THE QUANTIZATION OF SPACE TIME

It is possible to justify, or perhaps to rationalize, the quantization of space-time from prior work. The basic definition of force is

F º m a eq 22

and the basic definition of energy is

dE º F • ds eq 23

thus dE = m a • ds = (E/C²) a • ds eq 24

or dE/E = a • ds / C² eq 25a

E/E_o = e ^{ò(d2s /dt2 )•ds} /C ² eq 25b

Notice that there is energy and only that on the left side of eq 25a or 25b and space time variables and only those variables on the right. If energy is quantized, then that implies that space-time also is quantized.

As an induction it is proposed that the very fabric of space-time itself is quantized. This assumption implies that there is a fundamental smallest unit of space-time itself which determines the position of an event. "Position" implies the four dimensional coordinates of the event, including the x, y, z, and the time, t, of the change. This smallest unit of space-time will be called the SPASON.

The thing we call a vacuum is not "nothing". Indeed it does not have either mass or energy, and it does not exhibit momentum or in fact any mass related property. The most important thing is does have is the ability to convey position. In addition it has a capacitance, dielectric constant, e = -1, a speed of light (when paired with the attenuation per radian or the absorptive index, k, this defines the complex refractive index n* = n(1-ik) for other materials), permitivity¹⁴⁷ (8.85418782 x 10⁸ F/m) and permeability (4 p x 10^-7 H/m or 1.2566370614 ... x 10^-6 H/m)

The concept of the Spason is intended to subsume the older, and now semi-forgotten concepts of the HODON and the CHRONON previously proposed ca 1932 as the quanta of space and the quanta of time. The concepts of Hodons and Chronons fell into disrepute after their initial proposal since the early workers tried to apply energy quanta equations to them, and this failed. It failed since the concept of energy is irrelevant to these more fundamental quantum concepts. But there is a more basic flaw in the logic of having both separate quanta which caused the creation of the one new name Spason. Space and time are not really independent, but different aspects of a unified space-time continuum. Just like the electro-magnetic fields should not really be separated in the general case, space and time should not be separated in the general case.

TIME is CHANGE. To have change implies that there must be something there to change, that space itself must exist before time can exist. Time can not exist without space. Space might conceivably exist without time, but it would be unchanging and therefore rather un-interesting. Thus the concepts of space and time should not be separated, but the quantum of space and the quantum of time are linked, and the one word Spason is meant to remind us of this linkage. Where the spason exhibits mostly time like behavior the word chronon, which has priority, will be preserved to indicate this, and where the spason exhibits mostly distance like behavior, or the positional component is to be emphasized over the time like part, the word hodon will be preserved.

The basic quantum of the spason has been measured either in time units or in space (distance) units. But since it is one and only one concept, it is improper to measure it in both sets of arbitrary units at the same time. If the spason is measured in time units the chronon, T, would be expressed in seconds. If the spason is measured in distance units then the hodon, x, would be measured in centimeters or cm (or meters, but there will appear a reason for use of cm. as there is a coincidence that makes the cgs system of units very easy to use, and despite the recent domination of SI a return to cgs is here warranted).

The least possible change in distance, x, divided into the least change in time, T, yields a constant ratio for this universe (or any universe, but it may change for other universes). This least possible change ratio is C, the speed of the photon in that universe.

The speed of light in a vacuum, C, is defined as:

C= x/T cm/sec eq 26

THE FORMULATION OF A BASIS SET OF UNITS

The units used in "C" above violates the key statement that the spason should be measured only in either "time like" or "space like" units but not a mixture since this involves BOTH arbitrary units of time in seconds and arbitrary units in distance in centimeters. Here the basis will be taken in time like units alone and a new basis distance, x_b, equivalent to one second will be found. This equation will then be reformulated into a dimensionless set of BASIS units where the speed of light expressed only in consistent time like units is D and the distance in this basis frame of units, x_b, is expressed in units equivalent to seconds. The normalizing factor, b, that allows this is such that

x/b = x_b eq 27

C/b = D eq 28

D= x_b/T eq 29

Looking ahead the normalization factor b will be very close to one (in fact b = 1.04609857356) if time is in seconds converting to distance in centimeters. Thus this is the reason for reversion to the cgs system of units, since D for example comes out very near C in quantitative measure. It allows easy conceptualization of the system.

A BASIS VOLUME

In order to evaluate b it is necessary to introduce the concept of a "basis volume". The basis volume, B, is a small cell (this will turn out to be approximately one cc- in fact exactly 1.144768919 cc ) such that if all the mass and all the energy in the whole universe were divided by the total volume of the whole universe each photon would occupy one and only one basis volume. This implies that there will be one and only one set of changes happening in this basis volume. The photons thus are assumed to be the place where changes are occurring, i.e. as an induction the photon poles are being linked to space time itself as the place where changes occur. Since the poles change in two opposing ways these pairs of changes should not be de-coupled and a pair of changes is actually linked in the basis volume. [This pair of changes in fact are a sort of conservation of motion, one rotating left handed and the other right handed when the directly forward or the front option is blocked, thus this gives rise to several conservation laws at higher levels of interaction].

The basis volume, B, is related to the normalizing factor, b, such that

B = b³ eq 30

Let the total energy density of the universe, E_r, be defined such that if all the mass, m_u, and all the energy, E_u’, of the universe were converted to energy via equation 6

m_u C² = E_u" eq 6b

and if the total volume of the Universe, V_u , is divided into the sum of all types of mass or energy expressed in energy units, E_t then:

E_t= E_u’ + E_u" (= m_t C² ) eq 31a

r = m_t/V_u (gm/cc) eq 31b

and in a basis cell r/ B = r_b gm/basis volume eq 31c

or r /C² = E_r erg/cc eq 31d

then the energy density (also the total energy since there is one and only one photon involved) in the basis volume, E_b , will be

E_r /B = E_b erg/basis volume eq 32

and the number of disturbances, N_d , or the number of changes that take place, in this volume for the basis time of one second will require that the number times the duration of each quantum expressed in time, T be equal to 1 (second).

N_d T = 1 sec or N_d = 1/T eq 33

The next concept is that the number of spasons along one edge of this basis volume, N_e , can be given in terms of the quantum of distance, x, or more properly the quantum expressed in basis units, x_b, such that

N_e = b/ x = 1/ x_b eq 34

And the total possible number of spason in the basis volume, N, must also be the number along any edge cubed

N_e³ = N = 1/ ( x_b )³ eq 35

Note that these all must be in integer values, i.e. N, N_e, N_d , T and x_b all must be related by integer powers of some constant.

It is now possible to state all of these variables in terms of one key variable, the dimensionless speed of light, D. We already know that x_b

is related to T this way; x_b/T = D. These other variables must also be related by some integer power of D, and it is proposed that they must be related such that:

x_b = 1/D² eq 36

T = 1/D³ eq 37

N_d = D³ eq 38

N = D⁶ eq 39

N_e = D² eq 40

Note that the proportionality constants all are unity, since the events being described are also integer quantum events. There was early consideration that the quantum of time or the quantum of distance might be variable in differing parts of the universe, and might vary with, for example, photon energy. This was rejected since the photon energy is irrelevant and while the space-time quanta no doubt control the photon, it is less basic and can not control the more fundamental entity. Each change in space - time results in ONE - precisely one - no more no less - unit of time thus there is a one to one relationship in the number of disturbances, N_d, involved. What ever the number of possible spasons on the edge, N_e, of a basis cell is the total number clearly is N_e cubed.

Thus since N_d and T are inverse quantities, we must have some combination of integers that has powers of 3 and 2 in them and the lowest possible number for N is D⁶. Since 6 works with N_e coming out to the power 2, the whole system is the least power possible, and by Occam’s razor we cut the problem.

D is related to L₁ and the interpole distance to wavelength ratio implied in equation 3

l = L₁ d = 6 p⁶ d eq 3(repeated)

where

L₁ = 5768.3351 = 6 p⁶ eq 4(repeated)

The ratio of mass of electron to mass of proton has previously been stated in eq 13 as L₁/p. There are five poles involved and the space volume disturbed in the proton thus must be corrected to reach the linear portion of the motion by 5/3 for each photon. Thus the value of D should be

[(5/3) L₁/p ]³ = D eq 41

or in another form:

[(5/3) M_p/m_e]³ = D eq 42

or D= 1000 p¹⁵ = 2.8658145969 x 10 ¹⁰ eq 43

= 28,658,145,969 possibly as an exact integer.

C also can be calculated from theory, and is given by the equation below relating the mass of electron in a hydrogen atom plus the mass of the proton (6 p⁶ + p ) to the photon linear motion content involved as if the 30 photon poles, correcting for the restricted motion because of the electron- proton interaction accounting for 1/6750 total as if the poles were allowed to move in a linear (non- rotary or mass like) motion.

C= [ (30 + 1/ 2(15) ³ )( 6 p ⁶ + p )]²= eq 44

2.99792456193 x 10 ¹⁰ =29,979,245,619.3 compared to

29,979,245,620 (±110) = 2.997924562 x 10 ¹⁰ observed.

With the evaluation of D and C it is possible to evaluate the rest of the values.

b = C/D (eq 28) = 1.04609857356 eq 45

B = b³ (eq 30) = 1.144768919 eq 46

x_b =1/D² (eq 36) =1.2175977039 x 10 ^-21 (pure number) eq 47

T =1/D³ (eq 37) = 4.2486966942 x 10 ^-32 (pure number) eq 48

N_d = D³ (eq 38) = 2.35366295117 x 10 ³¹ (pure number) eq 49

N = D⁶ (eq 39) = 5.53972928467 x 10 ⁶² (pure number) eq 50

N_e = D² (eq 40) = 8.212893218 x 10 ²⁰ (pure number) eq 51

x = b( x_b ) = 1.2737272175 x 10 ^-21 cm equivalent to 1 sec eq 52

THE DENSITY OF MASS IN THE UNIVERSE

A TUBE MODEL INSTEAD OF THE CUBE MODEL ABOVE

There are at lest two quite different ways to look at the basis volume. It could be thought of as a very long tube down which the photon moved. Since it involved one second the photon at C would sweep out a length of roughly 3 x 10⁸ meters- or a small tube about 300,000 Km long and of such a diameter it would contain one photon.... in cross section roughly 3.81853 x 10 ^-11 cm sq or 3.486 x 10^-6 cm radius i..e. VERY small, particularly relative to the length. The other way to look at this basis volume was to set up a cubical volume (roughly 1 cc) such that it would contain one photon ON THE AVERAGE in each second. This means that as one photon leaves, on the average another photon will enter this volume. There are many of the tubes of spaghetti from the first model entering and leaving this cube, and each has one photon so that each tube contributes roughly a microsecond of residency to the basis volume. Thus the cube model overlaps many of the "tubes" above, and roughly a million photons will enter and leave this basis volume in the one second.

The second "cubical" model was, for me, easier to manipulate conceptually and to obtain quantitative meaningful results. Thus that will be presented here as the primary model. Below is a tube model.

The volume, V_d , of the average photon in the tube is

V_d = p r ² l’ = p d₁² l’/4 eq 53

letting L₁ (= 6 p⁵) be the constant relating the interpole distance, d, to the wavelength, l, then

L₁ d = l eq 4 or eq 53a

V_d =p d₁³ L₁ /4 = L₁ p d₁³ /4 eq 54

and converting the volume to a basis volume, V_db , and the diameter to a basis diameter, d_b, then

V_db = (L₁ p d_b³ )/4 eq 55

But V_db is the disturbed volume in the unit cell the fraction of the spasons actually involved in the change.

V_db= N_d /N = 1/ D³ = T = x_b x_b D eq 56

This then is the cross section of the photon times the length traveled, and note that, because to the paired nature of the poles, it is this rather than perhaps two times this value that also might perhaps be expected.

Because we know the value of D, that now allows us to compute the value of the energy of this "average" photon in the basis cell, it allows computation of the wavelength, and in fact all such properties including the frequency of this "average" photon and density of mass in the universe as a whole.

V_db = 1/ D³ = 4.2486966942 x 10 ^-32 (pure number) eq 57

solving eq 64 (V_db= L₁ p d_b³ /4) for d_b eq 58

d_b = (4/ p L₁)^1/3(1/D) =2.10881486229 x10^-12 (in b units) eq 59

=1/16.546978 D = [(2/3p)^1/3 (1/1000 p ¹⁷ ]=[(1/D³)(4/ p L₁)]^1/3

l_b =6 p⁶ d_b = 348.60775/D =1.21643511 x 10^-8 (in b units) eq 60

f = D/l_b = 2.3559124 x 10 ¹⁸(in cycles per second or Hz) eq 61

= D² /348.60775 = (3 p/2)^1/3 (106 p²⁶ /6)

(note f is independent of basis units)

E_b = hf = E_r/B = h D²/ 348.60775 =1.5610774 x 10^-8(erg/B) eq 62

r_b = r /B =E_b / C² = E_b/b² D²(per B) = h(3 p/2)^1/3 (6p⁴)(b²)=

=1.90075984 x 10 ^-29 /b² (in gms / B) eq 63

= 1.7369288x 10 ^-29 (in gms / B) eq 63a

r =1.90075984 x10^-29/b⁵(gms/cc)=1.517278x10^-29 gm/cc eq 64

The density of mass in the universe as shown in eq 64 is a function of the photon model, of h, and of b, but the D term has dropped out and thus the density is independent of the speed of light. This implies that the density of mass in the universe is an independent and basic property of the universe. It will be linked to the expansion of the universe, the radius and the age of the universe, but it is independent of C or D.

HUBBLE'S LAW

Based upon observation E. Hubble noted that far away objects appeared to be receding with a velocity, V, proportional to the distance, s, with the proportionality constant being the Hubble constant, H (note units in per second)

V = H s eq 65

The age of the universe, tu, and the radius of the universe, Ru, are related to H such that

tu = 1/H eq 66

Ru = C/H eq 67

Assuming that Hubble's constant is a true constant and

differentiating equation 65 we obtain the gravitational form of Hubble’s equation :

ds² /dt² = dV/dt = H ds/dt = HV = H² s eq 68

Taking the extremes ** of eq 68, the graviational form of Hubble’s equaion, this integrates to

s = so e^+Ht + B e^-Ht eq 69

The e^+Ht term is the expanding universe term, and the second term represents a contracting universe term usually ignored in our universe. This is a linked - time reversal - contracting contra-terrenne universe paired with our own universe.

** Taking the original Hubble equation V= Hs = ds/dt, this integrates directly to s = so e^+Ht (eq 69a) but lacks the second contracting universe term B e^-Ht (eq 69b) which is needed in the following derivation and should not be ignored so the extra step is really required to obtain the more general result.

Gravity is nothing more, or less, than the expansion of the universe. The expansion; however, is related directly to the presence of mass, or energy in the universe.

F₂ = m₂ a = m₂ d²r/dt² = G m₁ m₂ / r² eq. 70

d²r/dt² = G m₁ / r² eq. 71

r² d²r/dt² = G m₁ eq. 72

On the left are space-time and only space-time terms, on the right are mass terms. Thus mass is related ONLY to space-time.

substituting m₁ C² = E_{1 eq 6a}

C² r² d²r/dt² = G E₁

On the left are space-time terms and only space-time terms, on the right is energy. Thus if energy is quantized, then space-time also must be quantized, and the gravity term is directly related to space time, and also is quantized. The gravity thus occurs only where mass/ energy is present.

but d²r/dt² = H² r eq. 73

thus H² = G m₁ /r² =(4/3) p r³ r G/r² =4/3 pr G eq. 74

And finally the expansion of the universe must happen ONLY at the same point where gravity occurs, and all are really the same phenomena, just looked at in different ways.

There also is an energy of time flow. As time passes the photon poles are further and further apart relative to the distance at any given time. Thus the energy is less and less.

DE_t = t m Dt eq 75

where DE_t is the change in energy of mass, m, in a time interval Dt, and is the proportionality constant, but it also is:

t = H C² ( = 1850.908 ergs/ gm sec) eq 76

and eq 75 could also be rewritten as:

DE_t = t m Dt = H C²m Dt = mC² H Dt eq 77

The evaluation of t is 1850.908 ergs per gm second or 0.1850908 Joules per Kg sec. Starting with eq 78 the expansion term s_o e^+Ht taken for a "short time" relative to the 15 billion years total may be reduced to eq 75. For very long times, or near the start of the universe the exponential should really be used.