Review of 'Mpemba Effect- the Effect of Time'

The reviewer thinks that this paper presents an interesting idea that is not thorough enough both phenomenologically and theoretically. The author now intends to add the following to the article:

After careful consideration, the author now gives the following definition of time: “The time on a system (for example, the heat contained in an object in a vacuum, or the total deuterium tritium fusion energy contained in a star) is proportional to the energy per unit area outflow (or inflow) at unit observer’s time. The observer’s time is the time on the observer (such as an atomic clock on Earth).”

According to Stefan-Boltzmann's law, the total energy radiated from a unit area of a black body with absolute temperature T in unit time (observer’s time) is

B(T)=σT⁴                          （1）

Where σ is the Stefan-Boltzmann constant (also known as the blackbody radiation constant), equal to 5.67×10-8W/（m2·K4）, and T is the temperature of the blackbody.

If we define n as the total number of photons radiated by the unit area of the blackbody with absolute temperature T in all directions of space in unit observer’s time, ν is the average frequency of the photons. t is the time on the observer, s is the surface area of the black body, and t’ is the time on the black body, then we have:

t'∝ B(T)=σT⁴=nhν/st              （2）

For two black bodies at temperatures T₁ and T_2, we have:

t₁' ∝ B(T₁)=σT₁⁴=nhν₁/st        （3）

and

t₂'∝B(T₂)=σT₂⁴=nhν₂/st            （4）

From equations (3) and (4), it can be obtained:

t₁'/t₂'=(T₁/T₂)⁴                             (5）

If T₁= 373K and T₂= 293K, then we have:

t₁'/t₂'=(T₁/T₂)⁴  =2.626

If T₁= 10000K, T₂= 1000K, then:

t₁'/t₂'=(T₁/T₂)⁴  =10000

From the above analysis, we can conclude that the higher the temperature of the blackbody, the greater the average energy of the photons radiated by the blackbody, the faster the time passes on the blackbody, and the shorter the life of the system.

Although the blackbody is only an ideal object, any object can be approximated as a blackbody, and its thermal radiation will follow Planck's law. According to Planck's law, the following relation between the radiance rate and wavelength of electromagnetic waves at various temperatures has been plotted:

It is seen from the above figure that the higher the temperature, the shorter the wavelength of the maximum radiation rate λm, the greater the radiance I (λ, T) corresponding to the same wavelength. That is, the higher the temperature, the shorter the wavelength of the light quantum emitted by the object, the greater the radiation rate. The greater the radiation rate of the object, the faster the time on the object.

According to the gravitational redshift of light, the higher the energy density in space, the lower the frequency of the light quantum emitted by the objects in space, so the higher the spatial energy density, the slower the time on the object in space.

Because any system (such as the heat of an object, or the deuterium tritium nucleofusion energy of a star) is composed of quantums. So from the above analysis, we can conclude:

1. The higher the temperature of the object, the greater the average energy of the light quantums emitted, the faster the time passes on the object and the shorter the life of the system.
2. The higher the energy density of the space, the lower the average frequency of the light quantums radiated by the object in space, the slower the time on the object, and the longer the life of the system.