Quasi-Static Lapse Dynamics and Redshift Drift (TFH Path A2)

Quasi-Static Lapse Dynamics and Redshift Drift (TFH Path A2) Eric Levin Independent Researcher [email protected] Phone: 617.283.4468 November 17, 2025 Abstract We develop the quasi-static extension (“A2”) of the Time-Field Hypothesis (TFH), introducing minimal time dependence in the temporal potential of a static, spherically symmetric geometry. This yields a local energy-balance equation linking matter thermodynamics to the time variation of the lapse, and a falsifiable redshift-drift prediction. We present the full Einstein-equation system to leading quasi-static order, derive hydrostatic and energy-balance relations, and provide the exact redshift-drift observable. A small-amplitude, separable parametric deformation Φ( r, t ) = Φ 0 ( r ) + ϵf ( r ) h ( t ) gives a light-touch, phenomenologically testable extension of the static TFH baseline. This note is part of the three-paper EFE derivation trilogy (Path B, Path A, Path A2) supporting the TFH low-zanalysis. 1 Conventions and Quasi-Static Ordering We adopt: •Metric signature (−,+,+,+). •Explicit cand EFEs Gµν =8πG c4Tµν. •Perfect fluid at rest in these coordinates: Tµν= diag(−ρc2, p, p, p). The metric is ds2=−e2Φ(r,t)c2dt2+e2Λ(r,t)dr2+r2dΩ2.(1) Quasi-static assumptions: |∂tΦ|,|∂tΛ|≪|∂rΦ|, O(∂2 t) neglected.(2) These orderings will be explicitly indicated in the Einstein tensor. 1 2 Einstein Tensor to Leading Quasi-Static Order Inserting Φ(r, t) and Λ(r, t) into the standard static formulas yields: Gtt=e−2Λ r2(2rΛ′−1 + e2Λ)+O(∂2 t),(3) Grr=e−2Λ r2(−2rΦ′−1 + e2Λ)+O(∂2 t),(4) Gθθ=e−2ΛΦ′′ + Φ′2−Φ′Λ′+Φ′−Λ′ r+O(∂2 t).(5) Define the Misner–Sharp mass: e−2Λ(r,t)= 1 −2GM(r, t) c2r.(6) Then from Gtt: M′(r, t) = 4πr2ρ(r, t).(7) 3 Hydrostatic and Local Energy Balance 3.1 Hydrostatic balance ∇µTµr = 0 gives: p′=−(ρc2+p)Φ′.(8) 3.2 Local energy balance Project ∇µTµν = 0 along uν, where uµ= (e−Φ,0,0,0). We obtain: ∂tρ=−(ρc2+p)∂tΦ.(9) For p=wρc2: ∂tln ρ=−(1 + w)∂tΦ.(10) This serves as the quasi-static “Friedmann-like” relation of TFH. 4 Redshift and Redshift Drift On each quasi-static timeslice: ∂rln(1 + z)=Φ′(r, t),(11) evaluated along the past light cone. 4.1 Redshift drift For emission at (rem, tem) and observation at (0, t0) on the same null ray: ˙z= (1 + z) [∂tΦ(rem, tem)−∂tΦ(0, t0)]+O(∂tΛ).(12) The ∂tΛ term is suppressed by the quasi-static ordering (2). 2 4.2 Distance duality To first order in quasi-static evolution, the spatial slice obeys: dA=r, dL= (1 + z)2dA= (1 + z)2r. 5 Small-Amplitude A2 Deformation We propose: Φ(r, t) = Φ0(r)+ϵf(r)h(t),0< ϵ ≪1.(13) Example: f(r)=1−e−r/r⋆, h(t) = 1 −e−t/τ . Then: ∂tΦ=ϵf(r)˙ h(t),˙ρ=−(ρc2+p)ϵf(r)˙ h(t). The sign of ˙zin (12) follows from the sign of f(r)˙ h(t). 6 Regularity and Positivity Domain Regularity at r= 0 requires: e2Λ →1,Λ′(0) = 0,Φ(0, t) finite. From (6), geometry requires: 0≤2GM(r, t) c2r<1.(14) 7 Interface to Low-zData Data handshake. Φ 0 ( r ) is reconstructed from the TFH lowz Hubble-diagram analysis via r(z)∝dL(z)/(1 + z)2. The A2 parameters (ϵ, r⋆, τ) are then constrained by residual curvature and the redshift-drift prediction (12). 8 Discussion The A2 framework adds a controlled, falsifiable degree of freedom: time variation in the lapse. It preserves the static TFH spatial geometry while generating a measurable drift signal. SN data constrain Φ 0 ( r ); drift observations constrain ∂t Φ. A2 therefore offers an observational discriminator between TFH and FRW-based expansion. 3 Appendix: Unified Symbol Table for TFH EFE Trilogy Symbol Meaning / Definition gµν Metric tensor (signature −+ ++). ds2Line element of static or quasi-static spherical metric. cSpeed of light (kept explicit throughout). GNewton’s gravitational constant. TµνStress–energy tensor of perfect fluid. ρ(r, t) Energy density. p(r, t) Isotropic pressure. uµ4-velocity of static observers: (e−Φ,0,0,0). rAreal radius (defines 4πr2). θ, ϕ Angular coordinates. dΩ2Angular line element: dθ2+ sin2θ dϕ2. ψ(r) Baseline static temporal potential (pre-lapse). γ(r) Observational or embedded lapse factor. Φ(r, t) Effective temporal potential: Φ = ψ+ ln γ. Λ(r, t) Radial metric potential. ′Radial derivative: ∂r. ˙ Time derivative: ∂t. M(r, t) Misner–Sharp mass: e−2Λ = 1 −2GM/(rc2). M′(r, t) Mass closure: M′= 4πr2ρ. GµνMixed Einstein tensor components. Rµν Ricci tensor components. RRicci scalar. zGravitational redshift. ˙zRedshift drift (dz/dt0). dAAngular diameter distance: dA=r. dLLuminosity distance: dL= (1 + z)2r. f(r) Radial envelope for A2 perturbation. h(t) Temporal profile for A2 perturbation. ϵAmplitude of quasi-static deviation (ϵ≪1). r⋆Radial scale of A2 envelope. τTime scale of A2 evolution. H0Local Hubble parameter (from cosmographic expansion). q0Deceleration parameter. j0Jerk parameter. References [1] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, 1973). [2] R. M. Wald, General Relativity, University of Chicago Press (1984). [3] I. M. H. Etherington, “On the Definition of Distance in General Relativity,” Phil. Mag. 15, 761 (1933). 4