Law of Energy Conservation and the Doppler Effect
"It is thus shown that, although mechanical energy is indestructible, there is a universal tendency to its dissipation, which produces gradual augmentation and diffusion of heat, cessation of motion, and exhaustion of potential energy through the material universe"  Lord Kelvin ( 1824  1907 )
"Reality is merely an illusion, albeit a very persistent one"  Albert Einstein ( 1879  1955 )
note: be aware that the majority of these equations are derived for straight line of sight inertial motion versus transversal (perpendicular to sight) motion.
Atomic Doppler Effect
§ The Relative Motion of The Frame of Reference of a Moving and Oscillating Atom:
This above schema is simplified and equivalent to this emission diagram:
Just before the moment when the moving atom, velocity , excitedly emits a , total energy of the atom is
However just at the moment of emission, loses an amount of energy such that
where the moving atom experiences a recoil and acquires a new velocity and is the atom's natural oscillating frequency when observed at rest and the observer staying right at the reference frame of the atom source.
Furthermore,
§ The Law of Energy Conservation Dictates:
§ The Components of Linear Momentum:
From the above emission diagram, we can now deconstruct the following x, y  components of linear momentum in this following traditional manner:
Hence, the difference between and frequencies is wholly determined by the compression factor
which for gives and for which this compression factor for straight  line observation easily reduces down to
note: relative velocity is taken to be negative when the atom source is relatively moving away from the observer in stationary system ; otherwise, relative velocity is taken as a positive. This is extremely important when considering these equations!!
§ Red  Shift vs. Blue  Shift Wave Frequencies:
Notice that when the relative velocity of the oscillating atom source for increasingly moves away from the stationary observer in system at greater and greater amounts of , the observed frequency decreases with a concomitant increase in observed wavelength for . That is,
Conversely, for an oscillating atom source for moving towards a stationary observer in system
The ratio
is what produces the Doppler Effect for our source atom emitting !
In the non  relativistic limit where v << c,
Johann Christian Andreas Doppler ( born Salzburg, Austria, 1803  died Venice, Italy 1853 ) is the Austrian mathematician and physicist after whom this observed wave  frequency shift is named; ironically it was as a result of the 1866 Seven  Weeks War that "shifted" Johann Doppler's grave site from Austria to Italy.
§ Applying Relativistic Analysis to our Oscillating Atom:
We know from special relativity mathematics and Lorentz time dilation in the rate of observed passage of time that the observed frequency to an observer in system becomes
∗See: special relativity  time dilation
Notice here, again, that for a receding atom source where receding velocity is taken as ,
Finally, for
and in terms of energy
§ Straight  Line Observing Oscillating Photon  Emitting Atom:
That is, take up another perspective in stationary system so that the atom source for lies strictly along the x  axis and hence implying .
This then simplifies our equation for observed frequency
Please see part §7 bottom, page 16  17, of Einstein's 1905 paper, "On The Electrodynamics of Moving Bodies" for Einstein's ( abrupt ) presentation of this Doppler equation with the substitution of as being taken as .
Notice also that for
if velocity is taken as where the object is approaching, therefore
§ Some Further Mathematical Manipulation for Straight  Line Observation of Atom Emitting Photon:
Also,
§ Atomic Pair Production:
We know from an earlier Lorentz transformation that momentum, mass and energy are related as follows:
That is, momentum and mass transform from one system into another system according to the Lorentz transformation equations and the relativistic addition of velocities. Hence,
In the extreme case of , the atom source for no longer exists  i.e., it goes to zero and disappears!
For the other extreme situation where is extremely large but not quite at , becomes infinitely small as can also be seen from the inverse relation
but yet the numerator
never quite goes to zero.
Of course we know that
so we should not be surprised that
What this signifies is that because of Lorentz transformation of relativistic moving inertial frames of reference, that there are inherently analogous and different energies for equivalently different [ gamma  photon ] frequencies as revealed by the Doppler Effect!
On the other hand, we have seen that becomes extremely small whenever is extremely large but not at , however there surely must be a lower limit for when hits an atomic nucleus and thereby produces an elementary particle and its anti  particle!
That is, atomic pair production occurs whenever a hits an atomic nucleus and an elementary particle and its anti  particle  usually an electron and a positron  are created. For an electron  positron pair production to occur, the law of conservation of energy requires that the energy of , , be greater than the combined energies of the electron and positron. Or,
In any event, an oscillating atomic nucleus is required for any pair production to occur where any lost p.e. ( potential energy ) within the nucleus due to impact creates a compensating anti  particle; hence, the laws of conservation of momentum and energy are maintained.
Paul Dirac ( 1902  1984 ), British theoretical physicist and founder of modern quantum physics, produced his famous Dirac equation in 1928 which predicted atomic pair production and hence the existence of such anti  particles as the positron for which Carl David Anderson ( 1905  1991 ), American physicist and son of Swedish immigrants, together with Victor Hess ( 1883  1964 , Austrian  American physicist ), won the Nobel Prize in physics in 1936 after having confirmed its existence during one night in 1932. Paul Dirac also shared the 1933 Nobel Physics Prize with Erwin Schrödinger ( 1887  1961, Austrian  Irish physicist ), for their joint contribution in atomic theoretical physics, especially for the Schrödinger equation. Paul Dirac was an avowed atheist. Erwin Schrödinger, a born Catholic, nevertheless escaped Nazi Germany because of his opposition to German anti  Semitism and persecution of Jews and therefore finally fled with his wife in the summer of 1933 to South Tyrol, Italy. Later in 1940 Schrödinger helped establish the Institute for Advanced Studies in Physics, Dublin, Ireland, in which country he eventually adopted Irish citizenship.
Finally, in some sense therefore, we have come full circle to the beginning of our schema of a moving atom source for the emission.
Astronomical Doppler Effect
Doppler Effect:
Up until now our perspective has been from the point of view of an oscillating atom as the source of the light  emitting ; however, each and every one of the previous equations can be almost transferred whole to the macro universe. That is, receding and approaching bodies and galaxies, each by themselves, define a self  contained inertial frame of reference.
Of course, in the macro world we have the famous example of an approaching and then receding train whistle, where for one person standing alongside the tracks the whistle pitch rises and then falls as the train first approaches and then passes. However, for any person riding onboard the train, the frequency or pitch of the train's whistle remains constant throughout. This ratio, therefore,
is the Doppler factor giving a mathematical expression to the Doppler Effect.
§ Earth as a Platform for Astronomical Observations:
Because of earth's motion about the relatively stationary sun, all telescopes on "platform earth" which are imagined to reside on stationary "platform sun" owing to the great cosmic distances involved, must be constantly adjusted in terms of a deflection angle when pointing at a given [ relatively ] fixed star or galaxy. That is, as earth moves with relative velocity about [ relatively ] stationary sun, any telescope on "platform earth" must not point to the true star or galaxy position but rather must point to a position in the night sky which is displaced in the direction of earth's velocity ; and, notwithstanding our intuitive emotional feeling that earth is standing still, we must start to consider everything in this analysis from the point of view of [ relatively ] stationary sun owing to the Copernican Revolution ( Polish: Mikolaj Kopernik, 1473  1543 ) for the heliocentric model of our solar system.
Abstract #1
We now need to re  visit the equations for the relativistic addition of velocities as follows:
Abstract #2
From the perspective of [ relatively ] stationary system , sun, we thus have
On the other hand, from the perspective of [ relatively ] moving system , earth, we have
since we are observing the path of starlight just within earth's x  y plane.
§ Aberration Constant:
That is, the angle of deflection and hence the amount of relativistic aberration of light is determined by the angle of incoming starlight into the telescope and the relative motion between the sun's stationary frame of reference and earth's relatively moving frame of reference as can be seen in the above Abstract #1.
§ Placing Ourselves in the Sun's Frame of Reference for Earth's Relative Motion:
So, if we're standing in the frame of reference of [ relatively ] stationary sun, the path of starlight in the x  y plane is given by
However with respect to the moving frame of reference of earth, we must employ Lorentz inverse transformation equations as follows:
§ Angle of Deflection:
In fact
where is the angle of deflection for relativistic aberration of starlight which must be maintained for any telescope on "platform sun" pointing at a [ relatively ] fixed star or galaxy throughout earth's traverse about the [ relatively ] fixed sun!
What we further see is that
§ Relativistic Transversal Aberration on Platform Earth:
That is, is the angle of deflection for relativistic transversal aberration of starlight that must be maintained for any telescope on "platform earth" pointing at a [ relatively ] fixed star or galaxy throughout earth's traverse about the [ relatively ] fixed sun!
§ Star Aberration Calculation on Platform Earth:
Go to 'Star Aberration Derivation' to see how this 1905 Einstein calculation is accomplished.
§ Reformulated Doppler Equations:
Therefore, the reformulated Doppler equations incorporating relativistic aberration for starlight are
Hence,
Relativistic beaming occurs since
Relativistic red  shift z can now be determined as follows:
For "line of sight" relative motion, and hence
And, simplified
§ Using Relativistic Red  Shift to Determine Velocity and Time:
Relativistic red  shift z can now be utilized to estimate velocity of a receding galaxy and the time it took to arrive at its present position relative to earth's solar system.
Some Astronomical Spectral Lines ( Fraunhofer lines )  

Line Designation  Due to element  Wavelength ( Å )^{[a]} 
Wavelength ( nm )^{[b]} 
Line Designation  Due to element  Wavelength ( Å) ^{[a]} 
Wavelength ( nm )^{[b]} 
A band  O_{2}  7593.70  759.370  L  Fe  3820.44  382.044 
B band  O_{2}  6867.19  686.719  N  Fe  3581.21  358.121 
C (Halpha)  H  6562.81  656.281  P  Ti^{+}  3361.12  336.112 
a band  O_{2}  6276.61  627.661  T  Fe  3021.08  302.108 
D1  Na  5895.92  589.592  t  Ni  2994.44  299.444 
D2  Na  5889.95  588.995  e  Hg  5460.73  546.073 
E_{2}  Fe  5270.39  527.039  e  Fe  4383.55  438.355 
F (Hbeta)  H  4861.34  486.134  b_{1}  Mg  5183.62  518.362 
G  Fe  4307.90  430.790  b_{2}  Mg  5172.70  517.270 
G  Ca  4307.74  430.774  b_{3}  Fe  5168.91  516.891 
H  Ca^{+}  3968.47  396.847  b_{4}  Fe  5168.91  516.891 
K  Ca^{+}  3933.68  393.368  b_{4}  Mg  5167.33  516.733 

For example, Andromeda X [ satellite galaxy of Andromeda (M31) ] has been determined to lie at a distance of 2.9 M ly from the center of earth's solar system which would naturally be the sun where the entire Andromeda galaxy system is approaching Milky Way at a velocity of 300 km/sec. Two of the most prominent spectral lines are the H and K for ionized calcium. Choosing spectral K where its rest frequency is inverse to its rest wavelength provides the following:
We have = 393.368 nm for the observed lab rest  wavelength for spectral K. Let's say, hypothetically, that for this approaching Andromeda X satellite galaxy = 390.018 nm, the observed shorter wavelength by means of astronomical spectroscopy for element Ca^{+}. Therefore for "line of sight" relative motion [ see above ],
§ Relativistic Transversal Doppler Effect in Terms of Energy:
Analogous to Lorentz inverse transformation equations
is
Now,
Also, since
Again, what this signifies is that because of Lorentz transformation of relativistic moving inertial frames of reference, that there are inherently analogous and different energies for equivalently different [ gamma  photon ] frequencies as revealed by the Doppler Effect!
§ Lorentz, redshift and compression factors:
§ Hubble's Law and the recession expansion of the universe:
Read here Edwin Hubble's original paper "A Relation Between Distance And Radial Velocity Among ExtraGalactic Nebulae", by Edwin Hubble, Mount Wilson Observatory, Carnegie Institution of Washington, January 17, 1929
Hubble's Law is stated as follows with profound philosophical consequences:
Therefore at the very edge of the known universe ( 13.73 billion years, age of the universe ), the cosmic horizon is expanding at virtually the speed of light beyond which nothing is visible:
What Hubble's Law is demonstrating is that at the cosmic horizon, the entire edge of the known universe is disappearing! Our universe is literally evaporating into ( non  visible ) energy!! This fact is due no less than to Einstein's total energy and mass dilation equation
§ Brand new method ( 02/2013 ) for measuring cosmic distances using accreting Black Holes:
Using well established cosmology techniques for measuring black hole masses by the amount of accreting falling masses, a black hole's energy at the event horizon can be fairly well measured according to a recent paper by Prof. Hagai Netzer of Tel Aviv University, Israel, along with Jian  Min Wang, Pu Du and Chen Hu of the Institute of High Energy Physics of the Chinese Academy of Sciences and Dr. David Valls  Gabaud of the Observatoire de Paris. This energy can then be compared to the amount of energy radiation from a given black hole reaching earth in determining the universe's rate of expansion and hence very large cosmic distances with a very high degree of accuracy. See: "Super  Eddington Accreting Massive Black Holes as Long  Lived Cosmological Standards", publication date: 02/2013, reference: 2013PhRvL.110h1301W, Physical Review Letters, vol. 110, Issue 8, id. 081301.
Doppler Summary
Determining time dilation using Doppler and Minkowski's Spacetime Geometry
One admission that must be stated is that earlier Doppler was itself derived from time dilation and therefore there is a tautolgoy being expressed here. However, Doppler can and should be derived separately by means of the original Lorentz Transformation Equations which David Bohm's "The Special Theory of Relativity" accomplishes and to which in some later recapitulation of Relativity Science Calculator this author will submit a simplified version of this derivation. In the meantime, everything shown here is both mathematically and experimentally true and valid.
Cosmological Consequence
"People like us, who believe in physics, know that the distinction between past, present, and future is only a stubbornly persistent illusion"  Albert Einstein ( 1879  1955 )
Cosmological Consequence: One cosmological and perhaps philosophical consequence of these equations is how as the universe inexorably expands outwards at ever unimaginable velocities, that for some future intelligent cosmologists the night sky goes completely dark in The Big Freeze.
Here is the essential cosmological paradox: any future observer of the universe will observe a shrinking amount of the universe since it takes more and more time for the distant edges of light to reach an observer's eye in an ever more greatly accelerating and expanding fabric of space  time universe!! In other words, any future observer of the sky's universe will see less and less of an ever more expanding universe!
One main reason is because the wavelength of whatever light is received from the farthest edges of the cosmos ( the cosmic horizon at about 13.7 billion ≈ 14 billion light  years distance from earth ) will exceed the diameter of the observed universe. And, since it will be impossible to ever build a large enough telescope to receive an infinite wavelength of such light due to Doppler Effect and extremely attenuated energy and frequency of this light, it will, by any sort of cosmological mathematics, be impossible to see such light!!
Hence, the light goes out in the known observable universe for any sentient intelligent observer still alive in approximately 1 trillion earth  years, plus or minus another 100 million earth  years!!
In this situation also, the observed universe, if such a thing is even possible, will be flat since all observed wavelengths of whatever electromagnetic propagation will equivalently be observed as being essentially non  sinusoidal or flat. Just simply imagine looking at a drawn  out string without any crest  to  crest wavelength due to extreme Doppler or redshift and hence no frequency; this, then, will be whatever "light" reaches an observer's eye or telescope  neither the observer's eye nor his telescope will see anything due to infinite wavelength and extreme Doppler or redshift!
Additionally, any intervening ionized interstellar dust or gases will bounce any incoming electromagnetic propagation of extremely low energy content ( light, x  ray, whatever ) away from an observer's eye or telescope similar as to how presently earth's Ionosphere bounces AM radio waves back to earth at much longer distances than otherwise straight  line distances on earth would allow.
In other words, all observers and their telescopes would be "seeing" and "living" from within a black hole: an inside  out black hole!!
Finally, because there will be no stellar objects to see with everything having gone black for both "night and day", there will be no need for the human mind to create an accounting for the motion of bodies, which is precisely what time is ! In fact, distance and velocity will no longer be calculated since these mathematical quantities will no longer be required in this eventual and ultimate world of physics.
The mathematics of Doppler, the law of energy conservation, Hubble's equation and the inexorable expansion of the universe dictate this ultimate fate.
And there is absolutely no escaping this ultimate cosmological fate!!
It's called The Big Freeze.
Parallel Universes: See: Michio Kaku, Max Tegmark and Hugh Everett for articles giving cosmological reasoning for the existence of parallel universes.
Conclusion: The relativistic aberration of starlight only predicts a slight change in frequency depending upon the velocity of the approaching or separating star or galaxy whereas the speed of light forever remains unalterably constant!
§ References:
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