The Einstein-Hilbert Action
The action principle from which Einstein's field equations of General Relativity are derived.
Formulated by David Hilbert (1915) | Basis of General Relativity
Physical Interpretation
The Einstein-Hilbert action embodies the principle that matter tells spacetime how to curve, and spacetime tells matter how to move. The action is proportional to the total curvature integrated over spacetime.
Variational Principle
Varying the action with respect to the metric gμν and setting δS = 0 yields Einstein's field equations: Gμν = 8πG Tμν, where Gμν is the Einstein tensor and Tμν is the stress-energy tensor from matter.
Connection to Principia Metaphysica: 2T Physics Framework
In Principia Metaphysica, the Einstein-Hilbert action is generalized to 26 dimensions with signature (24,2), following the 2T physics framework of Itzhak Bars:
Complete Dimensional Reduction Pathway in 2T Framework
The Principia Metaphysica framework employs a multi-stage compactification using the 2T physics formalism, descending from the fundamental 26-dimensional bulk with signature (24,2):
The Complete Pathway: 26D (24,2) → 13D (12,1) → 6D → 4D
Each stage of dimensional reduction reveals new physical structure:
- 26D (24,2): Fundamental bulk with two timelike dimensions, signature (24,2)
- 26D → 13D: Sp(2,R) gauge fixing projects bulk to 13D shadow with signature (12,1)
- 13D → 6D: G₂ manifold compactification (7D compact, establishes SO(10) GUT, 3 generations)
- 7D×2 → 4D: Shared timelike dimensions unify the two 7D sectors to observable 4D (3,1)
Stage 0: 26D (24,2) Bulk with Sp(2,R) Gauge Symmetry
The 2T physics framework starts with a 26-dimensional bulk with signature (24,2): two timelike and 24 spacelike dimensions. This is the critical dimension for bosonic string theory, but now with two timelike coordinates protected by Sp(2,R) gauge symmetry:
The Sp(2,R) gauge symmetry acts on the two timelike coordinates (X0, X1), ensuring consistency and unitarity despite the unusual signature. Gauge fixing this symmetry reduces the manifest dimensionality and projects to physical 1T theories.
Stage 1: 26D → 13D (Sp(2,R) Gauge Fixing)
Applying Sp(2,R) gauge fixing to the 26D bulk projects it to a 13D shadow the two timelike dimensions:
The Sp(2,R) gauge fixing maintains the two timelike dimensions but splits the 24 spacelike dimensions into two sets of 12, yielding 14D1 (12,2) ⊗ 14D2 (12,2). The shared timelike structure ensures consistency between the two sectors.
Stage 2: 14D×2 → 7D×2 (Dual G₂ Compactification)
Each 14D half compactifies on a 7-dimensional G₂ manifold, reducing to two coupled 7D sectors:
The G₂ holonomy in each sector preserves supersymmetry and yields SO(10) gauge symmetry from D₅ ADE singularities. Each M⁷i has effective Euler characteristic χeff = 72, giving 3 fermion generations per sector via ngen = χ/24.
Stage 3: 7D×2 → 4D (Shared Time Unification)
The two 7D sectors share the two timelike dimensions, which unify to give the single observed time coordinate. The effective 4D theory emerges with signature (3,1):
Combined Formula in 2T Framework
Putting all stages together, the 4D Planck mass emerges from the full compactification:
This confirms the general Kaluza-Klein formula MPl2 = M*n+2 Vn with n=22 compact dimensions (24 spacelike - 2 projected by Sp(2,R) gauge fixing). However, V22 is not independently measurable - only the product M*24 V22 = MPl2 is constrained by observation.
Scale Hierarchy in 2T Framework
The multi-stage compactification in 2T physics naturally generates a hierarchy of scales:
- Mstring ~ 1018 GeV: 26D bosonic string scale
- M* ~ MPl ~ 1019 GeV: 26D (24,2) fundamental scale
- MSp(2,R) ~ MPl: Scale of Sp(2,R) gauge symmetry (26D → 13D projection)
- 1/RG₂ ~ MGUT ~ 1016 GeV: G₂ compactification scale (each sector)
- MEW ~ 100 GeV: Electroweak symmetry breaking (Higgs VEV)
The 2T framework naturally explains why the Sp(2,R) gauge fixing scale is close to MPl: the two timelike dimensions must decouple at or above the Planck scale to avoid observable violations of causality and unitarity.
Dimensional Reduction: Path Comparison
| Framework | Starting Dim | Internal Geometry | Final Dim |
|---|---|---|---|
| Original KK | 5D | S¹ (circle) | 4D + U(1) |
| Heterotic String | 10D | CY3 (6D Calabi-Yau) | 4D + gauge |
| F-Theory | 12D | CY4 (8D elliptic) | 4D + gauge |
| M-Theory (standard) | 11D | G₂ (7D) | 4D |
| Principia Metaphysica | 13D | G₂ (7D) + T² (2D) | 4D (via 6D) |
References & Further Reading
- Original Papers: Einstein (1915) & Hilbert (1915) "Die Feldgleichungen der Gravitation" [Einstein Papers]
- Wikipedia: Einstein-Hilbert Action | Ricci Curvature | Einstein Field Equations
- Textbook: Misner, Thorne & Wheeler, "Gravitation" (1973) [Wikipedia]
- Variational Principle: Palatini Variation | Principle of Least Action
Where Einstein-Hilbert Action Is Used in PM
This foundational physics appears in the following sections of Principia Metaphysica:
Where Einstein-Hilbert Action Is Used in PM
This foundational physics appears in the following sections of Principia Metaphysica: