkowski, Schwarzschild, Reissner-Nordstr¨om, Kerr. Spherically collapsing radiation and mat-
ter shells. (Anti-)de Sitter spaces*.
.
§Gravitational energy
Conserved quantities in gauge theories: the example of Yang-Mills theory. Covariant conservation and ordinary conservation. Asymptotically flat metrics. Gravitational energy-momentum pseudo-tensor. The superpotential. ADM energy and momentum. ADM energy of the Schwarzschild solution. The positive-energy theorem (without proof). Killing vectors and Komar integrals. Quadrupole radiation*.
.
§ The Cartan-Weyl formulation and Fermionic couplings
Local inertial frames. The frame field and its relation to the metric field. Non-coordinate bases. Local Lorentz transformations. The spin connection. The vielbein postulate. Torsion constraint and second-order formulation. The contorsion tensor. Local Lorentz curvature. Minimally coupled Fermionic matter. Dirac Lagrangian on curved manifolds. Gravity as a Yang-Mills theory of the Poincar´e algebra. Local Poincar´e transformations. Torsion and curvature over the Poincar´e algebra. Problems with quadratic actions. First-order formulation and Weyl’s action.
- Carroll S Spacetime and Geometry: An Introduction to General Relativity (Addison-Wesley 2014/Cambridge University Press, 2019)
Mutuazione: 20430009 Teoria dei Campi e Gravità in Fisica LM-17 R FRANCIA DARIO
Programme
The items marked with a ``*'' are assigned as homework problems.§ Gravity as a self-interacting spin-two field theory
sometries of flat spacetime. Lorentz transformations and the Poincar´e Group. Decomposition theorems and standard boost. Contractions: Galilei and Carroll limits. Symmetries: global vs local. Basics of Lie algebras. Noether’s theorem and conservation laws. The canonical stress-energy and angular momentum tensors. Non-canonical currents: improvements and their relevance for couplings. Belinfante’s symmetric energy-momentum tensor and angular momentum tensor. Scale invariance and traceless stress-energy tensor. Particles and fields in Special Relativity. Irreps of the Poincar´e group: Wigner’s induced representation method. Massless particles: ISO(D-2) little group and gauge invariance*. From relativistic massless spin-2 particles to full GR. Uniqueness of the quadratic Lagrangian. Noether method and non-linear completions. Noether’s construction of Yang-Mills Lagrangian*. The transverse-traceless gravitational cubic vertex. Weinberg’s Equivalence Principle from relativistic invariance of the S-matrix. Spin and sign of static forces*.
.
§ Isometries and maximally symmetric spaces
Lie derivative and isometries. Lie bracket. Conformal Killing vectors. Killing tensors. Integrability condition for Killing vectors. Maximal number of isometries. Homogeneous and isotropic spaces. Structure theorem for maximally symmetric spaces: identification of the curvature via the metric signature and a curvature constant. MSS as vacuum solutions to the EH equations with cosmological constant. Construction from embedding in (D+1) pseudo-Lorentzian spaces: metric and Christoffel coefficients. Conformally flat spaces. AdS in the Poincar´e patch. Weyl tensor. Riemann and Weyl in various dimensions: counting components for irreps of GL(D) and O(D). Conformal transformations of the metric tensor. Conformally coupled scalar fields*.
.
§ Elements of differential geometry
Topological spaces. Manifolds. Diffeomorphisms. Tangent spaces and vectors. Coordinate basis. Derivative operators on manifolds. Differential forms: definition, wedge product, interior and exterior derivatives, Cartan’s formula for the Lie derivative, Hodge dual. Integration. Maxwell and Yang-Mills theories in the language of forms. Stokes’s theorem.
§ Black holes
Spherically symmetric spaces. The Schwarzschild solution. Birkhoff’s theorem (partial proof). Singularities, definitions and criteria: curvature singularites and geodesic incompleteness. The tortoise coordinate. Extension of a space-time. Coordinates of Eddington-Finkelstein. Event horizons, black holes and white holes. Kruskal-Szekeres coordinates. Maximal extension of the Schwarzschild solution. Kruskal’s diagram and eternal black holes. The Reissner-Nordstr¨om solution. (A)dS-Schwarzschild space-time*. Killing horizons. The Kerr solution.
§ Conformal diagrams
Conformal compactifications and causal structure. Penrose diagrams of Minkowski, Schwarzschild, Reissner-Nordstr¨om, Kerr. Spherically collapsing radiation and mat-
ter shells. (Anti-)de Sitter spaces*.
.
§Gravitational energy
Conserved quantities in gauge theories: the example of Yang-Mills theory. Covariant conservation and ordinary conservation. Asymptotically flat metrics. Gravitational energy-momentum pseudo-tensor. The superpotential. ADM energy and momentum. ADM energy of the Schwarzschild solution. The positive-energy theorem (without proof). Killing vectors and Komar integrals. Quadrupole radiation*.
.
§ The Cartan-Weyl formulation and Fermionic couplings
Local inertial frames. The frame field and its relation to the metric field. Non-coordinate bases. Local Lorentz transformations. The spin connection. The vielbein postulate. Torsion constraint and second-order formulation. The contorsion tensor. Local Lorentz curvature. Minimally coupled Fermionic matter. Dirac Lagrangian on curved manifolds. Gravity as a Yang-Mills theory of the Poincar´e algebra. Local Poincar´e transformations. Torsion and curvature over the Poincar´e algebra. Problems with quadratic actions. First-order formulation and Weyl’s action.
Core Documentation
- Wald R, General Relativity (The University of Chicago Press, 1984).- Carroll S Spacetime and Geometry: An Introduction to General Relativity (Addison-Wesley 2014/Cambridge University Press, 2019)
Attendance
Attendance in person. Working students may attend the course remotely. Lecture recordings will also be made available for a period of one week following each lecture.Type of evaluation
Assessment is based on an oral examination. Alternatively, students may choose to solve the problems assigned during the course and submit their solutions as a single written report. In this case, the final assessment is based both on the quality of the written report and on an oral discussion of its contents, aimed at assessing the student's mastery of the course material and ability to establish connections between the different aspects of the theory. teacher profile teaching materials
§ Gravity as a self-interacting spin-two field theory
sometries of flat spacetime. Lorentz transformations and the Poincar´e Group. Decomposition theorems and standard boost. Contractions: Galilei and Carroll limits. Symmetries: global vs local. Basics of Lie algebras. Noether’s theorem and conservation laws. The canonical stress-energy and angular momentum tensors. Non-canonical currents: improvements and their relevance for couplings. Belinfante’s symmetric energy-momentum tensor and angular momentum tensor. Scale invariance and traceless stress-energy tensor. Particles and fields in Special Relativity. Irreps of the Poincar´e group: Wigner’s induced representation method. Massless particles: ISO(D-2) little group and gauge invariance*. From relativistic massless spin-2 particles to full GR. Uniqueness of the quadratic Lagrangian. Noether method and non-linear completions. Noether’s construction of Yang-Mills Lagrangian*. The transverse-traceless gravitational cubic vertex. Weinberg’s Equivalence Principle from relativistic invariance of the S-matrix. Spin and sign of static forces*.
.
§ Isometries and maximally symmetric spaces
Lie derivative and isometries. Lie bracket. Conformal Killing vectors. Killing tensors. Integrability condition for Killing vectors. Maximal number of isometries. Homogeneous and isotropic spaces. Structure theorem for maximally symmetric spaces: identification of the curvature via the metric signature and a curvature constant. MSS as vacuum solutions to the EH equations with cosmological constant. Construction from embedding in (D+1) pseudo-Lorentzian spaces: metric and Christoffel coefficients. Conformally flat spaces. AdS in the Poincar´e patch. Weyl tensor. Riemann and Weyl in various dimensions: counting components for irreps of GL(D) and O(D). Conformal transformations of the metric tensor. Conformally coupled scalar fields*.
.
§ Elements of differential geometry
Topological spaces. Manifolds. Diffeomorphisms. Tangent spaces and vectors. Coordinate basis. Derivative operators on manifolds. Differential forms: definition, wedge product, interior and exterior derivatives, Cartan’s formula for the Lie derivative, Hodge dual. Integration. Maxwell and Yang-Mills theories in the language of forms. Stokes’s theorem.
§ Black holes
Spherically symmetric spaces. The Schwarzschild solution. Birkhoff’s theorem (partial proof). Singularities, definitions and criteria: curvature singularites and geodesic incompleteness. The tortoise coordinate. Extension of a space-time. Coordinates of Eddington-Finkelstein. Event horizons, black holes and white holes. Kruskal-Szekeres coordinates. Maximal extension of the Schwarzschild solution. Kruskal’s diagram and eternal black holes. The Reissner-Nordstr¨om solution. (A)dS-Schwarzschild space-time*. Killing horizons. The Kerr solution.
§ Conformal diagrams
Conformal compactifications and causal structure. Penrose diagrams of Minkowski, Schwarzschild, Reissner-Nordstr¨om, Kerr. Spherically collapsing radiation and mat-
ter shells. (Anti-)de Sitter spaces*.
.
§Gravitational energy
Conserved quantities in gauge theories: the example of Yang-Mills theory. Covariant conservation and ordinary conservation. Asymptotically flat metrics. Gravitational energy-momentum pseudo-tensor. The superpotential. ADM energy and momentum. ADM energy of the Schwarzschild solution. The positive-energy theorem (without proof). Killing vectors and Komar integrals. Quadrupole radiation*.
.
§ The Cartan-Weyl formulation and Fermionic couplings
Local inertial frames. The frame field and its relation to the metric field. Non-coordinate bases. Local Lorentz transformations. The spin connection. The vielbein postulate. Torsion constraint and second-order formulation. The contorsion tensor. Local Lorentz curvature. Minimally coupled Fermionic matter. Dirac Lagrangian on curved manifolds. Gravity as a Yang-Mills theory of the Poincar´e algebra. Local Poincar´e transformations. Torsion and curvature over the Poincar´e algebra. Problems with quadratic actions. First-order formulation and Weyl’s action.
- Carroll S Spacetime and Geometry: An Introduction to General Relativity (Addison-Wesley 2014/Cambridge University Press, 2019)
Mutuazione: 20430009 Teoria dei Campi e Gravità in Fisica LM-17 R FRANCIA DARIO
Programme
The items marked with a ``*'' are assigned as homework problems.§ Gravity as a self-interacting spin-two field theory
sometries of flat spacetime. Lorentz transformations and the Poincar´e Group. Decomposition theorems and standard boost. Contractions: Galilei and Carroll limits. Symmetries: global vs local. Basics of Lie algebras. Noether’s theorem and conservation laws. The canonical stress-energy and angular momentum tensors. Non-canonical currents: improvements and their relevance for couplings. Belinfante’s symmetric energy-momentum tensor and angular momentum tensor. Scale invariance and traceless stress-energy tensor. Particles and fields in Special Relativity. Irreps of the Poincar´e group: Wigner’s induced representation method. Massless particles: ISO(D-2) little group and gauge invariance*. From relativistic massless spin-2 particles to full GR. Uniqueness of the quadratic Lagrangian. Noether method and non-linear completions. Noether’s construction of Yang-Mills Lagrangian*. The transverse-traceless gravitational cubic vertex. Weinberg’s Equivalence Principle from relativistic invariance of the S-matrix. Spin and sign of static forces*.
.
§ Isometries and maximally symmetric spaces
Lie derivative and isometries. Lie bracket. Conformal Killing vectors. Killing tensors. Integrability condition for Killing vectors. Maximal number of isometries. Homogeneous and isotropic spaces. Structure theorem for maximally symmetric spaces: identification of the curvature via the metric signature and a curvature constant. MSS as vacuum solutions to the EH equations with cosmological constant. Construction from embedding in (D+1) pseudo-Lorentzian spaces: metric and Christoffel coefficients. Conformally flat spaces. AdS in the Poincar´e patch. Weyl tensor. Riemann and Weyl in various dimensions: counting components for irreps of GL(D) and O(D). Conformal transformations of the metric tensor. Conformally coupled scalar fields*.
.
§ Elements of differential geometry
Topological spaces. Manifolds. Diffeomorphisms. Tangent spaces and vectors. Coordinate basis. Derivative operators on manifolds. Differential forms: definition, wedge product, interior and exterior derivatives, Cartan’s formula for the Lie derivative, Hodge dual. Integration. Maxwell and Yang-Mills theories in the language of forms. Stokes’s theorem.
§ Black holes
Spherically symmetric spaces. The Schwarzschild solution. Birkhoff’s theorem (partial proof). Singularities, definitions and criteria: curvature singularites and geodesic incompleteness. The tortoise coordinate. Extension of a space-time. Coordinates of Eddington-Finkelstein. Event horizons, black holes and white holes. Kruskal-Szekeres coordinates. Maximal extension of the Schwarzschild solution. Kruskal’s diagram and eternal black holes. The Reissner-Nordstr¨om solution. (A)dS-Schwarzschild space-time*. Killing horizons. The Kerr solution.
§ Conformal diagrams
Conformal compactifications and causal structure. Penrose diagrams of Minkowski, Schwarzschild, Reissner-Nordstr¨om, Kerr. Spherically collapsing radiation and mat-
ter shells. (Anti-)de Sitter spaces*.
.
§Gravitational energy
Conserved quantities in gauge theories: the example of Yang-Mills theory. Covariant conservation and ordinary conservation. Asymptotically flat metrics. Gravitational energy-momentum pseudo-tensor. The superpotential. ADM energy and momentum. ADM energy of the Schwarzschild solution. The positive-energy theorem (without proof). Killing vectors and Komar integrals. Quadrupole radiation*.
.
§ The Cartan-Weyl formulation and Fermionic couplings
Local inertial frames. The frame field and its relation to the metric field. Non-coordinate bases. Local Lorentz transformations. The spin connection. The vielbein postulate. Torsion constraint and second-order formulation. The contorsion tensor. Local Lorentz curvature. Minimally coupled Fermionic matter. Dirac Lagrangian on curved manifolds. Gravity as a Yang-Mills theory of the Poincar´e algebra. Local Poincar´e transformations. Torsion and curvature over the Poincar´e algebra. Problems with quadratic actions. First-order formulation and Weyl’s action.
Core Documentation
- Wald R, General Relativity (The University of Chicago Press, 1984).- Carroll S Spacetime and Geometry: An Introduction to General Relativity (Addison-Wesley 2014/Cambridge University Press, 2019)
Attendance
Attendance in person. Working students may attend the course remotely. Lecture recordings will also be made available for a period of one week following each lecture.Type of evaluation
Assessment is based on an oral examination. Alternatively, students may choose to solve the problems assigned during the course and submit their solutions as a single written report. In this case, the final assessment is based both on the quality of the written report and on an oral discussion of its contents, aimed at assessing the student's mastery of the course material and ability to establish connections between the different aspects of the theory. teacher profile teaching materials
§ Gravity as a self-interacting spin-two field theory
sometries of flat spacetime. Lorentz transformations and the Poincar´e Group. Decomposition theorems and standard boost. Contractions: Galilei and Carroll limits. Symmetries: global vs local. Basics of Lie algebras. Noether’s theorem and conservation laws. The canonical stress-energy and angular momentum tensors. Non-canonical currents: improvements and their relevance for couplings. Belinfante’s symmetric energy-momentum tensor and angular momentum tensor. Scale invariance and traceless stress-energy tensor. Particles and fields in Special Relativity. Irreps of the Poincar´e group: Wigner’s induced representation method. Massless particles: ISO(D-2) little group and gauge invariance*. From relativistic massless spin-2 particles to full GR. Uniqueness of the quadratic Lagrangian. Noether method and non-linear completions. Noether’s construction of Yang-Mills Lagrangian*. The transverse-traceless gravitational cubic vertex. Weinberg’s Equivalence Principle from relativistic invariance of the S-matrix. Spin and sign of static forces*.
.
§ Isometries and maximally symmetric spaces
Lie derivative and isometries. Lie bracket. Conformal Killing vectors. Killing tensors. Integrability condition for Killing vectors. Maximal number of isometries. Homogeneous and isotropic spaces. Structure theorem for maximally symmetric spaces: identification of the curvature via the metric signature and a curvature constant. MSS as vacuum solutions to the EH equations with cosmological constant. Construction from embedding in (D+1) pseudo-Lorentzian spaces: metric and Christoffel coefficients. Conformally flat spaces. AdS in the Poincar´e patch. Weyl tensor. Riemann and Weyl in various dimensions: counting components for irreps of GL(D) and O(D). Conformal transformations of the metric tensor. Conformally coupled scalar fields*.
.
§ Elements of differential geometry
Topological spaces. Manifolds. Diffeomorphisms. Tangent spaces and vectors. Coordinate basis. Derivative operators on manifolds. Differential forms: definition, wedge product, interior and exterior derivatives, Cartan’s formula for the Lie derivative, Hodge dual. Integration. Maxwell and Yang-Mills theories in the language of forms. Stokes’s theorem.
§ Black holes
Spherically symmetric spaces. The Schwarzschild solution. Birkhoff’s theorem (partial proof). Singularities, definitions and criteria: curvature singularites and geodesic incompleteness. The tortoise coordinate. Extension of a space-time. Coordinates of Eddington-Finkelstein. Event horizons, black holes and white holes. Kruskal-Szekeres coordinates. Maximal extension of the Schwarzschild solution. Kruskal’s diagram and eternal black holes. The Reissner-Nordstr¨om solution. (A)dS-Schwarzschild space-time*. Killing horizons. The Kerr solution.
§ Conformal diagrams
Conformal compactifications and causal structure. Penrose diagrams of Minkowski, Schwarzschild, Reissner-Nordstr¨om, Kerr. Spherically collapsing radiation and mat-
ter shells. (Anti-)de Sitter spaces*.
.
§Gravitational energy
Conserved quantities in gauge theories: the example of Yang-Mills theory. Covariant conservation and ordinary conservation. Asymptotically flat metrics. Gravitational energy-momentum pseudo-tensor. The superpotential. ADM energy and momentum. ADM energy of the Schwarzschild solution. The positive-energy theorem (without proof). Killing vectors and Komar integrals. Quadrupole radiation*.
.
§ The Cartan-Weyl formulation and Fermionic couplings
Local inertial frames. The frame field and its relation to the metric field. Non-coordinate bases. Local Lorentz transformations. The spin connection. The vielbein postulate. Torsion constraint and second-order formulation. The contorsion tensor. Local Lorentz curvature. Minimally coupled Fermionic matter. Dirac Lagrangian on curved manifolds. Gravity as a Yang-Mills theory of the Poincar´e algebra. Local Poincar´e transformations. Torsion and curvature over the Poincar´e algebra. Problems with quadratic actions. First-order formulation and Weyl’s action.
- Carroll S Spacetime and Geometry: An Introduction to General Relativity (Addison-Wesley 2014/Cambridge University Press, 2019)
Mutuazione: 20430009 Teoria dei Campi e Gravità in Fisica LM-17 R FRANCIA DARIO
Programme
The items marked with a ``*'' are assigned as homework problems.§ Gravity as a self-interacting spin-two field theory
sometries of flat spacetime. Lorentz transformations and the Poincar´e Group. Decomposition theorems and standard boost. Contractions: Galilei and Carroll limits. Symmetries: global vs local. Basics of Lie algebras. Noether’s theorem and conservation laws. The canonical stress-energy and angular momentum tensors. Non-canonical currents: improvements and their relevance for couplings. Belinfante’s symmetric energy-momentum tensor and angular momentum tensor. Scale invariance and traceless stress-energy tensor. Particles and fields in Special Relativity. Irreps of the Poincar´e group: Wigner’s induced representation method. Massless particles: ISO(D-2) little group and gauge invariance*. From relativistic massless spin-2 particles to full GR. Uniqueness of the quadratic Lagrangian. Noether method and non-linear completions. Noether’s construction of Yang-Mills Lagrangian*. The transverse-traceless gravitational cubic vertex. Weinberg’s Equivalence Principle from relativistic invariance of the S-matrix. Spin and sign of static forces*.
.
§ Isometries and maximally symmetric spaces
Lie derivative and isometries. Lie bracket. Conformal Killing vectors. Killing tensors. Integrability condition for Killing vectors. Maximal number of isometries. Homogeneous and isotropic spaces. Structure theorem for maximally symmetric spaces: identification of the curvature via the metric signature and a curvature constant. MSS as vacuum solutions to the EH equations with cosmological constant. Construction from embedding in (D+1) pseudo-Lorentzian spaces: metric and Christoffel coefficients. Conformally flat spaces. AdS in the Poincar´e patch. Weyl tensor. Riemann and Weyl in various dimensions: counting components for irreps of GL(D) and O(D). Conformal transformations of the metric tensor. Conformally coupled scalar fields*.
.
§ Elements of differential geometry
Topological spaces. Manifolds. Diffeomorphisms. Tangent spaces and vectors. Coordinate basis. Derivative operators on manifolds. Differential forms: definition, wedge product, interior and exterior derivatives, Cartan’s formula for the Lie derivative, Hodge dual. Integration. Maxwell and Yang-Mills theories in the language of forms. Stokes’s theorem.
§ Black holes
Spherically symmetric spaces. The Schwarzschild solution. Birkhoff’s theorem (partial proof). Singularities, definitions and criteria: curvature singularites and geodesic incompleteness. The tortoise coordinate. Extension of a space-time. Coordinates of Eddington-Finkelstein. Event horizons, black holes and white holes. Kruskal-Szekeres coordinates. Maximal extension of the Schwarzschild solution. Kruskal’s diagram and eternal black holes. The Reissner-Nordstr¨om solution. (A)dS-Schwarzschild space-time*. Killing horizons. The Kerr solution.
§ Conformal diagrams
Conformal compactifications and causal structure. Penrose diagrams of Minkowski, Schwarzschild, Reissner-Nordstr¨om, Kerr. Spherically collapsing radiation and mat-
ter shells. (Anti-)de Sitter spaces*.
.
§Gravitational energy
Conserved quantities in gauge theories: the example of Yang-Mills theory. Covariant conservation and ordinary conservation. Asymptotically flat metrics. Gravitational energy-momentum pseudo-tensor. The superpotential. ADM energy and momentum. ADM energy of the Schwarzschild solution. The positive-energy theorem (without proof). Killing vectors and Komar integrals. Quadrupole radiation*.
.
§ The Cartan-Weyl formulation and Fermionic couplings
Local inertial frames. The frame field and its relation to the metric field. Non-coordinate bases. Local Lorentz transformations. The spin connection. The vielbein postulate. Torsion constraint and second-order formulation. The contorsion tensor. Local Lorentz curvature. Minimally coupled Fermionic matter. Dirac Lagrangian on curved manifolds. Gravity as a Yang-Mills theory of the Poincar´e algebra. Local Poincar´e transformations. Torsion and curvature over the Poincar´e algebra. Problems with quadratic actions. First-order formulation and Weyl’s action.
Core Documentation
- Wald R, General Relativity (The University of Chicago Press, 1984).- Carroll S Spacetime and Geometry: An Introduction to General Relativity (Addison-Wesley 2014/Cambridge University Press, 2019)
Attendance
Attendance in person. Working students may attend the course remotely. Lecture recordings will also be made available for a period of one week following each lecture.Type of evaluation
Assessment is based on an oral examination. Alternatively, students may choose to solve the problems assigned during the course and submit their solutions as a single written report. In this case, the final assessment is based both on the quality of the written report and on an oral discussion of its contents, aimed at assessing the student's mastery of the course material and ability to establish connections between the different aspects of the theory. teacher profile teaching materials
§ Gravity as a self-interacting spin-two field theory
sometries of flat spacetime. Lorentz transformations and the Poincar´e Group. Decomposition theorems and standard boost. Contractions: Galilei and Carroll limits. Symmetries: global vs local. Basics of Lie algebras. Noether’s theorem and conservation laws. The canonical stress-energy and angular momentum tensors. Non-canonical currents: improvements and their relevance for couplings. Belinfante’s symmetric energy-momentum tensor and angular momentum tensor. Scale invariance and traceless stress-energy tensor. Particles and fields in Special Relativity. Irreps of the Poincar´e group: Wigner’s induced representation method. Massless particles: ISO(D-2) little group and gauge invariance*. From relativistic massless spin-2 particles to full GR. Uniqueness of the quadratic Lagrangian. Noether method and non-linear completions. Noether’s construction of Yang-Mills Lagrangian*. The transverse-traceless gravitational cubic vertex. Weinberg’s Equivalence Principle from relativistic invariance of the S-matrix. Spin and sign of static forces*.
.
§ Isometries and maximally symmetric spaces
Lie derivative and isometries. Lie bracket. Conformal Killing vectors. Killing tensors. Integrability condition for Killing vectors. Maximal number of isometries. Homogeneous and isotropic spaces. Structure theorem for maximally symmetric spaces: identification of the curvature via the metric signature and a curvature constant. MSS as vacuum solutions to the EH equations with cosmological constant. Construction from embedding in (D+1) pseudo-Lorentzian spaces: metric and Christoffel coefficients. Conformally flat spaces. AdS in the Poincar´e patch. Weyl tensor. Riemann and Weyl in various dimensions: counting components for irreps of GL(D) and O(D). Conformal transformations of the metric tensor. Conformally coupled scalar fields*.
.
§ Elements of differential geometry
Topological spaces. Manifolds. Diffeomorphisms. Tangent spaces and vectors. Coordinate basis. Derivative operators on manifolds. Differential forms: definition, wedge product, interior and exterior derivatives, Cartan’s formula for the Lie derivative, Hodge dual. Integration. Maxwell and Yang-Mills theories in the language of forms. Stokes’s theorem.
§ Black holes
Spherically symmetric spaces. The Schwarzschild solution. Birkhoff’s theorem (partial proof). Singularities, definitions and criteria: curvature singularites and geodesic incompleteness. The tortoise coordinate. Extension of a space-time. Coordinates of Eddington-Finkelstein. Event horizons, black holes and white holes. Kruskal-Szekeres coordinates. Maximal extension of the Schwarzschild solution. Kruskal’s diagram and eternal black holes. The Reissner-Nordstr¨om solution. (A)dS-Schwarzschild space-time*. Killing horizons. The Kerr solution.
§ Conformal diagrams
Conformal compactifications and causal structure. Penrose diagrams of Minkowski, Schwarzschild, Reissner-Nordstr¨om, Kerr. Spherically collapsing radiation and mat-
ter shells. (Anti-)de Sitter spaces*.
.
§Gravitational energy
Conserved quantities in gauge theories: the example of Yang-Mills theory. Covariant conservation and ordinary conservation. Asymptotically flat metrics. Gravitational energy-momentum pseudo-tensor. The superpotential. ADM energy and momentum. ADM energy of the Schwarzschild solution. The positive-energy theorem (without proof). Killing vectors and Komar integrals. Quadrupole radiation*.
.
§ The Cartan-Weyl formulation and Fermionic couplings
Local inertial frames. The frame field and its relation to the metric field. Non-coordinate bases. Local Lorentz transformations. The spin connection. The vielbein postulate. Torsion constraint and second-order formulation. The contorsion tensor. Local Lorentz curvature. Minimally coupled Fermionic matter. Dirac Lagrangian on curved manifolds. Gravity as a Yang-Mills theory of the Poincar´e algebra. Local Poincar´e transformations. Torsion and curvature over the Poincar´e algebra. Problems with quadratic actions. First-order formulation and Weyl’s action.
- Carroll S Spacetime and Geometry: An Introduction to General Relativity (Addison-Wesley 2014/Cambridge University Press, 2019)
Mutuazione: 20430009 Teoria dei Campi e Gravità in Fisica LM-17 R FRANCIA DARIO
Programme
The items marked with a ``*'' are assigned as homework problems.§ Gravity as a self-interacting spin-two field theory
sometries of flat spacetime. Lorentz transformations and the Poincar´e Group. Decomposition theorems and standard boost. Contractions: Galilei and Carroll limits. Symmetries: global vs local. Basics of Lie algebras. Noether’s theorem and conservation laws. The canonical stress-energy and angular momentum tensors. Non-canonical currents: improvements and their relevance for couplings. Belinfante’s symmetric energy-momentum tensor and angular momentum tensor. Scale invariance and traceless stress-energy tensor. Particles and fields in Special Relativity. Irreps of the Poincar´e group: Wigner’s induced representation method. Massless particles: ISO(D-2) little group and gauge invariance*. From relativistic massless spin-2 particles to full GR. Uniqueness of the quadratic Lagrangian. Noether method and non-linear completions. Noether’s construction of Yang-Mills Lagrangian*. The transverse-traceless gravitational cubic vertex. Weinberg’s Equivalence Principle from relativistic invariance of the S-matrix. Spin and sign of static forces*.
.
§ Isometries and maximally symmetric spaces
Lie derivative and isometries. Lie bracket. Conformal Killing vectors. Killing tensors. Integrability condition for Killing vectors. Maximal number of isometries. Homogeneous and isotropic spaces. Structure theorem for maximally symmetric spaces: identification of the curvature via the metric signature and a curvature constant. MSS as vacuum solutions to the EH equations with cosmological constant. Construction from embedding in (D+1) pseudo-Lorentzian spaces: metric and Christoffel coefficients. Conformally flat spaces. AdS in the Poincar´e patch. Weyl tensor. Riemann and Weyl in various dimensions: counting components for irreps of GL(D) and O(D). Conformal transformations of the metric tensor. Conformally coupled scalar fields*.
.
§ Elements of differential geometry
Topological spaces. Manifolds. Diffeomorphisms. Tangent spaces and vectors. Coordinate basis. Derivative operators on manifolds. Differential forms: definition, wedge product, interior and exterior derivatives, Cartan’s formula for the Lie derivative, Hodge dual. Integration. Maxwell and Yang-Mills theories in the language of forms. Stokes’s theorem.
§ Black holes
Spherically symmetric spaces. The Schwarzschild solution. Birkhoff’s theorem (partial proof). Singularities, definitions and criteria: curvature singularites and geodesic incompleteness. The tortoise coordinate. Extension of a space-time. Coordinates of Eddington-Finkelstein. Event horizons, black holes and white holes. Kruskal-Szekeres coordinates. Maximal extension of the Schwarzschild solution. Kruskal’s diagram and eternal black holes. The Reissner-Nordstr¨om solution. (A)dS-Schwarzschild space-time*. Killing horizons. The Kerr solution.
§ Conformal diagrams
Conformal compactifications and causal structure. Penrose diagrams of Minkowski, Schwarzschild, Reissner-Nordstr¨om, Kerr. Spherically collapsing radiation and mat-
ter shells. (Anti-)de Sitter spaces*.
.
§Gravitational energy
Conserved quantities in gauge theories: the example of Yang-Mills theory. Covariant conservation and ordinary conservation. Asymptotically flat metrics. Gravitational energy-momentum pseudo-tensor. The superpotential. ADM energy and momentum. ADM energy of the Schwarzschild solution. The positive-energy theorem (without proof). Killing vectors and Komar integrals. Quadrupole radiation*.
.
§ The Cartan-Weyl formulation and Fermionic couplings
Local inertial frames. The frame field and its relation to the metric field. Non-coordinate bases. Local Lorentz transformations. The spin connection. The vielbein postulate. Torsion constraint and second-order formulation. The contorsion tensor. Local Lorentz curvature. Minimally coupled Fermionic matter. Dirac Lagrangian on curved manifolds. Gravity as a Yang-Mills theory of the Poincar´e algebra. Local Poincar´e transformations. Torsion and curvature over the Poincar´e algebra. Problems with quadratic actions. First-order formulation and Weyl’s action.
Core Documentation
- Wald R, General Relativity (The University of Chicago Press, 1984).- Carroll S Spacetime and Geometry: An Introduction to General Relativity (Addison-Wesley 2014/Cambridge University Press, 2019)
Attendance
Attendance in person. Working students may attend the course remotely. Lecture recordings will also be made available for a period of one week following each lecture.Type of evaluation
Assessment is based on an oral examination. Alternatively, students may choose to solve the problems assigned during the course and submit their solutions as a single written report. In this case, the final assessment is based both on the quality of the written report and on an oral discussion of its contents, aimed at assessing the student's mastery of the course material and ability to establish connections between the different aspects of the theory. teacher profile teaching materials
§ Gravity as a self-interacting spin-two field theory
sometries of flat spacetime. Lorentz transformations and the Poincar´e Group. Decomposition theorems and standard boost. Contractions: Galilei and Carroll limits. Symmetries: global vs local. Basics of Lie algebras. Noether’s theorem and conservation laws. The canonical stress-energy and angular momentum tensors. Non-canonical currents: improvements and their relevance for couplings. Belinfante’s symmetric energy-momentum tensor and angular momentum tensor. Scale invariance and traceless stress-energy tensor. Particles and fields in Special Relativity. Irreps of the Poincar´e group: Wigner’s induced representation method. Massless particles: ISO(D-2) little group and gauge invariance*. From relativistic massless spin-2 particles to full GR. Uniqueness of the quadratic Lagrangian. Noether method and non-linear completions. Noether’s construction of Yang-Mills Lagrangian*. The transverse-traceless gravitational cubic vertex. Weinberg’s Equivalence Principle from relativistic invariance of the S-matrix. Spin and sign of static forces*.
.
§ Isometries and maximally symmetric spaces
Lie derivative and isometries. Lie bracket. Conformal Killing vectors. Killing tensors. Integrability condition for Killing vectors. Maximal number of isometries. Homogeneous and isotropic spaces. Structure theorem for maximally symmetric spaces: identification of the curvature via the metric signature and a curvature constant. MSS as vacuum solutions to the EH equations with cosmological constant. Construction from embedding in (D+1) pseudo-Lorentzian spaces: metric and Christoffel coefficients. Conformally flat spaces. AdS in the Poincar´e patch. Weyl tensor. Riemann and Weyl in various dimensions: counting components for irreps of GL(D) and O(D). Conformal transformations of the metric tensor. Conformally coupled scalar fields*.
.
§ Elements of differential geometry
Topological spaces. Manifolds. Diffeomorphisms. Tangent spaces and vectors. Coordinate basis. Derivative operators on manifolds. Differential forms: definition, wedge product, interior and exterior derivatives, Cartan’s formula for the Lie derivative, Hodge dual. Integration. Maxwell and Yang-Mills theories in the language of forms. Stokes’s theorem.
§ Black holes
Spherically symmetric spaces. The Schwarzschild solution. Birkhoff’s theorem (partial proof). Singularities, definitions and criteria: curvature singularites and geodesic incompleteness. The tortoise coordinate. Extension of a space-time. Coordinates of Eddington-Finkelstein. Event horizons, black holes and white holes. Kruskal-Szekeres coordinates. Maximal extension of the Schwarzschild solution. Kruskal’s diagram and eternal black holes. The Reissner-Nordstr¨om solution. (A)dS-Schwarzschild space-time*. Killing horizons. The Kerr solution.
§ Conformal diagrams
Conformal compactifications and causal structure. Penrose diagrams of Minkowski, Schwarzschild, Reissner-Nordstr¨om, Kerr. Spherically collapsing radiation and mat-
ter shells. (Anti-)de Sitter spaces*.
.
§Gravitational energy
Conserved quantities in gauge theories: the example of Yang-Mills theory. Covariant conservation and ordinary conservation. Asymptotically flat metrics. Gravitational energy-momentum pseudo-tensor. The superpotential. ADM energy and momentum. ADM energy of the Schwarzschild solution. The positive-energy theorem (without proof). Killing vectors and Komar integrals. Quadrupole radiation*.
.
§ The Cartan-Weyl formulation and Fermionic couplings
Local inertial frames. The frame field and its relation to the metric field. Non-coordinate bases. Local Lorentz transformations. The spin connection. The vielbein postulate. Torsion constraint and second-order formulation. The contorsion tensor. Local Lorentz curvature. Minimally coupled Fermionic matter. Dirac Lagrangian on curved manifolds. Gravity as a Yang-Mills theory of the Poincar´e algebra. Local Poincar´e transformations. Torsion and curvature over the Poincar´e algebra. Problems with quadratic actions. First-order formulation and Weyl’s action.
- Carroll S Spacetime and Geometry: An Introduction to General Relativity (Addison-Wesley 2014/Cambridge University Press, 2019)
Programme
The items marked with a ``*'' are assigned as homework problems.§ Gravity as a self-interacting spin-two field theory
sometries of flat spacetime. Lorentz transformations and the Poincar´e Group. Decomposition theorems and standard boost. Contractions: Galilei and Carroll limits. Symmetries: global vs local. Basics of Lie algebras. Noether’s theorem and conservation laws. The canonical stress-energy and angular momentum tensors. Non-canonical currents: improvements and their relevance for couplings. Belinfante’s symmetric energy-momentum tensor and angular momentum tensor. Scale invariance and traceless stress-energy tensor. Particles and fields in Special Relativity. Irreps of the Poincar´e group: Wigner’s induced representation method. Massless particles: ISO(D-2) little group and gauge invariance*. From relativistic massless spin-2 particles to full GR. Uniqueness of the quadratic Lagrangian. Noether method and non-linear completions. Noether’s construction of Yang-Mills Lagrangian*. The transverse-traceless gravitational cubic vertex. Weinberg’s Equivalence Principle from relativistic invariance of the S-matrix. Spin and sign of static forces*.
.
§ Isometries and maximally symmetric spaces
Lie derivative and isometries. Lie bracket. Conformal Killing vectors. Killing tensors. Integrability condition for Killing vectors. Maximal number of isometries. Homogeneous and isotropic spaces. Structure theorem for maximally symmetric spaces: identification of the curvature via the metric signature and a curvature constant. MSS as vacuum solutions to the EH equations with cosmological constant. Construction from embedding in (D+1) pseudo-Lorentzian spaces: metric and Christoffel coefficients. Conformally flat spaces. AdS in the Poincar´e patch. Weyl tensor. Riemann and Weyl in various dimensions: counting components for irreps of GL(D) and O(D). Conformal transformations of the metric tensor. Conformally coupled scalar fields*.
.
§ Elements of differential geometry
Topological spaces. Manifolds. Diffeomorphisms. Tangent spaces and vectors. Coordinate basis. Derivative operators on manifolds. Differential forms: definition, wedge product, interior and exterior derivatives, Cartan’s formula for the Lie derivative, Hodge dual. Integration. Maxwell and Yang-Mills theories in the language of forms. Stokes’s theorem.
§ Black holes
Spherically symmetric spaces. The Schwarzschild solution. Birkhoff’s theorem (partial proof). Singularities, definitions and criteria: curvature singularites and geodesic incompleteness. The tortoise coordinate. Extension of a space-time. Coordinates of Eddington-Finkelstein. Event horizons, black holes and white holes. Kruskal-Szekeres coordinates. Maximal extension of the Schwarzschild solution. Kruskal’s diagram and eternal black holes. The Reissner-Nordstr¨om solution. (A)dS-Schwarzschild space-time*. Killing horizons. The Kerr solution.
§ Conformal diagrams
Conformal compactifications and causal structure. Penrose diagrams of Minkowski, Schwarzschild, Reissner-Nordstr¨om, Kerr. Spherically collapsing radiation and mat-
ter shells. (Anti-)de Sitter spaces*.
.
§Gravitational energy
Conserved quantities in gauge theories: the example of Yang-Mills theory. Covariant conservation and ordinary conservation. Asymptotically flat metrics. Gravitational energy-momentum pseudo-tensor. The superpotential. ADM energy and momentum. ADM energy of the Schwarzschild solution. The positive-energy theorem (without proof). Killing vectors and Komar integrals. Quadrupole radiation*.
.
§ The Cartan-Weyl formulation and Fermionic couplings
Local inertial frames. The frame field and its relation to the metric field. Non-coordinate bases. Local Lorentz transformations. The spin connection. The vielbein postulate. Torsion constraint and second-order formulation. The contorsion tensor. Local Lorentz curvature. Minimally coupled Fermionic matter. Dirac Lagrangian on curved manifolds. Gravity as a Yang-Mills theory of the Poincar´e algebra. Local Poincar´e transformations. Torsion and curvature over the Poincar´e algebra. Problems with quadratic actions. First-order formulation and Weyl’s action.
Core Documentation
- Wald R, General Relativity (The University of Chicago Press, 1984).- Carroll S Spacetime and Geometry: An Introduction to General Relativity (Addison-Wesley 2014/Cambridge University Press, 2019)
Reference Bibliography
- Weinberg S, Gravitation and Cosmology - principles and applications of the general theory of relativity , (John Wiley & Sons, 1972). - Weinberg S, The Quantum Theory of Fields I, (Cambridge University Press, 1995). - Freedman D Z and Van Proyen A, Supergravity (Cambridge University Press, 2012). - Hawking S W and Ellis G F R, The Large Scale Structure of Space-Time (Cambridge University Press, 1973). - Feynman R P, Morinigo F B, Wagner W G Feynman Lectures on Gravitation (Westview Press 2003) - Dirac P A M General Theory of Relativity (Princeton University Press, 1996)Attendance
Attendance in person. Working students may attend the course remotely. Lecture recordings will also be made available for a period of one week following each lecture.Type of evaluation
Assessment is based on an oral examination. Alternatively, students may choose to solve the problems assigned during the course and submit their solutions as a single written report. In this case, the final assessment is based both on the quality of the written report and on an oral discussion of its contents, aimed at assessing the student's mastery of the course material and ability to establish connections between the different aspects of the theory. teacher profile teaching materials
§ Gravity as a self-interacting spin-two field theory
sometries of flat spacetime. Lorentz transformations and the Poincar´e Group. Decomposition theorems and standard boost. Contractions: Galilei and Carroll limits. Symmetries: global vs local. Basics of Lie algebras. Noether’s theorem and conservation laws. The canonical stress-energy and angular momentum tensors. Non-canonical currents: improvements and their relevance for couplings. Belinfante’s symmetric energy-momentum tensor and angular momentum tensor. Scale invariance and traceless stress-energy tensor. Particles and fields in Special Relativity. Irreps of the Poincar´e group: Wigner’s induced representation method. Massless particles: ISO(D-2) little group and gauge invariance*. From relativistic massless spin-2 particles to full GR. Uniqueness of the quadratic Lagrangian. Noether method and non-linear completions. Noether’s construction of Yang-Mills Lagrangian*. The transverse-traceless gravitational cubic vertex. Weinberg’s Equivalence Principle from relativistic invariance of the S-matrix. Spin and sign of static forces*.
.
§ Isometries and maximally symmetric spaces
Lie derivative and isometries. Lie bracket. Conformal Killing vectors. Killing tensors. Integrability condition for Killing vectors. Maximal number of isometries. Homogeneous and isotropic spaces. Structure theorem for maximally symmetric spaces: identification of the curvature via the metric signature and a curvature constant. MSS as vacuum solutions to the EH equations with cosmological constant. Construction from embedding in (D+1) pseudo-Lorentzian spaces: metric and Christoffel coefficients. Conformally flat spaces. AdS in the Poincar´e patch. Weyl tensor. Riemann and Weyl in various dimensions: counting components for irreps of GL(D) and O(D). Conformal transformations of the metric tensor. Conformally coupled scalar fields*.
.
§ Elements of differential geometry
Topological spaces. Manifolds. Diffeomorphisms. Tangent spaces and vectors. Coordinate basis. Derivative operators on manifolds. Differential forms: definition, wedge product, interior and exterior derivatives, Cartan’s formula for the Lie derivative, Hodge dual. Integration. Maxwell and Yang-Mills theories in the language of forms. Stokes’s theorem.
§ Black holes
Spherically symmetric spaces. The Schwarzschild solution. Birkhoff’s theorem (partial proof). Singularities, definitions and criteria: curvature singularites and geodesic incompleteness. The tortoise coordinate. Extension of a space-time. Coordinates of Eddington-Finkelstein. Event horizons, black holes and white holes. Kruskal-Szekeres coordinates. Maximal extension of the Schwarzschild solution. Kruskal’s diagram and eternal black holes. The Reissner-Nordstr¨om solution. (A)dS-Schwarzschild space-time*. Killing horizons. The Kerr solution.
§ Conformal diagrams
Conformal compactifications and causal structure. Penrose diagrams of Minkowski, Schwarzschild, Reissner-Nordstr¨om, Kerr. Spherically collapsing radiation and mat-
ter shells. (Anti-)de Sitter spaces*.
.
§Gravitational energy
Conserved quantities in gauge theories: the example of Yang-Mills theory. Covariant conservation and ordinary conservation. Asymptotically flat metrics. Gravitational energy-momentum pseudo-tensor. The superpotential. ADM energy and momentum. ADM energy of the Schwarzschild solution. The positive-energy theorem (without proof). Killing vectors and Komar integrals. Quadrupole radiation*.
.
§ The Cartan-Weyl formulation and Fermionic couplings
Local inertial frames. The frame field and its relation to the metric field. Non-coordinate bases. Local Lorentz transformations. The spin connection. The vielbein postulate. Torsion constraint and second-order formulation. The contorsion tensor. Local Lorentz curvature. Minimally coupled Fermionic matter. Dirac Lagrangian on curved manifolds. Gravity as a Yang-Mills theory of the Poincar´e algebra. Local Poincar´e transformations. Torsion and curvature over the Poincar´e algebra. Problems with quadratic actions. First-order formulation and Weyl’s action.
- Carroll S Spacetime and Geometry: An Introduction to General Relativity (Addison-Wesley 2014/Cambridge University Press, 2019)
Mutuazione: 20430009 Teoria dei Campi e Gravità in Fisica LM-17 R FRANCIA DARIO
Programme
The items marked with a ``*'' are assigned as homework problems.§ Gravity as a self-interacting spin-two field theory
sometries of flat spacetime. Lorentz transformations and the Poincar´e Group. Decomposition theorems and standard boost. Contractions: Galilei and Carroll limits. Symmetries: global vs local. Basics of Lie algebras. Noether’s theorem and conservation laws. The canonical stress-energy and angular momentum tensors. Non-canonical currents: improvements and their relevance for couplings. Belinfante’s symmetric energy-momentum tensor and angular momentum tensor. Scale invariance and traceless stress-energy tensor. Particles and fields in Special Relativity. Irreps of the Poincar´e group: Wigner’s induced representation method. Massless particles: ISO(D-2) little group and gauge invariance*. From relativistic massless spin-2 particles to full GR. Uniqueness of the quadratic Lagrangian. Noether method and non-linear completions. Noether’s construction of Yang-Mills Lagrangian*. The transverse-traceless gravitational cubic vertex. Weinberg’s Equivalence Principle from relativistic invariance of the S-matrix. Spin and sign of static forces*.
.
§ Isometries and maximally symmetric spaces
Lie derivative and isometries. Lie bracket. Conformal Killing vectors. Killing tensors. Integrability condition for Killing vectors. Maximal number of isometries. Homogeneous and isotropic spaces. Structure theorem for maximally symmetric spaces: identification of the curvature via the metric signature and a curvature constant. MSS as vacuum solutions to the EH equations with cosmological constant. Construction from embedding in (D+1) pseudo-Lorentzian spaces: metric and Christoffel coefficients. Conformally flat spaces. AdS in the Poincar´e patch. Weyl tensor. Riemann and Weyl in various dimensions: counting components for irreps of GL(D) and O(D). Conformal transformations of the metric tensor. Conformally coupled scalar fields*.
.
§ Elements of differential geometry
Topological spaces. Manifolds. Diffeomorphisms. Tangent spaces and vectors. Coordinate basis. Derivative operators on manifolds. Differential forms: definition, wedge product, interior and exterior derivatives, Cartan’s formula for the Lie derivative, Hodge dual. Integration. Maxwell and Yang-Mills theories in the language of forms. Stokes’s theorem.
§ Black holes
Spherically symmetric spaces. The Schwarzschild solution. Birkhoff’s theorem (partial proof). Singularities, definitions and criteria: curvature singularites and geodesic incompleteness. The tortoise coordinate. Extension of a space-time. Coordinates of Eddington-Finkelstein. Event horizons, black holes and white holes. Kruskal-Szekeres coordinates. Maximal extension of the Schwarzschild solution. Kruskal’s diagram and eternal black holes. The Reissner-Nordstr¨om solution. (A)dS-Schwarzschild space-time*. Killing horizons. The Kerr solution.
§ Conformal diagrams
Conformal compactifications and causal structure. Penrose diagrams of Minkowski, Schwarzschild, Reissner-Nordstr¨om, Kerr. Spherically collapsing radiation and mat-
ter shells. (Anti-)de Sitter spaces*.
.
§Gravitational energy
Conserved quantities in gauge theories: the example of Yang-Mills theory. Covariant conservation and ordinary conservation. Asymptotically flat metrics. Gravitational energy-momentum pseudo-tensor. The superpotential. ADM energy and momentum. ADM energy of the Schwarzschild solution. The positive-energy theorem (without proof). Killing vectors and Komar integrals. Quadrupole radiation*.
.
§ The Cartan-Weyl formulation and Fermionic couplings
Local inertial frames. The frame field and its relation to the metric field. Non-coordinate bases. Local Lorentz transformations. The spin connection. The vielbein postulate. Torsion constraint and second-order formulation. The contorsion tensor. Local Lorentz curvature. Minimally coupled Fermionic matter. Dirac Lagrangian on curved manifolds. Gravity as a Yang-Mills theory of the Poincar´e algebra. Local Poincar´e transformations. Torsion and curvature over the Poincar´e algebra. Problems with quadratic actions. First-order formulation and Weyl’s action.
Core Documentation
- Wald R, General Relativity (The University of Chicago Press, 1984).- Carroll S Spacetime and Geometry: An Introduction to General Relativity (Addison-Wesley 2014/Cambridge University Press, 2019)
Reference Bibliography
- Weinberg S, Gravitation and Cosmology - principles and applications of the general theory of relativity , (John Wiley & Sons, 1972). - Weinberg S, The Quantum Theory of Fields I, (Cambridge University Press, 1995). - Freedman D Z and Van Proyen A, Supergravity (Cambridge University Press, 2012). - Hawking S W and Ellis G F R, The Large Scale Structure of Space-Time (Cambridge University Press, 1973). - Feynman R P, Morinigo F B, Wagner W G Feynman Lectures on Gravitation (Westview Press 2003) - Dirac P A M General Theory of Relativity (Princeton University Press, 1996)Attendance
Attendance in person. Working students may attend the course remotely. Lecture recordings will also be made available for a period of one week following each lecture.Type of evaluation
Assessment is based on an oral examination. Alternatively, students may choose to solve the problems assigned during the course and submit their solutions as a single written report. In this case, the final assessment is based both on the quality of the written report and on an oral discussion of its contents, aimed at assessing the student's mastery of the course material and ability to establish connections between the different aspects of the theory.