Kurzgesagt – In a Nutshell
Kurzgesagt – In a Nutshell
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Sources – Survive Heat Death
Sources – Survive Heat Death
We thank the following expert for his comments:
Dr. Matthew Caplan
Illinois State University
The Noxans are a fictional alien species invented for this video. The biological temperature of the Noxans is 310 K, but the starting temperature of their artificial bodies is 300 K.
—Can we survive the heath death of the universe? One day, the last star will go out, galaxies will dissolve and black holes will evaporate. The cosmos will become an expanding, empty void where nothing happens. Forever. A place without life, purpose, or meaning.
Today, the most accepted cosmological model is the so-called “Lambda-CDM” model of the universe, in which dark energy is described by a cosmological constant (usually denoted by the greek letter Lambda, hence the name of the model):
#ESA (2020): “Mission Science: Euclid Mission”
A universe in which dark energy is given by a cosmological constant will expand forever and cool as it expands, ending in the Heat Death, or Big Freeze.
#NASA (2024): “WMAP's Universe: What is the Ultimate Fate of the Universe?”
https://wmap.gsfc.nasa.gov/universe/uni_fate.html
Quote: “There is a growing consensus among cosmologists that the total density of matter is equal to the critical density, so that the universe is spatially flat. Approximately 24% of this is in the form of a low pressure matter, most of which is thought to be “non-baryonic” dark matter, while the remaining 71% is thought to be in the form of a negative pressure “dark energy”, like the cosmological constant. If this is true, then dark energy is the major driving force behind the fate of the universe and it will expand forever exponentially.”
—Let’s travel 100 trillion years into the future, when the last stars are about to die and the cosmos starts turning dark forever.
It is thought that after some 100 trillion years, the “stelliferous era” (“star-forming age”) will come to an end:
#Busha, Michael T. et al. (2003): “Future Evolution of Cosmic Structure in an Accelerating Universe”, The Astrophysical Journal, Volume 596
https://iopscience.iop.org/article/10.1086/378043/meta
And since the longest-lived stars have a much lower expected lifespan (about 17 trillion years, see table above), this means that the last star will also die in about 100 trillion years from now.
—Here, in this dying universe we find the Noxans, the last civilization still alive.
This is a fictional civilization invented for this script. We assume their technology keeps improving over time.
—The first issue is energy. If you want to keep a civilization running you need a lot of it – but actually not that much in cosmic terms.
Civilizations run on energy. Human activity transforms sources of energy, like oil or uranium, into work, like moving people or squeezing the gas in an engine, that ultimately becomes waste heat. That heat must escape in the form of electromagnetic waves that cannot be recovered or reused.
In the future, a galactic civilization may transform white dwarfs or black holes into energy sources that can last for trillions of years. We even made a video about it:
#Kurzgesagt – In a Nutshell (2018): “The Black Hole Bomb and Black Hole Civilizations”
But in the extreme far future, all of these will eventually run out. The stars die out, the galaxies go dark, and black holes are exhausted.
So, the only way to keep operating a civilization is to store energy in a battery that can outlast the longest-lived natural source of energy. By its nature, it is a finite solution. A civilization with a constant or increasing energy use rate will eventually deplete it.
—With 5 hours of the full energy emitted by the Sun, we could power present day humanity for about 10 billion years.
The stellar structure shown here is meant to be a Dyson swarm, a version of the Dyson sphere, which captures a large part of the energy emitted by the star.
We have a video about Dyson spheres as well:
#Kurzgesagt – In a Nutshell (2018): “How to Build a Dyson Sphere - The Ultimate Megastructure”
Human civilization today runs on about 183 thousand terawatt-hours every year, which is equivalent to 21 terawatts.
#OWID (2024): “Energy Production and Consumption”
The Sun releases 3.8 × 1026 Watts in all directions, which is 380 trillion terawatts.
#NASA: “Sun Fact Sheet” (retrieved 2025)
https://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html
If we could collect five hours or 18,000 seconds’ worth of the Sun’s total energy, we’d have about 7 × 1030 joules. Divide this by the 21 terawatts needed to run human civilization and we get 3.3 × 1017 seconds, which is around 10 billion years.
—In principle, this energy could keep them alive for a few hundred trillion years, a long time but not even close to forever.
A galaxy like the Milky Way contains around a trillion stars
#ESA: “How many stars are there in the Universe?” (retrieved 2025)
Quote: “For the Universe, the galaxies are our small representative volumes, and there are something like 1011 to 1012 stars in our Galaxy, and there are perhaps something like 1011 or 1012 galaxies.”
Around 95% of all stars end their lives as white dwarfs.
#Harvard-Smithsonian Center for Astrophysics (2021): “Planetary Remnants around White Dwarf Stars”
https://www.cfa.harvard.edu/news/planetary-remnants-around-white-dwarf-stars
Quote: “White dwarf stars are common; over 95% of all stars will become white dwarfs.”
We will assume that the total number of stars is the same as today. Since we are at the end of the star forming era, almost all stars will be white dwarves (corpses of former stars) or relatively recent red dwarves. Both kinds of stars will have very low luminosities.
White dwarfs cool down over time. The physics controlling their cooling down is complicated, but a rough estimate for their temperature over time is given by the following expression:
#Maoz, Dan (2016): “Astrophysics in a Nutshell”. Princeton University Press.
(from chapter 4.2.3.3, “White-Dwarf Cooling”)
https://press.princeton.edu/books/hardcover/9780691164793/astrophysics-in-a-nutshell
(Where T is the temperature in kelvin and t is the elapsed time in years.)
This simple formula should be understood as only giving very rough orders of magnitude, but it still gives us an idea of how the cooling process works. Using it we find the following values for the temperature of a white dwarf of about 100 trillion years:
T ~ (1014/2.5 × 10 9)-1/310³K ~ 30 K
White dwarfs are approximately as big as Earth:
#NASA Goddard Space Flight Center (2021): “White Dwarf Stars”
https://imagine.gsfc.nasa.gov/science/objects/dwarfs2.html
Quote: “A typical white dwarf is half as massive as the Sun, yet only slightly bigger than Earth.”
So, using Stefan-Boltzman’s law, we can calculate how much radiation they emit
#Encyclopedia Britannica: “The Fermi Paradox: Where Are All the Aliens?” (retrieved 2025)
https://www.britannica.com/science/Stefan-Boltzmann-law
E = (σT4) × S = (σT4) × 4𝜋 × REarth2 = 2 ×1013 W
#NASA: “Earth Fact Sheet” (retrieved 2025)
https://nssdc.gsfc.nasa.gov/planetary/factsheet/earthfact.html
So these ultra-dim remnants across the galaxy could be harvested for a total of (2 ×1013 W per star) × (1012 stars) = 2 × 1024 W, or 6 × 1031 J every year.
Human civilization today runs on about 183 thousand terawatt-hours or 6.6× 1020 J every year.
#OWID (2024): “Energy Production and Consumption”
So harvesting the energy released by the last stars of their galaxy in the final thousand years of their lives is enough to run a human-like civilization for 1014 years, or the entire age of the Universe they inhabit up to that point.
If their energy requirements are higher, the efficiency of their energy collection is lower, or there are fewer stars available for extraction, they would have to collect energy for a proportionally higher number of years.
This is only one of many possible ways of harvesting energy for their battery. Noxans may use neutron star mergers or even use rotating black holes as energy generators, as we described in a previous Kurzgesagt video:
#Kurzgesagt – In a Nutshell (2018): “The Black Hole Bomb and Black Hole Civilizations”
But whichever method they use will eventually run out, whether it is in trillions, quadrillions, or decillions of years. At that point, the Noxans will have to consider storing energy in a battery to survive the long, dark future.
—There is an idea from a famous paper we’ll mercilessly simplify and call Dyson's Cold Thoughts – In a nutshell it the idea is that the lower your temperature, the less energy it takes to do something.
In this video, we will present the ideas developed by Freeman Dyson in the following paper:
#Dyson, Freeman J. (1979): “Time without end: Physics and biology in an open universe*”, Reviews of Modern Physics, vol. 51, 3, 447-460
https://suli.pppl.gov/2019/course/RevModPhys.51.447.pdf
His main aim is proving that, under certain hypotheses, a certain form of “life” could persist in the universe infinitely long, even if the universe undergoes Heat Death, with just a finite amount of energy previously stored in a sort of cosmic battery.
One of the main hypotheses that allow him to develop this argument is the following scaling law, which he postulates:
#Dyson, Freeman J. (1979): “Time without end: Physics and biology in an open universe*”, Reviews of Modern Physics, vol. 51, 3, 447-460
https://suli.pppl.gov/2019/course/RevModPhys.51.447.pdf
Quote: “For a quantitative description of the way life may adapt itself to a cold environment, I need to assume a scaling law that is independent of any particular material embodiment that life may find for itself. The following is a formal statement of my scaling law: Scaling Hypothesis. If we copy a living creature, quantum state by quantum state, so that the Hamittonian of the copy is
where H is the Hamiltonian of the creature, U is a unitary operator, and λ is a positive scaling factor, and if the environment is similarly copied so that the temperatures of the environments of the creature and the copy are respectively T and λT, then the copy is alive, subjectively identical to the original creature, with alI its vital functions reduced in speed by the same factor λ .”
This scaling law results in a quadratic temperature dependency of the metabolic rate
#Dyson, Freeman J. (1979): “Time without end: Physics and biology in an open universe*”, Reviews of Modern Physics, vol. 51, 3, 447-460
https://suli.pppl.gov/2019/course/RevModPhys.51.447.pdf
Quote: “A creature or a society with given Q and given temperature θ will dissipate energy at a rate (59)
Here m is the metabolic rate measured in ergs per second, k is Boltzmann's constant, and f [a constant]. It is important that m varies with the square of θ, one factor θ coming from the relationship between energy and entropy, the other factor θ coming from the assumed temperature dependence of the rate of vital processes.”
—Your brain runs at a temperature of 310 degrees above absolute zero. Zero is the temperature where nothing can happen anymore. At 310 degrees your brain needs 20 joules of energy to think a simple thought that takes one second to think, like “I should wear a funny hat”.
The normal temperature of the human body is around 37 degrees Celsius (°C) or 310 Kelvin (K). At this temperature, the human brain consumes around 20 watts of power, that is, it consumes about 20 joules per second.
#Vijay Balasubramanian (2021): “Brain power”, Proceedings of the National Academy of Sciences of the United States of America, vol. 118, 32
https://pmc.ncbi.nlm.nih.gov/articles/PMC8364152/
Quote: “The human brain is just 2% of the body’s weight, but 20% of its metabolic load (1–3), and 10 times more expensive per gram than muscle. On the other hand, the brain manages to produce poetry, design spacecraft, and create art on an energy budget of ∼20 W, a paltry sum given that the computer on which this article is being typed requires 80 W.”
—If your brain worked at colder temperatures, say 155 kelvin instead of 310, a few things would change. First of all, you’d think at half the speed and the same thought would now take two seconds. But in return, the energy you’d need to think the thought, now halves from 20 joules to only 10.
This is an immediate consequence of the scaling law postulated by Dyson:
#Dyson, Freeman J. (1979): “Time without end: Physics and biology in an open universe*”, Reviews of Modern Physics, vol. 51, 3, 447-460
https://suli.pppl.gov/2019/course/RevModPhys.51.447.pdf
Quote: “For a quantitative description of the way life may adapt itself to a cold environment, I need to assume a scaling law that is independent of any particular material embodiment that life may find for itself. The following is a formal statement of my scaling law:
Scaling Hypothesis. If we copy a living creature, quantum state by quantum state, so that the
Hamiltonian of the copy is
where H is the Hamiltonian of the creature, U is a unitary operator, and λ is a positive scaling factor, and if the environment is similarly copied so that the temperatures of the environments of the creature and the copy are respectively T and λT, then the copy is alive, subjectively identical to the original creature, with alI its vital functions reduced in speed by the same factor λ .[...]
From this point on, I assume the scaling hypothesis to be valid and examine its consequences for the potentialities of life. The first consequence is that the appropriate measure of time as experienced subjectively by a living creature is not physical time t but the quantity:
By differentiating this expression with respect to t, we find that the speed of thought is proportional to the temperature. That is, at half the temperature, you would need double the time to think the same thought.
One can also derive a formula for the metabolic rate from the scaling law postulated by Dyson:
#Dyson, Freeman J. (1979): “Time without end: Physics and biology in an open universe*”, Reviews of Modern Physics, vol. 51, 3, 447-460
https://suli.pppl.gov/2019/course/RevModPhys.51.447.pdf
Quote: “A creature or a society with given Q and given temperature θ will dissipate energy at a rate (59)
Here m is the metabolic rate measured in ergs per second, k is Boltzmann's constant, and f [a constant]. It is important that m varies with the square of θ, one factor θ coming from the relationship between energy and entropy, the other factor θ coming from the assumed temperature dependence of the rate of vital processes.”
Since the metabolic rate varies with the square of the temperature, if you work at half the temperature, you have a fourth of the metabolic rate. But, since you also work at half the speed, in total, it would take you half the energy to formulate the same thought.
—They have to leave biology behind and transfer their minds into some kind of mechanical or artificial brains
In his classic paper, Dyson reflects on how a civilization of conscious beings could exist infinitely by progressively lowering their temperature. However, Dyson is aware that this is not how living beings as we know them work. The progressive lowering of temperature and the scaling law do not apply to a single living being whose temperature is lowered, but rather to an ensemble of beings adapted to lower and lower temperatures:
#Dyson, Freeman J. (1979): “Time without end: Physics and biology in an open universe*”, Reviews of Modern Physics, vol. 51, 3, 447-460
https://suli.pppl.gov/2019/course/RevModPhys.51.447.pdf
Quote: “I should emphasize that the scaling law does not apply to the change of the metabolic rate of a given organism as a function of temperature. For example, when a snake or a lizard changes its temperature, its metabolic rate varies exponentially rather than linearly with T. The linear scaling law applies to an ensemble of copies of a snake, each copy adapted to a different temperature. It does not apply to a particular snake with varying T”
Further, he assumes that consciousness does not require a given material support and thus “life is free to evolve into whatever material embodiment best suits its purposes” and that a hyper-advanced form of consciousness could be “free to choose its temperature”. If the bodies of the Noxans are like the ones of life as we know it, they won’t be able to do this by themselves. The Noxans would need to transition to different bodies able to control their own temperature and change it in a way compatible with Dyson’s scaling law.
—Or even better, become completely virtual minds and build a digital world as good or better than the real one.
A virtual mind in a digital environment would be ideal for the Noxans as it could create the perfect environment for itself, while running on computer hardware that can resist very low operating temperatures.
For example, we have computer chips today that run at 0.003 Kelvin:
#University of Basel (2017): “The Coldest Chip in the World”
https://www.unibas.ch/en/News-Events/News/Uni-Research/The-coldest-chip-in-the-world.html
Quote: “Physicists at the University of Basel have succeeded in cooling a nanoelectronic chip to a temperature lower than 3 millikelvin.”
Note that even if Noxans can virtually insulate themselves from the outside world, they would need a physical computer to run the simulation. And, according to Landauer’s principle, there is a minimum amount of energy that must be consumed to perform irreversible operations that could be necessary in running such a simulation. If simulations that feel infinite require an infinite amount of operations, the Noxans will be limited by real-world energy consumption. So, virtual Noxans still have to worry about running out of energy, and so they still have to build their battery.
#Georgescu, Iulia (2021): “60 years of Landauer’s principle”, Nature Reviews Physics, vol.3, 12, 770
https://www.nature.com/articles/s42254-021-00400-8
Quote: “In 1961, Landauer showed that there is an absolute minimum thermodynamic cost associated with erasing one bit of information: kBTln2. Landauer observed that “logical irreversibility in turn implies physical irreversibility, and the latter is accompanied by dissipative effects”. Logical irreversibility means that an operation has multiple inputs, but a single output, a many-to-one map like an OR or AND. “Computation is inevitably done with real physical degrees of freedom, obeying the laws of physics,” Landauer later wrote in 1991. Thinking in terms of physical states, a logically irreversible operation will reduce the number of degrees of freedom and in so doing decrease entropy at the price of dissipating heat to the environment.”
For the original article by Landauer:
#Landauer, Rolf (1961): “Irreversibility and Heat Generation in the Computing Process”, IBM Journal of Research and Development, vol. 5, 3, 183 - 191
https://ieeexplore.ieee.org/document/5392446
—Well, this is at the core of the plan: Each time the Noxans get colder, they need less energy to stay alive – sure, they will be slower, but this is not relevant to them.
As we have explained above, thanks to the scaling law postulated by Dyson, the Noxans would require less and less energy and more and more time to think any thought as their temperature goes down.
As the speed of their thoughts slows down, their perception of the passage of time would slow down at the same rate, which means that the “slowing down” effect would be imperceptible to them.
—So from their perspective, as long as they can keep getting colder fast enough, their battery's energy is growing proportionally and could keep them alive forever.
After each hibernation period, the Noxans resume life even more slowly and at an even lower temperature. The energy they need to experience a moment of subjective time decreases further, so it is as if they had more energy left in their battery.
If the energy requirements of the Noxans decrease fast enough, the total energy spent across time will be finite. In more mathematical terms, the integral of their metabolic rate in time will converge.
As it turns out, there are choices of how to decrease the temperature that, in principle, would be fast enough and allow for the Noxans to exist forever and experience an indefinite amount of time.
#Dyson, Freeman J. (1979): “Time without end: Physics and biology in an open universe*”, Reviews of Modern Physics, vol. 51, 3, 447-460
https://suli.pppl.gov/2019/course/RevModPhys.51.447.pdf
Quote: “As an example of a possible strategy for a long-lived society, we ean satisfy the constraints (60) and (76) by a wide margin if we take:
where θ0 and t0 are the present temperature of life and the present age of the universe.
The exponent has to lie in the range
And for definiteness we take α = 3/8 [...]
The average rate of energy dissipation is by (75)
The total. energy metabolized over all time from t0 to infinity is
[...]
This example shows that it is possible for life with the strategy of hibernation to achieve simultaneously its two main objectives. First, according to (80), subjective time is infinite; although the biological clocks are slowing down and running intermittently as
the universe expands, subjective time goes on forever. Second, according to (83), the total energy required for indefinite survival is finite.”
The effects of hibernation, a process that we discuss in the following section, are already included in this calculation. The hibernation process is necessary to make the battery last forever. This cooling and hibernating strategy is compatible with the laws that dictate how fast a physical system can shed energy by radiating it to space, but, as we will see later, there are other factors that could prevent Noxans from actually enacting it.
—How do you cool down a light bulb? Well, you unplug it so it can radiate away its heat. So the Noxans switch off their brains and start to hibernate. They enter a state with no activity, thoughts or even dreams, and let their “bodies” radiate heat away to cool down a bit. When they activate again after some time, they are now colder than before. They think slower and each thought uses up less energy from their battery.
Dyson considered that beings of an advanced civilization like the Noxans would be “free to choose its temperature”, possibly thanks to their mechanical bodies. By our choice of the cooling rate, the Noxans cool down while they are awake as well thanks to this ability to choose their temperature. We have omitted this detail from the video to keep the explanation simple.
However, only cooling down while active, even at the maximum efficiency allowed by physics, would still not be enough to make the cosmic battery last forever. To be able to maintain a civilization of a given complexity forever, Noxans would have to alternate periods of activity with hibernation in which their metabolism effectively stops.
#Katherine Freese and William H. Kinney (2003): “The Ultimate Fate of Life in an Accelerating Universe”, Physics Letters B, vol. 558, 1–2, 1–8
https://arxiv.org/pdf/astro-ph/0205279
Quote: “The creature will fry to death unless it can dissipate the heat E that it creates; dissipation by radiation implies a lower limit on the operating temperature for the organism:
Since the ratio (Q/Ne) between the complexity of the society and the number of electrons at its disposal cannot be made arbitrarily small, there must be a finite minimum temperature for which computation is possible. Therefore Dyson’s condition cannot be satisfied, and the creature or society cannot survive indefinitely. However, Dyson proposes a strategy to avoid this sad conclusion: hibernation. Life may find a way to metabolize intermittently, yet continue to radiate waste heat into space during its periods of hibernation. The society can remain active for a fraction g(t) of its time while hibernating for the remaining 1 − g(t) fraction of the time. [...]
Therefore the temperature of the organism can drop below Tmin in Eq. (7) and the heat generated by the computation can still be dissipated: the condition (7) becomes:
The organism is free to choose g(t) and T (t) to satisfy Dyson’s condition.”
Adapting to a lower operating temperature after each hibernation period may require important changes to the body of the Noxians or even an entirely new body and way of living.
Note: To be able to show the normal flow of time, we have chosen to make the assistant robot Shirley (image below) move and act at a normal pace, as well as to stay awake during the hibernation process. This is a creative liberty we have taken. In reality any physical system, whether or not it has a biological origin, would have to follow the same slow-down and hibernation processes as the Noxans to stay active with finite energy for an infinite time.
—whatever type III civilizations think fun is in the future.
A Type III civilization on the Kardashev scale masters all the energy and resources of their galaxy.
#Encyclopedia Britannica: “The Fermi Paradox: Where Are All the Aliens?” (retrieved 2025)
https://www.britannica.com/story/the-fermi-paradox-where-are-all-the-aliens
Quote: “To break these numbers down further, scientists use the Kardashev scale, which splits intelligent life into three categories. Type I civilizations are able to use all the energy available on their home planet (we are approaching this; most scientists agree that we are currently at a 0.7 on the Kardashev scale, with a full Type I being about a century off). Type II civilizations can control and channel all the energy of their host star, and Type III civilizations have access to power equivalent to that of their host galaxy.”
—And during the night they are switched off, lost in dreamless non-existence, cooling down their brains even more.
During hibernation, the Noxans are completely unconscious, so they do not experience the passage of time.
—At the beginning the nights will be very short and the days extremely long. A day can last a million years while night is only a few hours.
#Dyson, Freeman J. (1979):”Time without end: Physics and biology in an open
universe*”, Reviews of Mode in Physics, vol. 51, 3, 447-460
https://suli.pppl.gov/2019/course/RevModPhys.51.447.pdf
Quote: “Suppose then that a society spends a fraction g(t) of its time in the active phase and a fraction [1-g(t)] hibernating. [...]
As an example of a possible strategy for a long-lived society, we ean satisfy the constraints (60) and (76) by a wide margin if we take
where θ0 and t0 are the present temperature of life and the present age of the universe.
The exponent has to lie in the range
And for definiteness we take α = 3/8”
We take t0 to be the time where the future civilization starts the process of cooling down, which we have chosen to take place when the Universe is 100 trillion years old. We have also chosen the starting temperature of the artificial bodies of Noxans at 300 K, or 300 °C above absolute zero. This is the temperature from which they start cooling down.
Though there are many configurations of how the cosmic day-night cycle could be set up, we have chosen it so that the Noxans always experience a million years of subjective time between night cycles.
In the first iteration, they are approximately at their original temperature, and physical and subjective time roughly correspond. The fraction of time awake is given by:
g=(time of first sleep/ time when the cooling begins)-3/8=((1014+106) /1014)-3/8=0.999999996
Since g is by definition:
g=duration of the “day”/(duration of the “day” + duration of the “night”)
Then, the duration of the “night” is:
duration of the night = duration of the day × ((1/g) - 1) =
= 106 years × ((1/0.999999996) - 1) = 0.004 years ~ 35 hours
—But warm things shed heat much faster than cold ones – cooling from 300 to 299 degrees takes much less time than going from 100 to 99. So as the Noxan brains become colder, the nights will have to grow longer and longer.
We have chosen the starting temperature of the artificial bodies of Noxans at 300 K. This is the temperature from which they start cooling down.
Ultimately, all energy is converted into waste heat that is removed by radiating it into the Universe. Dyson calculates an upper limit for the radiative power of any body:
#Dyson, Freeman J. (1979): “Time without end: Physics and biology in an open universe*”, Reviews of Modern Physics, vol. 51, 3, 447-460
https://suli.pppl.gov/2019/course/RevModPhys.51.447.pdf
Quote: “There is an absolute upper limit
on the power that can be radiated by a material radiator containing N electrons at temperature θ .”
This radiative power depends on the cube of the temperature, meaning that bodies at a higher temperature can radiate heat at a much higher rate than bodies at a lower temperature.
This imposes an upper limit on the metabolic rate of a conscious civilization of a given size, and as a consequence, a lower bound to its temperature, unless they resort to hibernation.
#Katherine Freese and William H. Kinney (2003): “The Ultimate Fate of Life in an Accelerating Universe”, Physics Letters B Volume 558, Issues 1–2
https://arxiv.org/pdf/astro-ph/0205279
Quote: “Life may find a way to metabolize intermittently, yet continue to radiate waste heat into space during its periods of hibernation. The society can remain active for a fraction g(t) of its time while hibernating for the remaining 1 − g(t) fraction of the time. [...]
Therefore the temperature of the organism can drop below Tmin in Eq. (7) and the heat generated by the computation can still be dissipated: the condition (7) becomes:
The organism is free to choose g(t) and T(t) to satisfy Dyson’s condition.”
Since the temperature needs to get smaller with time to use less and less energy, the fraction of the time that the civilization is “awake”, g, must also decrease with time.
—After a 100 trillion years of this cycle, their temperature has only dropped to 230 degrees above absolute zero. A simple thought now takes 1.3 seconds instead of 1. Each cosmic night is now 400,000 years long.
#Dyson, Freeman J. (1979): “Time without end: Physics and biology in an open universe*”, Reviews of Modern Physics, vol. 51, 3, 447-460
https://suli.pppl.gov/2019/course/RevModPhys.51.447.pdf
Quote: “Suppose then that a society spends a fraction g(t) of its time in the active phase and a fraction [1-g(t)] hibernating. The cycles of activity and hibernation should be short enough so that g(t) and θ(t) do not vary appreciably during any one cycle. [...]
As an example of a possible strategy for a long-lived society, we ean satisfy the constraints (60) and (76) by a wide margin if we take
where θ0 and t0 are the present temperature of life and the present age of the universe.
The exponent has to lie in the range
And for definiteness we take α = 3/8”
We take t0 to be the time where the future civilization starts the first hibernation cycle, which we have chosen to take place where the universe is 100 trillion years old. After another 100 trillion years, t is 200 trillion years. θ(t) is the temperature of the Noxans at that point in 200 trillion years, while θ0 is the starting temperature of 300 Kelvin, the temperature we have chosen to be the starting temperature of the artificial bodies of Noxans.
Then, from (77):
θ(t) = (200 trillion years / 100 trillion years) -3/8 × 300 Kelvin = 231 Kelvin.
Since by the hypothesis of Dyson's Cold Thoughts, the speed of their thoughts is proportional to their temperature, it takes them 300/231 ~ 1.30 seconds to think a thought they would have thought in a second before the cooling process started.
Though there are many configurations of how the cosmic day-night cycle could be set up, we have chosen it so that the Noxans always experience a million years of subjective time between night cycles.
By the previous calculation that means that the cosmic day must be 1.3 million years old when the universe is 200 trillion years old. Then, since
g= duration of the “day”/(duration of the “day” + duration of the “night”)
and by (77)
g(t) = (200 trillion years / 100 trillion years) -3/8 ~ 0.771.
the duration of the “night” is:
duration of the night = duration of the day × ((1/g) - 1) =
~ 1.3 ×106 years × ((1/0.771) - 1) ~ 390,000 years
or, approximately 400,000 years.
—But even accounting for their slower speeds and long hibernations, the Noxans have experienced the equivalent of 76 trillion years of life. Over 5000 times the age of the current universe.
#Dyson, Freeman J. (1979): “Time without end: Physics and biology in an open universe*”, Reviews of Modern Physics, vol. 51, 3, 447-460
https://suli.pppl.gov/2019/course/RevModPhys.51.447.pdf
Quote: “Subjective time [measured form the beginning of the universe] becomes:
is the present age of the universe measured in moments of consciousness.”
Note we use a different value of t0 from Dyson. For us, t0 is the moment where the Noxans start their cooling, when the universe is 100 trillion years old. This means that, for us, A=4×1014 years. The amount of subjective time that the Noxans have experienced since the start of the cooling is
u(t)-u(t0)= A((t/t0)1/4-1)= 4 ×1014 × (21/4-1) ~ 7.6×1013 years = 76 trillion years
Compared to the current age of the universe:
7.6×1013 years/14×109 years ~ 5400
#NASA’s Goddard Space Flight Center (2017): “Age & Size of the Universe Through the Years” (retrieved 2025)
https://imagine.gsfc.nasa.gov/educators/programs/cosmictimes/educators/guide/age_size.html
Quote: “Results from the WMAP satellite further confirmed and refined the age of the Universe to be 13.7 billion years.”
—After quadrillions of years, the corpses of former stars, white dwarfs, turn yellow, then orange, then red, then black.
#John D. Barrow and Frank J. Tipler (1986): “The anthropic cosmological principle”
https://global.oup.com/academic/product/the-anthropic-cosmological-principle-9780192821478
—Trillions of trillions of years later: All galaxies have dissolved like dandelions.
#John D. Barrow and Frank J. Tipler (1986): “The anthropic cosmological principle”
https://global.oup.com/academic/product/the-anthropic-cosmological-principle-9780192821478
—Decillions of decillions of years later: The first black holes evaporate in bursts of light like sad fireworks.
#John D. Barrow and Frank J. Tipler (1986): “The anthropic cosmological principle”
https://global.oup.com/academic/product/the-anthropic-cosmological-principle-9780192821478
—Their temperature is now a chilling 10 quintillionths of a kelvin. They’ve become so slow that a simple thought takes a trillion years. They spend 99.999999999999999997% of the time hibernating.
#Dyson, Freeman J. (1979): “Time without end: Physics and biology in an open universe*”, Reviews of Modern Physics, vol. 51, 3, 447-460
https://suli.pppl.gov/2019/course/RevModPhys.51.447.pdf
Quote: “Suppose then that a society spends a fraction g(t) of its time in the active phase and a fraction [1-g(t)] hibernating. The cycles of activity and hibernation should be short enough so that g(t) and θ(t) do not vary appreciably during any one cycle. [...]
As an example of a possible strategy for a long-lived society, we can satisfy the constraints (60) and (76) by a wide margin if we take
where θ0 and t0 are the present temperature of life and the present age of the universe.
The exponent has to lie in the range
And for definiteness we take α = 3/8”
We take t0 to be the time where the future civilization starts the first hibernation cycle, which we have chosen to take place where the universe is 100 trillion years old. The age of the universe at that point, t, is 1066 years. θ(t) is the temperature at that point, while θ0 is the starting temperature of 300 Kelvin, the starting temperature of the artificial bodies of Noxans.
By (77):
θ(t) = (1066 years / 1014 years)-3/8 × 300 Kelvin = 9.5 × 10 -18 Kelvin.
Since by the hypothesis of Dyson's Cold Thoughts, the speed of their thoughts is proportional to their temperature and by (77):
θ(t) /θ0 = (1066 years / 1014 years)-3/8 = 3.2 × 10 -20
it takes them 1/(3.2 × 10 -20) = 3.2 × 10 -19 seconds, or around one trillion years, to think a thought they would have thought in a second before the cooling process started. The fraction of the time they spend hibernating is about 1-g= 1- (3.2 × 10 -20)= 0.99999999999999999997, or 99,999999999999999997%.
—But for them, still nothing has changed. Their conscious experience is just as fluid and vibrant as on the first day.
As we have explained above, thanks to the scaling law postulated by Dyson, the Noxans would require less and less energy and more and more time to think any thought as their temperature goes down.
As the speed of their thoughts slows down, their perception of the passage of time would slow down at the same rate, which means that the “slowing down” effect would be imperceptible to them.
—And while the cosmic nights are now a quattuordecillion years long, it doesn’t matter to them.
Though there are many configurations of how the cosmic day-night cycle could be set up, we have chosen it so that the Noxans always experience a million years of subjective time between night cycles.
By the calculations of the previous section of this document, at this point the Noxans take around one trillion years to experience one second of subjective time. To experience a million years, the physical time that has to pass is:
1012 physical years per subjective second × 365 days/year × 24 hours/day × 60 minute/hour× 60 second/minute ×106 subjective years=
=3.2×1025 physical years
Since g is by definition:
g= duration of the “day”/(duration of the “day” + duration of the “night”)
and as calculated in the previous section:
g(t)=θ(t) /θ0 = (1066 years / 1014 years)-3/8 = 3.2 × 10 -20
Then, the duration of the “night” is:
duration of the night = duration of the day × ((1/g) - 1) =
= 3.2×1025 physical years × ((1/(3.2 × 10 -20)) - 1) ~ 1045 years
That is, a quattuordecillion years.
—At this point, if they could have lived, felt and experienced the equivalent of 4,000 trillion trillion years.
#Dyson, Freeman J. (1979): “Time without end: Physics and biology in an open universe*”, Reviews of Modern Physics, vol. 51, 3, 447-460
https://suli.pppl.gov/2019/course/RevModPhys.51.447.pdf
Quote: “Subjective time [measured form the beginning of the universe] becomes:
is the present age of the universe measured in moments of consciousness.”
Note we use a different value of t0 from Dyson. For us, t0 is the moment where the Noxans start their cooling, when the universe is 100 trillion years old. This means that, for us, A=4×1014 years. The amount of subjective time that the Noxans have experienced since the start of the cooling is
u(t)-u(t0)= A((t/t0)1/4-1)= 4 ×1014 × ((1066/1014)1/4-1) ~ 4 ×1027 years = 4,000 trillion trillion
—Their battery has only a tiny fraction of its original charge, but the Noxans now need so little energy that it seems larger than ever before.
#Dyson, Freeman J. (1979): “Time without end: Physics and biology in an open universe*”, Reviews of Modern Physics, vol. 51, 3, 447-460
https://suli.pppl.gov/2019/course/RevModPhys.51.447.pdf
Quote: “The average rate of energy dissipation is by (75)
Integrating over time, we can find the cumulative energy expense at a given point in time t.The result is:
M(t) =Minf(1-(t0/t)1/8)
Where Minf is the total energy spent in infinite time. Since we take to at the start of the cooling process, when the universe is 100 trillion years old, and the current time is 1066 years, we have used up:
M(t)/ Minf =(1-(1014/1066)1/8)=0.9999997=99.99997% of the battery.
Yet, because the energy expenses of the Noxans keep getting lower, they can do more and more with ever smaller portions of the battery.
—There are a few problems – dark energy might eventually make the universe hit a temperature limit of 10–30 degrees above absolute zero.
We talked about other consequences of the evolution of a cosmological-constant-dominated universe in this video:
#Kurzgesagt – In a Nutshell (2023): “Are You A Dream Of The Universe?”
This kind of universe reaches a point in its evolution where the very energy of empty space creates a sort of “background radiation” that heats space to a temperature that, while tiny, is strictly above zero Kelvin.
#Carroll, Sean (2020): “Why Boltzmann Brains Are Bad”. Dasgupta Shamik et al. (eds), Current Controversies in Philosophy of Science, Routledge.
https://arxiv.org/abs/1702.00850
Quote: “If the vacuum energy remains constant, the universe asymptotically approaches a de Sitter phase. Such a spacetime resembles, in certain ways, a box of gas in equilibrium, with a de Sitter temperature TdS = √(Λ/12π2) ~ 10−33 eV. (I will use units where ħ = c = k = 1 unless explicitly indicated.)”
The value above is expressed in units of energy since the calculation is done in “natural units”, in which the fundamental constants ħ = c = k = 1. To express the value in electronvolts (eV) in Kelvin, we just have to reinstate the Boltzmann constant by dividing by its value in eV/K, k ≈ 8.6 × 10–5 eV/K. This gives a temperature of the order of 10–29 K, which for simplicity has been rounded to 10–30 K .
Noxans wouldn’t be able to lower their temperature anymore once they reach this temperature, because they would receive just as much radiation from space as they emit to cool off.
—The Noxans will cool to that level after a googol years, but then they can’t cool any further and their battery will begin to run out.
#Dyson, Freeman J. (1979): “Time without end: Physics and biology in an open universe*”, Reviews of Modern Physics, vol. 51, 3, 447-460
https://suli.pppl.gov/2019/course/RevModPhys.51.447.pdf
Quote: “Suppose then that a society spends a fraction g(t) of its time in the active phase and a fraction [1-g(t)] hibernating. The cycles of activity and hibernation should be short enough so that g(t) and θ(t) do not vary appreciably during any one cycle. [...]
As an example of a possible strategy for a long-lived society, we can satisfy the constraints (60) and (76) by a wide margin if we take
where θ0 and t0 are the present temperature of life and the present age of the universe.
The exponent has to lie in the range
And for definiteness we take α = 3/8”
Form (77):
θ/θ0=(t/t0)-3/8
So, given that the starting temperature of artificial body of the Noxans is 300K, and that we take t0 —the moment where the cooling process starts— to be when the universe is 100 trillion years old, they reach the limit temperature at:
t= (θ/θ0)-8/3 × t0=(10-30/300)-8/3 × 1014 years = 4.0 × 10100 years
—Quantum noise might just destroy their brains, all matter may decay into iron or proton decay will destroy all atoms themselves.
On top of the temperature limit imposed by the cosmological constant, many other obstacles stand in the way of the Noxans.
For starters, quantum noise may destroy the data of whatever system they have chosen to use to simulate or “store” their brains.
We thank our expert Matt Caplan for the following comment:
Quote: “Eventually quantum noise will wreck whatever record keeping they're using.”
Further, even if protons do not decay via any of the mechanisms proposed by GUTs before the Noxans reach the temperature limit, they will decay via the Hawking process when the universe is 10122 years old, robbing Noxans of the atoms they would have built their computer or mechanical body with. Even if the Noxans somehow survive this, they might not survive the decay of all matter into iron at age 101500, assuming that ordinary matter has not been destroyed before.
#John D. Barrow and Frank J. Tipler (1986): “The anthropic cosmological principle”
https://global.oup.com/academic/product/the-anthropic-cosmological-principle-9780192821478
—But even if the Noxans don’t get a true forever, if Dyson's Cold Thought works in principle, then they could go on for something like a googol years old. This is the potential for consciousness to experience the equivalent of two trillion, trillion, trillion years.
#Dyson, Freeman J. (1979): “Time without end: Physics and biology in an open universe*”, Reviews of Modern Physics, vol. 51, 3, 447-460
https://suli.pppl.gov/2019/course/RevModPhys.51.447.pdf
Quote: “Suppose then that a society spends a fraction g(t) of its time in the active phase and a fraction [1-g(t)] hibernating. The cycles of activity and hibernation should be short enough so that g(t) and θ(t) do not vary appreciably during any one cycle. [...]
As an example of a possible strategy for a long-lived society, we can satisfy the constraints (60) and (76) by a wide margin if we take
where θ0 and t0 are the present temperature of life and the present age of the universe.
The exponent has to lie in the range
And for definiteness we take α = 3/8”
Form (77):
θ/θ0=(t/t0)-3/8
So, given that the starting temperature of the Noxans’s artificial bodies is 300K, and that we take t0 —the moment where the cooling process starts— to be when the universe is 100 trillion years old, they reach the limit temperature at:
t= (θ/θ0)-8/3 × t0=(10-30/300)-8/3 × 1014 years = 4.0 × 10100 years
The subjective time they have experienced by then is:
#Dyson, Freeman J. (1979): “Time without end: Physics and biology in an open universe*”, Reviews of Modern Physics, vol. 51, 3, 447-460
https://suli.pppl.gov/2019/course/RevModPhys.51.447.pdf
Quote: “Subjective time [measured form the beginning of the universe] becomes:
is the present age of the universe measured in moments of consciousness.”
Note we use a different value of t0 from Dyson. For us, t0 is the moment where the Noxans start their cooling, when the universe is 100 trillion years old. This means that, for us, A=4×1014 years. The amount of subjective time that the Noxans have experienced since the start of the cooling is
u(t)-u(t0)= A((t/t0)1/4-1)= 4 ×1014 × ((4.0 × 10100/1014)1/4-1) ~ 1.8×1036 years ~ 2 trillion trillion trillion
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