Einstein Field Equations with an Embedded Lapse (TFH Path A)

Einstein Field Equations with an Embedded Lapse (TFH Path A) Eric Levin Independent Researcher [email protected] Phone: 617.283.4468 November 17, 2025 Abstract We derive the Einstein field equations (EFEs) for a static, spherically symmetric metric in which an observational lapse γ ( r ) is embedded directly in the temporal metric coefficient, gtt → −γ2 ( r ) e2ψ(r)c2 . This “Path A” construction alters the geometric source terms by replacing ψ→ Φ ≡ψ + ln γ , modifying all occurrences of ψ′ and ψ′′ in the EFEs. We provide the explicit mapping ψ→ Φ through the Christoffel symbols, Ricci components, and mixed Einstein tensor; derive the corresponding hydrostatic-balance relation and mass-closure equation; and state the regularity and positivity-domain conditions required for physical sources. We also clarify the degeneracy between metric potential ψ and embedded lapse γ , which affects identifiability unless additional microphysics is specified. This document is one of three companion derivation notes supporting the TFH lowz analysis and the quasi-static A2 extension. 1 Conventions and Scope Throughout we adopt: •Metric signature (−,+,+,+). •Speed of light cretained explicitly; Newton’s constant Gappears in the EFEs as Gµν =8πG c4Tµν.(1) •Static, spherically symmetric geometry with area radius r. •Matter modeled as a perfect fluid at rest in these coordinates: Tµν= diag(−ρc2, p, p, p). We consider the embedded-lapse hypothesis: a smooth radial lapse factor γ ( r ) > 0 modifies the temporal metric coefficient inside the EFEs. This sharply contrasts with “Path B,” where lapse effects modify only the redshift inference without changing the EFEs. 2 Metric and Effective Temporal Potential The Path A metric is ds2=−γ2(r)e2ψ(r)c2dt2+e2Λ(r)dr2+r2dΩ2, dΩ2=dθ2+ sin2θ dϕ2.(2) 1 It is convenient to define the effective potential Φ(r)≡ψ(r) + ln γ(r),(3) so that gtt =−e2Φ(r)c2. All EFEs derived below depend only on Φ and Λ. This captures the central degeneracy of Path A: geometry cannot distinguish between choices of ψand γthat yield the same Φ. 3 Geometry: Christoffel Symbols and Ricci Tensor With primes denoting ∂r , the nonzero Christoffel symbols for the metric (2) are identical to the usual static case with ψ→Φ: Γttr = Φ′,Γrtt =c2Φ′e2(Φ−Λ),Γrrr = Λ′, Γrθθ =−re−2Λ,Γrϕϕ =−rsin2θ e−2Λ,Γθrθ =1 r,Γϕrϕ =1 r.(4) The Ricci components follow directly (standard static formulas with Φ): Rtt =c2e2(Φ−Λ)Φ′′ + Φ′2−Φ′Λ′+2Φ′ r,(5) Rrr =−Φ′′ −Φ′2+ Φ′Λ′+2Λ′ r,(6) Rθθ =e−2Λ(r(Λ′−Φ′)−1) + 1,(7) and Rϕϕ =Rθθ sin2θ. 4 Einstein Equations with Embedded Lapse The mixed Einstein tensor Gµνfor (2) has diagonal components: Gtt=e−2Λ r2(2rΛ′−1 + e2Λ),(8) Grr=e−2Λ r2(−2rΦ′−1 + e2Λ),(9) Gθθ=Gϕϕ=e−2ΛΦ′′ + Φ′2−Φ′Λ′+Φ′−Λ′ r.(10) Equating these with 8πG c4Tµνyields: 8πG c2ρ=e−2Λ r2(2rΛ′−1 + e2Λ),(11) 8πG c4p=e−2Λ r2(2rΦ′−1 + e2Λ),(12) with angular components automatically consistent via Bianchi identities. 2 5 Hydrostatic Balance and Mass Closure Because Φ appears in gtt, the force-balance equation for a perfect fluid becomes: p′=−(ρc2+p)Φ′.(13) This follows from ∇µTµr = 0. Explicitly: 0=∂rp+ (ρc2+p)∂rΦ. Introduce the Misner–Sharp mass via: e−2Λ = 1 −2GM(r) c2r.(14) Then the Gttequation gives the standard closure: M′(r) = 4πr2ρ(r).(15) 6 Regularity and Physical Domain Regularity at the center requires: e2Λ →1,Λ′(0) = 0,Φ(0) finite.(16) The Misner–Sharp mass must satisfy: 0≤2GM(r) c2r<1∀r, (17) to avoid trapped surfaces. These ensure ρ and p remain finite and the geometry is well behaved. 7 Gravitational Redshift and Distance Duality Because gtt =−e2Φc2, the gravitational redshift between rand the origin is: 1+z= exp[Φ(r)−Φ(0)].(18) The area distance dA = r follows from the definition of the area radius (regular center and no caustics at low z). Etherington reciprocity then gives: dL= (1 + z)2dA= (1 + z)2r. (19) Thus an embedded lapse modifies redshift via Φ and changes the inferred luminosity distance through the Φ-dependent null mapping. 8 Degeneracy Between ψand γ Because Φ = ψ + ln γ enters the EFEs, geometry depends only on Φ. Different splits ( ψ, γ ) that yield the same Φ produce the same curvature. This degeneracy is broken only if: •γ(r) is tied to additional microphysics or EoS assumptions, or • one adopts a dynamical extension (A2) where ∂t Φ becomes observable through redshift drift, or •independent probes constrain local clock-rate gradients. Path A therefore represents a stronger and more model-dependent hypothesis than Path B. 3 9 Companion Derivation Notes and Data Pointer This Path A note is one of three documents forming the EFE backbone for the TFH lowz analysis: 1. Path B (Static Baseline) — static EFEs, mass closure, hydrostatic balance, redshift, and dL= (1 + z)2r. 2. Path A (this document) — embedded lapse modifies geometric source terms via ψ→ Φ. 3. Path A2 (Quasi-Static) — introduces slow time dependence in Φ and derives the redshift-drift equation. All numerical analyses for the lowz paper and the A2 extension (including the PCHIP reconstruction, residual curvature tests, and redshift-drift forecasts) are available via the Zenodo repository accompanying the TFH low-zsubmission. 4 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] R. C. Tolman, Phys. Rev. 55, 364 (1939). [4] J. R. Oppenheimer & G. M. Volkoff, Phys. Rev. 55, 374 (1939). [5] I. M. H. Etherington, Phil. Mag. 15, 761 (1933). 5