The chapter "Theoretical Aspects of Massive Gravity" by Kurt Hinterbichler provides a comprehensive overview of the theoretical developments in massive gravity, a modification of general relativity that introduces a non-zero mass for the graviton. The author discusses the historical context, recent progress, and key challenges in this field. 1. **Introduction**: - General relativity (GR) is widely accepted as the correct theory of gravity at low energies, but its modification to include a massive graviton has been studied for over 70 years. - The possibility of a massive graviton was first explored by Fierz and Pauli in 1939, and recent progress has addressed traditional issues such as the vDVZ discontinuity and the Boulware-Deser ghost. 2. **The Free Fierz-Pauli Action**: - The Fierz-Pauli action describes a massive spin-2 particle and is derived to ensure it propagates a single massive graviton. - The action is analyzed in Hamiltonian form to count the degrees of freedom, confirming that it propagates the correct number of degrees of freedom for a massive spin-2 particle. 3. **Linear Response to Sources**: - The linear response of the Fierz-Pauli theory to external sources is discussed, including the vDVZ discontinuity, which arises from the failure of the theory to satisfy continuity in the parameters of the theory. 4. **The Stückelberg Trick**: - The Stückelberg formalism is introduced to address the vDVZ discontinuity and other issues in massive gravity. - This formalism involves introducing auxiliary fields to restore gauge invariance and resolve pathologies. 5. **Massive Gravitons on Curved Spaces**: - The behavior of massive gravitons in curved spacetime is explored, including the Vainshtein mechanism, which explains how non-linearities can cure the vDVZ discontinuity. - The Vainshtein radius, a length scale around which non-linear effects dominate, is discussed. 6. **Non-linear Interactions**: - The chapter discusses the non-linear interactions in massive gravity, including the Boulware-Deser ghost and the Λ3 theory, which aims to resolve these issues by tuning interactions to raise the cutoff. 7. **The Non-linear Stückelberg Formalism**: - The non-linear Stückelberg formalism is developed to handle the non-linearities in massive gravity, providing a more complete understanding of the theory. 8. **Stükelberg Analysis of Interacting Massive Gravity**: - The chapter analyzes interacting massive gravity theories, including the decoupling limit, ghost instabilities, and the resolution of the vDVZ discontinuity through the Vainshtein mechanism. 9. **The Λ3 Theory**: - The Λ3 theory is introduced as a promising framework that raisesThe chapter "Theoretical Aspects of Massive Gravity" by Kurt Hinterbichler provides a comprehensive overview of the theoretical developments in massive gravity, a modification of general relativity that introduces a non-zero mass for the graviton. The author discusses the historical context, recent progress, and key challenges in this field. 1. **Introduction**: - General relativity (GR) is widely accepted as the correct theory of gravity at low energies, but its modification to include a massive graviton has been studied for over 70 years. - The possibility of a massive graviton was first explored by Fierz and Pauli in 1939, and recent progress has addressed traditional issues such as the vDVZ discontinuity and the Boulware-Deser ghost. 2. **The Free Fierz-Pauli Action**: - The Fierz-Pauli action describes a massive spin-2 particle and is derived to ensure it propagates a single massive graviton. - The action is analyzed in Hamiltonian form to count the degrees of freedom, confirming that it propagates the correct number of degrees of freedom for a massive spin-2 particle. 3. **Linear Response to Sources**: - The linear response of the Fierz-Pauli theory to external sources is discussed, including the vDVZ discontinuity, which arises from the failure of the theory to satisfy continuity in the parameters of the theory. 4. **The Stückelberg Trick**: - The Stückelberg formalism is introduced to address the vDVZ discontinuity and other issues in massive gravity. - This formalism involves introducing auxiliary fields to restore gauge invariance and resolve pathologies. 5. **Massive Gravitons on Curved Spaces**: - The behavior of massive gravitons in curved spacetime is explored, including the Vainshtein mechanism, which explains how non-linearities can cure the vDVZ discontinuity. - The Vainshtein radius, a length scale around which non-linear effects dominate, is discussed. 6. **Non-linear Interactions**: - The chapter discusses the non-linear interactions in massive gravity, including the Boulware-Deser ghost and the Λ3 theory, which aims to resolve these issues by tuning interactions to raise the cutoff. 7. **The Non-linear Stückelberg Formalism**: - The non-linear Stückelberg formalism is developed to handle the non-linearities in massive gravity, providing a more complete understanding of the theory. 8. **Stükelberg Analysis of Interacting Massive Gravity**: - The chapter analyzes interacting massive gravity theories, including the decoupling limit, ghost instabilities, and the resolution of the vDVZ discontinuity through the Vainshtein mechanism. 9. **The Λ3 Theory**: - The Λ3 theory is introduced as a promising framework that raises