There is an interesting principle in physics, called the holographic principle. I will use the following extract as a description of the principle:
Is this picture worth a thousand words? According to the Holographic Principle, the most information you can get from this image is about 3x1065 bits for a normal sized computer monitor. The Holographic Principle, yet unproven, states that there is a maximum amount of information content held by regions adjacent to any surface. Therefore, counter-intuitively, the information content inside a room depends not on the volume of the room but on the area of the bounding walls. The principle derives from the idea that the Planck length, the length scale where quantum mechanics begins to dominate classical gravity, is one side of an area that can hold only about one bit of information. The limit was first postulated by physicist Gerard 't Hooft in 1993. It can arise from generalizations from seemingly distant speculation that the information held by a black hole is determined not by its enclosed volume but by the surface area of its event horizon. The term ‘holographic’ arises from a hologram analogy where three-dimension images are created by projecting light through a flat screen. Beware, other people looking at the featured image may not claim to see 3x1065 bits- they might claim to see a teapot.
[https://apod.nasa.gov/apod/ap170423.html]
In information theory, the information content of a system is defined as the logarithm of the states of the system. If we call the number of states n, then the information I is given as:
where the units of information (bits or nats) depend on the basis of the logarithm (base 2, or base e, respectively).
Information is closely related to entropy, through Boltzmann’s formula:
where kB is Boltzmann’s constant.
Alternatively, entropy is defined by the formula:
where E is the (thermal) energy and T is the temperature.
[https://en.wikipedia.org/wiki/Entropy]
In black hole thermodynamics, the previous formulas for the entropy take the equivalent forms,
where S is the entropy of the black hole, T its temperature, M its mass, R its radius, and A its area.
[https://en.wikipedia.org/wiki/Holographic_principle]
The meaning of the last formula is that since the entropy S of a black hole is proportional to its surface area A, then the information content of the black hole will be found on that area (instead of inside its volume). This is a way to express the holographic principle.
Here we will use the holographic principle in the context of the brachistochrone. First of all, the area of the brachistochrone, which we may call AB, in relation to its radius RB, is given by the following formula:
Thus, with respect to its area AB, the energy EB of the brachistochrone can be given as follows,
The previous formula can also be written in the following form, to express the state n,
The fact that the energy EB of the brachistochrone is proportional to its area AB, is a good indication that the holographic principle applies to the brachistochrone.
Now we will make the assumption that the state of the brachistochrone n directly corresponds to its configuration, so that its informational content IB, and entropy SB (where the index ‘B’ stands for ‘Brachistochrone’), will be given, respectively, by the formulas
where kB is Boltzmann’s constant (this index ‘B’ stands for ‘Boltzmann’).
Thus the relationship between the information In (or IB), the entropy Sn (or SB) and the energy En (or EB) at any state n, will be as follows,
Setting in the last equation
we take
where
If we make a comparison between the last formula,
and the formula for the entropy of black holes,
we see that we have reached a result which may be considered even more accurate, since, additionally to the fact that it gives the correct units (the units of entropy are the units of Boltzmann’s constant, while the logarithm of any quantity is dimensionless), it conveys more information about the system.
The general formula
will reappear in the next section, where the meaning of the constant 𝒞 will be identified with respect to the role of Consciousness in the universe.
Notes:
From the energy equations of the brachistochrone,
if we replace the wavelength λ of the reference photon with Planck length lP, we take that
This way the energy ε0 per wavelength will be Planck energy εP, while the states n of the brachistochrone will be identified with the number NP of Planck lengths lP which fit in the distance LB.
Additionally, if we set Planck energy equal to 1, as a reference energy, then for the entropy SB we have
This way we arrive at a similar result for the entropy SB in a simpler way.
Is this picture worth a thousand words? According to the Holographic Principle, the most information you can get from this image is about 3x1065 bits for a normal sized computer monitor. The Holographic Principle, yet unproven, states that there is a maximum amount of information content held by regions adjacent to any surface. Therefore, counter-intuitively, the information content inside a room depends not on the volume of the room but on the area of the bounding walls. The principle derives from the idea that the Planck length, the length scale where quantum mechanics begins to dominate classical gravity, is one side of an area that can hold only about one bit of information. The limit was first postulated by physicist Gerard 't Hooft in 1993. It can arise from generalizations from seemingly distant speculation that the information held by a black hole is determined not by its enclosed volume but by the surface area of its event horizon. The term ‘holographic’ arises from a hologram analogy where three-dimension images are created by projecting light through a flat screen. Beware, other people looking at the featured image may not claim to see 3x1065 bits- they might claim to see a teapot.
[https://apod.nasa.gov/apod/ap170423.html]
In information theory, the information content of a system is defined as the logarithm of the states of the system. If we call the number of states n, then the information I is given as:
[https://en.wikipedia.org/wiki/Holographic_principle]
where the units of information (bits or nats) depend on the basis of the logarithm (base 2, or base e, respectively).
Information is closely related to entropy, through Boltzmann’s formula:
Alternatively, entropy is defined by the formula:
[https://en.wikipedia.org/wiki/Entropy]
In black hole thermodynamics, the previous formulas for the entropy take the equivalent forms,
[https://en.wikipedia.org/wiki/Holographic_principle]
The meaning of the last formula is that since the entropy S of a black hole is proportional to its surface area A, then the information content of the black hole will be found on that area (instead of inside its volume). This is a way to express the holographic principle.
Here we will use the holographic principle in the context of the brachistochrone. First of all, the area of the brachistochrone, which we may call AB, in relation to its radius RB, is given by the following formula:
[https://en.wikipedia.org/wiki/Cycloid#Area]
Thus, with respect to its area AB, the energy EB of the brachistochrone can be given as follows,
Now we will make the assumption that the state of the brachistochrone n directly corresponds to its configuration, so that its informational content IB, and entropy SB (where the index ‘B’ stands for ‘Brachistochrone’), will be given, respectively, by the formulas
Thus the relationship between the information In (or IB), the entropy Sn (or SB) and the energy En (or EB) at any state n, will be as follows,
The general formula
Notes:
From the energy equations of the brachistochrone,
Additionally, if we set Planck energy equal to 1, as a reference energy, then for the entropy SB we have
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