Download The Calculus Reveals Special Properties of Light

The Calculus Reveals Special Properties of Light
Richard D. Sauerheber a,b
Submittted to Science, American Association for the Advancement of Science, “on lockdown”
a
Department of Chemistry, University of California, San Diego, La Jolla, CA 92037 U.S.A.
Department of Tutorial Services, Palomar Community College, San Marcos, CA 92069 U.S.A.
b
A previous article systematically presented, with geometric symmetric inverse functions, the validity of the
Fundamental Theorems of the Calculus and its important mathematical consequences1. Applications of the Calculus to
the physical world have been extremely diverse and productive. Here, the properties of the sine function and its
Calculus are described that delineate the intrinsic behavior of light. Light waves themselves may be both amplitude
and frequency modulated independently, where electronic acceleration and deceleration events in stars between
differing states of energy produce light of variable amplitude and frequency. However intrinsic propagation speed for
light is fixed at c, as determined by Maxwell in the 19th century from his famous equations of state for all massless
electromagnetic radiation. Thus, the wavelengths () of light resulting from such widely varying electronic transitions
must follow the relation  = c/f, where f is the frequency of the light, directly proportional to its intrinsic energy
content hf (h being Planck‘s constant).
It is important to emphasize the power of the Calculus in describing aspects of the physical world reliably. After
Maxwell’s major triumph, Albert Michelson in the San Gabriel Mountains, California performed difficult experiments
with a rotating slotted mirror for light beams traveling 70 km round-trip that confirmed to seven digit precision the
Maxwell value for light speed, as c = 1/()1/2, by direct measurement2,3,4. and  are the electrical permittivity and
magnetic permeability for the given medium. Maxwell's constitutive and field equations prove that light must travel as
sinusoidal waves of alternating self-induced perpendicular electric and magnetic fields, and we now know light waves
travel in vacuum in the complete absence of a material substance. Michelson’s data confirm that speed c reflects the
propagation speed along the wavelength axis.
A deep understanding of the physics of a light wave is possible by considering the properties of the sine function and
its multiple derivatives. From Newton we know that velocity and acceleration functions are respective sequential
derivatives of a position function. From Maxwell we know that the sin(x) graph delineates the variations in electric and
magnetic field intensities as a function of position along a light wave, where intensity is 0 at wave position 0 and
maximal at /2. The derivative of the sine function reports the rate of change of the field intensity along the wave,
where d[sin(x)]/dx = cos(x). Thus the cos(x) graph indicates the velocity with which the field intensity changes and is
maximum at 0 but slows to 0 at /2. Therefore a negative acceleration slows the velocity, from maximum to zero, and
this is borne out in the second derivative, where d2[sin(x)]dx2 = -sin(x). This upside down sin(x) curve indicates the
acceleration is negative between 0 and  /2, zero at 0, and maximum negative at /2. Thus the electric and magnetic
field intensity at position 0 has maximum velocity (with zero acceleration at this position) that is responsible for the
increasing value of intensity that proceeds along the wave direction.
The rate of increase in intensity however slows until position /2 where the velocity of intensity change is now zero
and the negative acceleration is maximal. The acceleration of electric field strength is due to the accompanying
perpendicular magnetic field induced coincidentally in time, and vice versa. After this, the acceleration reverses the
direction of field intensity with a negative velocity until position  at which time the velocity of the change is
maximal negative and the acceleration returns to zero. The magnitude of acceleration itself is also not constant but
sinusoidal, as indicated by the third derivative of sin(x), which is d 3[sinx)]/dx3 = -cos(x). Unlike gravity surrounding
mass, which is a fixed value at a particular distance, variable accelerations of magnetic and electric fields produce
varying velocities and intensities, all sinusoidal in shape, shifted in phase. This is analogous to a car accelerator pedal
being pushed variably rather than to a fixed position.
The fourth derivative of sin(x) reproduces the original function, d4[sin(x)]/dx4 = sin(x), which deserves special
comment. The fourth derivative is the second derivative of the second derivative and thus reflects the acceleration of
the acceleration. If an accelerator pedal is itself accelerated rhythmically back and forth, the car’s acceleration is
determined by the acceleration of the pedal. The return to the original function shape for the fourth derivative of the
sine indicates that the acceleration of the acceleration follows the same pattern as the original variation in electric and
magnetic field strengths themselves. The very fields that the variable accelerations produce are themselves responsible
for and pre-determine the magnitude of the changing accelerations. The original varying accelerating fields are first
produced when light forms by varying accelerating electrons undergoing energy transitions to produce the light unit.
The frequency, wavelength and amplitude of the produced wave are governed by the behavior of the electron and its
environment during its energy transition, and the induced field accelerations then mutually induce corresponding field
intensity changes in ever-repeating cycles.
From laws of thermodynamics, it is not possible to construct a perpetual motion machine to do work indefinitely on
its own. However, energy transmission in EM radiation, if transported through a perfect vacuum, millions of miles
from stars in space is a phenomenon that occurs in perpetuity, if matter does not interfere, that reflects conservation of
energy. Electric and magnetic field strengths both change in a regular fashion (sine) with a changing velocity (cosine)
due to changing acceleration effects (-sine) of the fields on each other that varies rhythmically (–cosine) with an
acceleration (sine) that reflects the originally produced changing field strengths. This is natural perpetual motion of
energy transport that is analogous to Newton’s first law of motion for masses that, in an ideal vacuum without gravity
(which only exists in theory), maintain travel velocity and direction in the absence of a perturbing force.
A stretched spring dangling from a roof that oscillates itself is a reasonable analogy, where the rebound force of the
spring is determined by the original input force of stretch, which itself reappears again because of the rebound force,
and so on. For an ideal spring oscillating as a sine function, the speed of the moving end is zero at the extreme
stretched and compressed positions, where acceleration is maximal due to the large restoring force when stretched. The
restoring force is itself variable with position, and thus the acceleration rhythmically changes in magnitude dictated by
the third derivative. The fourth derivative of the position of the spring returns to the sine function and represents the
acceleration of the acceleration. The acceleration of the end of the spring begins maximum at an extreme position of
stretch, decreases to zero at the midpoint of motion, and reverses direction and is maximal negative at the opposite
extreme. Thus the velocity of the acceleration change is not constant but exhibits a rate of change that is the
acceleration of the acceleration.
For light, the acceleration of a field, at any extreme amplitude, back toward the center of the wave axis is analogous
to the restoring force of the spring but is due to the effects of the mutually maximal (stretched) magnetic and electric
fields that must decelerate each other toward the wave axis again. Light is massless and does not have internal friction
and heating that a spring with mass does. Light oscillations are perfect fields not subject to gravity force fields, and
remain unperturbed in driving through space unless interacting with material substances, such as the sun’s corona
matter that refracts or reflects it7, or the earth’s oceans that refract, slow and absorb it, or any medium in which  and 
exhibit altered magnitudes.
These functions of sequential derivatives reveal the incredibly elegant, smooth order in the reversing and everchanging field magnitudes, and are fundamental. As shown here, ever-changing velocities and accelerations form a
harmonic regular cycling unit of field strengths that only can exist in this state of self-induced propagation. Light is
indeed special, where changing electric fields induce changing magnetic fields, and simultaneously, changing magnetic
fields induce changing electric fields (properties separately delineated by Oersted, Ampere, Lenz and Faraday) within
the same internal massless entity.
It is useful to think of the actual fundamental unit of light as being perpendicular electric and magnetic field vectors.
Oscillation of these vector fields, while propagating laterally, in a rhythmic pattern constitutes light. One cycle or wave
occurs when an increasing magnetic field from 0 to /2 induces an increasing electric field (or vice versa), and
subsequently a decreasing magnetic field (due to the electric field increase) decreases the electric field magnitude until
. At this point the decreasing electric field reverses direction and induces a reverse magnetic field (or vice versa) until
3/2, where the maximally decreased magnetic field begins to increase (as a result of the maximally decreased electric
field) (or vice versa) until 2 when the cycle starts again. The accelerating electrons on the sun, that lose energy
equivalent to that contained in the EM wave, produce the original electric and magnetic fields that subsequently induce
(accelerate), decrease and negate (de-celerate) and then again induce (accelerate) each other continuously while
propagating through space. Being massless radiation, light requires no medium in which to travel and in fact travels at
fastest speed in vacuum, dictated by constants in vacuum, c = 1/(oo)1/2.
While the light electric field accelerates and decelerates the simultaneous magnetic field, and vice versa, the sine
second derivative for acceleration reveals a maximum negative acceleration at position /2 and maximum positive
acceleration at (3/2). The acceleration itself is also changing and accelerating and it can be said that light accelerates
itself on an axis perpendicular to the wavelength axis along which it propagates. Light propagation speed is constant,
and since we know from the Calculus that the derivative of any constant is zero, light does not accelerate in the
propagation direction, while the field intensities accelerate and decelerate each other continuously. Think of a spring
that stretches only until its maximum potential energy position is achieved at which point it stops and then reverses,
accelerating and de-accelerating until another maximum position. Light also pulls electric and magnetic fields to
perpendicular maxima and must reverse again in an analogous manner. Here, forward acceleration a = dc/dt = 0.
Because a is a vector quantity, light propagation velocity can however be accelerated, for example by reflection from a
mirror, and then a = (vfinal – vinitial)/t = [+c – (-c)]/t = 2c/t where t is the time required to reverse direction.
From circular dichroism and chiral (light rotating) compounds in spectroscopy, we know that light waves can
actually rotate about its wavelength axis as it propagates through space. It is possible that any individual photon
produced from de-energized electrons on the sun always rotates since plane polarized light is the sum of two waves in
phase that rotate in opposite directions. Left or right-handed circularly polarized light represent the sum of electric
vectors for two waves rotating ¼ wavelength out of phase. This form of propagation is analogous to fired bullets that
spin while also traveling forward at muzzle velocity. Since the electric and magnetic fields oscillate perpendicular to
the axis of travel, the velocity of a leading edge of a field vector, as it rotates and also oscillates, exceeds magnitude c
so that the wavefront itself propels forward at fixed magnitude c. In absorption spectroscopy the net electronic vector
of this motion is referred to as the transition dipole moment that makes an angle with respect to the direction of light
polarization5.
The fundamental unit of light, the transverse electric and magnetic field intensity vectors are themselves responsible
for oscillating, rotating and propagating in a wave pattern of motion that repeats itself through space. This is analogous
to the de Broglie wave of an electron in an atomic orbital. The fundamental unit is the electron itself, but the unit of
motional activity is the de Broglie wave, the pattern in which the electron regularly and sequentially vibrates. For light,
the motional unit is also a wave, while the fundamental entity is the orthogonal electric-magnetic field axis pair. Useful
terms for light have included the corpuscle from Newton (as long as it is recognized the term refers to an entity that is
massless), the photon from Einstein's photoelectric effect and Compton's scattering phenomena, the quantum from
Planck's quantized emission energies of EM radiation, and the wave (Young, Maxwell) from phenomena of diffraction,
refraction, scattering and intrinsic sinusoidal behavior, where each term has advantages in its use that do not detract, as
a result of this discussion, from the usefulness of the other terms.
From the Calculus line integral for the sine function, the average speed for the field transverse oscillating edge
would be the arc length of the sine function from 0 to 2 divided by the time required for a wave to travel its own
wavelength. The line integral of the sine function computes the actual distance through space traveled by the field
edge where for a full wavelength of lateral travel is: arc length = ∫(1 + cos 2(x))1/2dx from 0 to 2 = 7.64… , the actual
distance spanned during the time for a wave of length 2 to pass itself at speed c. The actual intrinsic travel speed then
becomes [7.64/(2)]c = 3.6 x 108 m/s. Actual light waves are amplitude and frequency modulated independently, with
lower amplitude to wavelength ratios. An idealized wave, where the ratio of wavelength to amplitude is 2 and is
plane polarized without rotation, for simplicity, is used for simplicity.
The line integral for the sine function, calculated by electronic iteration to 3 digit precision, is 7.64. Evaluating with
several algebraic terms of area for rectangles drawn under the function, for three such rectangles with width 2/3,
when multiplied by the values from the integral formula are [1+cos 2(2/3)]1/2 + [1+cos2{2(2/3)}]1/2 +
[1+cos2{3(2/3)}]1/2 = 7.64 as well. Since the travel distance of the field edge as it oscillates is 7.64 units for 2 units
of travel for each wavelength, the total average speed traced by the field edge would be [7.64/(2)]c = 3.6 x 108 m/s.
The above speed, that exceeds propagation speed c, holds for any un-modulated idealized wave independent of
wavelength. For example 1.3 x 10-15 seconds is the time required for a 400 nm wave to travel one wavelength (t = 4 x
10-7 m/3 x 108 m/s). The average total speed of the changing electric field edge for such an un-modulated idealized
wave would be v = d/t = (7.64)(4.0 x 10-7 m/2(1.3 x 10-15 sec) = 3.6 x 108 m/s. The lengths of a fundamental unit of
electromagnetic radiation, first described as a wave by Young, of course vary tremendously, from femtometers for high
energy nuclear emission, to miles in length for radio waves, while the amplitudes depend on the physical arrangement
required to produce the radiation.
Electromagnetic field fluxes are related to the area of the sinusoidal hump and the total energy of the photon wave,
where equal amounts of energy occur in both the electric and magnetic fields. Since an electric field vector oscillates
while propagating laterally at speed c, from the 2nd Fundamental Theorem the field magnitude itself at any position
along the wavelength axis represents the slope derivative of the integral formula for the area the field sweeps out in
space. The instantaneous area dA swept through space at position x = /2, for a 1 meter amplitude radio wave of 2
meter wavelength is dA = (1m)(dx m) square meters. This rate only occurs for an instant in time since the field
intensity constantly changes, but for one second would sweep (1 m)(3x108 m/s)(1 sec) = 300 million square meters. At
x = 0, and the rate of area swept would be zero while the energy still travels laterally at speed c. The average
instantaneous area swept is (2/m)(dx m)since the average value of the sine function is 2/ The actual area swept
after one wave travels its own length is the absolute value area of two sine curve humps (4 m2 from integration, or
2(mm = 4 m2 from average value times wavelength). Dividing by the time required for a wave to travel a 2
m wavelength (t = 2m/c m/s), yields an average rate of area swept A = (4 m2)/[2m/c m/s] = 191 million square
meters per second.
It is generally accepted and borne out by experiment that physical objects with mass cannot exceed or achieve
speed c. But light having zero mass indeed propagates at this fast fixed speed that was determined to be c = 1/(
from coordinate of origin with the Calculus and is a universal constant in any frame of reference. The intrinsic speed
with which light propagates in the direction it travels, from the location in space at which it leaves its source, is c.
Increasing the velocity of the source cannot change the speed with which light emanates from it. Speed c is intrinsic to
light and is constant for a given medium. Intrinsic velocity for light however is a separate calculation, since velocity is
the rate of distance accumulation along any particular desired direction (rather than simply the total distance actually
traveled with time for intrinsic speed). Light reflected that returns to its origin has a velocity of zero, so to exceed light
velocity, simply do not make a u turn, or point the light in a direction other than the travel direction desired.
Experiments with laser light sources, observed while the earth revolves and orbits and produces variable lateral
source velocity, demonstrated that light pulses shift laterally along with the source and target while propagating at
speed c to intercept the target8. While intrinsic speed remains c in the propagation direction, the intrinsic velocity for a
light component depends on the bearing one uses in a computation, where component velocity may vary from +c
through 0 to –c. Light beams emanate from a lit candle at speed c only in their specific direction of travel. All
technically have different velocities, some velocity +c (East), others –c (West) and any traveling North have velocity
East of 0.
Maxwell’s equations for EM radiation demonstrate that intrinsic translational propagation speed for light is fixed at
c in a given medium, which is a factual tenet of special relativity. The relative speed and velocity however between a
moving detector and a light wave are not described by Maxwell’s equations. For example, Otis6 found that detectors on
earth when moving toward starlight detect increased light frequency, compared to that when detectors are receding.
Since c = f, it is the relative total velocity (between light front and the detector) that increased the observed frequency
of waves detected, with a Doppler effect, since c is fixed and the wavelength likewise is not altered by motion of a
detector toward light produced by a distant source. Also, two light beams propagating in opposite directions illuminate
twice the distance in unit time than can a single beam. Relative velocity of illumination is then magnitude 2c while
intrinsic velocity is c.
Much is special about relativity for light that differs from classical relativity for objects with mass. The Calculus
proved that propagation intrinsic speed for light must be fixed at c, and this is found for all light from any star
independent of the frequency and energy of the light wave. Wave length, amplitude, rotational speed, oscillation
frequency, color and energy vary widely for starlight from various sources, while all must travel at fixed propagation
speed c along the wavelength axis. Light produced from forward moving sources has higher frequency and energy, and
shorter wavelength, than light from a corresponding stationary source, but in both cases speed remains fixed at c from
the coordinate in space at which it departs the source.
For a light pulse to travel across a train boxcar of length L requires time tlight = L/c if the car and light source inside
(and for simplicity the earth and solar system also) were stationary. A completely different situation occurs however
when the boxcar and light source are moving. The time required to reach the other end is different because the boxcar
moves while the light travels across it, which must still travel at its same un-perturbed intrinsic speed c, albeit with
increased frequency and shortened wavelength. Thus, tlight = (L + d)/c where d is the extra small distance shifted by the
car during the time light travels across it. When car speed is small or zero and d = 0, the time then again returns to the
expression for the stationary car. This is special relativity for massless light which does not occur for projectiles with
mass.
Time (Stationary Car)
Attached
Source
Ground
Source
Time (Moving Car)
tlight = L/c
tbullet = L/v
L
tlight = L/c
tbullet = L/v
tlight = (L + d)/c
tbullet = L/v
tlight = (L + d)/c
tbullet = (L + d*)/v
The rectangle here is an overhead view of a railroad boxcar of length L on which an attached source fires either a
light pulse or a bullet indicated by the arrowhead. The adjacent source and arrowhead is on the ground, not attached to
the car. In the stationary case when the train is not moving the times are L/c for the light or L/v for the bullet with
imparted speed v, to reach the opposite position of the boxcar, whether the source is attached to the car or is on the
ground. In the case when the car is moving to the right, a pulse or bullet is fired from either the moving attached source
or the stationary source at the position shown. Here the time for light is the same for both stationary or moving
sources,( L + d)/c, due to constancy of intrinsic speed for light from either stationary or moving sources. The times for
the bullet differ for the moving gun vs. the stationary gun because of addition of train speed and bullet intrinsic speed.
Time is dilated to illuminate the moving object whether the flash is produced by a stationary or a moving source. For
the bullet to span the car, time is dilated due to motion of the car only for the ground gun. Unlike for light, time is not
dilated when the source gun moves along with the car.
In the stationary car, the time for a projectile with mass fired from a gun would be tbullet = L/v, where v is the intrinsic
speed of the projectile imparted by the gun above any speed of the gun. However, when the car and gun are moving at
speed s, the time required to cross the car remains the same as when stationary, as is found from the appropriate
algebra. Here
tbullet = (L + stbullet)/(v + s).
Note that speeds are additive for the bullet since it is a mass, so that the total ground speed of the projectile, v + s, is
sum of the intrinsic speed v imparted by the gun plus the speed of the train s. Algebraic rearrangement produces:
tbullet = L/(v + s) + stbullet/(v + s)
and tbullet = L/(v + s)/[1 – s/(v + s)] = L/v = t, as for the stationary car.
Thus, the time required for a projectile to span the length of a moving object is the same as when that object is at rest,
because the greater total relative velocity, v + s, makes up for the extra distance traveled to reach the target, L + d*,
where d* = stbullet. The relative velocity between the target and bullet remains v because the total intrinsic speed of the
bullet v + s, after subtracting the receding speed of the target s, is indeed v.
Light always requires a different time to illuminate a moving object than when that object is at rest, because light
intrinsic speed c is pre-determined by the Calculus Maxwell equations for light and cannot add to source speed. The
difference in time is due to the fact that a moving object shifts while light travels across it, requiring extra time for light
to accumulate the entire span. The effect would be detectable of course only at very large object speeds, where the total
relative velocity between target and light pulse is c – s when the target recedes from the pulse. References to time
dilation involve shifts in actual distance spanned by light beams, rather than shortening of absolute time, because light
must travel through free space at c independently from the moving object being illuminated.
Notice that the dilated time is associated with motion of the object that detects the light, which occurs whether the
source of light is stationary or at rest because light speed is fixed with respect to a specific coordinate in space at which
it departs its source, whether that source moves from that location or not. An analogous effect however can also be
produced for projectiles, by placing the gun on the stationary ground next to the boxcar. Here total bullet speed
remains v whether the boxcar is in motion or stationary. The time required to pass the stationary boxcar is t = L/v.
When the car is receding from the bullet, time is tbullet = (L + d*)/v. This effect confirms that the longer time for light to
illuminate a moving object is due to light being massless with always-fixed travel speed independent of motion of the
source, rather than to a dilation of absolute time or a bending of space. A bullet’s time also dilates if the rifle is not
attached to the moving object and has no alteration in velocity due to boxcar motion. The difference though is that
bullets have mass and the effect between moving and stationary car is not observed when the rifle is in the car, whereas
light is special in that the effect is observed in both cases because c cannot be altered by source motion. Sound speed is
also independent of source speed but is quite different than light in requiring a material medium in which to propagate.
The proposal that the total relative velocity between a light wave and a moving detector somehow warps to a fixed
value of c, an extrapolated part of special relativity theory, has been the source of much contention, disputed frequently
by Maxwell, Dirac, Dingle, Albert Michelson and his understudy Dorian Miller, Petr Beckmann, and Arthur Otis,
whose findings above confirmed that Maxwell fixed intrinsic light speed is not influenced by relative motion of source
or detector. Relative velocity between light front and a moving detector ranges from c + v to c – v, an important feature
of Einstein’s analysis of relativity in 19054, where v is the velocity of the detector. We now know that Doppler
motional effects caused by source motion produce corresponding changes in frequency and wavelength (higher
frequencies of solar radiation occur on the side of the sun spinning toward earth compared to frequencies on the side
spinning away from earth) while their product, intrinsic speed c = f, remains constant. Doppler effects due to detector
motion are different and alter the relative velocity and the relative frequency with which waves are detected, without of
course affecting either intrinsic frequency, wavelength or speed of the light produced at the source. Intrinsic frequency
is constant from the sun while the earth moves away in Spring and toward the sun in Fall, but observed relative
frequencies differ (under equal atmospheric conditions) by a small factor (the ratio of earth velocity to light, ± 30
km/sec/3x105 km/s).
Van Flandern4 reported that the earth always turns in its orbit toward the physical location of the sun, rather than the
perceived position of the sun seen from light waves arriving on earth from it. This indicates that gravity is sensed at a
speed greater than light. In the present example, the velocity of the electric field net average intensity vector is an
intrinsic property of light, where any vibration perpendicular to the line of travel must follow a path of distance greater
than the wave front, and thus must have an intrinsic velocity above c, the speed at which the wave itself must
propagate forward. A complete discussion of special and general relativity are beyond the intent of this article but has
been presented elsewhere4.
The simplest example in nature that compares in motional complexity to the light wave would be the earth, which
also propagates, rotates and oscillates. It propagates around the sun at average 30 km/sec while it rotates with angular
speed 2 radians per 24 hours and oscillates (or wobbles) its polar axis over a 26,000 year precession 8. Each motion
occurs at its own intrinsic speed and directional velocity along a particular axis (longitudinal orbital ecliptic for
propagation, polar axis about which rotation occurs, and precession occurs around an axis perpendicular to the
equator). The speed of a position on earth exceeds the propagation speed of the earth itself because of rotation and
oscillation. Likewise, an electric or magnetic field edge position in a light wave exceeds the velocity of light because
of oscillation and rotation.
Light propagates with intrinsic speed c, and if uninterrupted has intrinsic velocity +c along the direction of its own
wavelength axis. Unlike the earth, light rotation and oscillation are symmetric also about the same wavelength axis.
Any particular light fundamental unit (of electric and magnetic transverse field vectors) can have unique rotational and
oscillation intrinsic speeds, but always having fixed propagation forward intrinsic speed c. Spiral motion occurs when
propagation and rotation share the same axis without oscillation, as for a spinning bullet. Light can spiral forward (or
travel as a fixed plane if filtered and un-interrupted), but because of oscillation it is a vibrating, propelling spiral.
It is fortunate that we have the Calculus to add insights into this discussion, with its fully reliable properties.
Since the Calculus was definitive, Newton proceeded with liberty immediately after its discovery to solve pressing
and complex questions concerning the solar system and the behavior of gravity. He completed vast experimentation in
optics and first described physical properties of visible light, including the deduction that light is accelerated by
reflection from a surface without actual contact, similar to the earth that orbits around the sun by influence of gravity
from a distance. It should be noted that Newton did not make hypotheses, but rather tested questions and then stated
factual conclusions. However in writing in Opticks he was confident in discussing that electricity, magnetism and
gravity are natural forces that all act at a distance; material surfaces can reflect light without immediate contact, while
other materials of differing structure can emit light and others can absorb it.
Characterizing the mathematical structure of light waves has practical significance. Visible light with wavelengths
400-700 nm travels from the sun to the earth and stimulates the ejection of electrons from chlorophyll in plant
photosynthesis. The light reaction produces the world’s oxygen supply (as electrons from water replenish light-ejected
chlorophyll electrons) and the dark reaction removes carbon dioxide from the atmosphere to form carbohydrate,
continuously replenishing the world‘s food supply. Chlorophyll consists of pyrrole and other ring structures of
diameter approximately 0.2 - 0.4 nm. One would surmise that light energy compacted in sinusoidal humps of such size
would efficiently induce electronic oscillations in the molecule causing ejection of electrons to nearby cytochromes.
From Compton we can calculate the energy imparted to electrons by light absorption or scattering, and in the reverse,
an estimate of the electron velocity change involved in producing an EM wave and thus estimate the wave amplitude.
The estimate is crude since electronic transitions involve changes in potential energy as well as changes in translational
and rotational kinetic energy and magnetic energy, but for the sake of argument equating the electron kinetic energy to
the energy of the light wave formed, (1/2)mv2 = hc/ and (½)(9.1 x 10-31 kg)v2 = (6.6 x 10-34 J-sec)(3 x 108 m/s)/4 x 107
m) where v = 1.4 x 106 m/s, a velocity typical for electrons in atomic orbitals. Multiplying by the time for a quarter
wave to travel suggests that the electric field amplitude of such light waves, a = vt, could be on the order of 0.26-0.46
nanometer dimensions. Since this compares to the size of individual pyrrole rings in chlorophyll, light photons of such
structure, spinning forward with oscillations of proper frequency into the optically chiral molecule, would efficiently
couple absorption with electron ejection into cytochromes in the photosystem. In mammalian systems, the photoisomerization of 7-dehydrocholesterol to form pre-vitamin D3 with 296 nm light waves may also require an optimum
amplitude, where electrons in a non-aromatic cyclic conjugated chiral diene rotate to form an acyclic conjugated triene
isomer. The importance of circularly polarized light is well known in the natural world. Ants, bees, crickets and the
mantis shrimp have anatomic eye features that enable distinguishing between right and left circularly polarized light,
and beetle chitin, bird feathers and butterflies reflect circularly polarized light10.
References
1. Sauerheber, R. D., “Geometric Demonstration of the Fundamental Theorems of the Calculus,” International Journal
of Mathematical Education in Science and Technology, volume 41, 2010.
2. Giancoli, D., Physics for Scientist and Engineers, Prentice Hall, New Jersey, 2009.
3. Beiser, A., Concepts in Modern Physics, McGraw Hill Book Company, Inc., New York, N.Y., 1963.
4. Sauerheber, R. On the Nature of Light; Clarifying Relativity, copies at Library of Congress,
Washington, D.C. available at www.lulu.com.
5. Van Holde, Kensal Edward, Physical Biochemistry, Foundations of Modern Biochemistry Series, Prentice Hall
Incorporated, Englewood Cliffs, New Jersey, U.S.A., 1971, p. 165
6. Otis, A., Light Velocity and Relativity, Burckel and Associates, Yonkers-on-Hudson, N.Y., U.S.A., 1960.
7. McCausland, Journal of Scientific Exploration, vol 113, p. 271, 1999
8. “Solar System”, The New Encyclopaedia Brittanica, vol. 27, Encyclopaedia Britannica, Inc., Gwinn, R P., ed.,
Chicago, ILL, U.S.A., 1989.
9. Einstein, A., “On the Electrodynamics of Bodies in Motion”, 1905, in: The Principle of Relativity, Methuen and
Co., Ltd., London, 1923.
10. Roberts, N.W., Chiou, T., Marshall, N.J. and Cronin, T.W., A biological quarter-wave retarder with excellent
achromaticity in the visible wavelength region, Nature Photonics, 2009, 189.
Acknowledgements
One could only wish that Newton and Maxwell could still be here to continue their work. Thanks to friend Larry
Rosefeld and many students for patient discussions and for personal communications from Tom VanFlandern,
California Institute of Technology before his passing. I remain very grateful to Andrew A. Benson, discoverer of the
Calvin-Benson cycle of photosynthesis, for my first laboratory position (Scripps Institution of Oceanography, La Jolla,
CA), who is sorely missed.