Chapter 6: The Entropic Principle

Faith Across the Multiverse by @AndyWalsh, 2018 by Hendrickson Publishers, Peabody, Massachusetts. Used by permission. All rights reserved.

The Dying and Living of the Light

Having discussed entropy and information and their relationship to sunlight, we are now ready to consider how that informs our understanding of Jesus. To start, we go all the way back to the beginning, when the world was described as “without shape” (Genesis 1:2), which sounds an awful lot like an unstructured, high entropy state. Creation introduces asymmetry so that transformations become meaningful, the universe becomes dynamic, and life becomes possible. We don’t know precisely what the very first moment of the universe looked like or how it came to be that way. The earliest state of the universe we can describe had intermediate entropy and proceeded to higher overall entropy states, which nevertheless show signs of macroscale organization.

The key to understanding this observation is not conflating all forms of organization and actual thermodynamic entropy. The famous cosmic microwave background radiation actually indicates that kinetic energy was distributed symmetrically with little structure, which is not the lowest possible entropy state. With respect to mass and gravity, however, this even distribution is low in entropy because matter will tend to go from such an arrangement toward more clumps and clusters. Raising entropy with respect to gravity creates opportunities for kinetic energy to be gathered and put to further use. In other words, creation stems from a balance of the attractive force of gravity and the tendency of kinetic energy to spread everything further apart.

From there, let’s look at some of my favorite bits in the Bible, the first chapter of the Gospel of John. It also provides an account of the beginning of the universe, albeit in more abstract terms. “The Word was with God in the beginning. All things were created by him, and apart from him not one thing was created that has been created. In him was life, and the life was the light of mankind” (John 1:2–4). How intriguing that the world is described as being created by the Word, or alternatively by information. As we noted earlier, processing information is a way to go from a disordered state to an ordered one, and here we see the source of that information identified as the Word of God. And then we see the connection to light and life, just as the light from the sun brings the order needed to preserve life here on Earth.

I want to reiterate that I am not asserting that these Bible passages offer a detailed account of the exact process by which the world came to be as it is now or was then. In particular, I would expect there to be particulars of the mechanism of universe-formation that can’t be directly mapped to verses or phrases from the Bible. Detailed scientific accounts, in terms we would recognize today, would have been incomprehensible to the original audience, so I wouldn’t really expect to find them in the Bible. The purpose of both the Genesis and John passages is primarily to indicate that God’s agency was involved; figuring out the exact details of the mechanisms he used are left as an exercise to the reader. I would expect to see general themes that can be expressed in different ways to different people at different times. Here the theme is what we need and how God provides it, and I find it striking that we can talk about that in terms of Λόγος (logos), the Greek concept rendered as “Word” in most English translations, and that we can talk about it in terms of order, information, and entropy.

And the connections don’t stop there. As we explored with our description of the sun’s activities, the low entropy photons, or light, from the sun are required to sustain life on earth. And recall that this can be viewed as an influx of information. So when we read “Man does not live by bread alone, but by every word that comes from the mouth of God” (Matthew 4:4) we can appreciate a new sense of what that means. An ongoing input of order and information are indeed necessary to sustain life biologically.

Then we get to the death of Jesus. Put simply, Jesus died so that we might live, just as the sun is exhausting itself to sustain us. In addition, we described death as a process of disseminating stored information. Likewise, the death of Jesus was a key reason why the gospel message spread as widely as it did.

I think this entropic perspective helps us get our heads around one of the more challenging ideas of the gospel—that we are called to die just as Jesus did. Jesus said, “If anyone wants to become my follower, he must deny himself, take up his cross, and follow me” (Matthew 16:24). That’s strong language, and yet most of his followers have not literally been crucified, nor faced a martyr’s death of any sort. It doesn’t take much reflection to realize that if every Christian died in that way, Christianity probably wouldn’t have made it past a couple of generations at best. But I don’t think that gets us out of having to take that command seriously.

Or consider the words of the apostle Paul, who encourages believers “to present your bodies as a sacrifice—alive, holy, and pleasing to God” (Romans 12:1). An animal presented for sacrifice would have been killed and either eaten or burnt up. Grain offerings would likewise be consumed in one fashion or the other. So what exactly would it mean to be a sacrifice and yet remain alive?

A discussion of entropy here, which will certainly prompt what you precisely mean by it @AndyWalsh. Help us out?

I understand that @AndyWalsh is writing a book for popular audience from the point of view of a biologist, but as a physicist I have to pick at this a little. This is less of a criticism but more an input to make @AndyWalsh’s book and arguments to be as strong as possible.

I don’t know what is meant by “with respect to mass and gravity”, but for the typical definition of entropy, this statement is wrong. It is well known that a ball of gas decreases in entropy as it undergoes gravitational collapse. That’s why in cosmological structure formation, gravitational collapse of baryons is mediated by interactions with photons that can cool the gas and store entropy. Also, note that the situation is much different for dark matter, which do not interact with photons, but also do not provide pressure support by colliding with each other.

Further complicating the issue, the picture of gravitational collapse of gas from some evenly distributed (in space) initial condition is ill-defined in Newtonian theory, and only makes sense in cosmology due to the general relativistic expansion of spacetime.

Hmm, I don’t know what is meant by “Raising entropy with respect to gravity”, or how it “creates opportunities for kinetic energy to be gathered and put to further use”. Usually gravitational potential is not put in the entropy but in the chemical potential. Perhaps this is what @AndyWalsh meant.

Again, the key part in cosmological structure formation is that fields play a role. Typically this is the E&M fields that can carry kinetic energy and entropy away from the collapsing gas. The story is not complete with just gravity (that collapses things) and the kinetic energy of particles (that spread things apart).

Further, in cosmology, gravity is not entirely attractive. The expansion of spacetime and its acceleration is a purely gravitational effect. The expansion allows the gas to cool adiabatically and promotes collapse in a manner that is completely different than the usual gravitational attraction. In contrary, the gravitational acceleration of the expansion actually hinders collapse.

This statement is wrong, the photons from the sun is blackbody, i.e. it is in thermal equilibrium. It has the highest possible entropy for its amount of energy. In a quantum language, it is a mixed state with a thermal density matrix. Perhaps this could be rescued by saying that it is still lower entropy than typical entropies of systems on Earth, but I don’t known if that’s actually a true statement. Nevertheless, it is wrong to call blackbody photons “low entropy photons”.

The physical picture is not “low entropy” photons being exported to the Earth from the Sun, thus allowing useful energy to be generated, but rather that there is a huge difference in temperature between the Earth and the Sun, which allows energy from the Sun to be converted to useful energy on Earth with minimal loses.


Well, despite what it might seem based on the feedback from PdotdQ above, I mean (or intend to mean) the same thing as everybody else. In the book, I rely most on the definition of thermodynamic entropy in terms of state space volume and number of macroscopically equivalent states. This relates to information in that the more equivalent states there are, the more information is necessary to specify the full state given the macroscopic state.

For analogy purposes, I mainly focus on the more general concept of symmetry, in an attempt to avoid associating thermodynamic entropy with nonthermodynamic phenomena.

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Thanks for the feedback, and for this framing. For what it’s worth, and I know this is probably and author cliché, but given the complexity of the topic I think this chapter in particular works best as a whole.

On this topic, my initial source is the writing of Roger Penrose in books like Cycles of Time. He refers to the early state of the universe as intermediate with respect to entropy, given the thermal equilibrium (high entropy) and evenly distributed mass (low entropy). Perhaps language focused on gravitational degrees of freedom rather than gravitational entropy could be clearer?

Here is a more convenient Internet reference for the same topic that I just came across. I will note that the author probably agrees with you; still, his discussion of the issue lays out the kind of reasoning that led me to write what i wrote.

Perhaps unraveling the “gravitational entropy” / “gravitational degrees of freedom” issue from above will clarify this also; if not, let me know.

To this & the subsequent discussion, I can’t really add much more than to say that you are correct and that I was attempting to simplify, and thus perhaps oversimplify, in the interest of not getting too technical. I don’t think the additional details would change any of my larger points, but yes I could have provided a more complete picture.

All usage of “high” and “low” are intended as relative, so yes, this is what I am aiming for. If it is still not accurate, that’s obviously an issue.

This is not correct. Gravity changes entropy in two ways: through its own degrees of freedom, and by causing interactions in the collapsing matter. In cosmological structure formation, the gravity is completely Newtonian, and thus gravitational degrees of freedom is completely neglected. In other words, in structure formation, the change in entropy due to gravity has nothing to do with the gravitational degrees of freedom, but everything to do with interaction.

If you want to include “gravitational degrees of freedom”, then the work is cut out for you, as while entropy for gravitational horizons are known, the entropy of gravitational fields (i.e. spacetime) away from horizons is still an active area of research.

Note that in cosmological structure formation, there are two collapses:

  1. Collapse of dark matter, in which photons (EM fields) barely plays a role (it does a little bit)
  2. Collapse of baryonic matter, in which photons plays a large role

It might be useful (or at least interesting) to distinguish how entropy works in these two types of collapses:

  1. For dark matter, the entropy curve is maximum when the system obeys virial equilibrium. In terms of collapse, the entropy goes up as the system collapses, reaches its peak when the system reaches a radius that is uncreatively named the virial radius, and is thus halted from collapse. These “virialized” systems are nevertheless really large (much, much larger than a galaxy), and are not the structures that we usually associate with the night sky (i.e. galaxies, clusters, stars, etc).
  2. Baryons at first follows the dark matter, and reaches a size that is comparable to this virial radius. Then, interaction with E&M fields occur as I laid out in my previous post. This allows the storing of entropy in electromagnetic degrees of freedom, which allows the collapse of baryons to proceed to produce the wonderful structures that we have.

All of this is also made more complicated by the fact that spacetime is expanding during this collapse.

First of, I am not sure that it is true that photon from the Sun in general is lower entropy than typical systems on Earth per energy, but this seems intuitively true. However, this picture of photons on Earth being able to be used to produce useful energy on Earth due to its low entropy is not the correct physical picture (or at least not how a physicist would think of it).

This picture is not complete without referring to the massive temperature difference between the Earth and the Sun. By the 1st law, the 2nd law penalty to create useful energy from heat is temperature dependent. Stating “low entropy” photons really doesn’t say anything in terms of the useful energy that can be harvested from sunlight for the purposes of life.

Again @AndyWalsh, this is not a criticism, but something that I thought might make your book more accurate and compelling, especially to physicists and engineers who are over-represented amongst your audience: people who enjoy things like axioms, sci-fi, and comic books.

Thanks for the further clarifications on the relationship of gravity and entropy. I’m still not entirely sure how to reconcile all of that with what I’ve read elsewhere. But I appreciate that you have much more insight into this topic. So I will keep trying to sort it all out.

As for the relationship of the sun and the earth as mediated via light, I’m wondering if you can comment on the observation that the light arriving from the sun is via fewer, higher energy photons from a specific spatial direction, while the light departing the earth is via more lower energy photons in all directions. I understand this to mean that the arriving photons have fewer degrees of freedom. Is that correct? If so, why can that not be connected to the relative entropy of those photons?


I will be happy to help you make this reconciliation. From a personal standpoint, I am also curious of your sources if they claim that the entropy of “gravitational degrees of freedom” is important in cosmological structure formation - this is well known to be false.

First of, photons are quantum objects, so their entropy is given by the von Neuman entropy instead of the usual Boltzmann S= k ln Ω (where Ω is the number of microstate). The entropy of a photon is therefore related to how mixed the quantum state of the photon is. Therefore, the proper way to answer this question is to compare the quantum states of incoming photons from the Sun with the quantum states of emitted photons from Earth. A shortcut might be made due to the photons being blackbodies.

I did not do this exercise, but I don’t need to to know what you will get. Because I believe in the 2nd law, I am sure that in the end you will get that all re-radiated photons from the Earth will have greater entropy than all the received photons from the Sun. Edit: actually, I forgot about the entropy stored in the Earth itself, so one cannot use the 2nd law as a shortcut here. I edited the required calculation at the end of this post.

Regardless, my point is that comparing entropies is not useful to assess the ability for systems on Earth to use energy from the Sun. It is hard to explain this without a concrete example in mind, so try computing the amount of energy that is necessarily wasted due to the 2nd law from a heat pump with the Sun as the high temperature reservoir and the Earth as the low temperature reservoir.

Now do the problem again with a “faint sun” of temperature only 500K. I hope you will then be convinced that the huge difference in temperature between the Earth and the Sun plays a major role in the ability for life to harvest useful energy from the Sun. It’s not all in the entropy difference, and that without a reference to the temperatures, this exercise of “comparing entropies” is really physically meaningless!

Edit: @AndyWalsh, here is the calculation for whether arriving photons from the Sun have less entropy than emitted photons from the Earth

  1. Imagine a point on the surface of the Sun. The photons leaving this point are blackbody, therefore the entropy is given by (4/3)U/Ts, where U is the energy of the photon gas, and Ts is the effective temperature of the Sun (~5777K)
  2. Photons arrive on Earth from a particular direction. In particular, from the point of view of the Sun it is confined to a solid angle 4Re²/AU², where Re is the radius of the Earth and AU the astronomical unit. Therefore the flux of energy being sent to Earth is: F= σTs⁴ x (4Re²/AU²)/(4π).
  3. Remember, that was flux from a single point on the surface of the Sun. We integrate over the Earth-facing surface of the sun to get the total flux to be F= σTs⁴ x (4Re²/AU²)/(4π) x (2πRs²), where Rs is the radius of the Sun.
  4. From 1. the flux of entropy received on Earth is therefore: F_S(Sun) = (4/3)σTs³ x (4Re²/AU²)/(4π) x (2πRs²). This quantity has units of “rate of entropy change”, Entropy/(Time).

Now, what is the rate of entropy change due to the Earth’s radiating thermal blackbody radiation?

  1. The luminosity of the Earth is Le = 4πRe²σTe⁴. Te is the effective temperature of the Earth (~300K).
  2. Since this is also blackbody, then the rate of entropy change is F_S(Earth) = -(16/3)πRe²σTe³. The negative sign is there because this is entropy loss from the Earth.

Is the entropy received from the Sun larger or smaller than entropy loss from the Earth’s infrared radiation?

  1. The net rate of change of entropy is given by R_S = F_S(Sun) + F_S(Earth)
  2. R_S = (4/3)σTs³ x (4Re²/AU²)/(4π) x (2πRs²) - (16/3)πRe²σTe³
  3. Collecting like terms, R_S = (16/3)Re²σπ x [Ts³2Rs²/(4πAU²) - Te³]
  4. The entropy emitted by the Earth is therefore larger than that received from the Sun if Te³>Ts³2Rs²/(4πAU²)
  5. The temperature term signifies that larger temperature photons have larger entropies, while the 2Rs²/(4πAU²) term signifies the entropy penalty due to the photons coming from a particular source (a small angular area)
  6. 2Rs²/(4πAU²) is ~10⁻⁶, which means that Ts³2Rs²/(4πAU²)~664000 cubic Kelvin
  7. Te³~27 million cubic Kelvin
  8. Therefore, Te³>Ts³2Rs²/(4πAU²) and the entropy emitted by the Earth is larger than that received by the Sun

This was a fun exercise, but as I talked about in the body of this post, comparing “entropies” this way really does not have any physical meaning with regards to converting the Sun’s heat to usable energy without a reference to the large temperature difference between the Earth and the Sun.

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As far as sunlight and life on earth Denton’s new book Children of Light presents some interesting information.

Well, @PdotdQ, you definitely win the “above and beyond the call of duty” award. I appreciate you putting in all of that effort to educate me (and hopefully some other folks in the forum). Clearly my descriptions were incomplete–which was always likely, given the nature of the project, but it’s helpful to have more details on exactly what is missing. At the video chat on Saturday, I will be sure to encourage folks to check out this thread so they can benefit from your work as well.

It would be interesting to play around with the different components to see under what conditions the Sun/Earth relationship flips. In particular, I wonder how big of an angular area the Sun would have to occupy to reverse things. Maybe it will tell us something about why the Sun is the same apparent size as the moon. (I am very skeptical that it actually would and am content for the Sun/Moon size ratio to be a coincidence, but it would be interesting if there were more of an explanation.)

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Can you elaborate?

You meant I won the “biggest procrastinator” award :wink:

This is an interesting line of thought. Indeed, it will flip when the ratio of the temperatures to the third power, (Te/Ts)³, becomes smaller than θ²/4π, where θ is the angular size of the Sun in the sky.

Ah, yes. I know that gambit well. The math I don’t have to do is often more fun than the math I do.

In that spirit, if I’m doing all the math right* and using all the correct values for the constants**, I get a critical value for \theta of \sim 0.06. At that point, \left (\frac{T_e}{T_s}\right )^3 = \frac{\theta^2}{8\pi}. To a first order, that’s 6x the actual value. So probably nothing to do with the size of the moon, but it still answers how much bigger/closer the Sun would have to be to change that entropy relationship.

*Going by your formula in #8 above ( T_e^3 > \frac{T_s^3 2 R_s^2}{4 \pi AU^2} ) I feel like there is an extra factor of 2 that I’m not certain I’m accounting for. My understanding is that \theta \approx \frac{R_s}{AU} because AU \gg R_s. Edit: Mathing while tired is hazardous; Diameter = 2 * Radius; that’s where the extra factor of 2 goes. \theta \approx \frac{2R_s}{AU}.

** For reference, I’m using T_e = 300K, T_s = 5777K, R_s = 695,700km and AU = 149.6 gigameters.

Edit: Corrected some calculation errors.

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Hmm, I didn’t realize that this forum supports latex.

Interesting! I might assign this to my future students.

I don’t guarantee that my formulae are correct to every factors of 2π’s, but I also defined θ as the full angular size of the Sun ~(2Rs/AU) instead of ~Rs/AU. Edit: yup, I dropped a 1/2, it should be θ²/8π.

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Note that after updating for the correct value of \theta and using 8\pi instead of 4\pi, it comes out to roughly 6x the actual value, not 10x.

Yup, radius is not diameter, so I was correct that I wasn’t properly handling a factor of 2. I have edited the post accordingly.

Just a reminder that I’ll be on Facebook Live tomorrow night (11/17) at 7:30pm EST to chat all things Faith across the Multiverse. This will be the final video chat for this year, although I welcome ongoing discussion here in the forum over the holidays.

This should be the link for the video stream when it starts:

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How much bigger would it need to be?