C-0006 Verified HIGH certainty
The symbol r_c in Alcubierre's energy-density equation denotes the spherical radius r_s. Recomputing the density from the metric via the Hamiltonian constraint reproduces the published equation exactly, and only, under that reading.
Standing our derivation · derivation under assumptions · TRL 1 Assumes the metric of eq. 5 with unit lapse and flat spatial slices; K_ij as given by the paper's own eq. 7; a GENERAL shape function f(r_s) — the result does not use the tanh profile Stops applying metrics whose spatial slices are not intrinsically flat, where the three-Ricci scalar no longer drops out of the Hamiltonian constraint Computed in compute/verify_energy_density.py, tests/test_q1_rc.py Note Resolves open question Q1. Three rival readings of r_c (R, rho, x) leave non-zero residuals.
What it rests on
E-0009 · S-0002 — The warp drive: hyper-fast travel within general relativity · equation
…pidly a "top hat" function: With the above definitions, the metric () can be rewritten as: d s^2 = - d t^2 + ( d x - v_s f ( r_s ) d t )^2 + d y^2 + d z^2 . It is easy to understand the geometry of our spacetime from the previous expressions. Firs…
EXACT · re-found in source
E-0012 · S-0002 — The warp drive: hyper-fast travel within general relativity · equation
…s is given by: then one can show that these observers will see an energy density given by: T^{\alpha \beta} n_{\alpha} n_{\beta} = \alpha^2 T^{ 0 0} = \frac{1}{8 \pi} G^{ 0 0} = - \frac{1}{8 \pi} \frac{v_s^2 \rho^2}{4 {r_c}^2} ( \frac{d f}{d r_s} )^2 . The fact that this expression is everywhere negative implies that the weak and dominant en…
EXACT · re-found in source
Attacks run against it
X-0005 SURVIVED recomputation · by straz
Attacked the reading by trying to make a different one work. Built the extrinsic curvature from the paper's eq. 7 with a general shape function, contracted it through the Hamiltonian constraint with R^(3) = 0 (valid because eq. 1 gives flat spatial slices), and compared the result against the published expression under four candidate readings of r_c: the spherical radius r_s, the bubble radius R, the perpendicular distance rho, and the axial distance x. Only r_s gives a zero residual; the other three leave residuals that are explicitly printed rather than summarised. The derivation never uses the tanh profile, so the identification is not an artefact of the shape function.