Near-Horizon Dynamics of a Black Hole

A black hole's event horizon is a region where GR predicts extreme spacetime curvature.

Near the event horizon:

• GR alone fails due to infinite curvature (singularities).
• QM fails to describe how quantum fluctuations influence spacetime geometry at the macro scale.

We are tasked with understanding:

• How quantum fluctuations (e.g., Hawking radiation) dynamically interact with macroscopic curvature near the event horizon.
• How the combined dynamics evolve the black hole’s observable properties (mass, spin, evaporation).

Einstein Field Equations:

• Describes macroscopic spacetime curvature: Gμν+Λgμν=κTμν,G_{\mu\nu} + \Lambda g_{\mu\nu} = \kappa T_{\mu\nu},Gμν​+Λgμν​=κTμν​,where:
  • GμνG_{\mu\nu}Gμν​: Einstein tensor (spacetime curvature).
  • Λgμν\Lambda g_{\mu\nu}Λgμν​: Cosmological constant term.
  • TμνT_{\mu\nu}Tμν​: Stress-energy tensor.

GR Limitations:

• At the event horizon, TμνT_{\mu\nu}Tμν​ diverges due to quantum effects, making GμνG_{\mu\nu}Gμν​ undefined.
• GR cannot incorporate probabilistic quantum fluctuations directly.

1. Schrödinger Equation:
   
   • Describes the probabilistic evolution of quantum states: iℏ∂ψ∂t=H^ψ,i\hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi,iℏ∂t∂ψ​=H^ψ,where ψ(x,t)\psi(x, t)ψ(x,t) is the wavefunction, and H^\hat{H}H^ is the Hamiltonian operator.

1. Hawking Radiation:
   
   • Near the event horizon, virtual particle pairs are generated due to quantum fluctuations: ∣ψ(x,t)∣2∝Hawking radiation flux.|\psi(x,t)|^2 \propto \text{Hawking radiation flux}.∣ψ(x,t)∣2∝Hawking radiation flux.

1. QM Limitations:
   
   • QM assumes a fixed background spacetime and cannot dynamically adjust the geometry near the event horizon.
   • Fails to describe how spacetime curvature evolves due to quantum processes.

Relational Tensor Model:

• Encodes quantum fluctuations (T(1)T^{(1)}T(1)) and spacetime curvature (T(3)T^{(3)}T(3)) as nested tensors: Ti,j,kUnified(t)=⋃n=13Ti,j,k(n)(t),T^{\text{Unified}}_{i,j,k}(t) = \bigcup_{n=1}^{3} T^{(n)}_{i,j,k}(t),Ti,j,kUnified​(t)=n=1⋃3​Ti,j,k(n)​(t),where T(n)T^{(n)}T(n) captures relational dynamics at scale nnn:
  • T(1)T^{(1)}T(1): Quantum fluctuations.
  • T(2)T^{(2)}T(2): Local interactions (e.g., particle ensembles).
  • T(3)T^{(3)}T(3): Macro-scale curvature.

Unified Relational Field Equations:

• Coupled dynamics of quantum fluctuations and spacetime curvature: Ti,j,k(3)+ΛTi,j,k(3)+Qi,j,k=κTi,j,k(2),T^{(3)}_{i,j,k} + \Lambda T^{(3)}_{i,j,k} + \mathcal{Q}_{i,j,k} = \kappa T^{(2)}_{i,j,k},Ti,j,k(3)​+ΛTi,j,k(3)​+Qi,j,k​=κTi,j,k(2)​,where:
  • Qi,j,k=ℏc3∇μ∇νTi,j,k(1)\mathcal{Q}_{i,j,k} = \frac{\hbar}{c^3} \nabla_\mu \nabla_\nu T^{(1)}_{i,j,k}Qi,j,k​=c3ℏ​∇μ​∇ν​Ti,j,k(1)​: Quantum correction tensor.

Dynamic Cross-Scale Propagation:

• Quantum fluctuations propagate upward: ΔTi,j,k(3)=g(Tl,m,n(1)),\Delta T^{(3)}_{i,j,k} = g(T^{(1)}_{l,m,n}),ΔTi,j,k(3)​=g(Tl,m,n(1)​),modifying macroscopic curvature.
• Spacetime curvature imposes constraints downward: ΔTi,j,k(1)=f(Tl,m,n(3)),\Delta T^{(1)}_{i,j,k} = f(T^{(3)}_{l,m,n}),ΔTi,j,k(1)​=f(Tl,m,n(3)​),influencing quantum probabilities.

Feedback Mechanism:

• Near the event horizon, a feedback loop between quantum and macroscopic scales stabilizes the geometry: ∂Ti,j,kUnified∂t=F(T(1),T(2),T(3)).\frac{\partial T^{\text{Unified}}_{i,j,k}}{\partial t} = F(T^{(1)}, T^{(2)}, T^{(3)}).∂t∂Ti,j,kUnified​​=F(T(1),T(2),T(3)).

Emergent Observables:

• Hawking radiation flux (ψ(x,t)\psi(x,t)ψ(x,t)) evolves dynamically with spacetime curvature (GμνG_{\mu\nu}Gμν​): ∣ψ(x,t)∣2=h(Ti,j,k(3),Ti,j,k(1)).|\psi(x,t)|^2 = h(T^{(3)}_{i,j,k}, T^{(1)}_{i,j,k}).∣ψ(x,t)∣2=h(Ti,j,k(3)​,Ti,j,k(1)​).

• GR cannot handle singularities or incorporate quantum fluctuations.
• GμνG_{\mu\nu}Gμν​ becomes undefined at extreme curvatures.

• QM cannot dynamically adjust spacetime curvature.
• Assumes a fixed spacetime background incompatible with evolving geometries.

• UCF/GUTT integrates both GR and QM using relational tensors, which:
  • Dynamically couple quantum states and spacetime curvature.
  • Stabilize near-horizon dynamics through multi-scale feedback.
  • Predict emergent phenomena, such as time-dependent Hawking radiation flux.

Input:

• Initial quantum fluctuations (T(1)T^{(1)}T(1)) near the horizon.
• Macro-scale curvature tensor (T(3)T^{(3)}T(3)).

Unified Tensor Dynamics:

• Quantum fluctuation evolution: iℏ∂Ti,j,k(1)∂t=∑l,m,nHi,j,k,l,m,nTl,m,n(1).i\hbar \frac{\partial T^{(1)}_{i,j,k}}{\partial t} = \sum_{l,m,n} H_{i,j,k,l,m,n} T^{(1)}_{l,m,n}.iℏ∂t∂Ti,j,k(1)​​=l,m,n∑​Hi,j,k,l,m,n​Tl,m,n(1)​.
• Macro-scale curvature adjustment: Ti,j,k(3)+Qi,j,k=κTi,j,k(2),T^{(3)}_{i,j,k} + \mathcal{Q}_{i,j,k} = \kappa T^{(2)}_{i,j,k},Ti,j,k(3)​+Qi,j,k​=κTi,j,k(2)​,with: Qi,j,k=ℏc3∇μ∇νTi,j,k(1).\mathcal{Q}_{i,j,k} = \frac{\hbar}{c^3} \nabla_\mu \nabla_\nu T^{(1)}_{i,j,k}.Qi,j,k​=c3ℏ​∇μ​∇ν​Ti,j,k(1)​.

Output:

• Time-evolving Hawking radiation: ∣ψ(x,t)∣2∝h(Ti,j,k(3),Ti,j,k(1)).|\psi(x,t)|^2 \propto h(T^{(3)}_{i,j,k}, T^{(1)}_{i,j,k}).∣ψ(x,t)∣2∝h(Ti,j,k(3)​,Ti,j,k(1)​).

Using the UCF/GUTT framework, the near-horizon dynamics of a black hole are described through a unified relational tensor model that:

1. Dynamically integrates quantum fluctuations and spacetime curvature.
2. Handles feedback loops that stabilize singularities.
3. Produces emergent observables like Hawking radiation flux.

This capability goes beyond GR and QM, which cannot independently handle such multi-scale, dynamic interactions. UCF/GUTT provides a truly unified approach to the most challenging problems in modern physics.