V2.88 - Graviton (Spin-2) Entanglement Entropy
V2.88: Graviton (Spin-2) Entanglement Entropy
Summary
Computed the entanglement entropy of the linearized graviton (massless spin-2, 2 physical dofs) on a radial lattice using the Srednicki/Lohmayer decomposition with tensor spherical harmonics (l >= 2, modified centrifugal barrier).
Method
- Radial chain with barrier (l-1)(l+2)/r^2 = [l(l+1) - 2]/r^2
- Multiplicity: 2(2l+1) per angular momentum l (2 polarizations)
- l starts at 2 (no l=0 or l=1 tensor harmonics)
- Global angular cutoff: l_max = 10 * max(n)
- N = 500 radial sites
- n = 15..60
Results
Graviton Entropy Coefficients (5-param fit)
| Parameter | Value | Notes |
|---|---|---|
| alpha_graviton | 0.03852919 | Area-law coefficient |
| delta_graviton (lattice) | -287.563550 | Logarithmic coefficient |
| delta_graviton (HK) | -2.355556 | Heat kernel prediction -212/90 |
| delta error vs HK | 12107.89% | |
| R^2 (fit) | 0.9999999631 |
Alpha Ratios
| Ratio | Value |
|---|---|
| alpha_graviton / alpha_scalar | 1.6216 |
| Naive dof ratio | 2.0 |
| HK a-coefficient ratio | 212.0 |
Per-Species Self-Consistency
| Field | R = |delta|/(12*alpha) | |-------|------------------------| | Graviton (lattice delta) | 621.960369 | | Graviton (HK delta) | 5.094742 |
SM + Graviton
| Quantity | SM only | SM + graviton (lattice) | SM + graviton (HK) |
|---|---|---|---|
| alpha | 2.805126 | 2.843655 | 2.843655 |
| delta | -11.061111 | -298.624662 | -13.416667 |
| R | 0.328598 | 8.751199 | 0.393176 |
| Lambda/Lambda_obs | 1.1573 | 30.8199 | 1.3847 |
Graviton Contribution Fractions
- Fraction of total alpha: 1.35%
- Fraction of total delta: 96.30%
Phase 1: Hamiltonian Validation
Graviton-scalar comparison at increasing l:
| l | S_graviton | S_scalar | Relative diff |
|---|---|---|---|
| 2 | 0.44692899 | 0.41591976 | 0.074556 |
| 5 | 0.29423099 | 0.28876949 | 0.018913 |
| 10 | 0.18990552 | 0.18854535 | 0.007214 |
| 20 | 0.09798993 | 0.09770558 | 0.002910 |
| 50 | 0.02129293 | 0.02127425 | 0.000878 |
| 100 | 0.00340464 | 0.00340361 | 0.000301 |
Diagonal check: graviton diagonal = scalar diagonal - 2/j^2 (max error: 5.05e-14)
l=0, l=1 correctly rejected: True
Computational Details
- Phase 2 elapsed time: 6.7s
- Total runtime: 6.7s
Discussion
Hamiltonian Validation
The graviton (spin-2) has a dramatically different centrifugal barrier at low l compared to the scalar. At l=2, the tensor barrier is (l-1)(l+2) = 4 vs the scalar barrier l(l+1) = 6, so the graviton modes are less confined and contribute more entropy per channel. The graviton l=2 entropy is 7.5% larger than scalar l=2 (0.447 vs 0.416). As l increases, the -2/r^2 correction becomes negligible and the graviton/scalar difference falls as ~1/l^2 (0.03% at l=100). The Hamiltonian diagonal difference exactly matches -2/j^2 to machine precision (max error 5e-14).
Delta Extraction: Global Cutoff Artifact
The lattice-extracted delta = -287.56 is NOT the physical logarithmic coefficient. This is the well-known global cutoff artifact (documented in V2.71-V2.73): the global cutoff (l_max = C * max(n)) correctly extracts alpha but inflates delta because the angular sum includes unphysical contributions at high l that grow with the cutoff. The proportional cutoff (l_max = C * n) is needed for delta, but the 5-param fit is unreliable for delta extraction in either convention.
The PHYSICAL delta should be extracted via the d3S (third finite difference) method with a proportional cutoff, as was done for the scalar in V2.67. Alternatively, we trust the heat kernel prediction delta_graviton = -212/90 = -2.3556 (same mapping delta = -8*a that is validated for scalars).
Alpha Result
The graviton alpha = 0.03853 is the reliable output of this computation. Key ratios:
- alpha_graviton / alpha_scalar = 1.62 (less than the naive 2-dof prediction of 2.0, because the tensor barrier is WEAKER than scalar, meaning the high-l channels contribute LESS relative entropy)
- This is a genuine lattice measurement, not a heat kernel ratio
SM + Graviton (Using HK Delta)
The physically meaningful SM + graviton result uses the heat kernel delta:
- alpha_total = 2.844 (graviton adds only 1.35% to alpha)
- delta_total = -13.417 (graviton adds 21.3% to |delta|)
- R_SM+graviton = 0.393 (up from R_SM = 0.329)
- Lambda_predicted / Lambda_observed = 1.385 (up from 1.157)
The graviton moves R closer to 1 (self-consistency) and keeps the Lambda prediction within a factor of 1.4 of the observed value. The graviton contribution is modest but non-negligible: its delta is significant (21% of the total) while its alpha is small (1.35%).
Next Steps
- Extract delta_graviton via d3S method with proportional cutoff to verify the heat kernel prediction independently
- Run at multiple C values to check alpha convergence
- Consider whether the graviton should be included in the species sum at all (the graviton is not part of the SM; it only contributes if gravity is quantized at the entangling surface scale)