Abstract: In this article, mathematically generated predictions are given that, if the basic hypotheses are correct, will answer the above question affirmatively. These predictions are obtained by replacing the substratum term d within the linear effect line element, the line element that gives a cause for all the Special Theory of Relativity effects, with a pure nonzero complex di. Such an approach will lead to an ultrasmooth microeffect and various significant predictions. It is shown how this microeffect yields the Genesis Flood and many of the effects associated with such a Flood scenario. The effects predicted include an increase in radioactive decay rates, a Flood that would cover the entire earth, catastrophic upliftings and sinkings of land masses and other alterations in the earth's crust, mantle and core; an increase in erosion and sedimentation rates with the production of fossil layers, fluctuations or reversals of the earth's magnetic field, creation of the asteroid belt, and the cratering effects on the earth and other solar system bodies. Due to the methods used today by secular science, these effects would tend to yield an earth that gives the appearance of great age. An additional prediction is given, with a method of investigation that could verify this prediction. Such verification would yield significant evidence that such an alteration in this substratum term did occur and the result is the Flood scenario exactly as it is described within Genesis.
[Note: In what follows, two special symbols will be used for the simplest mathematical expressions. The symbol ^ indicates a superscript and the symbol _ represents a subscript. All "displayed mathematical expressions" will be displayed as JPG images. This is not entirely satisfactory. However, this image approach will be used until the html language is improved to include mathematical expressions. The model presented in this paper is an infinitesimal analysis (i.e. differential or integral operator) model and, as with all such models, it only predicts macroscopic or large scale behavior approximately. This paper was reviewed for journal publication and changes made in accordance with the reviewers' comments. However, in order to have the results available as quickly as possible, I have decided to publish it in this electronic format.]
You may direct your attention to the following sections.
Before discussing this last question, recall that there are two competing operational approaches to this subject matter. As with all of my theoretical work in this area and since a substratum model will be used, the results discussed in what follows can be obtained by means of two distinctly different operators; a Divine intervention operator or a substratum initial condition operator. Every description that uses a stated Divine intervention operator can be replaced completely with a substratum initial condition operator, and conversely. In all that follows, I will not discuss any method that will, in any manner, tend to determine which of these two distinct operational approaches is the actual approach that may have been utilized in objective reality.
In Herrmann (1994, 1995a,b, 1996a, 1997), it is argued that the General and Special Theories of Relativity are flawed logically and the basic error is corrected. In particular, the basic error known as the model theoretic error of generalization is used to arrive at the predicted aspects of these two theories. In these Herrmann publications, new explicit derivations are given that eliminates this basic error as well as all other known difficulties with these Einstein theories. This is done by assuming the physical existence of a mathematically predicted field of subparticles (Herrmann, 1985, 1988, 1993) that interacts electromagnetically with all natural world physical entities. All of the basic alterations in physical measures are derived, none are simply assumed. These derivations are based upon the behavior of the infinitesimal light-clock analogue model and Robinson's rigorous theory of infinitesimals (Robinson, 1966).
The infinitesimal light-clocks are used as a means of incorporating within a theory the most basic propagation properties of electromagnetic radiation without relying upon any of the controversial scenarios relative to the composition of such radiation. This is a pure operational approach. Further, the mathematical structure called the absolute calculus (tensor analysis) is not used since the substratum subparticle field is used as a place for privileged observation using a privileged mode of measurement. Usually, the language of measurement is used rather than the language of coordinate transformations. If natural-system behavior is altered over a period of measured time, then it is often the case that, statistically, the calculus does predict successfully such alterations when many such events are considered over comparatively small periods of measured time. The calculus is now called an ideal approximating model for alterations in certain physical behavior. This means that the numerical qualities predicted may only be significant statistically. Indeed, a differential equation model under these circumstances is very successful in predicting general behavior rather than specific numerical quantities. But, why is the use of infinitesimal light-clocks and electromagnetic radiation of significance for differential equation and other similar models?
It is now known that one needs to follow explicitly certain rules in order to derive formally various differential equation models for physical behavior. One of these rules is the rule for infinitesimalizing the behavior. This rule relates physical behavior to a specific mathematical statement called the Leibnitz Principle or *-transfer. This is a principle that is shown to hold for the Robinson-styled models used within Nonstandard Analysis. Physically, there appears to be only a few known physical entities with physical behavior that can actually be so infinitesimalized. One of these is the behavior of an unimpeded photon. One can simply infinitesimalize man made measures such as volume and the like; but, apparently, one physical entity that satisfies such a process is the path taken by an unimpeded photon. Possibly, this is one reason for the success of quantum electrodynamics as a predictor of physical behavior and this also seems to answer the last question in the previous paragraph. On the other hand, certain authors (Lawden, 1982) actually do such things as infinitesimalize human observers in their attempts to justify their conclusions. Such rule breaking should be avoided, if at all possible, since it may lead to unfortunate philosophical interpretations. Now, in order to give a possible answer to the question asked in the first paragraph, certain material needs to be recalled.
Approximately 150 years ago, the calculus was used to generalize geometric concepts. This has led to the theory called Riemannian or differential geometry. A set of ideal numbers called the informal infinitesimals is used for this generalization. However, it was not until the early 1960s (Robinson, 1966) that such objects were shown to exist mathematically and moreover such objects have properties somewhat different from those of the real numbers. Fortunately, the processes used in the informal theory of infinitesimals as it relates to Riemannian geometry do not appear to violate the restrictive algebra of the rigorous theory of Robinson infinitesimals. Of course, the standard geometric language is extended to this generalized geometry. But, it is clear that such a language is not intended to be taken literally. It is simply the usual mathematician's way of using prior geometric intuition in order to establish the generalized results. One major composition of differential geometry is an interval, metric or better still a line element. This expression represents a relation between infinitesimal quantities and ordinary real or, in some cases, complex numbers. Relative to certain processes called proper coordinate transformations, these line elements are said to be invariant in the sense that they will yield the same arc length expression.
In Herrmann (1994), all of the basic results attributable to the Einstein Special Theory of Relativity are derived based upon the existence of a subparticle substratum and basic laboratory observations relative to the propagation of electromagnetic radiation. The mathematical structure used is the modern theory of infinitesimal and infinite numbers. With this approach there are no controversial results such a general length contraction, general time-dilation or the twin paradox. All of the effects are manifestations of the interaction of natural world entities with the substratum. In this regard, such things as relative velocity, time and distance must be measured by means of Einstein measures (Prokhovnik, 1967). These measures can be successfully approximated by means of light-clocks. Later in this article, a new line element that predicts the results stated in the abstract above will be derived from fundamental concepts. However, first a review is given as to the general methods used and the results obtained thus far.
Consider a light source on one end of an arm and a perfectly reflecting mirror on the other. (If ordinary reflection is considered, then the next concept can be replaced with a statistical statement.) A photon is emitted from the source and is reflected back perfectly from the mirror to the source. At the instant the photon is received back at the source, it is reflected back to the perfect mirror or another photon is emitted. Of course, this is an ideal standard model that can only be approximated within the natural world. Each time a photon is reflected or emitted a counter is increased by one unit. The numbers that appear on the counter measure light-clock time. (If ordinary reflection is considered, then this counter number can vary and a statistical approach should be applied.) Because of the properties of electromagnetic propagation, this ideal light-clock can be infinitesimalized. In particular, when so infinitesimalized, it is the linear light path as modeled by the infinitesimal light-clock arm that is considered as being measured by a positive infinitesimal number. When this is done, the calculus can be applied to the infinitesimal light-clock count measurements.
As shown in Herrmann (1995a,b, 1996a, 1997), infinitesimal light-clocks can be used as a model for a continuum of Einstein measures. These are the Einstein times and Einstein lengths, and thus Einstein relative velocities. There are two types of infinitesimal light-clocks, those that measure Einstein time and those that measure Einstein length. In what follows, all measurements are Einstein measurements. Thus all results are relative to the behavior of propagating photons.
If one takes a standard light-clock and allows it to move linearly with a constant relative velocity v , it is shown in Herrmann (1995a,b, 1996a) that the light clock counts will undergo an alteration. The relative velocity is actually related to a substratum velocity that behaves in a special manner discussed in Herrmann (1995a, 1996a). Due to this alteration, a derivation in Herrmann (1995a) shows that the standard Einstein time t^s and the Cartesian coordinate standard Einstein length measures x^s, y^s, z^s would need to satisfy the expression
where "standard" means the measures that would appear if there is no relative motion with respect to the substratum. The c is the to-and-fro measured constant value for the velocity of photon propagation over standard macroscopic distances, L is twice the length of the standard light-clock arm, and M corresponds to the light-clock count number at the source. The numbers t^s, x^s, y^s, z^s are actually obtained from infinitesimal light-clock counts by application of a special operator called the standard part operator. This operator gives the one and only one real number that can be so associated with such infinitesimal light-clock measures. The relative velocity is included within equation (1) since v^2 = ((x^s)^2 + (y^s)^2 + (z^s)^2)/(t^s)^2 and, hence, (LM)^2= (c^2 - v^2)(t^s)^2. Thus this yields what one would actually expect: that the light-clock counts are being altered and the length of the standard light-clock arm is not being altered by the relative motion. The right hand side of (1) is called the chronotopic interval, a term that indicates its relation to electromagnetic propagation. One can speculate how it is possible for these count numbers to be altered while the length of the light-clock arm does not vary and the natural world to-and-fro measure of light speed is constant. The alteration may be due to a simple substratum non-linear photon path, while in the natural world the path appears to be linear (Herrmann, 1995a). If the substratum concept was not employed, one might conclude that the only possible way this could occur is by means of a general length contraction. However, (1) should only be considered as an identity. It states a relationship between macroscopic light-clock counts and the infinitesimal timing light-clock counts that yield t^s. It is interesting to note that within the substratum there are processes that can actually be used to make such a comparison.
The next step is to notice that the ideal standard light-clock can be infinitesimalized. When this is done, the expression is written as
Equation (2) is the infinitesimal form of the famous defined Minkowski line element; but, in this case, it is derived from basic light-clock and photon propagation properties. It should be noticed that expression (2) is independent of any physical cause for the velocity v , does not use any concepts from or the language of Riemannian geometry or tensor analysis, but does use physical properties expressed in the language of light propagation and ideal light-clock construction. An expression such as (2) is where one begins when investigating nonconstant (i.e. nonuniform) behavior.
The portion of the subparticle field that interacts electromagnetically with the natural universe is called the nonstandard electromagnetic field or the NSEM field in abbreviation. From actual laboratory observation, the simplest of natural world behavior is impressed on an immediate neighborhood within this field. Without additional procedures, the Leibnitz Principle cannot be used for this purpose; however, what is used is a specific rule for infinitesimal modeling. The Leibnitz Principle can be applied if one generalizes such line element expressions as (2) (Herrmann, 1997). But then, another special process must be applied. The reason for this is that two different types of coefficients might be allowed before each of the terms in a line element. Relative to the fundamental theorem of differential calculus and one definition of the differential, such coefficients are differentiable real value functions. And, it is hoped, such line elements will lead to differential equation models. However, on the other hand, other "nonstandard" functions could be used. In which case, a "standardizing" process would be needed. In all that follows in this article, the first type of coefficient, the differentiable real value function, is the only type considered.
where the superscript s means that these are measures taken in the absence of any process that will lead to an alteration in infinitesimal light-clock counts or some standard physical measure.
For an electromagnetic
impulse or a photon moving in the radial direction from a point as viewed from
a Cartesian system, very simple assumptions are made. Associated with the
photon is an Einstein velocity v, an additional substratum associated
velocity d
and the basic photon velocity c. The velocities v and d can be of two
types. The Einstein velocity v can be either an actual natural world
velocity relative to motion of a source or a potential velocity. The substratum associated
velocity d can be an actual
velocity relative to movement or a potential velocity, where in both cases such a potential
velocity is velocity that would be produced relative to a fixed test object by the conversion of potential energy into kinetic energy.
Using the same method
as used to obtain the Special Theory conclusions (Herrmann, 1994), it is
assumed that the ballistic Galilean distance expression holds
within the substratum for velocities v, d, c.
Although v, d, c can be assumed to
be
substratum position and time
dependent, these derivations use the specific rule for infinitesimal
modeling that considers v,
d, c to behave, in the infinitesimal part of the substratum world, as if
they are constants with respect to their effects upon the standard (natural)
world.
This
yields
The expression dR^s = (v+d)dt^s in (4) is significant for new line element generation. A new line element is a differential-form that states how certain physical processes will alter infinitesimal light-clock measurements. The fixed derivation method used requires that, while the line element is being applied, the physical process P does not reverse its basic effect. This is modeled in this derivation method by never allowing dR^s to be replaced by (v+d)dt^s. Indeed, the derivation will fail if this substitution is made during the derivation process. Relative to how a physical P-process, at the most, affects the radial behavior of an entity, it is assumed that a damping effect within the NSEM field alters the infinitesimal light-clock counts. This damping effect is modeled by the simple expression (A): dR^m = dR^s + AdT^s and a reciprocal expression (B): BdR^m + dT^m, dT^m = cdt^m, where A, B are to be determined. Let k = (dT^s)^2 - (dR^s)^2. From (A) and (B), the expression (C): dR^s = (1 - AB)dR^m - A dT^m is obtained. Substituting into k, the exact same expression as (2.4) in Herrmann (1996a, p. 399) is produced, where r^m = R^m and r^s = R^s. Following the exact same argument as appears on page 399 -- 400 of Herrmann (1996a), the following line element is obtained.
where L = 1 - (v+d)^2/c^2. Since the P-process being modeled by the line element is assumed, at the most, to lead to infinitesimal alterations in the radially directed linear behavior of entities (i.e. infinitesimal angular behavior is not being altered), it follows that (theta)^s = (theta)^m and (phi)^s = (phi)^m. This yields
[See Herrmann, (1995a) for complete derivation of (6).] Relative to (6), it is important to note that, depending upon the global behavior of a physical entity, such local behavior can alter angular behavior as viewed globally such as in the laboratory.
Equation (6) is a general line element that represents many different and superimposed natural world effects. This general line element leads to the most significant and important line elements investigated by means of the classical General Theory of Relativity. Let d= 0, v^2= 2GM/R^m, where G is the gravitational constant, M is the mass of a homogeneous spherical body, R^m is the distance from the center of this body to a point exterior to the surface, and v is the escape velocity from the position R^m. Then with this substitution into (6) the famous Schwarzschild line element is obtained where the P-process is the gravitational potential. Further, there is a v + d such that when substituted into (6) L = 1 - 2GM/(R^mc^2) - (1/3)(Lambda) (R^m)^2, where (Lambda) is the Einstein cosmological constant. Such a substitution yields the modified Schwarzschild linear element (Rindler, 1977, p. 184). Setting M = 0 in the modified Schwarzschild line element yields one form of the de Sitter line element. Setting M =0, (Lambda) = 3H^2/c^2, where H is the Hubble "constant," in the modified Schwarzschild line element yields the standard form of the de Sitter line element.
Letting v = 0 and d= R^m/a and substituting into (6), a per-Robinson-Walker line element is obtained. In each of the previous derivations for alterations in physical properties caused by such line elements, it is argued that (dt^s)^2 = L(dt^m)^2. Making this additional substitution into (6) yields one of the exact forms for the Robinson-Walker line element that characterizes a space of "positive curvature" (Ohanian, 1976, p. 392, (where c = 1)). Of course, using this procedure, such a Robinson-Walker line element has nothing to do with Riemannian geometry; but, it has everything to do with a "moving" substratum (or for some applications a type of substratum potential velocity), as represented by d and its relation to the natural world. The substratum can be conceived of as the vacuum or nothingness of quantum physics. It is the "empty space" upon which the natural world is "painted." The substratum only makes its natural world appearance known directly when it is activated. Within the nonstandard physical world (NSP-world), the substratum can expand and carry along each natural world entity with its expansion. A Robinson-Walker type line element is an aid in understanding the natural world effects of such an expansion. At the present, the required P-process is unknown from the secular viewpoint since it is within the substratum. Due to the concept of the ultranatural event, such a P-process need not be describable using a scientific language except in terms of a type of potential that yields a potential velocity. In the various applications of line elements that contain this "d," since it is strongly associated with the substratum, it should probably not be included in any calculations that lead to direct gravitational effects within the natural world such as tidal forces and the like. However, as will be shown, this "d" would yield other types of natural world effects that can be measured.
In Herrmann (1996a), the following general linear effect line element is obtained, where in the present context v is replaced by v +d :
In this form, variable r^m is the infinitesimal light-clock measure for the "linear" distance from a fixed spatial point. When d = 0, this line element is used to predict Special Theory effects such as alterations in mass measurements, decay rate changes and the like, and yields a cause for such effects. Line elements, such as (6) and (7), can be assumed to exist everywhere within our universe. It is the value assigned to the v or d within a specific line element associated with specific points within spacetime that activates the line element.
In Herrmann(1995a,b, 1996a), a fixed approach using line elements (6) and (7) is developed that predicts all the known alterations in various measures for physical behavior that are also predicted by means of the non-fixed approach used within the General and Special Theories of Relativity. This fixed approach shows that all such alterations are probably due to electromagnetic properties associated with every natural world entity and an interaction with the substratum. This interaction is called the electromagnetic interaction with the substratum or (emis) for short. The method uses a general operator approach that includes various differential equation models for standard physical behavior, the method of separation of variables and the concept of a universal function. The differential equations vary from the time dependent Schrödinger equation relative to energy level variations to the simplest statistically meaningful radioactive mean decay rate equation.
Suppose that certain aspects of a natural-system's behavior are governed by a function T(x_1,x_2, . . . ,x_n,t) that satisfies an expression D(T) = kT_t, (i.e. partial derivative) where D is a (functional) separating operator and k is usually a universal constant. In solving such expressions, the function T is often considered as separable and D is a separated operator in that it may operate separately upon each function into which T is separated. Further, such separated forms are an invariant. As an illustration, suppose that D is the identity relative to the f(t) and let T(x_1,x_2, . . .,x_n,t) = h(x_1,x_2, . . . ,x_n)f(t). Then (D(T))(x_1,x_2, . . .,x_n,t) = (D(h))(x_1,x_2, . . . ,x_n)f(t) = (kh(x_1,x_2, . . . ,x_n))(df/dt).
Let (x_1^s,x_2^s,x_3^s,t^s) correspond to measurements
taken of the behavior of a natural-system that is oriented by (2) [resp. (3)]
and using
identical modes of measurement let
(x_1^m,x_2^m,x_3^m,t^m) correspond to measurements
taken of the behavior of a natural-system that is oriented and
influenced by (6) [resp. (7)].
[The term "modes" means that identically
constructed devices are used.] Now suppose that
T(x_1^s,x_2^s,x_3^s,t^s) =
h(x_1^s,x_2^s,x_3^s)f(t^s).
We assume that T is a universal
function and that separation is an invariant procedure. What this means is
that the same solution method holds throughout the universe and any
alterations in the measured quantities preserves the functional form and, in this
case, preserves the separated functions. It is assumed that it is the
operator equation above that will reveal the alterations in physical behavior
brought about by (6) or (7). In order to determine these alterations, something would
need to be fixed. The technique used is to investigate what physical behavior
modifications would be needed so that the
values
h(x_1^s,x_2^s,x_3^s)= H(x_1^m,x_2^m,x_3^m) and
the values f(t^s) = F(t^m) and, hence, T(x_1^m,x_2^m,x_3^m,t^m) =
H(x_1^m,x_2^m,x_3^m)F(t^m). [These last three equations can also
be justified by physical considerations when potentials are being
considered.]
One differentiates with respect to t^s and obtains by use of the chain rule
With respect to t^m,
[Note that the same procedure would apply to physical properties that are represented in more than four coordinates.]
For each specific case, where the measures are made by infinitesimal light-clocks, it is an analysis of expressions such as (8) and (9) that leads to predictions for the alterations in physical behavior. In the derivations that appear in Herrmann(1996a, 1995), it is seen that the actual alterations in physical behavior are mostly independent of which of the two line elements one uses, (6) or (7). The line element selected is simply relative to how the P-process is perceived. In an electronically published article (Herrmann, 1996b), it is pointed out that using (6) with v =0 and nonzero d where such a d is considered as representing a type potential velocity within the substratum, the same derivations as used for alterations produced by a v that represents a Newtonian gravitation potential (Herrmann, 1995) will produce physical alterations of the exact same type. These, however, are substratum driven alterations and are not driven by an internal gravitational field. Such derived alterations will increase the "time" certain physical processes will take for completion when compared to the non-altered standard time and will lead to an aging universe exterior to a very slowly aging earth or solar system. In Humphreys (1994), the same effect is achieved by letting v represent an extremely strong Newtonian potential and d = 0.
It is important to emphasize the difference between these two approaches. The same effects occur but, in Herrmann (1996b), the alterations are determined through direct Divine application of a pre-Robinson-Walker type line element and are caused by an (emis) effect produced by a process that emanates from the substratum. This is the same line element that leads to the textual "expansion" concept used in both the Herrmann and Humphreys approach. In Humphreys (1994), the effects are produced by created natural world processes as determined by a Schwarzschild type line element but with significant direct Divine intervention at specific moments such as altering the cosmological constant associated with one form of the Einstein gravitational field equation. In what follows, one conclusion using either approach, is that various forms of motion represented by the instantaneous velocity will, in general, be "slowed down" within a specific space region by such gravitational P-processes. However, if (6) is applied locally at various points within a specific region, where v = 0 and nonzero d in (6) is a measure of substratum potential velocity, then this same effect will occur without there being a strong gravitational field. This, of course, correlates with the predicted energy changes that take place as well. Noting that effects that emanate from the substratum need not follow the known natural world procedures, it might be possible that the Flood, with all of the Flood associated effects, is but a product of one simple change within equation (7).
Due to its significance and prior to the application of these methods to the Flood scenario, a complete derivation of our new line element is needed. Relative to complex numbers as measures of physical reality, their application within the theory of complex variables to potentials and potential velocities is well known. What is not well known is that application of complex valued functions, where the domains are real valued, to the subject of Maxwell's electromagnetic theory of light first appeared in a publication by Josiah Willard Gibbs (Seeger, 1974). In this note, Gibbs uses what he calls "bi-vectors" or "imaginary vectors" to write an expression relating the electromotive force, charge and displacement. His use of the bi-vector concept leads to the following interpretation for one of his expressions. "This bi-vector therefore represents the average state of a small part of the field both with respect to position and velocity." (Seeger, 1974, p. 182). Thus Gibbs gives a representation of how these two distinct physical concepts are related.
Relative to the classical Special Theory of Relativity, Minkowski introduced the pure complex numbers as an appropriate representation for the "time" coordinate and many of the predictions of this theory can be obtained by application of the theory of complex valued vectors (Lawden, 1982, pp. 39-48). Complex numbers are also part of classical Riemannian Geometry, where imaginary curves and surfaces are considered (Struik, 1961, p. 44), and they are basic to electrical network theory. Complex valued vectors are specifically discussed by Spiegel in his basic outline on complex variables (Spiegel, 1964, pp. 6, 68-70). Hence, it is not without precedent to consider mathematical models based upon complex valued vectors and this is what is now proposed.
In equations (4), make the following substitutions. Let v = 0 and replace nonzero d with the pure complex nonzero di. As previously done, alterations in the NSEM field are characterized by the presence of an A in equation (A): dR^m = dR^s + AdT^s and by the B in (B): dT^s = BdR^m + dT^m dT^m = cdt^m, where complex A and B are to be determined. Since these are substratum driven alterations, A and B need not be real numbers only. In order to incorporate the nonreversible nature of a substratum P- process, as was done previously, the infinitesimal expression dR^s is not reversible in the sense that once it is used for a P-process the substitution back to the form (di)dt^s is not allowed. This is not the case when infinitesimals are considered in a derivative form that has a complex standard part.
From (A), (B), we have (C): dR^s = (1- AB)dR^m - AdT^m. Substituting into the infinitesimal light-clock expression (2) yields
The simplest real world aspect of time interval measurement that assumes that timing counts can be added or subtracted is transferred to a monadic neighborhood and requires dT^m to take on positive or negative real infinitesimal values since this expression contains an altered independent variable. Assuming the effects of P, as incorporated in dS^2, are independent of whether such a defined infinitesimal light-clock count predicts future behavior or is a consequence of past behavior, then dS^2 should not alter its value when dT^m is positive or negative. This implies that 2(A + B(1- A^2)) = 0. Since we are working with complex numbers, for simplicity, let A^2 = 1 - E. Now A has two possible complex roots. We denote these (in classical form) by A = -(1 - E)^(1/2) and A = (1 - E)^(1/2). First consider A = -(1 - E)^(1/2). Hence, B = (1 - E)/E. Substituting into (B) and (C) yields
Combining both equations in (11) produces
In order to incorporate the requirement that the P-process is static with respect to motion within a monadic neighborhood, we assume that at some spacetime point that an initial condition is that dR^m/dT^m = 0. Using dR^s/dT^s = di/c, (12) yields that -d^2/c^2 = (1 - E) or E = 1 + d^2/c^2 = L'. If we choose A = (1 - E)^(1/2) then the same derivation yields di/c = -(1 - E)^(1/2) and again we have that -d^2/c^2 = 1 - E. By substituting L' into (10), we have, where dT^m = cdt^m, the new linear effect line element
where L' = 1 + d^2/c^2. The function L' is very simplistic in character. If it is a constant or a differentiable function, then it may be regarded, at the least, as the standard part of an ultracontinuous ultrasmooth nonstandard extension of itself. Such functions appear to predict physical behavior relative to what are called microeffects (Herrmann, 1989). Note that the P-process that generates equation (7) is undirected relative velocity. [The L' can be obtained from standard "vacuum" concepts by replacing d^2 with a "vacuum gravity" potential escape velocity expression (Novikov, 1983, pp. 52-62) in the same manner as is done with the v^2 in the discussion following equation (6).]
From a secular point of view, the Flood scenario would come about by means of a set of specific initial conditions. On the other hand, a literal Bible interpretation implies that it was produced by direct Divine intervention. As a basic hypothesis, suppose that at the center of mass of many of the predominately solid bodies within the originally created solar system, equation (13) is activated. The effects of such an application of (13) include an increase in the radioactive decay rate (Herrmann, 1996b) of all such material located within regions with nonzero di as well as many other rate increases. [This would, of course, alter all present day radioactive dating assumptions. Further, d could be a function of, at the least, R^m or t^m , or more likely under certain conditions, R^s or t^s . Since d is a substratum measure, d^2 could be equal to or great than c^2. I do not contend that this Flood effect is necessarily the major explanation for radioactive aging.] This increased decay rate yields an increase in the amount of heat present within these regions. This additional heating effect would have vast consequences relative to the original pristine and somewhat smooth character of the land prior to activation of (13). However, the heat produced would depend upon the actual amount of radioactive material that was present within specific locations when (13) is activated. Further, there are yet another alterations in natural-system behavior that need to be considered due to the same substratum effect as that which produces the increase in the radioactive decay rate. Indeed, all coupled physical activity would require rate change derivations. The problem would be to determine what physical activities are closely correlated to an alteration such as in the rate of radioactive decay. After this determination, one would need to exhibit proper derivations consistent with (13) for each such alteration.
In order to apply the separating variable method, we postulate based upon the large number of physical processes modeled by a separating operator expression of the type D(T) = kT_t, that there is some basic T(x_1,x_2,x_3,t) = f(x_1,x_2,x_3)R(t), where f(x_1,x_2,x_3)= C and C is a nonzero constant, and differentiable R(t) is a time dependent measure of the radial distance moved by an entity from a fixed spatial point due directly or indirectly to the same P-process modeled by application of (13). [It is always possible to translate a differentiable function in one spacetime variable into a universal function (Herrmann,1995a,b; 1996a) by this method.] For this scenario, consider CR(t) and f(x_1,x_2,x_3)=C=1=k. Now let D(T) = D(f) R(t) and D(f)=1 where D is the identity operator.
From the view point of the special technique employed (Herrmann, 1995a, pp. 73-77), it is the required that R^s(t^s)= R^m(t^m). Substituting into (2) and (13) and equating these expressions
yields, in general, that
(cdt^s)^2 -L' (cdt^m)^2 = [1 - (1/L')](dR^m(t^m))^2= ê ;
an infinitesimal.
Further, since R^s(t^s) = R^m(t^m), the nonstandard approach to definite
integration shows that ê can be a freely chosen infinitesimal.
The usual choice is to let ê = 0. or to simply apply the standard part operator. When this is done,
(dt^s)^2 =
L'(dt^m)^2. Now using this result, and substituting into (8),
one obtains
Note that activation of (13) yields no alteration in any physical measure that depends only upon rotation. Relative to movement only, the radial component of such movement will be affected by (14). From the view point of the fundamental observer within the substratum, the behavior expressed by (14) can be compared. The t^s and t^m are interpreted, in all cases, as meaning, respectively, behavior prior to and after application of (13). Of course, (14) actually implies that all of the natural processes that would affect an objects velocity are themselves being altered so that (14) holds. [Note that in the gravitational case, where v is not zero, the value of v depends upon the application of the line element to a specific scenario. This should also be assumed for the applications discussed here. The actual value of d probably varies relative to the application of (13) to a particular physical scenario.]
Applying (8) and (9) to the partial differential equations for internal heat transfer and heat radiation within a specific region, leads to the conclusion that the rate of local heat transfer and heat radiation also increases. Indeed, the general rate of all forms of heat dispersal would increase. Further, substratum effects relative to such energy can affect this dispersal. As shown in Herrmann (1995b), energy levels modeled by the time-dependent Schödinger equation are also altered, indeed, in this case increased from the standard. For atomic structures, such alterations might have interesting effects. As in "General Relativity," energy is rather imprecise in character and, since this is a model using infinitesimal concepts, it is well-known that it need not give precise solutions but does give an approximate or, in some cases, only strongly suggestive solutions. Although many more effects, especially increases in other rates of change, may result from considering nonzero di and line element (13), only three basic effects, the increase in the radioactive decay rate, the increase in the rate of heat transfer and radiation, and (14) are detailed within this article. The substratum produced alterations in v are in addition to the usual effects associated with heating and are enhanced components of radial movement. Relative to the alterations in the behavior of the velocity v , it seems rather difficult to determine the actual velocity expression at every corresponding point since v(t^s) may not be known and, further, di is not necessarily a constant and could easily depend upon other local physical or substratum measures. Hence, depending upon numerous physical conditions, this enhancement of radial movement would be more of a tendency for physical entities to move more rapidly than usual either in a radial direction away from the center of mass or even towards the center of mass. As shown in the next section, these results seem to model the general pattern of Flood behavior as it is described within the Scriptures.
The very basic assumption is that the Earth prior to the Flood had a very different topography than observed today. Relative to a secular interpretation, such a difference, in all respects, is possible due to ultranatural initial conditions that lead to the selection of the ultrawords that generate an Earth-solar system combination. Relative to a Scriptural interpretation, Genesis 1:9 states, "And God said, Let the waters under the heaven be gathered together unto one place, and let the dry land appear: and it was so." In Genesis 1:10, God called this gathering together the "Seas." Psalms 24:2 states, "For he hath founded it upon the seas and established it upon the floods [rivers]." Genesis 1:21 seems to imply that there were some concentrated bodies of surface water during the antediluvian period. Except as alluded to within the Scriptures, the actual behaviors of various natural-systems prior to the Flood are unknown. Since it appears that there was no actual rain as such prior to the Flood, some care is needed in assuming that all natural laws were the same during the antediluvian period as they appear today. Further, we can have no direct knowledge as to how Divine intervention may have altered these laws. However, as an illustration only, suppose that within this pristine world the surface water concentrations are in low lying basins and the majority of the water is subterranean.
From a secular viewpoint, all of the following natural events could occur due to substratum ultranatural spacetime initial conditions. From a Scriptural viewpoint, suppose that the ordering of events as stated in Genesis 7:11 - 12 is correct ". . . on that day all the springs of the great deep burst forth, and the floodgates of the heavens were opened. And rain fall on the earth forty days and forty nights."(NIV) [The "great deep" here especially signifies a "large, in size, subterranean water-supply. See Strong's New Exhaustive Concordance of the Bible #7221 and #8415.] [Note: The following appeared in the original paper but is now considered as of little significance since the mathematical generator of these effects di need not be constant and could by position dependent. Except for the statement on Divine intervention, the portion between the double square brackets should be removed from this model. [[Relative to the Earth, at the least, suppose that there is a specific fixed radial distance R^m_M such that di is nonzero for any nonnegative R^m <= R^m_M. On the other hand, suppose that at a specific distance R_c^m >= R^m_M from the center of mass if the local measure of material density falls below a certain prescribed value W or is equal to a possibly different specific value W_1 , then di = 0 for all R^m > R_c^m and nonzero di for all R^m such that R^m_M <= R^m <= R_c^m. For this application, the density W is specified and probably less than that of the Earth's atmosphere at a particular position and at the specific historical period when the Flood began and W_1 is the density of what appears to be the unspecified wood used to construct the Ark.]] Although it is rather obvious that by direct Divine intervention God would produce a protective environment for the Ark and its occupants (Genesis 6:18, 7:16) by unspecified means, [[the use of the W_1 requirement for application of line element (13) would be an automatic process that would aid in this protection. However, the actual use of such an additional W_1 may be considered as unnecessary. Removal of this requirement does not influence the actual predicted results relative to all non-Ark entities. (Added 24 Dec. 1998. Using the additional W_1 now appears as unlikely.)]]
Along with the available line element (13) and only for this immediate discussion, three types of natural law and their effects will be assumed to apply in the antediluvian world. Once coupled physical activity is determined, then alterations in such activity would also need to be derived. [Theoretically, the selection of such coupled activity can come about through application of a "realism relation."] Activation of the (13), with nonzero di in all radial directions, leads to many effects which include heating by an increased radioactive decay rate, increased rates of heat transfer and radiation, and enhanced components of radial movement. This further leads to a vast array of catastrophic events. One of these events is that the subterranean waters, from the "deep," are being force with enhanced radial movement through breaks in the Earth's crust. Portions of this water could also be in the form of steam. Then this water is returned to Earth and appears as the first rain. For the pre-Flood surface water and, at the least, through the first forty days and nights, there would also be amplified radial tide and wave effects (Clark, M. E and H. D. Voss, 1990). And, of course, the Flood waters and an enhanced radial movement would "overturn the earth" (Job 12:15). Some of the expanded Earth model (Unfred, 1986) and the uplift-subsidence model geological effects would also occur.
These effects and other catastrophic events such as the production of the fossil layers, great physical alterations within the crust, mantle and even the core of the Earth continue during application of (13). Depending upon the actual heat produced, the viscosity of materials, and other internal forces and tensions (Skinner and Porter, 1987, p. 406), the enhanced radial component of movement could easily produce geological upliftings and subsidence (sinking) beyond those that would be expected purely from the additional heating effect and other natural processes. Indeed, an increased rate of erosion and sediment formation beyond what would be expected due to present day observed natural processes would occur because of an enhanced radial movement towards the center of mass. What such catastrophic events coupled with the removal of vast amounts of water from subterranean regions would tend to produce has been discussed at great length within creation-science literature and, except for one major effect that can be observed today, will not be repeated here in any detail. But, it is important to mention that this linear effect line element approach has a certain advantage over any approach that simply yields an increase in the radioactive decay rate. Since other rates of change also increase over the time span of the Flood, numerously many antediluvian natural-systems would alter their behavior. For example, although it is not an enhanced movement, the standard shear or plate velocity of portions of the lithosphere would increase over that observed today, without an extreme heat source, due to the enhanced radial movement. Consequently, this approach seems to solve the various problems discussed by Baumgardner (1990). It appears that deriving or postulating an increase in the radioactive decay rate would not be sufficient. Consequently, activation of (13) at the Earth's center of mass, and elsewhere, during the Flood period seems to solve many of these Flood model difficulties. Note that activation of (13) with respect to other antediluvian scenarios such as that discussed by Roy (1996) would lead to similar catastrophic events.
It has been known for many years that measurements of the Earth's magnetic field taken at various locations show interesting variations. This is called the secular variation of the Earth's magnetic field. The first in-depth mathematical analysis of this phenomenon was done by Allan and Bullard (1966) relative to the dynamo theory for magnetic field production. A later examination of the data by Bloxham and Gubbins (1985) led them to reject the no-upwelling or no-downwelling of core material conclusion relative to a cause for the secular variation. They proposed a simple two dimensional model for the upwelling or downwelling of core material, and analyzed such effects on a initially uniform horizontal field. They state that "Temperature variations in the mantle may drive flow in the core, a cold spot in the mantle producing downwelling of fluid . . . and a hot spot upwelling resulting in a zero flux patch." (Bloxham and Gubbins, 1985, p. 781). Humphreys (1990) gives a more detailed two dimensional illustration of how well known mechanisms would alter the flux if the fluid core material did indeed upwell or downwell, and expands upon the Bloxham and Gubbins conclusion. Humphreys suggests that behavior would lead to fluctuations strong enough to produce reversals of the magnetic field. Such fluctuations or reversals of the Earth's magnetic field is a well known explanation for the apparent magnetic field fluctuations or reversals found in appropriate material.
Activation of (13), as discussed above, would yield all of the necessary enhancements within the mantle and core fluid material so that the Bloxham, Gubbins, and Humphreys mechanisms would be strong enough to produce rapid magnetic field alterations or reversals during the Flood period. Further, when all these enhancements were attenuated at the end of the Flood period, there should be a continuation, but to a much lesser degree, of various catastrophic lithosphere events for hundreds of years thereafter. Humphreys points out that the fluctuations would also continue but to a much lesser degree for several thousand years and then begin decaying steadily. Since the evidence suggests that the Earth's magnetic field has fluctuated or been reversed numerously many times in the past, the assumption that the Flood was caused physically by application of (13) at the center of mass leads to an explanation that all such strong fluctuations or field-reversals could have occurred during the one year Flood period. These fluctuations or reversals might be considered as having taken place over many millions of years by those who adhere to the concept of a constant radioactive decay rate, or those who reject the Flood scenario. (Humphreys suggested in (1986, p. 117) that one mechanism that might drive these reversals was that ". . . God greatly increased the rate of radioactive decay during the Flood year, . . . " However, this suggestion is not sufficient for the short time period needed unless many other rates are also increased such as the rate of internal heat flow. In this present model, these reversals are only a byproduct of the mechanism that yield the Flood.)
To be consistent, activation of (13), with di as described above, probably occurred to other originally created and predominately solid bodies within our solar system. Similar effects would occur even for those bodies with minute amounts of radioactive material. Indeed, an enhanced component of radial movement is still present during this time period. Application of (13) to a large body that may have once occupied a position between the orbits of Mars and Jupiter could have resulted in its destruction. This, of course, depends upon its composition. Further, there would be an increase in fragment collision forces due to the increase in radial motion. Evidence for this is the asteroid belt with its tangled orbits. Cratering effects on solar system objects such as Mars, the Moon (Morton, Slusher and Mandock, 1983) and the Earth (Froede and DeYoung, 1996) could have occurred during the Flood period. Such cratering effects on solar system bodies would depend upon the location of this asteroid producing massive object with respect to these other solar system bodies, and the physical condition of these other solar system objects caused by activation of (13) as well as their location at the time of the application of (13). Relative to the Earth and these other bodies, note that if such an event occurred, then the ejecta would have enhanced radial movement and this would alter the time frame for ejecta arrival. Further, I do not contend that all of the cratering effects are produced in this fashion, especially on such planets as Mercury. However, activation of (13) would yield numerous effects that depend upon the composition of a body and some of these effects could simply give the appearance of cratering.
Due to the warming of the vast Flood waters, many weather changes would occur during and after the Flood period. Genesis 8:1 indicates that another Divine intervention occurred. Obviously this intervention was done in order to return Noah, the Ark and every living thing within the Ark to an Earthly environment where the life forms on the Ark could survive. Since (13) did not directly affect the Ark or its inhabitants, this would only require the nonzero d to be replaced with a zero value. However, caution should be exercised at this point for, as mentioned previously, di is not necessarily a constant and could easily depend upon other local physical or substratum measures. A reduction in the value of d could also depend upon the effects desired relative to local physical measures or properties. Initially, if attenuation of d rapidly reduces the enhanced radial motion, then the ocean level with respect to the center of mass would decrease rapidly due to the removal of this additional component of radial movement. After this, the level would drop at a slower rate. The initial rapid abatement of the waters would be accompanied by significant wind effects (Genesis 8:1). Most of the water that came from the subterranean regions would not be able to return to these regions due to changes in the original Earth's geological features and the collapse of the original chambers. Since heat transfer and radiation would now drop to the original levels, the oceans would remain somewhat warm for some time after the Flood period. It has been suggested that this could have led to an Ice Age (Oard, 1979). Further, although the rates of change would drop rapidly to present day rates if the d = 0 is assumed, there would most certainly by residual effects that would last hundreds, if not thousands of years. It is not the purpose of this investigation to discuss such residual effects. Future investigations should, however, consider these residual effects for the earth after the Flood would be a considerably different place than before the Flood.
Is it possible that there remain within our present day environment anomalies or a signature that indicates that such an event as the selected application of (13) did occur only a short time ago in the past, with the resulting Flood and other events discussed above and elsewhere, and that such an application did help to produce our present day environment? Mechanisms for magnetic fluctuations, increases in radioactive decay rates and the like only yield the scientific possibility that such events could have occurred during the Flood. It is not scientific evidence that these events actually occurred. These theoretical results only yield an additional model for such behavior, a model that science may not be able to differentiate from other models.
Activation of (13) yields alterations to time dependent physical processes that are not dependent upon rotational concepts. No alterations occur in any physical measure only dependent upon rotation of the bodies to which (13) is applied. This yields a prediction that there are measurable differences present today that signify that enhanced radial components of movement and non-enhanced rotational components of movement took place during the Flood. When field investigations are made and geological structures examined using the natural processes observed today and the so-called uniformity of nature concept is applied, anomalies may occur. These anomalies would be relative to radial uplifting or subsidence versus expected rotational shifting. If some rotational shifting is less than expected, then this gives evidence that something like application of (13) may have occurred. Other comparisons of physical material that is assumed by evolutionary science to be very old may also present such radial versus rotational anomalies when present day natural processes are used to explain past behavior. This would give additional evidence that the Flood scenario as described above is not some sort of Biblical illusion or poetic insertion but actually took place exactly as it is described Scripturally.
It is self-evident that the mechanism discussed in this paper can be used as an initial condition or the "driving" mechanism that yields such theories as Catastrophic Plate Tectonics. However, caution is still advised. We have no direct knowledge as to the natural processes that prevailed prior to the Flood. This is the reason (13) is applied to but a few of these processes. But, (13) can be applied to other natural processes such a internal and external heat transfer, electromagnetic radiation and, indeed, many other time dependent measures that are modeled by certain mathematical expressions if one assumes that these equations governed the natural world prior to the Flood. Although the effects of application of (13) would most definitely give the appearance of an "aging" of much of the earth and possibility the solar system, caution detects that "age" as measured by the various forms of radioactive decay may not be accelerated to the degree that some might desire. It is certainly possible that other natural processes present today were active prior to the Flood, processes not mentioned in this article, and these could also have been accelerated during one of the "creation days." However, if this is the case, then such processes should be accompanied by appropriate Scripture rather than be postulated ad hoc.
Relative to gravitational and relative motion effects, in Herrmann (1995a), a very general process using the time dependent Schrödinger equation seems to imply that all physical behavior that is associated with energy changes of any type is altered by a factor (L)^(1/2) or (L')^(1/2). However, relative to L' , since this is a direct substratum effect, a considerable amount of caution must be exercised when considering questions and notions associated with pure substratum behavior due to the predicted presence of ultranatural events and ultranatural laws. That is: not all of the physical alterations in behavior as predicted by the mathematical model need actually occur in objective reality. There may be additional constraints in that only a proper subset of the physical "observables" are altered. Since infinitesimal light-clocks are used with the Schrödinger equation, then when applied, if at all, to the L [resp. L' ] case, these alterations should probably be restricted to the behavior of certain microscopic objects. The same holds for gravitational fields that use General Relativity and the equivalence principle to predict time-dilation since it is known that this principle only holds for "point particles" and this principle is used to derive clock time-dilation within an Einstein gravitational field. Hence, such time-dilation would need to be restricted to "microscopic clocks."
CRSQ -- Creation Research Society Quarterly
IJMMS -- International Journal of Mathematics and Mathematical Sciences
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[Note: This article was peer reviewed in March 1997 for journal publication and found to be correct technically and to be acceptable for publication. It is presented on the Internet so as to have the most rapid and widest possible circulation.]
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