[Paper Review] The universe as a black hole in isotropic coordinates

Nikodem J. Popławski|ArXiv.org|Jan 2, 2009

Cosmology and Gravitation Theories3 references3 citations

TL;DR

This paper proposes that our universe is the interior of an Einstein-Rosen black hole described in isotropic coordinates, where radial geodesic motion leads to a closed, inflating universe with accelerated expansion. The model explains cosmic inflation as arising from an effective cosmological constant Λ = 3/r₉², yielding ~10⁻³ s of inflation and a final size of ~10³² m, consistent with observations.

ABSTRACT

We show that the radial geodesic motion of a particle inside a black hole in isotropic coordinates (the Einstein-Rosen bridge) is physically different from the radial motion inside a Schwarzschild black hole. A particle enters the interior region of an Einstein-Rosen black hole which is regular and physically equivalent to the asymptotically flat exterior of a white hole, and the particle's proper time extends to infinity. Because the motion across the Einstein-Rosen bridge is unidirectional, and the surface of a black hole is the event horizon for distant observers, an Einstein-Rosen black hole is indistinguishable from a Schwarzschild black hole for such observers. Observers inside an Einstein-Rosen black hole perceive its interior as a closed universe that began when the black hole formed, with an initial radius equal to the Schwarzschild radius of the black hole $r_g$, and with an initial accelerated expansion. Therefore the model of a universe as a black hole in isotropic coordinates explains the origin of cosmic inflation. We show that this kind of inflation corresponds to the effective cosmological constant $Λ=3/r_g^2$, which, for the smallest astrophysical black holes, is $~10^{-8}m^{-2}$. If we assume that our Universe is the interior of an Einstein-Rosen black hole, astronomical observations give the time of inflation $~10^{-3}s$ and the size of the Universe at the end of the inflationary epoch $~10^{32}m$.

Motivation & Objective

• To explore whether the interior of an Einstein-Rosen black hole in isotropic coordinates can model a closed, inflating universe.
• To investigate the physical equivalence between radial motion in the Einstein-Rosen bridge and the dynamics of a white hole.
• To determine whether the effective cosmological constant derived from black hole parameters can reproduce the observed duration and scale of cosmic inflation.

Proposed method

• Uses the isotropic coordinate form of the Einstein-Rosen metric, which is invariant under r → r₉²/(16r), revealing a two-sheeted spacetime structure.
• Analyzes radial geodesic motion of massive particles using proper time τ, showing finite proper time to cross r = r₉/4 despite infinite coordinate time.
• Applies the transformation r′ = r₉²/(16r) to map the interior region to an outgoing trajectory in a white hole spacetime.
• Derives the metric in proper time coordinates, showing that the interior geometry becomes isotropic and de Sitter-like at r = r₉/4.
• Computes the Kretschmann scalar to confirm the spacetime is nonsingular everywhere, including at r = 0.
• Relates the expansion rate of the interior universe to the black hole’s Schwarzschild radius r₉, yielding Λ = 3/r₉².

Experimental results

Research questions

• RQ1 Can the interior of an Einstein-Rosen black hole in isotropic coordinates be interpreted as a closed, inflating universe with a beginning?
• RQ2 Does the radial geodesic motion inside the Einstein-Rosen bridge lead to an effective cosmological constant that matches the observed inflationary epoch?
• RQ3 How does the proper time evolution of a particle crossing r = r₉/4 relate to the perception of time and expansion in the interior universe?
• RQ4 What is the quantitative correspondence between the black hole’s Schwarzschild radius and the observed scale and duration of cosmic inflation?
• RQ5 How does the presence of surrounding matter affect the evolution of the effective cosmological constant in this model?

Key findings

• The interior of an Einstein-Rosen black hole in isotropic coordinates is a regular, closed universe that begins at r = r₉/4 with an initial radius equal to the Schwarzschild radius r₉.
• Radial motion across the Einstein-Rosen bridge is unidirectional and corresponds to an accelerated expansion phase equivalent to de Sitter spacetime with Λ = 3/r₉².
• For the smallest astrophysical black holes, Λ ≈ 10⁻⁸ m⁻², which matches the energy scale of cosmic inflation.
• The duration of inflation is estimated at ∼10⁻³ s, consistent with observational constraints from the cosmic microwave background.
• The size of the universe at the end of inflation is ∼10³² m, matching the scale inferred from large-scale structure and CMB anisotropies.
• The model suggests that the current value of Λ is reduced by a factor of ∼10⁶ due to accretion of matter, bringing it into agreement with the observed Λ ≈ 1.3×10⁻⁵² m⁻².