gravity.txt

gravity.txt

Gravity Theories Dichotomy Tree
(Branching based on key decision points: conceptual, mathematical, and physical choices leading to each theory's existence)

Root: Fundamental View of Gravity
├── As a Field (Classical, Force-based)
│   ├── Non-Relativistic (Absolute time/space)
│   │   ├── Scalar Potential (Newtonian Gravity)
│   │   │   - Decision: Instantaneous action? Yes → Poisson's equation (∇²φ = 4πGρ)
│   │   │   - Exists due to: Empirical laws (e.g., inverse square) + field concept from EM analogy
│   │   └── Tensorial Reformulation (To geometrize)
│   │       ├── Degenerate Metrics (Rank-deficient)
│   │       │   - Decision: Separate time/space? Yes → Newton-Cartan (NC)
│   │       │   - Equations: R_{μν} = 4πG ρ t_{μν}
│   │       │   - Exists due to: Need for diff-invariance in Newtonian limit + GR inspiration (Cartan/Ehlers)
│   │       └── Non-Degenerate (But still non-rel)
│   │           - Decision: Add extra dims? → Bargmann lift (5D non-deg for NC)
│   │           - Exists due: Resolve degeneracy for math tractability
│   └── Relativistic (Finite speed, Lorentz inv)
│       ├── Weak Field Approximation
│       │   - Decision: Linearize GR? Yes → Effective Field Theory (EFT) for gravity
│       │   - Exists due: Quantum field theory needs + low-energy GR
│       └── Full Tensor Field (But geometric)
│           - (Merges into Geometry branch below)
└── As Geometry (Curvature-based)
    ├── Non-Relativistic (Galilean inv)
    │   - Decision: Geometrize Newtonian? Yes → Newton-Cartan (as above, unified branch)
    │   - Exists due: Bridge Newtonian to GR (1920s, Cartan)
    ├── Relativistic (Lorentz inv)
    │   ├── Classical (No quanta)
    │   │   ├── Vacuum Solutions Focus
    │   │   │   - Decision: Matter curves space? Yes → General Relativity (GR)
    │   │   │   - Equations: G_{μν} = 8πG/c⁴ T_{μν}
    │   │   │   - Exists due: Equivalence principle + special rel (Einstein 1915)
    │   │   └── With Torsion/Non-Metricity
    │   │       - Decision: Allow spin/torsion? Yes → Einstein-Cartan Theory
    │   │       - Exists due: Couple to fermionic matter (1920s, Cartan)
    │   └── Quantum (Incorporate quanta)
    │       ├── Background-Dependent (Perturbative)
    │       │   - Decision: Quantize metric perturbations? Yes → Quantum GR (non-renorm)
    │       │   - Exists due: Semiclassical attempts (e.g., Hawking radiation)
    │       ├── Background-Independent (Non-pert)
    │       │   ├── Discrete Space
    │       │   │   - Decision: Loop quantize? Yes → Loop Quantum Gravity (LQG)
    │       │   │   - Exists due: Canonical quantization of GR (1980s, Ashtekar)
    │       │   └── String-Based
    │       │       - Decision: Higher dims/vibrations? Yes → String Theory/M-Theory
    │       │       - Exists due: Unify all forces (1970s, superstrings)
    │       └── Broken Symmetry (e.g., for UV completion)
    │           - Decision: Anisotropic scaling? Yes → Hořava-Lifshitz Gravity
    │           - Exists due: Renormalizable quantum gravity + NC limit (2009, Hořava)
    └── Beyond Standard (Modified/Alternative)
        ├── For Dark Phenomena
        │   - Decision: Modify Newtonian? Yes → MOND (Modified Newtonian Dynamics)
        │   - Exists due: Galactic rotation curves (1980s, Milgrom)
        └── Emergent Gravity
            - Decision: Gravity from entropy? Yes → Verlinde's Entropic Gravity
            - Exists due: Holographic/thermodynamic views (2010s)