III Geometric Description of a Uniform Expansion The following section will develop the formulas that describe a “simple” uniform expansion of space-time, as represented in Figure 1.   Later the same relationships will be used to “expand the expansion”.   Notation   The following notation will be used to describe the various measures of distance and time used in the development of the geometric model describing the uniform expansion of space-time.   1.      Uppercase and Lower case letters.  Uppercase letters, such as T,D, V,A refer to absolute measures, and t,d,v,a refer to relative measures of time, distance, velocity and acceleration. 2.      Sequential notation.  Following the first designator letter, such as T, or t, with a “1” or “2”  demarcates when a measurement is made with “1” referring to an earlier age and “2” referring to a later age.  For example, T1 = 2 billion absolute years, T2 = 9 billion absolute years and t2 = 12 billion relative years.  (Sometimes, where necessary instead of a 1 or 2,  the letters A and B will be used). 3.      Intervals or periods.  Adding an “i” means that the term is associated with a specific interval or unit of measure. For example, T1i  equals a unit of measure of time associated with an absolute age of T1. Usually a year will be the fundamental dimensional unit. 4.      If an additional repeated Capitol A letter is added to an absolute measure, , such as VA1 or AA2, this correlates to motion along the “unobserved” dimension. This notation standard will be used in the section on “expanding the expansion”.    5.      Tu = The present age of the Universe. S = a volume of Absolute space (D^3).   The fundamental Equation describing the “simple” expansion of Space-time   dS/dT=T        Eq. III-1   The d stands for the standard mathematical derivative operator associated with rates of change; the S stands for a common three-dimensional volume of Space; the capitol T term stands for  “Absolute Time".  This simple relationship describes the rate that space expands with the passage of Absolute time.  (Philosophically, this equation is stating that because space changes, time exists. The epistemological considerations of this extra dimension of time and it’s relationship to space will be discussed in more detail later, once the basic relationships are formulated.)   Those who may be objecting to the proposed relationship because of the dimensional imbalance, (distance cubed = time squared) should know that this defines a relationship between space and time, just as the speed of light does in Special Relativity.   Integrating Equation 1 yields,               S = k(1)  T^2 + C              Eq. III-2    This formula predicts that the Absolute volume of an object or region of space would increase 4 times if the age of the universe were to double, and Figure 2 shows an object uniformly expanding.

Figure 12 shows two points on the expanding region of space (or object) with the notation used to describe the various dimensional relationships between absolute measures and relative measures.   Since it is usually more meaningful to describe relationships by actual distances between points rather than the volume, it is possible to establish how points on an expanding object, or region of space, will vary, given the above relationship between the volume of an object and a linear distance tied to the object, such as the width.               S= k(a) x D^3             Eq. III-3   This equation states that the absolute volume of any object or region of space is related to the specific Absolute distance, (D) between two points on the object or region of space cubed, then multiplied by some constant k(a).  Remember, the relative distance, d, still remains the same measure in a uniformly expanding space-time field so there is no locally measurable increase in the relative volume of the object or region of space.    Substituting the equation for the relationship for points on an object, Eq 3 into equation 2 yields,             k(a)/k(1) x D^3 = T^2                            Eq. III-4 D = k T^(2/3)                                          Eq. III-5                                                                         Equation 5 states that for a specific volume of space, the distance between the points on that volume of space will vary to the 2/3 power of the Absolute or Cosmic time elapsed.  If the age of the universe were to double, the distance between two points would increase by 2^(2/3).  Again, this is an absolute change in distance, relative measures of distance between the two points has remained the same.   Differentiating the distance between the two points D, with respect to time, yields the absolute velocity V associated with the expansion of space. Again, since there is no observed relative measure of a change in distance, there is no locally observed measure of velocity; the observed velocity change is only observed from the “absolute” reference frame.                V = k x (2/3) x T^(-1/3)                      Eq. III-6   If the age of the universe were to double, the velocity of the two points would decrease by 1/(2^(1/3)), as observed from the absolute reference frame.   Also, since this velocity is based upon a relationship between absolute time, which is perpendicular to relative time, the direction the velocity vector will be orthogonal to the line connecting the two points.  This change in velocity is associated with affecting the “intrinsic” velocity.             Differentiating the velocity between the two points D, with respect to time, yields the relative acceleration A, between the two points. (Two orthogonal rotations in the same plane, or again, a change in the “intrinsic” acceleration between the two points.)               A = k (-2/9) x T^(-4/3)                        Eq. III- 7   The k term can be removed from the above relationships by making the relationships comparative for two measures of Absolute time,  T1 and T2. .               D = k T^(2/3);             D1 = k T1^(2/3);        D2 = k T2^(2/3)  Eq. III-8 (A,B)             D2/D1 = (T2/T1)^(2/3)          Eq III-8   Similarly for Velocity and acceleration               V2/V1= (T1/T2)^(1/3)                        Eq III-9             A2/A1 = (T1/T2)^(4/3)                       Eq III-10   The absolute velocity of an object also has absolute Kinetic Energy (K.E) by K.E. = 1/2 m V^2.  This leads to the following expression for energy that can be associated to a mass at a given location.   E2/ E1 =  (T1/T2) ^(2/3)        Eq. III-11   These derived formulas are called the Ratios of Time   The Ratios of Time   D2/D1 =  (T2 /T1)^(2/3)        Eq III-8 V2/ V1 =  (T1/T2) ^(1/3)        Eq III-9 A2/A1  =  (T1/T2) ^(4/3)        Eq III-10 E2/ E1 =  (T1/T2) ^(2/3)        Eq III-11     The ratios of time relationships as developed so far describe the relationship between two points in an expanding space-time field.  Later these formulas will be proposed to additionally represent the actual motion of space-time itself.