Toy metric for an old looking young universe

Abstract

This paper proposes a gravitational-time-dilation–based explanation for the observed disparity between cosmological age indicators and the apparent youth of galactic structures. By introducing a “toy metric” incorporating a radially dependent potential and hypothesized gravity leakage from subspace, we derive a first-order relationship between time flow and spatial position within a bounded, galaxy-scale frame. The result offers a mechanism by which peripheral regions of the universe may appear older than they truly are, providing a natural reconciliation between observed galactic maturity and a younger absolute cosmological age.

1. Introduction

Conventional cosmology assumes a uniform flow of time throughout the expanding universe. Yet observational evidence—such as unexpectedly mature galaxies at extreme redshifts—suggests that local time dilation effects could have a more profound cosmological role.

Here we explore a simplified, weak-field metric in which gravitational potential gradients across cosmic scales cause differential time flow. Extending classical potential theory, we introduce an additional subspace leakage term representing gravity that escapes from normal spacetime into an external manifold, analogously to dark matter effects.

2. Theoretical Basis

2.1 Weak-Field Gravitational Time Dilation

For slowly varying potentials, the proper-time increment dτd\taudτ relates to coordinate time dtdtdt as:

dτ≈dt(1+2Φ(r)c2),(∣Φ∣≪c2)d\tau \approx dt \left(1 + \frac{2\Phi(r)}{c^2}\right), \quad (|\Phi| \ll c^2)dτ≈dt(1+c22Φ(r)​),(∣Φ∣≪c2)

where Φ\PhiΦ is the Newtonian potential. Deeper potentials slow local clocks.

2.2 Potential Within a Uniform Spherical Mass

Assume a uniform-density region of comoving radius RRR containing effective gravitating mass MMM. The potential is:

Φ(r)=−GM2R3(3R2−r2),0≤r≤R\Phi(r) = -\frac{GM}{2R^3}(3R^2 - r^2), \quad 0 \le r \le RΦ(r)=−2R3GM​(3R2−r2),0≤r≤R

The central potential is Φ(0)=−3GM/(2R)\Phi(0) = -3GM/(2R)Φ(0)=−3GM/(2R). The local clock rate relative to the center is then:

γ(r)=1+2Φ(r)c21+2Φ(0)c2=1−GMRc2(3−r2R2)1−3GMRc2\gamma(r) = \frac{1 + \frac{2\Phi(r)}{c^2}}{1 + \frac{2\Phi(0)}{c^2}} = \frac{1 - \frac{GM}{Rc^2}\left(3 - \frac{r^2}{R^2}\right)}{1 - \frac{3GM}{Rc^2}}γ(r)=1+c22Φ(0)​1+c22Φ(r)​​=1−Rc23GM​1−Rc2GM​(3−R2r2​)​

For small compactness μ=GM/(Rc2)≪1\mu = GM/(Rc^2) \ll 1μ=GM/(Rc2)≪1,

γ(r)≈1+μ(r22R2−1)\gamma(r) \approx 1 + \mu\left(\frac{r^2}{2R^2} - 1\right)γ(r)≈1+μ(2R2r2​−1)

Hence, clocks accelerate outward, giving

Δγ≈μ2\Delta \gamma \approx \frac{\mu}{2}Δγ≈2μ​

between the center and the edge.

3. Apparent Age Gradient

Let the central proper time since creation be T0T_0T0​. At radius rrr:

T(r)=γ(r)T0T(r) = \gamma(r) T_0T(r)=γ(r)T0​

Thus the same cosmic event appears older when viewed from the periphery and younger near the center—a natural “old-looking young universe.”

4. Incorporating Subspace Leakage

We postulate a distributed “gravity leakage” density ρℓ(r)\rho_\ell(r)ρℓ​(r) representing gravitational influence bleeding into adjacent subspace regions (dark-matter analogue). The effective enclosed mass becomes:

Meff(r)=4π∫0r[ρb(r′)+ρℓ(r′)]r′2dr′M_{\text{eff}}(r) = 4\pi \int_0^r [\rho_b(r') + \rho_\ell(r')] r'^2 dr'Meff​(r)=4π∫0r​[ρb​(r′)+ρℓ​(r′)]r′2dr′

This modifies Φ(r)\Phi(r)Φ(r) and amplifies time-rate differences. If ρℓ\rho_\ellρℓ​ rises with radius, outer clocks accelerate further, accentuating the apparent age gradient.

5. Fibonnaci-Scaling Time Field

To parameterize nonlinear compounding across galactic units (GU), we map the time-dilation gradient using the Fibonacci sequence FnF_nFn​ such that:

dTdR∝Fnμ\frac{dT}{dR} \propto F_n \mudRdT​∝Fn​μ

with nnn indexed by distance in galactic-unit steps (1 GU = galactic diameter). This naturally creates exponential-like scaling that matches observed rapid change near cosmological boundaries while remaining smooth near the local group.

6. Derived Implications

1. Galaxy Maturity: Peripheral galaxies, where time runs faster, would appear billions of years older than the central epoch.

2. CMB Uniformity: Subspace leakage acts as an isotropizing field, diffusing potential gradients and maintaining large-scale homogeneity.

3. Dark Matter Analogue: Apparent mass excess is reinterpreted as curvature leakage between time-dilated strata.

4. Cosmic Age: For observed 13.8 Gyr mean light-travel time, central proper time may be as young as ~7.5 Gyr in this model.

7. Discussion

This toy framework is not presented as a full cosmological metric but as a conceptual scaffold linking gravitational time dilation, radial energy leakage, and nonlinear scaling effects. By tying local time flow to position in a self-gravitating manifold, it reframes dark matter, expansion, and apparent age as emergent rather than intrinsic.

Future work could couple this potential to relativistic hydrodynamics, exploring whether galactic rotation curves, gravitational lensing, and early-galaxy formation align quantitatively with the time-gradient model.

8. References

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