Einstein Field Equations for a Static Lapse Mapping (TFH Path B)

Einstein Field Equations for a Static Lapse Mapping (TFH Path B) Eric Levin Independent Researcher [email protected] Phone: 617.283.4468 November 17, 2025 Abstract We present a first-principles derivation of the Einstein field equations (EFEs) for a static, spherically symmetric spacetime, emphasizing mixed components suitable for direct comparison with physical density and pressure. Within this geometric setting we articulate a purely observational “cosmic lapse” mapping, γ ( r ), that modifies the relation between the observed redshift and the underlying static metric without altering the EFEs themselves (“Path B”). This mapping is motivated by the operational distinction between clock rates in different environments and provides a kinematic handle that can mimic dark-energy–like effects at low redshift. We establish explicit relations between derivatives of an effective potential and the standard cosmographic parameters ( H0, q0, j0 ), thereby yielding testable lowz predictions. We also contrast this observational mapping with alternatives where γ is part of the physical metric (“Path A”), and briefly discuss connections to Wiltshire’s Timescape framework. The formalism is designed to be falsifiable against Hubble-diagram data (e.g. Pantheon/Pantheon+), and it clarifies where additional physics (beyond pressureless fluids) would be required. 1 Conventions and Scope We work in a metric theory of gravity with: •Metric signature (−,+,+,+). • Speed of light c retained explicitly; Newton’s constant G appears with the usual 8 πG/c4 prefactor in the EFEs. •Einstein equations in covariant form Gµν =8πG c4Tµν,(1) with Gµν the Einstein tensor and Tµν the stress–energy tensor. • Greek indices µ, ν, . . . run over ( t, r, θ, ϕ ). We consider only diagonal configurations and a perfect fluid at rest in the chosen static coordinates. Throughout this paper we assume a static, spherically symmetric geometry and restrict attention to the low-redshift regime where null geodesics remain well behaved (no caustics) and the angular diameter distance is monotonic along the line of sight. Path B, our primary focus here, modifies the redshift mapping but does not change the EFEs themselves. 1 2 Metric, Christoffel Symbols, and Ricci Tensor We assume a static, spherically symmetric metric with signature (−,+,+,+), ds2=−e2ψ(r)c2dt2+e2Λ(r)dr2+r2dθ2+ sin2θ dϕ2,(2) where ψ ( r ) and Λ( r ) are dimensionless functions of the area radius r . Nonvanishing Christoffel symbols (primes denote d/dr) are Γttr =ψ′,Γrtt =c2ψ′e2ψ−2Λ,Γrrr = Λ′, Γrθθ =−re−2Λ,Γrϕϕ =−rsin2θ e−2Λ,Γθrθ =1 r, Γϕrϕ =1 r,Γθϕϕ =−sin θcos θ, Γϕθϕ = cot θ. (3) The Ricci tensor components follow by contraction (see, e.g., standard GR texts): Rtt =c2e2ψ−2Λψ′′ +ψ′2−ψ′Λ′+2ψ′ r,(4) Rrr =−ψ′′ −ψ′2+ψ′Λ′+2Λ′ r,(5) Rθθ =e−2Λr(Λ′−ψ′)−1+ 1, Rϕϕ =Rθθ sin2θ. (6) 3 Mixed-Component EFEs and TOV Structure Let Gµν = Rµν−1 2δµνR be the mixed Einstein tensor. In the metric (2) the nonzero diagonal components are Gtt=e−2Λ r22rΛ′−1 + e2Λ,(7) Grr=e−2Λ r2−2rψ′−1 + e2Λ,(8) Gθθ=Gϕϕ=e−2Λψ′′ +ψ′2−ψ′Λ′+ψ′−Λ′ r.(9) For a perfect fluid at rest in these coordinates, Tµν= diag(−ρc2, p, p, p),(10) the EFEs Gµν=8πG c4Tµνyield 8πG c2ρ(r) = e−2Λ r22rΛ′−1 + e2Λ,(11) 8πG c4p(r) = e−2Λ r22rψ′−1 + e2Λ,(12) together with hydrostatic balance from ∇µTµν = 0, p′(r) = −ρc2+pψ′(r).(13) Eqs. (11) – (13) are the TOV-like structure adapted to the static metric (2) . They close once an equation of state (EoS) p=p(ρ) is specified. 2 Derivation of hydrostatic balance. For completeness, one may project ∇µTµν = 0 along the static fluid 4-velocity uµ = ( e−ψ, 0 , 0 , 0) and in the radial direction. The r -component yields ∂rp+ (ρc2+p)∂rψ= 0, which is Eq. (13). This confirms that the force balance is governed by ψ′in the static case. Regularity and Physical Domain Regularity at the center requires e2Λ →1,Λ′(0) = 0, ψ(0) finite,(14) so that Gµν and hence ρ, p remain finite as r→ 0. It is also convenient to introduce the Misner–Sharp mass function M(r) via e−2Λ(r)= 1 −2GM(r) c2r,(15) implying the standard closure M′(r) = 4πr2ρ(r).(16) Physical configurations require 0≤2GM(r) c2r<1∀r, (17) to avoid trapped surfaces within the domain of interest. 4 Null Geodesics and Gravitational Redshift In a static metric, the timelike Killing vector ∂t leads to energy conservation along photon geodesics. For emission at radius r and observation at r = 0, one obtains the purely gravitational (static) redshift 1+z=νem νobs =eψ(r) eψ(0) = exp ψ(r)−ψ(0).(18) Differentiating with respect to rgives dz dr = (1 + z)ψ′(r),dr dz =1 (1+z)ψ′(r).(19) In this baseline (no extra fields), the slope ψ′(r) directly controls the z(r) relation. Area Distance and Luminosity Distance The area radius r in Eq. (2) is defined so that 2-spheres of constant ( t, r ) have geometric area 4 πr2 . For a bundle of radial null geodesics intersecting such a sphere, the angular diameter distance is therefore dA(r) = r, (20) provided there are no caustics along the ray in the lowz regime considered here and the center is regular. In any metric theory with photon number conservation and standard light propagation, Etherington’s reciprocity relation holds: dL= (1 + z)2dA.(21) 3 Combining Eqs. (20) and (21) yields dL(z) = (1 + z)2r(z),(22) so that once the null path r7→ z is known from Eq. (19) , the luminosity distance is fixed. An overall multiplicative constant in dL can be absorbed into calibration (e.g. via Cepheids or absolute SN magnitudes). 5 An Observational Cosmic Lapse Mapping (Path B) We now introduce a phenomenological lapse mapping γ ( r ) > 0 that rescales the operational rate at which an observer would infer times, without modifying the EFEs themselves. Define the effective potential ψeff(r):=ψ(r) + ln γ(r).(23) The observed redshift is then 1+z= exp ψeff(r)−ψeff (0),(24) implying the differential law dz dr = (1 + z)ψ′ eff(r) = (1 + z)ψ′(r) + d dr ln γ(r).(25) Crucially, Eq. (25) affects only the inference of z ( r ) from data; the dynamical content of Eqs. (11) – (13) is left intact. This “Path B” choice is deliberately conservative: it tests whether an observational lapse can mimic dark-energy–like kinematics in Hubble-diagram data, before promoting γto a dynamical field inside the EFEs. Degeneracy and identifiability. Because ψeff = ψ + ln γ enters the redshift mapping, there is a degeneracy between the underlying metric potential ψ and the observational lapse ln γ at the level of kinematics. In Path B we treat ψ as the metric potential determined by the EFEs, and γ as a phenomenological mapping that alters how observers interpret redshifts. Geometry (and thus curvature) is still fixed by ψ, Λ via Eqs. (7) – (9) ; the lapse γ changes only the z ( r ) relation and hence the inferred kinematics. 6 Small-zExpansion and Cosmography Define the effective potential Ψ(r)≡ψeff(r) = ψ(r) + ln γ(r),(26) and expand around r= 0: Ψ(r) = Ψ0+ Ψ1r+1 2Ψ2r2+1 6Ψ3r3+O(r4).(27) From Eq. (25), integrating term-by-term yields (to cubic order in r) z(r) = Ψ1r+1 2Ψ2+ Ψ2 1r2+1 6Ψ3+ 3Ψ1Ψ2+ Ψ3 1r3+O(r4).(28) Inverting perturbatively to obtain r ( z ) and using the static reciprocity dL ( z ) = (1 + z ) 2r ( z ) from Eq. (22), one recovers the standard low-zluminosity-distance series H0 cdL(z) = z+1−q0 2z2−1−q0−3q2 0+j0 6z3+O(z4),(29) 4 with kinematic parameters ( H0, q0, j0 ) encoded by derivatives of Ψ. Matching coefficients identifies H0∝Ψ1, q0= 1 −Ψ2 Ψ2 1 , j0= 1 + Ψ3 Ψ3 1 −3Ψ2 Ψ2 1 + 3 Ψ2 Ψ2 12 .(30) Thus any nontrivial γ ( r ) alters Ψ k and therefore ( q0, j0 ), even if the underlying metric potential ψwould have produced the flat-ΛCDM values. Remark on identifiability. Because the overall distance scale is fixed observationally (e.g. by Cepheid/SN calibration), Ψ 1 (and hence H0 ) is set by lowz anchors; higher derivatives (Ψ 2, Ψ 3 ) control ( q0, j0 ) and are constrained by the Hubble-diagram curvature. This offers a clean, falsifiable route to testing lapse hypotheses: any proposed γ ( r ) must induce Ψ( r ) whose derivatives reproduce the observed (q0, j0) within uncertainties. 7 Path A (for Completeness): Embedding γin the Metric If instead one postulates a physical lapse that modifies the metric time factor, gtt → −γ2(r)e2ψ(r)c2,(31) then ψ→ψ + ln γ within the EFEs themselves. All occurrences of ψ′ and ψ′′ in Eqs. (11) – (9) must then be replaced by ψ′→ψ′+ (ln γ)′, ψ′′ →ψ′′ + (ln γ)′′ +(ln γ)′2+ 2ψ′(ln γ)′.(32) This is a stronger—and riskier—assumption: it changes the source terms required by Eqs. (11) – (12) . We focus on Path B in this paper to isolate the kinematic question first. Path A demands additional stress–energy (or a specific Lagrangian for a time field) to remain physically viable. 8 Observational Strategy and Model Testing We summarize a minimal observational programme for testing the Path B lapse mapping. 1. Lowz cosmography. Fit Eq. (29) to calibrated SNe with z≲ 0 . 3 and infer ( H0, q0, j0 ); map the results to (Ψ 1, Ψ 2, Ψ 3 ) via Eq. (30) . Evidence for j0 = 1 at high significance signals kinematic deviations from flat ΛCDM or supports a nontrivial lapse. 2. Nonparametric Ψ( r )from data. Reconstruct r ( z ) from the Hubble diagram using a monotonic spline; then compute Ψ′(r) = d dr ln(1 + z(r)),(33) and integrate to obtain Ψ( r ) up to an additive constant. Comparing with models for ψ ( r ) from Eqs. (11)–(13), any residual can be attributed to ln γ(r) under Path B. 3. Consistency with EFEs. If one adopts Path A, use Eqs. (11) – (13) with the replacements in Eq. (32) to recover ( ρ ( r ) , p ( r )) and check physicality (positivity, reasonable EoS, subluminal sound speed). Failures here strongly disfavour Path A. 5 9 Relation to Timescape (Wiltshire) Wiltshire’s Timescape framework distinguishes regional clock rates (e.g., voids vs. walls) in an inhomogeneous expanding universe via a phenomenological lapse ¯γ ( t, x ) tied to volume partitioning and backreaction. Our Path B is philosophically adjacent—a controlled, observational lapse applied to the redshift mapping—but technically distinct: we do not modify the EFEs nor invoke Buchert averaging in this paper. Nevertheless, the notion that differing clocks can systematically bias inferred kinematics is shared. We therefore cite Timescape as motivation for considering lapse effects while keeping the present analysis static and spherically symmetric. 10 Discussion and Outlook We have isolated a minimal, falsifiable hypothesis: a static metric governed by Eq. (2) and EFEs (11) – (13) , combined with an observational lapse mapping (23) – (25) . The lapse alters the effective potential derivatives (Ψ 1, Ψ 2, Ψ 3 ) and therefore ( H0, q0, j0 ) at low z , enabling darkenergy–like kinematics without immediately changing the dynamical sources. Three outcomes are possible when confronted with data: 1. A small, smooth γ ( r ) improves the Hubble-diagram fit and yields sensible ( H0, q0, j0 ); Path B remains viable. 2. No reasonable γ ( r ) can reconcile the data; Path B is disfavoured and one must either adopt Path A (with new stress–energy/EoS) or abandon staticity. 3. The inferred ( ρ, p ) under Path A are unphysical (e.g., negative densities, superluminal sound speeds), arguing against embedding γin the metric. Future work includes introducing mild radial pressure or anisotropy, exploring Pad´e approximants for Ψ( r ) to stabilize higher-order cosmography, and confronting the framework with joint probes (BAO, strong lensing time delays, and local distance ladders). A quasi-static extension with explicit time dependence in the lapse (A2) is treated in a companion note; together, these documents provide the formal backbone for testing the Time Field Hypothesis against lowz Hubble-diagram data and, eventually, redshift-drift measurements. Acknowledgements. The author thanks the broader community for open supernova compilations and constructive debate around alternative kinematic interpretations of the Hubble diagram. 6 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. 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