Hyperbolic growth

When a quantity grows towards a singularity under a finite variation (a "finite-time singularity") it is said to undergo hyperbolic growth.^[1] More precisely, the reciprocal function ${\displaystyle 1/x}$ has a hyperbola as a graph, and has a singularity at 0, meaning that the limit as ${\displaystyle x\to 0}$ is infinite: any similar graph is said to exhibit hyperbolic growth.

The reciprocal function, exhibiting hyperbolic growth.

Description

If the output of a function is inversely proportional to its input, or inversely proportional to the difference from a given value ${\displaystyle x_{0}}$, the function will exhibit hyperbolic growth, with a singularity at ${\displaystyle x_{0}}$.

In the real world hyperbolic growth is created by certain non-linear positive feedback mechanisms.^[2]

Comparisons with other growth functions

Growth equations

Like exponential growth and logistic growth, hyperbolic growth is highly nonlinear, but differs in important respects.

These functions can be confused, as exponential growth, hyperbolic growth, and the first half of logistic growth are convex functions; however their asymptotic behavior (behavior as input gets large) differs dramatically:

• logistic growth is constrained (has a finite limit, even as time goes to infinity),
• exponential growth grows to infinity as time goes to infinity (but is always finite for finite time),
• hyperbolic growth has a singularity in finite time (grows to infinity at a finite time).

Applications

Global macrodevelopment

A 1960 issue of Science magazine included an article by Heinz von Foerster and his colleagues, P. M. Mora and L. W. Amiot, proposing an equation representing the best fit to the historical data on the Earth's population available in 1958:

Fifty years ago, Science published a study with the provocative title "Doomsday: Friday, 13 November, A.D. 2026". It fitted world population during the previous two millennia with P = 179 × 10⁹/(2026.9 − t)^0.99. This "quasi-hyperbolic" equation (hyperbolic having exponent 1.00 in the denominator) projected to infinite population in 2026—and to an imaginary one thereafter.

—Taagepera, Rein.^[3]

The global population is equal to ${\displaystyle {\tfrac {179000000000}{2026.9-t}}}$ and hyperbolically grows as t approaches 2026.9. When t surpasses 2026.9, the number of people living on the planet Earth suddenly becomes negative—on 13 November 2026 AD, all humans instantaneously disappear.

In 1975, von Hoerner suggested that von Foerster's doomsday equation can be written, without a significant loss of accuracy, in a simplified hyperbolic form (i.e. with the exponent in the denominator assumed to be 1.00):

${\displaystyle {\text{Global population}}={\frac {179000000000}{2026.9-t}},}$

where

• 2026.9 (more precisely, 13 November 2026 AD) is the date of the so-called "demographic singularity"^[4] and von Foerster's 115th anniversary;
• t is the number of a year of the Gregorian calendar.^[4]

Despite its simplicity, von Foerster's equation is very accurate in the range from 4,000,000 BP^[4] to 1997 AD. For example, the doomsday equation (developed in 1958, when the Earth's population was 2,911,249,671^[5]) predicts a population of 5,986,622,074 for the beginning of the year 1997:

${\displaystyle {\frac {179000000000}{2026.9-1997}}=5986622074.}$

The actual figure was 5,924,787,816.^[5]

On Kurzweil's graph, the speed of living cells' sociotechnological progress is equal to ${\displaystyle {\tfrac {2054}{2029-t}}}$ and hyperbolically increases as t approaches 2029. When t surpasses 2029, the extremely (but not infinitely) positive speed of sociotechnological progress suddenly becomes infinitely negative—in the beginning of 2029 AD, the entire multibillion-year-long progress of life becomes instantaneously reset to the initial state.

Having analyzed the timing of the events plotted on Ray Kurzweil's "Countdown to Singularity" graph, Andrey Korotayev arrived at the following best-fit equation:

${\displaystyle {\text{Speed of sociotechnological progress}}={\frac {2054}{2029-t}},}$

where

• "speed of sociotechnological progress" is the number of sociotechnological phase transitions per a unit of time;
• 2029 (more precisely, the beginning of the year 2029 AD) is the date of sociotechnological singularity;
• t is the number of a year of the Gregorian calendar.^[4]

Korotayev also analyzed the timing of the events on the list of sociotechnological phase transition points independently compiled by Alexander Panov, and arrived at the following best-fit equation:

${\displaystyle {\text{Speed of sociotechnological progress}}={\frac {1886}{2027-t}},}$

where

• "speed of sociotechnological progress" is the number of sociotechnological phase transitions per a unit of time;
• 2027 (more precisely, the beginning of the year 2027 AD) is the date of sociotechnological singularity;
• t is the number of a year of the Gregorian calendar.^[4]

Korotayev discovered that these two equations are entirely identical with von Foerster's doomsday equation describing the world population growth. Both empirical and mathematical analyses indicate that all the three hyperbolic equations describe the same global macrodevelopmental process, in which demography is indivisibly combined with technology.^[4] It can be set forth as follows: technological advance → increase in the carrying capacity of the Earth → population growth → more potential inventors → acceleration of technological advance → faster growth of the Earth's carrying capacity → faster population growth → faster growth of the number of potential inventors → faster technological advance → faster growth of the Earth's carrying capacity, and so on.^[1][6]

Lorentz factor

The Lorentz factor γ is defined as^[7] $${\displaystyle \gamma ={\frac {1}{\sqrt {1-\left({\frac {v}{c}}\right)^{2}}}}={\frac {1}{\sqrt {1-\beta ^{2}}}},}$$ where

• ${\displaystyle v}$ is the relative velocity between inertial reference frames;
• ${\displaystyle c}$ is the speed of light in vacuum;
• β is the ratio of ${\displaystyle v}$ to c, or the speed in units of ${\displaystyle c}$.

Proxima Centauri is approximately 4.27 light-years away from the Earth. From a terrestrial observer's perspective, a traveller would cover the distance to Proxima Centauri in approximately 8.54 years at half the speed of light. However, due to the Lorentz factor, the time experienced by the traveller would be shorter:

${\displaystyle {\text{Proper time}}={\frac {\text{Time}}{\text{γ}}}={\frac {\text{8.54 light-years}}{1.155}}{\text{≈ 7.39 light-years}}.}$

The following graph shows the journey times for twenty runs to Proxima Centauri from the ship viewpoint. Notice that as speeds approach the speed of light, the journey times reduce dramatically, even though the actual increments in speed appear slight. On the 20th run, at ^1048575/_1048576 of the speed of light, the distance shrinks to 0.0059 light-years and the traveller experiences a journey time of 2.15 days. Whereas to those on Earth the ship looks almost "frozen" and the journey still takes 4.27 years, plus a couple of days.

The equation describing the growth of the Lorentz factor with speed is unmistakably hyperbolic, so the Lorentz factor of a spaceship, subjected to even a small but constant accelerating force, must become infinite in a finite proper time. This requirement is met by assuming that a translationally accelerating spaceship loses its rest mass (which is the spaceship's resistance to its further translational acceleration along the path of flight): $${\displaystyle \gamma ={\frac {m_{\text{rel}}}{m_{0}}},}$$ where

• ${\displaystyle {m_{0}}}$ is rest mass (resistance to longitudinal acceleration);
• ${\displaystyle {m_{\text{rel}}}}$ is relativistic mass (resistance to transverse acceleration).^[8]

The initial magnitude of the Lorentz factor γ is 1 (when v = 0). As the velocity of a rest-massive object approaches the speed of light (v → c), γ hyperbolically reaches infinity in a finite proper time, at which moment the object turns into a beam of photons.^[9]

At ${\displaystyle v}$ = 0, the magnitude of the Lorentz factor is$${\displaystyle \gamma ={\frac {m_{\text{rel}}}{m_{0}}}={\frac {1}{1}}=1.}$$ At ${\displaystyle v}$ = 0.5 c, the magnitude of the Lorentz factor is$${\displaystyle \gamma ={\frac {m_{\text{rel}}}{m_{0}}}={\frac {1.072}{0.928}}=1.155.}$$ At ${\displaystyle v}$ = 0.999 c, the magnitude of the Lorentz factor is$${\displaystyle \gamma ={\frac {m_{\text{rel}}}{m_{0}}}={\frac {1.914}{0.086}}=22.366.}$$ Following this pattern, the spaceship will, after a finite proper time, turn into a beam of photons:

Photons may be regarded as limiting particles whose rest mass has become zero while their Lorentz factor has become infinite.

—Rindler, W.^[9]

The light-speed spaceship will then cover the remaining distance to its destination in zero proper time:

Since when traveling at the speed of light no apparent time elapses, the spacecraft would arrive instantly and simultaneously at all locations along the path of flight. Thus to the crew on the spacecraft, all spatial separations would collapse to zero along this path‑of‑flight. There is no relativistic dilatation, as all spatial separations are transverse to a light-speed spacecraft's flight. <...> Thus the spacecraft would disappear after reaching light speed, followed immediately by its reappearance trillions of miles away in the proximity of the target star, when the spacecraft returns to sub-light speed, Figure 9.6.

—Czysz, Paul A.; Bruno, Claudio.^[10]

The universe's matter is falling into the universe's gravitational field:

Gravity rules. The moon orbiting Earth, matter falling into black holes, and the overall structure of the universe are dominated by gravity.

—Seeds, Michael A.; Backman, Dana.^[11]

Consequently, the universe's matter accelerates to ever greater speeds, so that its Lorentz factor hyperbolically increases to infinity, while its rest mass hyperbolically vanishes:

As we go forwards in time, material weight continually changes into radiation. Conversely, as we go backwards in time, the total material weight of the universe must continually increase.

—Jeans, James Hopwood.^[12]

At the end of the hyperbolic growth of its Lorentz factor, the universe's matter attains the speed of light:

'It all just seemed unbelievably boring to me,' Penrose says. Then he found something interesting within it: at the very end of the universe, the only remaining particles will be massless. That means everything that exists will travel at the speed of light, making the flow of time meaningless.

—Brooks, Michael.^[13]

So, the universe will eventually consist of relativistic kinetic energy, which is negative, i.e. hierarchically binding/enslaving:

A beam of negative energy that travels into the past can be generated by the acceleration of the source to high speeds.

—Skinner, Ray.^[14]

It is seen that the relativistic kinetic energy is always negative and therefore will lower the energy levels of a bound system.

—Ruei, K. H.^[15]

This hierarchically binding/enslaving negative energy is the universe's spirit or information:

Remember, more binding energy means the system is more bound—has greater negative energy.

—Shu, Frank.^[16]

The Spirit is the binding energy expressed by the word re-ligio/religion—a word that itself reflects the brokenness and fragmentation of the universe, that God is trying to heal.

—Grey, Mary C.^[17]

Szilard's explanation was accepted by the physics community, and information was accepted as a scientific concept, defined by its statistical‑mechanical properties as a kind of negative energy that introduced order into a system.

—Aspray, William.^[18]

Thus, the hyperbolic growth of the Lorentz factor of the universe's matter hierarchically binds/enslaves or, which is the same, animates/informs the universe's matter. The sociotechnological singularity of the terrestrial animated/informed matter, expected at the end of the year 2026 AD (see Global macrodevelopment) will signify that the Lorentz factor of the universe's matter has become infinite—since the end of the year 2026 AD, the universe's matter will be falling into the universe's animating/informing gravitational field (which is the funnel-shaped gradient of matter's negative-energiedness, animateness, informedness) at the speed of light:

The negative energy of the gravitational field is what allows negative entropy, equivalent to information, to grow, making the Universe a more complicated and interesting place.

—Gribbin, John.^[19]

"Well, then, if you picture what I'm describing, it's a funnel of some sort, which begins with an extremely wide mouth, but which has now narrowed to an extremely small and fast-moving kind of situation. And this is why history is a self-limiting process. It isn't that we have broken away from the slow-moving processes of ordinary nature. It's that we represent nature at a different time frame. And I think this is why history is ending. Because it's going so much faster than it used to go that it's going to finish very soon. There may be as much experience ahead of us as there is behind us, but we're moving through it so much faster than we used to that we're literally approaching the end of time at a faster and faster speed."
       —McKenna, Terence. Reality is Complexifying 1992

"It's this idea that we represent some kind of singularity, or that we announce the nearby presence of a singularity. That the evolution of life and cultural form and all that is clearly funneling toward something fairly unimaginable."
       —McKenna, Terence. A Weekend with Terence McKenna August 1993

"In other words, we end the whole thing. We collapse the state vector and everything goes into a state of novelty. What happens then I think is the universe becomes entirely made of light."
       —McKenna, Terence. Appreciating Imagination 1997

"The conventions of relativity say that time slows down as one approaches the speed of light, but if one tries to imagine the point of view of a thing made of light, one must realize that what is never mentioned is that if one moves at the speed of light, there is no time whatsoever. There is an experience of time zero. <...> One has transited into the eternal mode. One is then apart from the moving image; one exists in the completion of eternity. I believe that this is what technology pushes toward."
       —McKenna, Terence. New Maps of Hyperspace 1984

"What exactly is immortality? It's the negation of time. How do we negate time? By getting close to, and perhaps matching, the speed of light. If you ARE light, everything is instant."
       —Time fUSION Anomaly, 1999 10 11

"And the angel that I saw standing upon the sea and upon the land lifted his hand up to heaven, and swore by him who lives forevermore, who created heaven and the things that are in it, and the sea and the things that are in it, that time shall be no more, but in the days of the voice of the seventh angel, when he begins to blow, even the mystery of God shall be finished, as he preached by his servants the prophets."
       —Revelation 10:5-7 New Matthew Bible

Queuing theory

Another example of hyperbolic growth can be found in queueing theory: the average waiting time of randomly arriving customers grows hyperbolically as a function of the average load ratio of the server. The singularity in this case occurs when the average amount of work arriving to the server equals the server's processing capacity. If the processing needs exceed the server's capacity, then there is no well-defined average waiting time, as the queue can grow without bound. A practical implication of this particular example is that for highly loaded queuing systems the average waiting time can be extremely sensitive to the processing capacity.

Enzyme kinetics

A further practical example of hyperbolic growth can be found in enzyme kinetics. When the rate of reaction (termed velocity) between an enzyme and substrate is plotted against various concentrations of the substrate, a hyperbolic plot is obtained for many simpler systems. When this happens, the enzyme is said to follow Michaelis-Menten kinetics.

Mathematical example

The function

${\displaystyle x(t)={\frac {1}{t_{c}-t}}}$

exhibits hyperbolic growth with a singularity at time ${\displaystyle t_{c}}$: in the limit as ${\displaystyle t\to t_{c}}$, the function goes to infinity.

More generally, the function

${\displaystyle x(t)={\frac {K}{t_{c}-t}}}$

exhibits hyperbolic growth, where ${\displaystyle K}$ is a scale factor.

Note that this algebraic function can be regarded as an analytical solution for the function's differential:^[1]

${\displaystyle {\frac {dx}{dt}}={\frac {K}{(t_{c}-t)^{2}}}={\frac {x^{2}}{K}}.}$

This means that with hyperbolic growth the absolute growth rate of the variable x in the moment t is proportional to the square of the value of x in the moment t.

Respectively, the quadratic-hyperbolic function looks as follows:

${\displaystyle x(t)={\frac {K}{(t_{c}-t)^{2}}}.}$

See also

• Exponential growth
• Logistic growth
• Mathematical singularity