Cosmology in a Nutshell: God’s Equation

· cosmology, Newtonian mechanics, physics

Here is a clue to a solution of the Christmas and New Year puzzle in the previous post in the form of the Euler equations for a self-gravitating gas:

God’s Equation: Matter and Antimatter

According to the mathematician Pierre-Simon Laplace, the World is governed by the differential equation:

  • \rho =\Delta \varphi

where \varphi is the gravitational potential depending on space-time coordinates (x,t), \Delta is the 2nd order Laplace differential operator. If you are a fan of relativity theory, you can think if this equation as Einstein’s equation and if you have a religious inclination, as God’s equation. Recall that Laplace upon the question by Napoleon about the absence of God in his mathematics, responded: Sire, I have no need of that hypothesis.

The right hand side \rho of God’s equation expresses the presence of matter/antimatter given as the Laplacian of the gravitational potential \rho=\Delta\varphi:

  • where \rho =\Delta\varphi > 0 (\varphi convex), there is matter with density \rho
  • where \rho =\Delta\varphi < 0 (\varphi concave), there is antimatter with density \vert\rho\vert =-\rho.
We identify (see figure below):
  • matter around points of local minimum of \phi
  • antimatter around points of local maximum of \phi.
We here think of the gravitational potential \varphi as given with matter/antimatter \rho =\Delta\varphi as a derived quantity. This is in contrast to the more common view with \rho given and \varphi the derived quantity as the solution of the differential equation \Delta\varphi =\rho.  Notice that \Delta\varphi results from differentiation which is a local operation (like digging where you stand), while the solution \varphi results from \rho by integration/summation which is a global operation reflecting action at distance. See The Hen and the Egg of Gravitation.

Newton’s 2nd Law

  • \vert\rho\vert\frac{Du}{Dt}= -\rho\nabla\varphi
where u and \frac{Du}{Dt} is matter/antimatter velocity  and acceleration.

Conservation of Matter/Antimatter

  • \frac{D\rho}{Dt}=-\rho\nabla\cdot u
where \nabla\cdot u is the divergence of u indicating contraction where \nabla\cdot u <0 and expansion where \nabla\cdot u >0.


Starting from a smooth gravitational potential \varphi with a smooth distribution of matter \rho =\Delta\varphi where \Delta\varphi >0 and antimatter where \Delta\varphi <0, we have

  • \frac{D\rho}{Dt}>0 where \nabla\cdot u<0 and \rho >0
  • \frac{D\rho}{Dt}>0 where \nabla\cdot u >0 and \rho <0

which combined with the direction of the force -\nabla\varphi shows

  • concentration of matter around points of minimum of \varphi
  • thinning of antimatter around points of maximum of \varphi.

We thus identify a tendency of concentration of matter and thinning of antimatter, which we can associate with the formation of galaxies separated by large almost voids.

Dark Energy and Matter

The gravitational potential generates a repulsive force between regions of matter separated by almost voids of antimatter, which can be interpreted as an effect of dark energy. We conclude that God’s equation \rho =\Delta\varphi appears to come out of a truly heavenly inspiration. It is natural to identify real visible matter as delta-functions of \Delta\varphi connected with -\frac{1}{r} singularities of \varphi, dark matter with smooth \Delta\varphi >0, and antimatter with smooth \Delta\varphi < 0, see the above figure for a schematic illustration.

Visible matter may develop from dark matter by concentration into delta-functions, while antimatter is subject to thinning rather than concentration.

Sire, I Have No Need of Big Bang

The initial distribution of matter and antimatter can be viewed as resulting from a perturbation of a zero potential, where the Laplacian acts as an amplifier of the perturbation. The Universe with its distribution of matter and antimatter would thus be created by the Laplacian operator acting on a little innocent perturbation of the gravitational potential as expressed by God’s equation \rho =\Delta\varphi. Laplace could thus say about Big Bang: Sire. I have no need of that hypothesis.

Comnpare with illuminating illustration by Joel Primack:


Comments RSS
  1. Richard T. Fowler

    You are scaring me, Claes.

    “The Universe with its distribution of matter and antimatter would thus be created by the Laplacian operator [. . .]”

    Is the “Laplacian operator”, by which I assume you mean “the rule(s) that this operator describes”, are these not part of the universe??

    They certainly ARE part of the universe!

    Who created them? Or, where did they come from?

    “[. . .] acting on a little innocent perturbation of the gravitational potential [. . .]”

    Is the gravitational potential, and the field in which it operates, not part of the universe??

    Both of these things most certainly ARE part of the universe!

    Who created them? Where did they come from?

    “Laplace could thus say about Big Bang: Sire. I have no need of that hypothesis.”

    This can be simplified to: He could say about the idea that matter, energy, space and time had a beginning, “I have no need of the hypothesis that they had a beginning. To me it is self-evident that they have always existed.”

    Well he is entitled to his opinion about that. (Though we are presuming a bit, aren’t we, since he wasn’t actually talking about the Big Bang.) But his opinion is absurd, and I pity anyone who thinks that Laplace’s presumed opinion on this matter makes sense.

    Claes, if you do not believe that either matter, energy, space, or time had a point of beginning within a larger construct of Godly or supernatural “time” that exists outside the universe, then it is very easy for you to go from that to, “There is no need of a supernatural Creator.” You are simply starting from the ad-hoc (and ridiculous) assumption that there was never any supernatural creation to begin with.

    In English this is called the logical fallacy of “begging the question”: one assumes a priori the conclusion that one is supposed to be proving; one then states a number of unrelated “premises”, and then at the end, one simply declares as proven the very conclusion that one had already assumed (ad hoc) to be true.

    In this case, the “question” (i.e. proposition) being begged is, “There is no supernatural creator”. From this a-priori assumption flows the unproven (and false) conclusion, “If there is no supernatural Creator, then there cannot have been a supernatural creation of anything,” and this conclusion becomes the implied premise of your final conclusion, which is, “There is no need of a supernatural creation in order to explain the existence of the observed universe.”

    To boil it down, you go from the ad-hoc a-prior assumption, “There has never been a supernatural creation of anything” to “Therefore, there is no need of a supernatural creation of anything in order to explain the existence of the observed universe.”

    “There has never been, therefore there is no need of ….” But you never proved that little part about “There has never been”!

    It’s amazing what one can do with logic if one puts one’s mind to it!

    But man is smarter than this, and God expects better of him.

    The Wikipedia article you cite above states that Newton agreed with my position on the matter, and the last time I checked, he did actually know some calculus.

    Let us remember that we are told in the Bible that “In the beginning, God created the Heaven and the Earth.”

    The meaning of “Earth” is, I think, reasonably straightforward. Surely, if there is a consensus about anything, there is one about that!

    But what is meant by the term, “the Heaven”? Well, this is a complex question involving many Bible passages, as well as physical inquiry and experiment. But fortunately, the question can be simplified for the purpose of a casual discussion. Simply look at Genesis 1:2-4.

    Verse 2: “And the Earth was without form, and void; and darkness was upon the face of the deep. And the Spirit of God moved upon the face of the waters.”

    Clearly, with this verse, the terms “Earth” and “waters” are not referring to the things that we know in this universe by those names. So we might in some sense see them as metaphors. But clearly, for the believer in the Bible, these terms must also refer to things which predated the existence of the universe; and there is no reason to think that these things do not still exist. Thus, I think of them as being “spiritual Earth” and “spiritual waters”, and they must obviously have some spiritual connection with the physical Earth and some physical water, somewhere.

    Verse 3: “And God said, Let there be light: and there was light”.

    There’s your bang, right there. Only an idiot could fail to see it. And you are no idiot.

    Verse 4: “And God saw the light, that it was good: and God divided the light from the darkness.”

    You see, we are told that this happened after the bang. Not at the same time. So what we are told, clearly, is that within the confines of the newly created universe, there at first was only light, and only subsequently was darkness (PHYSICAL darkness, not the extra-universal SPIRITUAL darkness that was referenced before) allowed into the “space” (the PHYSICAL space) that represents “universal” space, i.e. what we commonly perceive as space from our intra-universal point of view.

    So, verse 4 clearly describes the differentiation of stars and galaxies from a universial primordium of intense energy and possibly some matter.

    I do not see how any of this necessitates general relativity or quantum physics, nor do I see how it makes classical Maxwellian physics not the best model we have had to date. These are separate questions, which require very different lines of argument than that which appears in this post.

    For the record, I do not believe that the “laws” of physics have ever been static. I believe that they are all in an asymptotic state of change, and I believe that the rate of change was rapid at first, but very slow now, so that we have not been able to clearly observe the change experimentally — or if some of us have, the fact has been covered up. I believe that intra-universal time, the time that we commonly perceive, is one of those “laws” that are in continuous asymptotic flux, parallel with the expansion of the space, which I believe was fast at first and is now progressing more slowly. Thus, it is possible to have a relatively large amount of natural history in space occur during what we would perceive as a relatively short period of time, compared to the relatively slow passage of time today.

    I would also like to suggest that if you truly believe that we face a political catastrophe that is at least partly the making of the AGW alarmists and green goons/conspirators who have tried to appropriate the mantle of environmentalism for their own evil use, you might try to focus a little less on the areas such as this where there will be massive disagreement even from AGW skeptics, and a little more on the areas of physics that are of interest to all. For that same reason, I try to avoid discussing my cosmological beliefs in the context of climate and earth-science discussions, because it is clearly not God’s will for us to get caught up in debating each other over these issues at such a sensitive time in history. This amounts to rearranging the deck chairs while the ship is sinking.

    One final thought: have a look at Proverbs 17:14. The King James translation (which is the best English translation we have) reads, “The beginning of strife is as when one [lets] out water [. . . .]”

    There’s that word, “beginning”, again. Thus I maintain that the word “strife”, in this context, can be seen as a metaphor for the laws of physics, and physical interactions. And I maintain that this entire phrase constitutes God communicating to us, subtly, about the nature of the beginning of this universe of His. Certainly, He is known from other parts of His Word to have a sense of humor! I believe it is on display in this verse.


  2. claesjohnson

    It is not my intention to discuss religion. The reference to Einstein’s equation as God’s equation is not my invention. But there are close connections between science and religion in the sense that both express “beliefs in ghosts”. The idea that the World arises as light is divided from darkness, or matter divided from antimatter, is central to both religion and science.

    • Richard T. Fowler


      I appreciate your further clarifications. Well, I suppose if there is a silver lining to this discussion, it is that it will probably get you more traffic! (I don’t know if that is a welcome development, but I hope it is! 😉 )

      You state, “But there are close connections between science and religion in the sense that both express “beliefs in ghosts”.”

      Not all religious people believe in “ghosts” per se (I do not), but I understand you to mean “the supernatural generally”. There is of course the problem that not everyone’s perceptions of nature are always the same, and science does a particularly poor job of addressing that fact. Many religions do a very poor job, as well. Science is not a “religion”, but I was taught by a PhD natural scientist in university that it is a “truth system”, as are religions and philosophies, and certainly they all demand certain beliefs that cannot always be proven, as a condition of acceptance by other group members. This is not inherently problematic for science, as long as scientists avoid the temptation to deny this commonality between science and other truth systems. I’m glad to see that you admit certain commonalities between science and religion, for there can be no progress in knowledge without admission of obvious realities, e.g. that one.

      As far as my belief in supernatural things, I will close with one last thought about this.

      A famous thinker said, “I think, therefore I am”. I agree with this conclusion, and with its line of reasoning, or “rationale”. But I take it a step further: “I am; therefore, I was created.” If I were not created, I would be indistinguishable from the infinite void, and I am definitely NOT indistinguishable from the infinite void. I am decidedly distinguishable from it. Therefore, I cannot be the void, or any part of it. Consequently, if only the void can be without beginning (i.e., if all that is “something” must arise from “nothing”), then I must have a beginning, before which time I must not have existed. And if I did not exist before my creation, then I could have had no role in that creation; I could not have created myself or played any role in my own creation. Therefore someone or something created me. The same logic is applicable to anything which is not the infinite void or part of it. Physical laws, and/or rules of physical interactions, are not the void, and are not part of it. Therefore, someone or something created them. QED


  3. cementafriend

    RF you are making an assumption that god existed and the people who wrote the bible around the year 1000AD knew what they were doing. I do not agree with Buddhists but they are a very large group ,out of which Judeo-Christianity grew. From my reading they do not believe in a god but they believe in a spiritual state (maybe another dimension) that has always existed. Of course as with Jews, Moslems and Christians there is a range of Buddhist sects (with a variety of intellectual leaders or patriarchs eg the Dali Lama) and it is difficult to determine the most comprehensive belief system. However, I gather the majority believe in a spiritual state and reincarnation. Buddha who lived some 500BC was regarded as an enlightened teacher responsible for the spread of Buddhism throughout Asia, and the middle east (where it was changed to evolve into Judeo-Christianity). I think the religous beliefs date back to about 3000BC. The early Chinese had a good knowledge of Astronomy and mathematics. They did not think that the Earth was the centre of the Universe. Only the stupid Christians believe that the sun went around the earth.

  4. mkelly

    Only the stupid Christians believe that the sun went around the earth.

    As you use the present tense “believe” am I to assume you are unware that vast majority of person like myself do not believe the sun goes around th earth.

    Please also explain the violation of the first law of thermodynamics in the creation of something from nothing or as Newton talked the cause and effect. The universe as we see it is the effect but what is the cause. Spontaneous generation of the universe? Strings? Branes?

  5. kuhnkat


    your idea that the Bible was written around 1000AD is a rather Catholic centric view. Some of the Tora, which is the basis for the Old Testament, was around before writing. The Christian New Testament, of course, being primarily about Christ, was not written until later.


    blaming a particular wrong cosmological idea on all Christians insted of a particular sect which gained power through the Roman empire is particularly useless other than to bash something you do not understand.

  6. Richard T. Fowler

    CaF states that I am “making an assumption that god existed “.

    How do you know?? Maybe I have proof that He does, and I am simply choosing not to share it at this time. Also, I may have additional proof that I’d be willing to share, but cannot for some reason. Anyway, I did prove that someone or something created the universe, and I did so without assuming the existence of God, or one anyone or anything else, prior to the universe. I did assume the existence, prior to the universe, of an infinite void, which could be described with the word “nothing”, and which may also, in some sense, be divisible.

    None of the previous paragraph is inherently Christian. I believed in a Creator for more than seven years before I believed in the Gospel. And when I say “believed” here, I mean that in the sense of “knew of, had personal knowledge of, had positive proof of.” There is belief that is with doubt, and there is belief that is without doubt. These are two entirely different kinds of belief. The latter kind is synonymous with personal knowledge, and it is in no sense an assumption.

    Complete copies of the New Testament exist that are dated unequivocally to the first century AD. If you are trying to claim that these are forgeries, you have the burden of proof, and you may not assume that there is no God in order make that proof. If I were you, I’d retract the statement about the Bible’s provenance.

    I would also like to suggest that further new points to be made about this be made elsewhere, since our host has suggested that he had not wanted to discuss religion. Thank you.


  7. Ace Rosales del Pilar

    Dark energy in the Universe is a part of the Universe, and a part of the nature of God is to be the creator of the Universe. The Universe is made up of Space, Stuff, and Time, and because we are in it we cannot observe or discover what it outside of it.

    So most theists believe God is a creative and perhaps controlling force outside of the Universe, and most atheist’s believe nothing is outside the Universe. Personally I don’t see how you can have a Universe with nothing outside it, so I’m not an atheist.

    • Richard T. Fowler

      I offer the following in my own defense, and not out of a desire to prolong the matter.

      Ace, you seem to believe that you disagree with me, but I don’t think you do. I think you are caught up in a minor matter of semantics. Can “nothing” be, in some sense, “something”? That is to say, can something that exists outside the universe, but is not commonly perceptible from within the universe, be properly characterized as being, in at least one sense, “nothing” by observers within that universe? I say yes, and if you agree, then there is no reason to question what I have written above; we are apparently in agreement. But it should be clear, then, that I am in fact a theist who believes (and knows) that, in some sense, “nothing” is outside the universe, and that that “nothing” includes God.

      My message is intended for atheists, in the hope that they will come understand that a Creator is not only possible, but proven and therefore an unavoidable part of the conduct of science. Short of God himself giving the athesit a booming message (which appears not to be His style, these days) my approach of careful semantics and precise logic (which is always subject to being misinterpreted and taken out of context by other Theists) is the only way I can see for a true atheist to be shown the truth. Your approach, while not false, is dependent on your own perceptions (“Personally I don’t see [. . . .]”) and thus it cannot serve as proof. My argument isn’t merely true (when taken in the context it was intended); it also is the only way to irrefutably prove the existence of a Creator.


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