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Saturating Scalar Warp Model (SSW-1)A rewritten, action-first replacement of IPT–Warp v0.10. What this isSSW-1 asks one question:Can a real scalar field with a saturating effective coupling source a warp-like ADM shift whose Einstein tensor is accounted for by that field’s stress-energy, and can the resulting metric perturbation be bounded well enough to be tested? It does not assume superluminal travel. It does not treat intention, consciousness, or a “regulator of will” as physical fields. Those words in v0.1 are replaced by an ordinary scalar Φ\Phi\Phi and a bounded scalar functional R[Φ]\mathcal{R}[\Phi]\mathcal{R}[\Phi] .Established pieces: Einstein gravity, ADM kinematics, canonical scalar stress-energy, Alcubierre-type and physical-warp (Bobrick–Martire / Fuchs-type) shift geometries.Speculative pieces: the saturating coupling F(R)\mathcal{F}(\mathcal{R})\mathcal{F}(\mathcal{R}) that modulates the shift sector, and the claim that laboratory coherence can drive a measurable δgμν\delta g_{\mu\nu}\delta g_{\mu\nu} beyond ordinary TμνT_{\mu\nu}T_{\mu\nu} .1. Variables and signatureMetric signature (−+++)(-+++)(-+++) . Units kept explicit until the geometric section, then c=1c=1c=1 where noted.Fields:spacetime metric gμνg_{\mu\nu}g_{\mu\nu}

real scalar Φ\Phi\Phi

optional ordinary matter TμνmT_{\mu\nu}^{\mathrm{m}}T_{\mu\nu}^{\mathrm{m}} (fluid, shell, EM, ...)

Derived scalars:X:=−12∇αΦ∇αΦ,N:=X+12m2Φ2.X := -\frac12\nabla_\alpha\Phi\nabla^\alpha\Phi, \qquad N := X + \frac12 m^2\Phi^2.X := -\frac12\nabla_\alpha\Phi\nabla^\alpha\Phi, \qquad N := X + \frac12 m^2\Phi^2. (N) is a local, non-negative combination in the mostly-plus convention only after the kinetic sign is checked on a given solution; in the action below the kinetic term is written canonically, and (N) is used only as a phenomenological density for saturation.Saturating functional (dimensionless):R[Φ]=1−e−kN/N∗,0≤R<1,\mathcal{R}[\Phi]

1-e^{-k N/N_*}, \qquad 0\le\mathcal{R}<1,\mathcal{R}[\Phi]

1-e^{-k N/N_}, \qquad 0\le\mathcal{R}<1, where k>0k>0k>0 is dimensionless and N∗N_N_*

sets the saturation density. No extra “surge” prefactor is introduced. If a measured susceptibility later demands a correction (1+εS)(1+\varepsilon S)(1+\varepsilon S) , ε\varepsilon\varepsilon must be derived or fitted, not inserted as (0.06).2. ActionThe theory is defined byS=∫d4x−g[c416πG R+Lm−12∇μΦ∇μΦ−V(Φ)−μ2 F(R) KijKij]+SGHY.S

\int\mathrm{d}^4x\sqrt{-g}\Biggl[ \frac{c^4}{16\pi G},R +\mathcal{L}{\mathrm{m}} -\frac12\nabla\mu\Phi\nabla^\mu\Phi -V(\Phi) -\frac{\mu}{2},\mathcal{F}(\mathcal{R}),K_{ij}K^{ij} \Biggr] +S_{\mathrm{GHY}}.S

\int\mathrm{d}^4x\sqrt{-g}\Biggl[ \frac{c^4}{16\pi G},R +\mathcal{L}{\mathrm{m}} -\frac12\nabla\mu\Phi\nabla^\mu\Phi -V(\Phi) -\frac{\mu}{2},\mathcal{F}(\mathcal{R}),K_{ij}K^{ij} \Biggr] +S_{\mathrm{GHY}}. Here:(R) is the Ricci scalar, SGHYS_{\mathrm{GHY}}S_{\mathrm{GHY}} is the Gibbons–Hawking–York term, KijK_{ij}K_{ij} is the extrinsic curvature of a chosen foliation, F(R)\mathcal{F}(\mathcal{R})\mathcal{F}(\mathcal{R}) is a bounded coupling, μ\mu\mu has dimensions of energy density (it sets the strength of the shift-sector coupling).

This is the central change from v0.1. The geometry is not edited by hand. The coupling sits in the action, so TμνT_{\mu\nu}T_{\mu\nu} and the constraints follow from variation.A conservative choice:F(R)=η RpRcp+Rp,η≥0,p≥1,0<Rc<1.\mathcal{F}(\mathcal{R})

\eta,\frac{\mathcal{R}^p}{\mathcal{R}_c^p+\mathcal{R}^p}, \qquad \eta\ge 0,\quad p\ge 1,\quad 0<\mathcal{R}_c<1.\mathcal{F}(\mathcal{R})

\eta,\frac{\mathcal{R}^p}{\mathcal{R}_c^p+\mathcal{R}^p}, \qquad

\eta\ge 0,\quad p\ge 1,\quad 0<\mathcal{R}c<1. Then F(0)=0\mathcal{F}(0)=0\mathcal{F}(0)=0 and F→η\mathcal{F}\to\eta\mathcal{F}\to\eta as R→1\mathcal{R}\to 1\mathcal{R}\to 1 . Low coherence: standard GR plus a free scalar. High coherence: a finite renormalization of the extrinsic-curvature kinetic term in the chosen foliation.Caveat. Coupling to KijKijK{ij}K^{ij}K_{ij}K^{ij} prefers a foliation. That is acceptable as an effective description on a laboratory or bubble rest frame, but it is not a fundamental diffeomorphism-invariant completion. A covariant rewrite replaces KijKijK_{ij}K^{ij}K_{ij}K^{ij} by a scalar built from ∇μuν\nabla_\mu u_\nu\nabla_\mu u_\nu for a dynamical unit timelike field uμu^\muu^\mu (Einstein–Aether / khronometric style). SSW-1 uses the ADM form only as the working effective theory.3. Field equationsVarying with respect to gμνg^{\mu\nu}g^{\mu\nu} givesGμν=8πGc4(Tμνm+TμνΦ+TμνF).G_{\mu\nu}

\frac{8\pi G}{c^4} \Bigl( T_{\mu\nu}^{\mathrm{m}} +T_{\mu\nu}^{\Phi} +T_{\mu\nu}^{\mathcal{F}} \Bigr).G_{\mu\nu}

\frac{8\pi G}{c^4} \Bigl( T_{\mu\nu}^{\mathrm{m}} +T_{\mu\nu}^{\Phi} +T_{\mu\nu}^{\mathcal{F}} \Bigr). Canonical scalar piece:TμνΦ=∇μΦ∇νΦ−gμν(12∇αΦ∇αΦ+V(Φ)).T_{\mu\nu}^{\Phi}

\nabla_\mu\Phi\nabla_\nu\Phi -g_{\mu\nu}\Bigl(\tfrac12\nabla_\alpha\Phi\nabla^\alpha\Phi+V(\Phi)\Bigr).T_{\mu\nu}^{\Phi}

\nabla_\mu\Phi\nabla_\nu\Phi -g_{\mu\nu}\Bigl(\tfrac12\nabla_\alpha\Phi\nabla^\alpha\Phi+V(\Phi)\Bigr). TμνFT_{\mu\nu}^{\mathcal{F}}T_{\mu\nu}^{\mathcal{F}} is the stress-energy of the F(R)KijKij\mathcal{F}(\mathcal{R})K_{ij}K^{ij}\mathcal{F}(\mathcal{R})K_{ij}K^{ij} term. It contains:terms ∝F KijKij\propto\mathcal{F},K_{ij}K^{ij}\propto\mathcal{F},K_{ij}K^{ij}

terms from δKij/δgμν\delta K_{ij}/\delta g^{\mu\nu}\delta K_{ij}/\delta g^{\mu\nu}

terms from δR/δgμν\delta\mathcal{R}/\delta g^{\mu\nu}\delta\mathcal{R}/\delta g^{\mu\nu}

through N[g,Φ]N[g,\Phi]N[g,\Phi] Those last terms are essential. They are how saturation backreacts on geometry. They were missing when v0.1 wrote βx∝W(R)\beta^x\propto\mathcal{W}(R)\beta^x\propto\mathcal{W}(R) by hand.The scalar equation is∇μ∇μΦ−V′(Φ)=μ2 F′(R) δRδΦ KijKij.\nabla_\mu\nabla^\mu\Phi-V'(\Phi)

\frac{\mu}{2}, \mathcal{F}'(\mathcal{R}), \frac{\delta\mathcal{R}}{\delta\Phi}, K_{ij}K^{ij}.\nabla_\mu\nabla^\mu\Phi-V'(\Phi)

\frac{\mu}{2}, \mathcal{F}'(\mathcal{R}), \frac{\delta\mathcal{R}}{\delta\Phi}, K_{ij}K^{ij}. Geometry sources Φ\Phi\Phi ; Φ\Phi\Phi sources geometry. No independent “regulator field equation” is added.4. Search class: warp-like ADM metricsTo search for solutions one may restrict to the ansatzds2=−α2c2dt2+γij(dxi−βidt)(dxj−βjdt).\mathrm{d}s^2

-\alpha^2 c^2\mathrm{d}t^2 +\gamma_{ij}\bigl(\mathrm{d}x^i-\beta^i\mathrm{d}t\bigr)\bigl(\mathrm{d}x^j-\beta^j\mathrm{d}t\bi gr).\mathrm{d}s^2

-\alpha^2 c^2\mathrm{d}t^2 +\gamma_{ij}\bigl(\mathrm{d}x^i-\beta^i\mathrm{d}t\bigr)\bigl(\mathrm{d}x^j-\beta^j\mathrm{d}t\bi gr). Minimal working slice (not a solution by itself):α=1,γij=δij,βx=vwf(rs),βy=βz=0,\alpha=1, \qquad \gamma_{ij}=\delta_{ij}, \qquad \beta^x=v_w f(r_s), \qquad \beta^y=\beta^z=0,\alpha=1, \qquad \gamma_{ij}=\delta_{ij}, \qquad \beta^x=v_w f(r_s), \qquad \beta^y=\beta^z=0, rs=(x−xs(t))2+y2+z2,r_s=\sqrt{\bigl(x-x_s(t)\bigr)^2+y^2+z^2},r_s=\sqrt{\bigl(x-x_s(t)\bigr)^2+y^2 +z^2}, f(rs)=tanh [σ(Rb+rs)]−tanh [σ(rs−Rb)]2tanh (σRb).f(r_s)

= \frac{ \tanh\bigl[\sigma(R_b+r_s)\bigr] -\tanh\bigl[\sigma(r_s-R_b)\bigr] }{2\tanh(\sigma R_b)}.f(r_s)

\frac{ \tanh\bigl[\sigma(R_b+r_s)\bigr] -\tanh\bigl[\sigma(r_s-R_b)\bigr] }{2\tanh(\sigma R_b)}. RbR_bR_b is the bubble radius, σ−1\sigma^{-1}\sigma^{-1} the wall thickness, vwv_wv_w a coordinate shift amplitude.This metric is a trial function. It is admitted only if the Einstein equation is satisfied for some Φ\Phi\Phi and TμνmT_{\mu\nu}^{\mathrm{m}}T_{\mu\nu}^{\mathrm{m}} in the action of §2. In particular one does not replace vwv_wv_w by vwW(R)v_w\mathcal{W}(\mathcal{R})v_w\mathcal{W}(\mathcal{R}) . If saturation modifies the shift, that must appear as a solution of the momentum constraint, not as a redefined ansatz.Effective coordinate speed of the bubble center:veff:=dxsdt.v_{\mathrm{eff}} := \frac{\mathrm{d}x_s}{\mathrm{d}t}.v_{\mathrm{eff}} := \frac{\mathrm{d}x_s}{\mathrm{d}t}. For the standard Alcubierre identification one often sets vw=veffv_w=v_{\mathrm{eff}}v_w=v_{\mathrm{eff}} . That identification is optional and must be checked against the constraints.5. ClosureDefineTμνreq:=c48πGGμν[g],ΔTμν:=Tμνreq−Tμνm−TμνΦ−TμνF.T_{\mu\nu}^{\mathrm{r eq}} := \frac{c^4}{8\pi G}G_{\mu\nu}[g], \qquad \Delta T_{\mu\nu} := T_{\mu\nu}^{\mathrm{req}} -T_{\mu\nu}^{\mathrm{m}} -T_{\mu\nu}^{\Phi} -T_{\mu\nu}^{\mathcal{F}}.T_{\mu\nu}^{\mathrm{req}} := \frac{c^4}{8\pi G}G_{\mu\nu}[g], \qquad \Delta T_{\mu\nu} := T_{\mu\nu}^{\mathrm{req}} -T_{\mu\nu}^{\mathrm{m}} -T_{\mu\nu}^{\Phi}

-T_{\mu\nu}^{\mathcal{F}}. A configuration (g,Φ)(g,\Phi)(g,\Phi) is a solution iffΔTμν=0\Delta T_{\mu\nu}=0\Delta T_{\mu\nu}=0 everywhere, with Φ\Phi\Phi obeying its field equation and with finite total energy on each slice (or a controlled ADM mass).This is the only consistency condition that matters.6. Energy conditionsFor every timelike uμu^\muu^\mu and null kμk^\muk^\mu ,WEC:Tμνuμuν≥0,NEC:Tμνkμkν≥0,DEC:Tμνuμuν≥0 and Tμνuν non-spacelike,SEC:(Tμν−12Tgμ ν)uμuν≥0.\begin{aligned} \mathrm{WEC}:&\quad T_{\mu\nu}u^\mu u^\nu\ge 0,\ \mathrm{NEC}:&\quad T_{\mu\nu}k^\mu k^\nu\ge 0,\ \mathrm{DEC}:&\quad T_{\mu\nu}u^\mu u^\nu\ge 0 \text{ and }T^\mu{}\nu u^\nu\text{ non-spacelike},\ \mathrm{SEC}:&\quad \bigl(T{\mu\nu}-\tfrac12 T g_{\mu\nu}\bigr)u^\mu u^\nu\ge 0. \end{aligned}\begin{aligned} \mathrm{WEC}:&\quad T_{\mu\nu}u^\mu u^\nu\ge 0,\ \mathrm{NEC}:&\quad T_{\mu\nu}k^\mu k^\nu\ge 0,\ \mathrm{DEC}:&\quad T_{\mu\nu}u^\mu u^\nu\ge 0 \text{ and }T^\mu{}\nu u^\nu\text{ non-spacelike},\ \mathrm{SEC}:&\quad \bigl(T{\mu\nu}-\tfrac12 T g_{\mu\nu}\bigr)u^\mu u^\nu\ge 0. \end{aligned} These must be tested for a dense sample of observers, not only Eulerian ones.Two regimes:I. Physical subluminal. Seek 0≤veff<c0\le v_{\mathrm{eff}}<c0\le v_{\mathrm{eff}}<c with WEC and NEC intact, finite ADM mass, and a matter shell plus Φ\Phi\Phi that can in principle be assembled from known fields. This is the

Physical subluminal. Seek 0≤veff<c0\le v_{\mathrm{eff}}<c0\le v_{\mathrm{eff}}<c with WEC and NEC intact, finite ADM mass, and a matter shell plus Φ\Phi\Phi that can in principle be assembled from known fields. This is the only regime close to existing published constructions.II. Superluminal trial. Increase R\mathcal{R}\mathcal{R} through Rc\mathcal{R}c\mathcal{R}c and/or increase vwv_wv_w . If veffv{\mathrm{eff}}v{\mathrm{eff}} crosses (c) only when min Tμνkμkν\min T_{\mu\nu}k^\mu k^\nu\min T_{\mu\nu}k^\mu k^\nu becomes negative, SSW-1 has not removed the standard obstruction. That outcome is the expected default.No claim is made that Regime II exists.7. Weak-field laboratory limitWrite gμν=ημν+hμνg_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu} , ∣h∣≪1|h|\ll 1|h|\ll 1 . Linearized Einstein equations source hμνh_{\mu\nu}h_{\mu\nu} from Tμνm+TμνΦ+TμνFT_{\mu\nu}^{\mathrm{m}}+T_{\mu\nu}^{\Phi}+T_{\mu\nu}^{\mathcal{F}}T_{\m u\nu}^{\mathrm{m}}+T_{\mu\nu}^{\Phi}+T_{\mu\nu}^{\mathcal{F}} .The ordinary gravitational field of a lab energy density ρ\rho\rho

at distance (r) is∣h∣∼2Gρ Vc2r∼10−27(ρ V1 kg c2)(1 mr)|h|\sim\frac{2G\rho,\mathcal{V}}{c^2 r} \sim 10^{-27} \left(\frac{\rho,\mathcal{V}}{1,\mathrm{kg},c^2}\right) \left(\frac{1,\mathrm{m}}{r}\right)|h|\sim\frac{2G\rho,\mathcal{V}}{c^2 r} \sim 10^{-27} \left(\frac{\rho,\mathcal{V}}{1,\mathrm{kg},c^2}\right) \left(\frac{1,\mathrm{m}}{r}\right) in order of magnitude for compact sources (rest energy of a kilogram). Any claim that “coherence” produces a larger (h) requires F\mathcal{F}\mathcal{F} to amplify TμνFT_{\mu\nu}^{\mathcal{F}}T_{\mu\nu}^{\mathcal{F}} far above the scalar’s own energy density. That amplification is bounded by existing tests unless it is arranged to vanish in the Solar System and appear only in a tuned lab geometry — which must be stated and justified.Useful observables, all standard:atom-interferometer phase Δφ\Delta\phi\Delta\phi

clock comparison Δν/ν∼12h00+...\Delta\nu/\nu\sim \tfrac12 h_{00}+\ldots\Delta\nu/\nu\sim \tfrac12 h_{00}+\ldots

optical delay Δt\Delta t\Delta t

Sagnac / one-way h0ih_{0i}h_{0i} phase

Predicted SSW-1 deviation from GR+ordinary matter:δhμν=hμν[Tm+TΦ+TF]−hμν[Tm+TΦ]F=0.\delta h_{\mu\nu}

h_{\mu\nu}\bigl[T^{\mathrm{m}}+T^{\Phi}+T^{\mathcal{F}}\bigr]

h_{\mu\nu}\bigl[T^{\mathrm{m}}+T^{\Phi}\bigr]{\mathcal{F}=0}.\delta h{\mu\nu}

h_{\mu\nu}\bigl[T^{\mathrm{m}}+T^{\Phi}+T^{\mathcal{F}}\bigr]

h_{\mu\nu}\bigl[T^{\mathrm{m}}+T^{\Phi}\bigr]{\mathcal{F}=0}. The null hypothesis is δhμν=0\delta h{\mu\nu}=0\delta h_{\mu\nu}=0 once the ordinary TμνT_{\mu\nu}T_{\mu\nu} of the apparatus is subtracted.A numerical prediction requires SI values of {μ,η,p,Rc,k,N∗,m,V}{\mu,\eta,p,\mathcal{R}c,k,N,m,V}{\mu,\eta,p,\mathcal{R}c,k,N,m,V
} . Without those, the model is not yet falsifiable in the laboratory.8. Diagnostic plotsThe two plots that decide the model:veff/cv_{\mathrm{eff}}/cv_{\mathrm{eff}}/c versus max R/Rc\max\mathcal{R}/\mathcal{R}_c\max\mathcal{R}/\mathcal{R}_c

min observersTμνkμkν\displaystyle\min_{\text{observers}}T_{\mu\nu}k^\mu k^\nu\displaystyle\min_{\text{observers}}T_{\mu\nu}k^\mu k^\nu on the same axis

Secondary diagnostics: ADM mass, peak ∣Φ∣|\Phi||\Phi| , constraint violation ∥ΔT∥|\Delta T||\Delta T| , wall compactness σRb\sigma R_b\sigma R_b .If the first curve crosses (1) exactly where the second curve crosses (0), SSW-1 is a dressed restatement of the Alcubierre obstruction.9. Numerical programRestrict first to axial symmetry, constant coordinate velocity, α=1\alpha=1\alpha=1 , γij=δij\gamma_{ij}=\delta_{ij}\gamma_{ij}=\delta_{ij} only as an initial guess; then relax α\alpha\alpha and γij\gamma_{ij}\gamma_{ij} .Parameters:{Rb,σ,vw,m,V0,μ,η,p,Rc,k,N∗}.{R_b,\sigma,v_w,m,V_0,\mu,\eta,p,\mathcal{R}c,k, N}.{R_b,\sigma,v_w,m,V_0,\mu,\eta,p,\mathcal{R}c,k,N}. Loop:Choose a trial βi\beta^i\beta^i and a trial Φ\Phi\Phi . Compute KijK_{ij}K_{ij} , (N), R\mathcal{R}\mathcal{R} , F\mathcal{F}\mathcal{F} . Compute GμνG_{\mu\nu}G_{\mu\nu} , TΦT^{\Phi}T^{\Phi} , TFT^{\mathcal{F}}T^{\mathcal{F}} , optional shell TmT^{\mathrm{m}}T^{\mathrm{m}} . Measure ΔTμν\Delta T_{\mu\nu}\Delta T_{\mu\nu} . Relax Φ\Phi\Phi with the scalar equation; relax α,β,γ\alpha,\beta,\gamma\alpha,\beta,\gamma with the constraints (or use a 3+1 GR solver). Evaluate energy conditions and ADM mass. Scan η\eta\eta and R/Rc\mathcal{R}/\mathcal{R}_c\mathcal{R}/\mathcal{R}_c .

Level of success:Level Criterion 0 Code conservation: constraints stable, ΔT→0\Delta T\to 0\Delta T\to 0 numerically 1 Smooth subluminal solution, finite energy

2 Level 1 plus WEC/NEC for sampled observers 3 veff>cv_{\mathrm{eff}}>cv_{\mathrm{eff}}>c with finite energy and intact NEC

Level 3 is not expected. Finding it would require independent reproduction and a covariant completion of the KijKijK_{ij}K^{ij}K_{ij}K^{ij} term.10. What was discarded from v0.1, and whyv0.1 SSW-1 Universal Intent Field Real scalar Φ\Phi\Phi

IPT Regulator with φintent2(1+0.06S)\phi_{\mathrm{intent}}^2(1+0.06S)\phi_{\mathrm{intent}}^2(1+0.06S)

Bounded R=1−e−kN/N∗\mathcal{R}=1-e^{-kN/N_}\mathcal{R}=1-e^{-kN/N_}

βx=vwf W(R)\beta^x=v_w f,\mathcal{W}(R)\beta^x=v_w f,\mathcal{W}(R) by hand Shift determined by constraints; F(R)\mathcal{F}(\mathcal{R})\mathcal{F}(\mathcal{R}) in the action Nonlocal kernel left unspecified Omitted until a causal kernel is written 6% surge as a sharp test Removed; an ε\varepsilon\varepsilon -correction is allowed only if derived ∂g/∂R≠0\partial g/\partial R\neq 0\partial g/\partial R\neq 0 as a postulate True only if F≠0\mathcal{F}\neq 0\mathcal{F}\neq 0 and ΔT=0\Delta T=0\Delta T=0 both hold FTL as an open hope attached to saturation FTL remains the standard obstruction unless Level 3 is actually solved

Saturation is kept because it is a legitimate effective-field idea: a coupling that cannot run past a finite value. It is not a mechanism that manufactures negative energy or a horizon-free shortcut.11. Falsifiable statementAfter {μ,η,p,Rc,N∗}{\mu,\eta,p,\mathcal{R}c,N}{\mu,\eta,p,\mathcal{R}c,N} and a concrete source for Φ\Phi\Phi are fixed:Relative to general relativity sourced by the same ordinary matter and the same canonical scalar, SSW-1 predicts a specific additional metric perturbation δhμν\delta h_{\mu\nu}\delta h_{\mu\nu} localized where R\mathcal{R}\mathcal{R}

is appreciable the predicted amplitude lies below existing gravimetry and interferometry for all parameters that also satisfy Solar-System bounds, the model is not laboratory-testable with current methods; it then survives only as a numerical GR exercise.12. One-paragraph summarySSW-1 is general relativity plus a canonical scalar plus a saturating, foliation-tied coupling to extrinsic curvature. Warp-like metrics are trial geometries, not definitions. The theory lives or dies on ΔTμν=0\Delta T_{\mu\nu}=0\Delta T_{\mu\nu}=0 , energy conditions, and a finite ADM budget. Subluminal, positive-energy configurations are the only serious target. Superluminal solutions are not assumed and are not made more likely by saturation. “Intent” is not part of the model.That is the version I would work with. If you want the next artifact, it should be a constraint-evaluation notebook for the α=1\alpha=1\alpha=1 , γij=δij\gamma_{ij}=\delta_{ij}\gamma_{ij}=\delta_{ij} slice: compute GμνG_{\mu\nu}G_{\mu\nu} and TμνΦT_{\mu\nu}^{\Phi}T_{\mu\nu}^{\Phi} for a given wall and a given Φ(r)\Phi(r)\Phi(r) , and show ΔT\Delta T\Delta T before any talk of W(R)\mathcal{W}(R)\mathcal{W}(R) .