Laplace’s Equation as God’s Equation

· cosmology, physics
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In recent posts we have met the following differential equation, referred to as Poisson’s or Laplace’s equation in mathematics literature:

  • \rho =\Delta\varphi

where \varphi is a gravitational potential and \rho represents the presence of matter (where \rho >0 and antimatter (where \rho < 0). This equation is the basis of Laplace’s monumental treatise in 5 volumes on Celestial Mechanics (1799-1825) and thus may qualify to be named God’s equation.

The equation is derived from two basic postulates:

  1. \varphi is a gravitational potential in the sense that the gravitational force field is given as -\nabla\varphi. This means that the gravitational energy of an object only depends on its location (e.g. altitude) and not on the path to be brought there.
  2. The flow of the gravitational force field into any volume is equal to the amount of matter/antimatter in the volume.

Here 2. can be taken as a definition of matter/antimatter and thus only 1. is an actual real assumption. This assumption says that a gravitational force field is conservative and thus is free of friction.

One way to express this requirement is to say that gravitational energy can be transformed into kinetic energy without losses into heat energy. Assumption 1. is thus a necessary component of a mechanics which does not come to a stop by friction, and can thus be viewed as a necessary requirement of true celestial mechanics.

We thus come to the conclusion that the equation \rho =\Delta\varphi is a gift to humanity from the Creator, and as such has to accepted and used in as clever way as possible.

2 Comments

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  1. Richard T. Fowler

    I am reading the Wikipedia article about Georges Lemaître, and it contains some timely information, some of which contains some rather rich irony within the present context.

    The article reports that in 1927, at the time of the publication of Georges Lemaître’s work proposing a universe of growing radius,

    “[. . .] Einstein, while not taking exception to the mathematics of Lemaître’s theory, refused to accept the idea of an expanding universe; Lemaître recalled him commenting “Vos calculs sont corrects, mais votre physique est abominable”[8] (“Your math is correct, but your physics is abominable.”) The same year, Lemaître returned to MIT to present his doctoral thesis on The gravitational field in a fluid sphere of uniform invariant density according to the theory of relativity.

    Later, we read that Lemaitre’s proposal of an expanding universe …

    “met skepticism from his fellow scientists at the time. Eddington found Lemaître’s notion unpleasant. Einstein found it suspect because he deemed it unjustifiable from a physical point of view.

    [. . .]

    “Einstein at first dismissed Friedmann, and then (privately) Lemaître, out of hand, saying that not all mathematics lead to correct theories. After Hubble’s discovery was published, Einstein quickly and publicly endorsed Lemaître’s theory, helping both the theory and its proposer get fast recognition.”

    [All bold-face in the above is mine.]

    RTF

  2. Richard T. Fowler

    Hello Claes,

    How about this as an alternative to your and Laplace’s idea?

    * A force can be considered conservative in isolation, but when looked at as part of a system of forces acting in concert — over a finite period of time, within a finite region of space, and involving a finite number of objects — the force cannot be considered conservative. I will explain.

    * Gravity is a force which results from a perturbation of an invisible field. The perturbation is caused by a type of energy which I will refer to as gravitational energy, which is created, i.e., added freshly into the universe, continuously for as long as a massive particle exists.

    * The gravitational energy, at some point after its addition to the particle, perturbs the field, which is synchronous with the exit of the energy from the particle to the field. There is no loss of energy at this event.

    * The gravitational energy travels through space as a wave, with loss of energy as a function of distance traveled and speed (assuming speed is subject to change, which in an expanding universe I would argue it must be.) The wave also contains information (due to its various properties) about the state of its emitter at the time that it was emitted — just like your Maxwellian EM waves do.

    * When the gravitational wave hits another massive particle, your same finite precision computation is used to determine how much gravitational energy the particle absorbs, and how much it transmits.

    * Absorbed gravitational energy, at some point after the absorption, factors into the equation of momentum for the particle.

    * The change in momentum of the particle is synchronous with a change in the pressure on the EM and subatomic force vectors of the particle. These vectors determine both the net kinetic energy and net heat production of the particle, as well as the net EM and subatomic pressure on the nearest particles in each direction.

    * In accordance with your revised “second law of thermodynamics”, gravitational flux can occur only in one direction between a pair of bodies in informational contact — that is, from the body of higher gravitational density to the one of lower gravitational density.

    * Therefore, the gravitational force cannot be considered in isolation, because if one follows a certain quantity of energy from its creation (which is a function of the existence of mass in a particle at a point in time) through to the time that motion is produced, there is a net increase in energy in the universe, without any corresponding decrease in mass. Thus, there must some heat dissipation that ultimately results from the creation of gravitational energy; otherwise, there would be a continuous increase in gravitation of massive objects over time.

    * Thus, if my reasoning is correct, then gravity, when looked at as part of a system of forces acting in concert (together) over a finite period of time; in a finite region of space; on, in, and with a finite number of objects; cannot possibly be conservative. It can only appear conservative when treated in isolation from other forces which are simultaneously present in the study region and inextricably bound together with the gravity during the study time period. And this appearance is an artifact of incorrect calculation, which in turn is based on false assumptions about the nature of the universe. And the assumptions are apparently based on a (conscious or unconscious) insistence that the universe must be a certain way, because for it to be otherwise would cognitively unpleasant to a particular observer. Such a line of reasoning is not valid, and can result in ad hoc propositions about the structure of the universe.

    * Since gravity, in this model, is not actually conservative but merely appears to be, Laplace’s equation cannot be inferred, and thus there is no basis for “negative particle density” which makes no physical sense, anyway.

    I don’t know if you’ve already thought of all this or part of it, and rejected it, but I personally cannot recall ever reading all of these ideas, put together in one complete hypothesis. So I’m throwing it out there. There it is, people of the world. Have at it.

    So, is there a line for the Nobel prize for physics, or is the room presently unoccupied?

    RTF

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