Idea holography the time. Perhaps time can be expressed as
[math]$$ t=\frac{Gh}{c^4}\int\frac{dS}{r} $$[/math]
Where S is the entropy of entanglement of an arbitrary closed surface. r is the radius to the surface point. Integration over a closed surface.
This is very similar to the analogy. Time behaves as a potential, and entropy as a charge.
From this formula there are several possible consequences.
1.Bekenstein Hawking entropy for the event horizon. Light cone case
[math]$$ r=ct $$[/math]
[math]$$ S=\frac{c^3}{Gh}r^2$$[/math]
2.Gravitational time dilation. The case if matter inside a closed surface processes information at the quantum level according to the Margolis-Livitin theorem.
[math]$$ dI=\frac{dMc^2 t}{h} $$[/math]
[math]$$ \Delta t=\frac{Gh}{c^4}\int\frac{dI}{r}=t\frac{GM}{c^2r}$$[/math]
3.The formula is invariant under Lorentz transformations.
4.If this definition is substituted instead of time, then the interval acquires a different look, which probably indicates a different approach of the Minkowski pseudometric with a complex plane
[math]$$ s^2=(l^2_{p}\frac{S}{r})^2-r^2 $$[/math]
Where is the squared length of Planck
[math]$$ l^2_{p}=\frac{Gh}{c^3} $$[/math]
Quantum tunneling of noncommutative geometry gives the definition of time in the form of holography, that is, in the form of a closed surface integral.
Ultimately, the holography of time shows the dualism between quantum mechanics and the general theory of relativity.
[math]$$ t=\frac{Gh}{c^4}\int\frac{dS}{r} $$[/math]
From here, the definition of time is obtained, as the ratio of the entropy at the boundary of the sphere to its radius. This is the definition of arising time. Where the entropy at the boundary of the sphere should be considered as entropy of entanglement between the boundary of the sphere and the point inside, where the moment of time is determined.In general, the resulting time will be as a closed surface integral. In this form, you can come to the general formula for any closed arbitrary surface.In this formula, time is determined at a certain point, where a closed surface is taken around through the integral of entropy of entanglement on a given surface
https://osf.io/8nzwd/download
https://frenxiv.org/3muny/download