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Gerardus D. Bouw, Ph.D.




          This paper was started back in 2002 and was originally entitled “Inconstant Constants.”  However, no matter how exciting and stimulating its start, the original attempts quickly disintegrated into boredom. 

          This paper focuses on the speed of light.  The universe appears to be immense.  We speak of billions of light years as if it really took billions of years for the light to reach earth from the most distant objects observed.  That such long travel times are not required has been demonstrated numerous times by such luminaries as Parry Moon and Dominica Spencer and John Byl.  In 1956 Moon and Spencer showed that the light from a 10-billion light-year distant galaxy could reach earth in as little as 15.6 years.[1]  About twenty years ago, Dr. John Byl reduced that to less than ten years. 

Still, many geocentrists and creationists think that the universe cannot possibly be so large without making a liar of God in the physical realm.  They assume that we know all things perfectly—that the speed of light is sacrosanct—so that God would have to invent a fictitious history of the universe in order to make it appear that light took billions of years to reach earth whereas light could have traveled only as far as 6,000 light years since the creation. 

To address that concern, Barry Setterfield postulated that the speed of light was much greater during the creation week.  Later he teamed with Lambert Dolphin to reassess and confirm his conclusion that historic evidence shows a steady decline in the speed of light until 1960.  But therein lies the problem with his theory.  Why would the speed of light stop decreasing in 1960?  It is too much of a coincidence to believe that we would detect the decrease in the speed of light just after the time it stopped decreasing.  It seems much more likely that our measuring technology is much better than in prior decades. 

In 1982 the inflationary universe was introduced into the study of cosmology.  According to that theory, for a brief instant in time, the universe’s size inflated some thirty orders of magnitude[2] while the speed of light was equally increased.  The inflationary theory was first proposed some ten years earlier, in 1972, but was ignored because it showed the entire universe to be at most 100,000 years old instead of the “scientifically acceptable” ten billion or more.  By moving the time of the inflation back, closer to the universe’s origin, the billions of years supposition was saved and the theory was rescued from obscurity. 

The inflationary model demonstrated that a rapid stretching of space increases the speed of light without affecting time.  Prof. James Hanson notes that modern science views time as the ultimate independent variable.[3]  The net effect is that the universe “ages” even though the length of a second of time remains the same.  In a sense, this is the four-dimensional counterpart of the expanding universe.  The Big Bang is sometimes described as “an explosion of space instead of an explosion in space.”  Likewise, inflation can be likened to an explosion of time instead of an explosion in time. 

And that brings us to the essence of this paper.  What happens if our units of measure, the inch, the second, the pound, the kilogram, the meter, etc. changes over time?  The question is related to how the wavelength of light and radio waves changes as the universe expands.  Although the mathematics is algebra with a little bit of multivariate calculus notation thrown in, it should never be forgotten that we are not describing how things normally happen.


Technical Introduction


          The analysis presented here is not to be thought of as an attempt to predict the behavior of normal interactions in space and time.  Nor is it an order-of-magnitude study (meaning the use of gross approximations).  This study deals with fundamental units, namely, units of mass, length, and time.  It describes what would happen to the speed of light, say, if the first law of thermodynamics—also called conservation of energy and often described as “Energy can neither be created nor be destroyed”—is inviolate and the length of an inch or centimeter were to shrink or expand.  In order to conserve energy, other units such as the second or the gram would have to adjust.  In effect we say that the unit of energy, the erg, is a true constant.[4]  The analysis shows how those units will adjust to any such fundamental change.


Conservation of Energy


          We have all seen the formula E = mc2.  It is the most famous of equations and the foundation of our analysis.  Unit-wise this formula can be stated as erg=gm cm2/sec2.  Doing so is rather confusing so we shall designate the unit of energy as <E>, the unit of mass as <m>, the unit of length as <l>, and the unit of time as <t>.  That way we are not bound to cgs (centimeters, grams, seconds) or mks (meters, kilograms, seconds) units but can deal with any units.  Our famous formula now fades from view when we rewrite it as:


<E> = <m> <l>2 <t>-2.                                         (1)


Remember that this is not the same as E = mc2.  It is a statement about the units we use to express that formula. 

          The changes in units for expression (1) relate as follows ( reads “change in” and d as “the total change in”):


d<E>=<l><t>-2 <m> + 2<m><l><t>-2 <l> – 2<m><l>2<t>-3 <t>.


In what follows, we shall drop the unit notation unless it is necessary to the understanding.  Doing so for the above statement gives:


dE = l t -2 m + 2m l t -2 l – 2m l 2 t -3 t.


          Conservation of energy tells us that the total change to the unit of energy dE must be zero.  In turn, that makes the above restatement read:


 l t -2 m + 2m l t -2 l – 2m l 2 t -3 t = 0.                            (2)


Physicists avoid this complication by assuming the solution m = l = t = 0; a trivial and boring solution.  We can simplify equation (2) quite a bit by multiplying both sides by (t2 l-2):


l m + 2ml – 2m l t/t = 0                                      (3)


Equation (3) relates changes in the units of mass, length, and time under the constraint that energy must be conserved.  Thus an increase in the centimeter (l>0) must be counteracted by either a decrease in the gram (m<0) or an increase in the second (t>0) or some fit combination of the latter two changes.


Planck’s Constant Considered Unit-wise


          Planck’s Constant is usually denoted as h or ħ (h-bar).[5]  It comes into play when we need to compute the quantum energy of a photon or the spin of a particle.  It is sometimes called “central motion” and Planck, himself, labeled it “linear harmonic oscillator.”  We can write its unit-wise relationship as:


<h> = <m> <l>2 <t>-1


We can now write any change in the unit of h as, again dropping the unit notation, taking the partial (), and multiplying both sides by (t/l):


(t / l) h = l m + 2m lm(l / t) t.


          Subtracting (m l / t) t from both sides gives:


(t / l) h  - m(l / t) t = l m + 2m l – 2m(l / t) t


From (3) we see that the rhs (right hand side) is zero.  It follows then that after a bit of algebra and rearranging terms:


h = m l2 t –2 t.


Converting this to unit notation for a moment we get:


<h> =<m> <l>2 <t>-2 <t>.                                      (4)


This is a particularly important result because it says that any changes detected in h over time means that the unit of time must have changed.  Conversely, if we find no change in h in the history of the universe, then time’s unit has not changed and time has “flowed evenly” since the creation.  Thus, if <t>=0, equation (3) becomes:


l m = -2ml


which says that any change in  the unit mass will be countered by twice as large a change in the unit length.  In simpler terms, if the gram were to double, then the centimeter would be reduced to a quarter of its current length.  (For now we beg the question as to how we could know that happened as there would be no noticeable change.) 

          Comparing equation (4) with equation (1) shows us that <m> <l>2 <t>-2 = <E> so we can rewrite (4) as:


<E><t> = <h>.                                              (5)


If we replace the partial change symbol, by the uncertainty or error symbol, Δ we can rewrite (5) as:


<E> Δ<t> = Δ<h>.                                               (6)


Usually, physicists assume Δ<h> = 0, that is, they assume that h is constant.  So assuming means that any change in time, t must be counterbalanced by a change in energy, E.  In other words, modern physicists hide a change in h with a change in E or a change in t.  Thus we arrive at the usual form presented in physics texts:


Δ<E> Δ<t> = <h>.                                             (7)


Converting back from unit notation to regular notation, we can rewrite (7) in its regular form,

ΔE Δt ħ/2,                                                (8)


which is called the “Energy Uncertainty Principle” or EUP for short.  It is somewhat related to the famous Heisenberg Uncertainty Principle and we shall have much more to say about this mysterious expression in Part III of our paper.  For the time being, we shall confine ourselves to the relationship between the expression (5) and the inequality (8). 

          The classical, albeit erroneous interpretation of this form of the Energy Uncertainty Principle says that no experiment can ever determine both energy and time to any greater accuracy than half a Planck Constant.  The Uncertainty Principle has to do with uncertainties in experimental measurements, not in units.  In (8) it is assumed that there is no change in h.  In (5), on the other hand, there is no “uncertainty” in the energy, E, because we approached the problem from the assumption that energy is conserved; that is, from the perspective of a closed system instead of an individual particle which may have energy imparted to it from the outside.  That is, expression (5) translates (8) to


                                                        2E Δt Δħ.                                               (9)


In the parlance of physics, (8) is local physics while (5) is universal.  The reader must not infer from this that there is here a contradiction of some sort.  Uncertainty in a measurement is not the same as changes in the lengths of the units used to record the measurement.  In other words, any inaccuracy in a measurement of one’s height (local physics) is far more likely due to uncertainties in the measuring process than any uncertainty in the exact length of an inch or centimeter (universal or global physics). 




          In this first of three papers, we looked at the relationship between energy and time.  We started with the assumption of conservation of energy: that energy can neither be created nor destroyed by natural processes.  Conservation of energy is also known as the First Law of Thermodynamics.  We next examined what would happen if the fundamental units of length, mass, and time were changed under the constraint of the First Law.  Although we presented the units of length as the gram, centimeter, and second, they could be any set of units, even the Planck mass, Planck time, and Planck length. 

          In the course of the analysis, we derived a form of the Energy Uncertainty Principle (9), which does not exactly correspond to the standard EUP (8) because the latter is generally interpreted as statistical instead of physical.  However, cosmologists have long recognized that the standard EUP cannot be interpreted statistically.  The reason is that the standard uncertainty principles require vectors or operators on the left-hand side of their respective statements.  Energy can be an operator, e.g. as a Hamiltonian, but time cannot.  Our analysis thus exposes a flaw in our concept of time as the ultimate independent variable.  To put it bluntly, there is a problem with our linear notion of time.  There is a problem with the common view that time flows in a straight line from the past to the future and that the border between the two is the present.  However, before we can solve that problem, we need to examine time as used in our so-called natural languages; that is, we need to look at the linguistics of time. 

[1] What Moon and Spencer did was in the same vein as what we shall do in this paper.  Moon and Spencer proposed a Riemanian metric (curved space) along which light travels.  We shall confine ourselves to Euclidean (flat) space because we are dealing in a realm where space is absolute and relativity is irrelevant. 

[2] In mathematics, an order of magnitude is a factor of ten.  Thus two orders of magnitude is a factor of 100 and thirty orders of magnitude is a one followed by thirty zeroes.

[3] Independent variables are quantities that drive the dependent variables.  Usually the dependent variable is found on the left hand side of an equal sign while the independent ones are on the right side.  Philosophically, treating time as the ultimate independent variable means that scientists will have to look to time to make their theories work.  For instance, those who do not like the Bible’s account of creation will look to time (billions and billions of years) to account for the creation as a chance event.  Sometimes that appears as, “In time we will discover how it ‘really’ happened.”  For instance, 25 years ago I spoke with a biologist who thought Joshua’s long day was a hallucination.  When I mentioned that Joshua’s long day was a long day for half the world and a long night for the other half and that there was even an account of a long sunset he was flustered.  He had assumed that all tales of a long day were of a long period of daylight hallucinated by one man or a mass hallucination.  All he could do was to blurt out, “Well, the study of phenomenology is a just new science.  In time we’ll know how it happened.”  In other words, “I don’t want the Bible to be true, so I’ll put my hope in the thought that in the future someone will come up with an explanation for such a mass hallucination.” 

[4] This is probably the weakest part of my argument.  Even though the conservation of energy is called a law, viz. the first law of thermodynamics, yet a certain degree of ambiguity (fiction?) enters the picture when we consider potential energy, which is a relative energy, not absolute.  Ludwig Boltzmann and Ernst Mach argued whether energy is real or not, Mach claiming it to be real.  From Boltzmann’s point of view, I should work not from the stance of the first law of thermodynamics but from the second law, entropy.  The second law can be derived from the first, however, so we continue under the assumption that the only absolute potential energy is bound in the rest mass of a particle and is thus measured relative to the firmament.  We shall also ignore the “force times distance” definition of energy. 

[5] Basically, ħ corresponds to a radius equal to the smallest amount of energy that can be passed from one object to another and h is the corresponding circumference, i.e., h=2π ħ.