[Paper Review] The zero point energy and gravitation

Hristu Culetu|ArXiv.org|Oct 12, 2004

Cosmology and Gravitation Theories6 references3 citations

TL;DR

This paper proposes a gravity-based ultraviolet cutoff to regularize the infinite zero-point energy of quantum fields, using the Schwarzschild radius as a natural high-momentum cutoff. It finds that below ~1 mm, vacuum fluctuations behave like dark matter (pressure = 1/3 energy density), while above 1 mm, they mimic dark energy (pressure = −energy density), with energy scaling as W ∝ V^{1/5} and exhibiting a transition tied to quantum gravity scales.

ABSTRACT

A possible connection between the energy W of the vacuum fluctuations of quantum fields and gravity in "empty space" is conjectured in this paper using a natural cutoff of high momenta with the help of the gravitational radius of the vacuum region considered. We found that below some "critical" length $L = 1 mm$ the pressure $sigma$ is one third of the energy density $epsilon$, as for dark matter, but above $1 mm$ the equation of state is $sigma = -(epsilon)$ (dark energy). In the case of a massive field, W does not depend on the mass of the field for $L<<1 mm$ but for $L>>1 mm$ it does not depend on the Planck constant. In addition, when the Newton constant tends to zero, W becomes infinite. The energy density is also a function of the volume V of the vacuum region taken into account.

Motivation & Objective

• To resolve the divergence of zero-point energy in quantum field theory by introducing a gravitational cutoff based on the Schwarzschild radius of the vacuum region.
• To explore the connection between vacuum energy, gravity, and the cosmological constant using a finite, gravity-regulated zero-point energy model.
• To determine how the equation of state of vacuum fluctuations transitions from radiation-like (σ = ε/3) to cosmological-constant-like (σ = −ε) at a critical length scale.
• To investigate the dependence of vacuum energy W on volume V, Planck constant ℏ, and field mass m, particularly in the macroscopic limit.

Proposed method

• Introduces a gravitational UV cutoff by requiring the linear size L of a vacuum region to exceed its Schwarzschild radius rg = 2W, thus preventing black hole collapse.
• Uses the condition L ≥ 2W to derive a maximum frequency ω_max = π/W, which limits the integral over field modes in the energy density calculation.
• Computes the finite zero-point energy W for a massless scalar field via ∫₀^{ω_max} ω³ dω, leading to W ∝ V^{1/5} in geometrical units.
• For massive fields, applies a similar cutoff and derives W(V) in two regimes: L << 1 mm (W independent of m, σ = ε/3) and L >> 1 mm (W independent of ℏ, σ = −ε).
• Uses the holographic principle and black hole energy bounds to justify the cutoff, linking it to the Planck scale and quantum gravity phenomenology.
• Analyzes the equation of state σ + ε = 0 in the macroscopic limit, showing it leads to a Lorentz-invariant vacuum with a cosmological constant Λ ∝ ε.

Experimental results

Research questions

• RQ1 How can the infinite zero-point energy of quantum fields be regularized using gravitational physics rather than ad hoc cutoffs?
• RQ2 What is the physical significance of the critical length scale L ≈ 1 mm in the context of vacuum energy and quantum gravity?
• RQ3 How does the equation of state of vacuum fluctuations (σ/ε) evolve with the size L of the vacuum region?
• RQ4 Why does the zero-point energy W become independent of ℏ for large L, and what does this imply for the quantum-to-classical transition of vacuum energy?
• RQ5 Can the observed dark energy and dark matter behaviors emerge from a single vacuum energy model with a gravity-based cutoff?

Key findings

• The zero-point energy W scales as W ∝ V^{1/5} for a massless field, with the cutoff determined by the condition that the region’s size L must exceed its Schwarzschild radius.
• For L << 1 mm, the vacuum energy density ε and pressure σ satisfy σ = ε/3, mimicking radiation or dark matter, and W is independent of the field mass m.
• For L >> 1 mm, the equation of state becomes σ = −ε, characteristic of dark energy, and W becomes independent of Planck’s constant ℏ.
• At the critical length L ≈ 1 mm, the transition between the two regimes occurs, with W ≈ m²_P / (2m) and V/V_P ≈ 50 (m_P/m)^5.
• The energy density ε depends on the volume V and decreases with increasing V, leading to a cosmological constant Λ ≈ 10^{-48} erg/cm³, corresponding to a de Sitter radius R_c ≈ 10^{12} cm.
• In the macroscopic limit (L >> 1 mm), the vacuum energy produces a Lorentz-invariant stress-energy tensor with Λ ∝ ε, and the spacetime curvature is negligible compared to the system size.