| Email: Ali A. Faraj |
Absolute Velocities:
In this exposition, the predictions of the
Emission Theory of light, concerning absolute velocities of isolated systems,
have been worked out and discussed in detail. In addition, the related aspects
of this theory have been reviewed and investigated at length. The aim is to
facilitate the task of possible experimental testing, in the future, and to
dispel long-standing and appalling misconceptions surrounding this important
subject.
Keywords: Doppler effect; light aberration; relative velocity;
absolute space; superluminality
Absolute velocity can be defined as the common uniform linear velocity of the various components of a physical system, relative to absolute space. Absolute space, in turn, can be defined as the immobile three-dimensional physical void that exists independent of material bodies.
In spite of all the claims to the contrary, absolute space is one of the most intuitive and self-evident concepts ever encountered. Rational beings that lack a clear perception of absolute space, simply, do not exist. In fact, if all the obscurities of matter vanished, and the notion of matter were as simple and clear-cut as the notion of absolute space, there would have been no need, in that case, for physics at all.
Since Newton’s time, it has been recognized that measuring rotational and non-uniform velocities of isolated systems, on the basis of Galilean kinematics, presents no great difficulty.
By contrast, the determination of uniform linear velocities, with respect to absolute space, is fraught with all sorts of difficulties. The main stumbling blocks are, undoubtedly, the immense blind spots induced by theoretical prejudices and historical controversies.
Right from the start, the Copernicans set out to destroy the Ptolemaic astronomy. However, they encountered a startling but otherwise erroneous objection, namely, "a moving Earth flings non-attached bodies away into space"!
Galileo responded to this major objection of the Ptolemaists by expounding, in great detail, the kinematics of relative velocities.
As it is often the case with great truths, in the end, dogmatism prevails. Some of Galileo's successors have, actually, gone too far in this direction to assert the absolute impossibility of finding out uniform linear motions relative to absolute space under any conceivable circumstances. This is despite the fact that such determination, is not only possible; but it's, also, an immediate and straightforward consequence of Galilean and Newtonian Relativity.
The widely publicized failure of the Wave Theory’s prediction, regarding velocities relative to the Ether, has made this dogmatism even stronger. So much so, that the supposed impossibility of measuring absolute velocities has become a fundamental principle of physics, i.e. the ‘Relativity Principle’. And eventually, A. Einstein has chosen this principle to be one of the two pillars of his Special Theory. Ironically, even here, absolute velocities, ‘beautifully unexpected’, show up as a consequence of this Relativity-Principle-based theory.
Nonetheless, there are more daunting aspects of this notorious problem than mere collective prejudices and conventional blind spots. Here, we shall discuss, primarily, only those aspects related to the Emission Theory of light.
First and foremost, wrestling with the kinematical Principle of Relativity can be very unpleasant experience, and it is not recommend for the faint-hearted! Like a whirlpool, it draws you deeper and deeper into an incredible abstract world where infinity is the limit and where the most brilliant breakthroughs are made and destroyed in matter of minutes.
The Relativity Principle, in this respect, is quite unique among other principles of physics. Try, for instance, to disprove Newton’s First Law; and you won’t go far in that direction, before your attempt at disproof is crushed under a heavy load of internal contradictions. That is not the case with the much weaker Relativity Principle where a successful disproof always appears to be just over the horizon.
Furthermore, the theory under discussion does not invalidate the Relativity Principle. It, merely, restricts the scope of its application.
As it will be demonstrated in this discussion, the common linear velocity of an isolated system in uniform motion with respect to absolute space, can be determined, on the basis of the current theory, if and only if two or more of its parts are in relative motion with respect to each other. And accordingly, the Principle of Relativity must be recast, on the basis of this theory, as follows:
[A] It is impossible, within the framework of the Emission Theory, to determine the common uniform linear velocity of an isolated system whose components are at rest with respect to each other.
[B] It is impossible, within the framework of the Emission Theory, to determine the common uniform linear velocity of an isolated system whose components have only relative-velocity vectors along the line of the common-velocity vector of that system.
In addition, the minuteness of the predicted magnitudes of observable effects of comparatively low absolute velocities, based upon this theory, can be, depressingly, Einsteinian. For example, the observable variations, as predicted by the most generalized form of the Emission Theory, concerning two bodies with absolute velocity of (300 kms-1 ) and relative velocity of (30 kms-1), are no greater than (30 ms-1). The predicted magnitudes of observable effects are, of course, more significant and much easier to measure, if the given absolute velocities are about or greater than (1,000 kms-1).
The Emission Theory, whose predictions are the main topic of this paper, is itself an obscure and less known theory. And it’s essential, therefore, to begin this exposition with a brief discussion of the relevant aspects of this theory.
1. General Remarks
The synonymous terms "emission theory", "ballistic theory", and " corpuscular theory" of light have been used in the published literature to refer to a collection of closely related theories based firmly on Galilean and Newtonian kinematics.
To avoid redundancy, the term 'emission theory’ will be used consistently throughout this discussion to denote the most generalized version of those theories.
The published work on the Emission Theory is substantial and it goes as far back as Newton’s ‘Opticks’ in the 17th century. Consequently, no attempt at the survey of such extensive literature is made here. And only references, to it, will be specified when appropriate.
Although the Emission Theory has its roots in studies of optical and electromagnetic phenomena, its revolutionary nature is more conspicuous in the field of astronomy. It, literally, turns the science of astronomy on its head. Not a single bit of modern astronomy can remain unchanged, if the speed of light is taken to be variable in the Galilean sense and in accord with the generalized version of the Emission Theory. Thus, if ever there is a second Copernican revolution in astronomy, it will most likely be based on the Emission Theory or something very close to it. That is because right from the beginning, astronomy has been built entirely upon the assumption of constant speed of light. Just place the old Assumption of ‘Earth at rest’ beside the current Postulate of ‘constant light speed’; and it should be obvious at once what the overthrow of the latter assumption means for the future of modern astronomy.
In its current stage of development, astronomy has become increasingly cluttered with useless kludges and artificial hypotheses to such a degree that the Ptolemaic astronomy, in comparison, looks like a paragon of science and rational thinking itself. And so a second revolution may not be out of the question. But again, the tenacity of the astronomers is proverbial. And so it should come as no surprise, if they still have enough inertia left in them to keep astronomy on its status-quo trajectory for the next 1,500 years.
By then, however, the idea of developing general consensus around one single theory could become obsolete. Such a development has occurred already in the field of politics, which at present is far more advanced than any other field in this regard. Likewise, it may well be found necessary, in the long run, that the optimal arrangement and the best mechanism for advancement, in the field of astronomy and science in general, is to have two or more opposing schools take control of shared resources in a periodic and organized fashion. That is, of course, still a long way off, perhaps 1,500 years or more! For the time being, let’s examine next the fundamentals of the Emission Theory of light, and hope that this theory is destined to rule over physics someday.
2.
Change of Velocities upon Reflection
The Stewart-Thomson Law:
In the reference frame of the laboratory, light is always reflected from a moving reflecting surface with the resultant velocity of its relative velocity with respect to the reflecting surface and the velocity of the reflecting surface relative to the laboratory, [Ref. #1].
This important law is a generalization of the law of reflection. And it plays an important role in the treatment of optical phenomena on the basis of the Emission Theory. In its precise mathematical form, the Stewart-Thomson law can be derived and formulated by treating reflection of light as a special case of elastic collision and applying the conservation laws of linear momentum and kinetic energy, for moving bodies, to the incident light and the reflecting surface.
The quantitative treatment of this subject can be significantly simplified by assuming that the ratio between the mass of the incident light and the mass of the reflecting surface is infinitesimally small and practically equal to zero. And therefore, the recoil caused by the incident light on the reflecting surface can be neglected without affecting the exactness and precision of the quantitative treatment.
Consider the simple case of a plane mirror approaching or receding from a stationary light source along the normal to its reflecting surface with a uniform linear velocity, v. If the angle of incidence with the normal, i, is measured counterclockwise with respect to the velocity vector of the mirror, then the magnitude of the relative velocity of the incident light, c', can be computed by applying the law of cosines:
c' = [ c2 + v2 + 2vccosi ]1/2 [2.1].
Where c is the Maxwellian speed of light in vacuum.
The direction of this relative velocity, i', can be obtained by applying the law of sines to the above arrangement:
sini' =
[c/c' ]sini
[2.2].
The angle i’ is the true angle of incidence as measured in the reference frame of a moving mirror.
From the law of reflection, the angle of reflection for reflected light, in the reference frame of a moving mirror, must be equal to the direction of the relative velocity of the incident light, i', and as a result:
cosi' = [
1 -
sin2i' ]1/2
= [ 1 - c2/c'2
sin2i ]1/2 = (ccosi + v)
/ c’ [2.3].
And by applying the law of cosines once more to compute the
speed of the reflected light in the reference frame of the laboratory, we
obtain c'':
c'' =
[ c'2 + v2 + 2vc'cosi' ]1/2 [2.4].
By combining equations [2.1], [2.3], & [2.4], we obtain for the general case:
c'' = c[1 + 4v2/c2 + (4v/c)cosi]1/2
[2.5].
The direction of this relative velocity, i”, can be obtained by applying the law of sines once more:
sini” =
[c’/c” ]sini’ = [c/c” ]sini =
sini / [1 + 4v2/c2 + (4v/c)cosi]1/2 [2.6].
The angle i” is the true angle of reflection as measured in the reference frame of the laboratory.
When a mirror approaches directly a light source along the normal to its reflecting surface, Equation #[2.5] is reduced to:
c'' =
c + 2v
[2.7].
And when it recedes from the source along the same line, we obtain:
c'' = c - 2v
[2.8].
If one insists on taking account of the vanishingly small recoil of an approaching or receding mirror under the effect of incident light, the following exact equations of linear elastic collision should be used for both cases:
c'' =
c[(mm – mc)/(mc + mm)] +
v[2mm/(mc + mm)] [2.9],
c'' =
c[(mm – mc)/(mc + mm)] -
v[2mm/(mc + mm)] [2.10].
Where mc and mm are the mass of the incident light and the mass of the mirror, respectively.
Equation #[2.5] can be generalized further, through the rotation of the normal to the surface of the plane mirror by an angle, j, around the velocity vector of the mirror and applying the laws of cosines and sines in each case.
One important case is when j = 90o.
Consider the case of a plane mirror approaching or receding from a stationary light source along its reflecting surface, i.e. j = 90o, with a uniform linear velocity v. If the angle of incidence with the normal, i, is measured counterclockwise with respect to the velocity vector of the mirror, then the angle between the velocity vector of the incident light and the velocity vector of the mirror is (q = 90o – i) when (j = 90o) & the mirror is approaching the source; and (q = 90o + i) when (j = 90o) & the mirror is receding from the source.
We, now, compute the resultant of (c & v) in the case of approach:
c' = [ c2 + v2 +
2vccosq ]1/2 =
[ c2 + v2 +
2vcsini ]1/2 [2.11].
sinq’
= (c/c' )sinq =
(c/c' )cosi
[2.12].
Using q’, we obtain the angle q’’ between the velocity vector of the reflected light and that of the mirror:
q” = 180o - q’ [2.13].
Then we use (c’ & q”) to obtain the velocity of the reflected light c” with respect to the reference frame of the laboratory:
c” = [ c’2 + v2 +
2vc’cosq” ]1/2
[2.14].
From equations [2.12] & [2.13], we calculate cosq”:
cosq” = - cosq’
= -[(csini + v)/c’] [2.15].
Inserting the values of (c’ & cosq“) in Equation #[2.14], we obtain c”:
sini’ =
(c’/c”)sinq “ [2.17].
By using equations [2.12], [2.16], & [2.17], we obtain:
sini’ = cosi [2.18].
To obtain the angle of reflection i” with respect to the normal to the mirror surface, we use the relation (i” = i’ - 90o) & Equation #[2.18]:
sini” = sin(i’ – 90o) = cosi’ =
sini
[2.19].
Where i” is the angle of reflection as measured in the reference frame of the laboratory.
Thus, in the case of (approach & j = 90o), we have (c” = c & sini” = sini), which are the same values as in the case of reflection from a stationary mirror.
Finally, to calculate the resultant of (c & v) in the case of a receding mirror whose (j = 90o), we use the relation (q = 90o + i):
c' = [ c2 + v2 +
2vccosq ]1/2 =
[ c2 + v2 -
2vcsini ]1/2 [2.20].
sinq’
= (c/c' )sinq =
(c/c' )cosi
[2.21].
Accordingly, cosq’ = -[(csini - v)/c’].
By the use of q’, we obtain the angle q’’ between the velocity vector of the reflected light and that of the mirror:
q” = 180o - q’ [2.22].
Then we use (c’ & q”) to obtain the velocity of the reflected light c” with respect to the reference frame of the laboratory:
c” = [ c’2 + v2 +
2vc’cosq” ]1/2
[2.23].
From equations [2.21] & [2.22], we calculate cosq”:
cosq” = - cosq’
= (csini - v)/c’
[2.24].
Inserting the values of (c’ & cosq” ) in Equation #[2.23], we obtain c”:
sini’ =
(c’/c” )sinq “= (c’/c” )sinq ‘
[2.26].
By using equations [2.21], [2.25], & [2.26], we obtain:
sini’ = cosi [2.27].
To obtain the angle of reflection i” with respect to the normal to the mirror surface, we use the relation (i” = i’ - 90o) & Equation #[2.27]:
sini” = cosi’ = sini [2.28].
Hence, in the case of (recession & j = 90o), we have (c” = c & sini” = sini), which are the same results as in the case of reflection from a stationary mirror.
As it is clear from Equations #[2.7] & #[2.8], the speed of reflected light, each time, is increased by twice the speed of a directly approaching mirror, and decreased by twice the speed of a directly receding mirror. The process can be repeated indefinitely. This, in fact, is one of the most astonishing predictions of the theory under discussion. In principle, at least, it implies no less than the complete control over speed of light by increasing or decreasing it to any desirable level through the use of multiple reflection from moving mirrors in the reference frame of the laboratory. Because of the high speed of light, the desired results can be achieved by the use of this mechanism, even in the case of slowly moving mirrors, in a fraction of a second. Consider, for example, a mirror moving directly with velocity of (100 ms-1) towards a stationary mirror 20 m away. By making multiple passes, through this optical loop, light can reach a superluminal speed of 2c in less than 0.2 of a second.
In practice, however, for superluminal speeds over 2c, Doppler effect presents a serious problem. As it will be shown later in this discussion, each time the speed of the reflected light is doubled, its frequency is doubled as well due to Doppler effect on light whose speed is boosted to a superluminal level by the use of multiple reflection from approaching mirrors. In practical terms, this Doppler shift to higher frequencies means that every part of the visible light will be shifted to the ultra-violet region of the spectrum; by the time its superluminal speed is close to 3c.
Ordinary mirrors are inefficient ultra-violet reflectors; and so the process of speed boosting through the use of multiple reflections from those mirrors must fail for speeds greater than 2c. Is there a way out of this Doppler-boosting problem?
Despite their horrific miasma of “Tunneling”, “Exiting-before-entering”, and similar theoretical nonsense, experimenters of the so-called 'Photonic Revolution’ have, in the last decade or so, amassed an impressive array of experiments, techniques, and data on superluminality, through the use of anomalous dispersion in carefully-prepared materials.
Their finding, in itself, is undoubtedly one of the most striking discoveries in experimental physics in recent years, and most certainly bound to sweep away, in due time, long-held false beliefs and errors in this field. See, for instance, [Ref. #9].
Nonetheless, neither anomalous dispersion, nor photonic crystals, nor specially prepared refractive media, have any potential use within the context of the Doppler-boosting problem encountered here. Refractive indices, simply, cannot be used to boost speed of light outside their artificially prepared media. That is because light, upon exiting the prepared medium, very simply, restores its standard Maxwellian speed in vacuum.
It’s possible, however, that the superluminal speeds of light, in those experiments, are caused by multiple reflections from moving layers of atoms inside the artificial media. If that is indeed the case, then upon exit, light can retain its superluminal speeds in vacuum. But this possibility, of course, does not solve the Doppler-boosting problem.
In a nutshell, multiple reflection from approaching mirrors is the only technically feasible method for achieving long-range superluminal speeds of light in free space. But Doppler boosting to higher frequencies and the unavoidable absorption of light by the materials of mirrors, impose severe limitations on the practicality of this method.
Suppose, for a moment, that a perfect mirror capable of reflecting electromagnetic radiation of any frequency repeatedly and with 100% efficiency is practically feasible.
Can the superluminal speeds of light produced by that mirror be used in long-range communications?
If such a perfect mirror can be found, and if the Emission Theory is correct and universally applicable, then the answer to the above question is, absolutely and unequivocally, ‘yes’.
Not only that, but one also can be absolutely sure that advanced civilizations, across the cosmos, are talking to each other over our heads. But thanks to the Einsteinian slumber of our physics, we, Homo sapiens, are completely oblivious of their communications!
So let's hope that, someday, our Homo sapiens shall live up to their illustrious name! And let us try to figure out how superluminal signaling can be achieved in practice.
Given a perfect mirror, it's quite possible to start with the radio portion of the spectrum; and to code your message using standard methods in radio communications.
After that, you can use the perfect mirror to boost the speed of the coded radiation to the desirable superluminal level.
To minimize the effect of the Inverse Square Law on the coded radiation, maser and laser techniques have to be used. However, to maximize the chances of reception, the Inverse Square Law must be allowed to work as required, during travel time, to make the cross section of the carrying beam large enough to encompass the entire targeted area by the time of its arrival. In brief, your tasks as a sender, in superluminal telecommunications, are to code, boost, collimate, and send.
Receivers of superluminal signals must, in turn, be equipped with a perfect mirror, as well, to be able to tune in. The mirror, here, is used to convert the received superluminal radiation to its original Maxwellian form, which can be fed to a regular radio receiver to decode the sender's message in a standard fashion.
It should be mentioned, in this context, that a team of the Italian Council of Research reported, recently, achieving superluminal speeds in air by bouncing a microwave beam off a mirror, [Ref. #6]. From the viewpoint taken in this section, their experiment is very interesting, although it is not clear from their report how exactly and how efficiently a microwave mirror works.
3. The
Characteristics of the Corpuscular Photon
The concept of the Corpuscular Photon
is an integral part of the Emission Theory of light. In the published
literature, the Corpuscular Photon is defined in two different ways:
[A] The Corpuscular Photon is defined as one single corpuscle whose mass
is in direct proportion with frequency across the electromagnetic spectrum.
This concept of the Corpuscular Photon is extremely rigid and inadequate in
dealing with the wave aspects of light. For this reason, it is often used as a
'straw-man' argument against the Emission Theory. How on Earth, its opponents ask, can it account correctly for
interference and diffraction phenomena?
However, this old objection sounds increasingly hollow after the
discovery of similar phenomena related to electrons and other subatomic
particles. Nevertheless, the inadequacy of the 'One-Corpuscle Photon’ is
obvious. And its fundamental flaws cannot be weeded out by merely pointing out
more serious flaws in the concept of the Conventional Wave Photon.
[B] The Corpuscular Photon can, also, be defined as a group of corpuscles
whose number and spatial separation (wavelength) vary in direct proportion with
frequency across the electromagnetic spectrum. The total number of corpuscles,
in a photon, is determined only by its frequency and the duration of its pulse
during the time of emission. Its energy and momentum, in turn, depend solely on
the number and the speed of its corpuscles. For a non-polarized photon, the
linear trajectories of its corpuscles form randomly its cross section. Thus, it
should be clear that a corpuscular photon whose cross section consists of only
one single geometrical point, though appealing, is a fictional idealization
that cannot be realized in actual situations.
The
Corpuscular Photon, when defined in terms of distinct groups of corpuscles on
the basis of their frequencies, is a versatile and powerful concept. It,
literally, transforms the Emission Theory from a weak and timid hypothesis to a
revolutionary tool of the first order. It, no longer, suffices for the critics
of this theory to throw on it Poisson's ‘Bright Spot’, de Setter's ‘Circular
Orbits’, and similar obsolete objections to score a point. The table has,
really, been turned on them. And this is how:
(1) The notion of distinct groups of corpuscles allows the theory, under
discussion, to account for interference, diffraction, polarization, and related
phenomena, in a natural and satisfactory manner, and to dispose of the old
objections, along with the Wave-Particle Duality, at once.
(2) Since subatomic particles emit photons, and since photons consist of
corpuscles, it follows as a natural consequence of this theory that the
subatomic particles themselves are structured aggregates of corpuscles. The
idea of particles composed of aggregates of corpuscles, analogous to aggregates
of stars in galaxies, has the potential of restoring order to the current
chaotic state, opening new avenues for probing deeper into the nature of
matter, and revolutionizing the stagnant field of elementary particles.
(3) The concept of corpuscular photons enables the Emission Theory to
include precise formulae for the Doppler effect, the Fresnel Convection, and
the Law of Aberration derived on a basis more solid and intuitive than that of
any other theory in this field.
(4) This concept of photons, in conjunction with the Stewart-Thomson Law,
enables the Emission Theory to explain clearly and correctly the
Michelson-Morley Experiment and related experiments, giving it a decisive edge
over a variety of unrealistic kludges such as the notorious Lorentz-Fitzgerald
Contraction Hypothesis.
(5) Redefining the photons in terms of distinct groups of corpuscles leads
to the inclusion of the J. G. Fox ‘Re-radiation’
hypothesis as a special case, and disposing, at the same time, of its
undesirable consequences, [Ref. #4].
Notwithstanding its success in the special case, the application of the
Re-radiation hypothesis, generally, has the misfortune of making no prediction
at all. That is because the re-radiation mechanism, as used by Fox in his
Extended Ritz Theory, leads to de-facto constant speed of light, banishes
superluminality, and renders the application of the Galilean Transformations to
electromagnetic phenomena useless. See [Ref. #7],
and, [Ref. #8]. Furthermore, except in the special
case, where the refractive medium absorbs and re-emits the incident light with
different frequencies, the Re-radiation hypothesis is in contradiction with the
conservation laws of energy and momentum. Where does the truncated portion of
the initial energy and momentum go? The Re-radiation hypothesis gives no
answer. It’s true that the Conventional theories, in the field, have violated
these same laws at more basic levels. But that is no excuse for committing
another clear violation of them. In brief, the Re-radiation mechanism must be
restricted and its unwarranted generalization has to be abandoned, in order to
apply the Galilean Equations in a productive and self-consistent manner.
(6) The definition of photons, as independent groups of corpuscles,
enables the Emission Theory to give a clear and natural explanation of the
Cosmological Red-shift of distant galaxies and to do away with the hypothesis
of ‘Expanding Universe’ along with its bizarre and absurd consequences.
By merely taking accelerations of the light source into account, one can easily
obtain the exact formula of the Hubble Law as a natural consequence of this
theory. And as a bonus, the ‘Big Bang’ can more readily be consigned to
the dustbin of history along with its monstrous offspring.
(7) This is the most controversial part of the Emission Theory.
The exact
quantitative details of this topic have been elaborated elsewhere.
See [Ref. #1]. Here, it suffices to restate the
main problem in more general terms. Without doubt, the empirical classification
of stars on the basis of their luminosity, size, and spectral type, is the most
concrete part of modern astronomy. In addition, it's the foundation upon which
Stellar Evolution; one of the most attractive theories in astrophysics has been
built. The Emission Theory does not contradict outright the Theory of Stellar
Evolution. It, simply, creates, on purely kinematical grounds, an exact replica
of its foundation. It's easy to see that systematic and continuous variations
in the velocity of a light source, lead to systematic and continuous variations
in the velocity of its radiation output. These variations in the velocity of
the output, inevitably, lead to changes in the spectrum and the radiant flux of
the source, which vary linearly with distance. Thus, it's possible to produce
the entire phenomena included in stellar classification by merely using a 'hypothetical
sun' and varying its orbital configurations at various distances from the
observer. Take, for instance, the spectacular phenomenon of Supernova
Explosion. As spectacular as it is, the
Supernova Phenomenon is the easiest to duplicate using nothing more than our 'hypothetical
sun'. Increasing its velocity by an extremely tiny but continuous amount,
which the far half of its orbit around some distant galaxy can do, our 'hypothetical
sun' can pour its output of radiation, during one hundred million years, in
just only few days, and fools naive Einsteinian observers that they have just
witnessed a stellar explosion on a gigantic scale! In the same way, the variability of speed of light can make out
of our 'hypothetical sun' every star of every size and spectral type in
the Hertzprung-Russel Diagram. And it can replicate every star in the Variable
Stars' Catalogue. In short, the observational basis of modern astronomy has
been undercut. And serious doubts have been cast upon its dynamical reality.
This, of course, does not mean that all the stars are identical to the sun, nor
their classification is physically baseless. All what the above considerations
imply is that if speed of light is variable in accord with the Emission Theory,
then a thick cocoon of mirages and optical illusions must be removed first,
before the real dynamical phenomena underneath can be unveiled. Then, and only
then, a true understanding of the physical universe can follow the observation.
4. The Doppler Effect
As discussed earlier, the Corpuscular Photon of the Emission Theory is composed of a finite number of corpuscles whose primary source emits them one after the other at regular intervals of time. This definition makes the task of deriving the exact formulas of the Doppler Principle straightforward and simple. Compare the following easy steps with the convoluted methods of deducing the Doppler Equations on the basis of the One-Corpuscle Emission Theory or on the basis of Einstein Theory:
[1] The Case of Direct Approach:
Consider the simple case of a light source approaching directly with velocity vs an observer at rest in the reference frame of the laboratory. Let the period of the emitted corpuscles of a photon be T as measured in the inertial frame of their source, and T’ as measured in the reference frame of the laboratory.
Since the frequency f, by definition, is the reciprocal of the period T, we obtain:
f = 1/T [4.1],
f’ = 1/T’ [4.2].
Where f and f’ are the frequencies in the inertial frame of the source and the reference frame of the laboratory, respectively.
Since the source is approaching directly, then the velocity of its light c’, relative to the laboratory:
c’ = c + vs [4.3].
Where c is the velocity of light in the inertial frame of the source.
Next, we use T, vs, and c’ to compute T’:
T’ = [T(c + vs
) - T vs ] / (c
+ vs ) = Tc
/ (c
+ vs ) [4.4].
From Equations # [4.2] and # [4.4], we obtain the Doppler formula for this special case:
f’ = 1/T’ = f[1
+ vs/c] [4.5].
If the observer approaches with velocity vo a stationary source of light, we obtain:
T’ = (Tc - T’
vo )/ c = Tc/(c
+ vo ) [4.6],
f’ = 1/T’ = f[1
+ vo /c] [4.7].
Where f’ is the frequency in the inertial frame of the observer.
[2] The Case of Direct Recession:
Repeating the above steps, we obtain for a source receding directly from an observer at rest:
f’ = f[1 - vs /c] [4.8].
And we obtain for an observer receding directly from a source of light at rest:
f’ = f[1 - vo /c] [4.9].
[3] The General Case:
In order to obtain the Doppler formula in the general case, let the angle i, between the line of sight and the velocity vector of the source vs, be measured counter-clockwise.
Let the angle j that the velocity vector of the observer vo makes with the line of sight be measured clockwise and corrected for Light Aberration. Since the line of sight is the direction of the resultant velocity of light c' from a moving source, we obtain:
c' = c[1 - (vs2/c2)sin2i]1/2 + vscosi [4.10].
For light emitted with period T and frequency f in the inertial frame of the source, we compute the period as measured by the observer:
T' = [Tc' - Tvscosi - T'vocosj] / c' = T(c' -vscosi) / (c' + vocosj) [4.11].
By taking the reciprocal of T', we obtain the general formula for Doppler effect in the reference frame of moving observer:
f' = f[1 + {(vs /c)cosi + (vo/c)cosj} / {1 - (vs2/c2)sin2i}1/2] [4.12].
Let's now compare the Doppler formulas for the first two simple cases of direct approach and direct recession, according to the Emission Theory, with the Doppler formulas for the same two cases according to Maxwell's Theory and Einstein's Special Relativity, respectively.
[A] Maxwell’s Theory:
1. For a source approaching directly an observer at rest, the theory gives:
f' = f[1 + vs/(c - vs)] [4.13].
By comparing this equation with Equation #[4.5], we find that the Doppler shift of approaching sources i.e., (f' - f)/f, as computed on Maxwell's Theory, is always greater than that of the Emission Theory by a factor of [1 - vs/c]-1.
2. For an observer approaching directly a stationary source of light, the Maxwell Doppler formula is:
f' = f[1 + vo/c] [4.14].
By comparing this equation with Equation #[4.7], we conclude that the Doppler effect in this case is the same as calculated on both theories.
3. For a source receding directly from an observer at rest, Maxwell’s Theory gives:
f' = f[1 - vs/(c + vs)] [4.15].
By comparing this equation with Equation #[4.8], we find that the Doppler shift of receding sources, as calculated on Maxwell's Theory, is always less than that deduced from the Emission Theory by a factor of [1 + vs/c]-1.
4. For an observer receding directly from a stationary source of light, the Maxwell Doppler formula is:
f' = f[1 - vo/c] [4.16].
By comparing this equation with Equation #[4.9], we conclude that the Doppler effect, in this case, is the same as calculated on both theories.
[B] Einstein’s Special Theory:
This theory has two different sets of equations for computing Doppler effect:
[1] According to
Einstein:
Special Relativity, as expounded in Einstein’s 1905 paper, takes the Maxwellian Doppler formulas for the moving observer, and divides them by the factor {1 – v2/c2 }1/2, where v stands for vs , vo , or both. And then, it uses these new formulas in all the four simple cases above:
f' = f[(c + vs ) / (c - vs)]1/2 (for directly approaching source) [4.17],
f' = f[(c + vo ) / (c - vo)]1/2 (for directly approaching observer) [4.18],
f' = f[(c - vs )/(c + vs)]1/2 (for directly receding source) [4.19],
f' = f[(c - vo ) / (c + vo)]1/2 (for directly receding observer) [4.20].
f' = f[{1 - (v/c)cosf} / {1 – v2/c2 }1/2] (for the general case) [4.21].
Where v stands for vs, vo, or both, and f stands for i, j, or both, [Ref. #3].
[2] According to
Ives & Stilwell:
Special Relativity, according to Ives & Stilwell, takes the Maxwellian Doppler formulas for the moving source, and multiplies them by the factor {1 – v2/c2 }1/2, where v stands for vs , vo , or both. Then, it uses these new formulas in all the simple cases above and gives in the general case:
f' = f[{1 – v2/c2 }1/2 / {1 - (v/c)cosf}] [4.22].
Where f stands for i, j, or both [Ref. #5].
The Einstein and the Ives-Stilwell general formulae give the same numerical results at f = 0o and f = 180o; but they make contradictory predictions at f = 90o with regard to the transverse Doppler effect.
Now by comparing the above
equations with those of the Emission Theory, we obtain:
f'E/f'R = [1
- vs 2/c2]1/2 [4.23]
Where f'E & f 'R are the observed frequencies as predicted by the Emission Theory and Einstein's Relativity, respectively.
Therefore, we conclude that in all cases of approach, Einstein’s Theory predicts Doppler shift i.e. (f'-f)/f greater than the one predicted by the Emission Theory. And in all cases of recession, it predicts Doppler shift less than that predicted by the Emission Theory. Thus, from the perspective of the Emission Theory, Einstein’s Special Relativity makes the correct Maxwell formulas of the moving observer erroneous by a factor of {1 – v2/c2}-1/2, but, at the same time, it restores the symmetry and reduces the error in the case of the moving source by using the same formulas for both the source and the observer.
5. The Law of
Aberration
Within the framework of the Emission
Theory, Light Aberration is defined as the angle between the true position of
the source, at the time of emission, and the direction of the resultant relative
velocity of the velocity of the incident light, from that source, and the
velocity of the observer, at the time of reception.
Thus, if
the direction of the resultant relative velocity of the incident light and the
observer is j, and the true position
of the source, at emit time, is j',
then the Light Aberration b is the
difference between j' and j:
b = j' -
j = Dj [5.1].
Notice that
the angle j' can be computed, but can
never be observed in the inertial frame of a moving observer.
Two forms
of the Law of Light Aberration will be discussed here:
[1] The
Standard Form of the Law of Light Aberration:
Let's
consider, first, the simple case of a stationary source of light and an
observer moving with uniform linear velocity vo. By applying the law of sines to this case, we obtain
the well-known form of Bradley's Law of Aberration:
sinDj