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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 )

Atomic Doppler Effect

§ The Relative Motion of The Frame of Reference of a Moving and Oscillating Atom:

oscillating_atom.png

This above schema is simplified and equivalent to this gamma_photon2.png emission diagram:

ocsillating_atom_simplified.png

Just before the moment when the moving atom, velocity velocity_v.png, excitedly emits a gamma_photon2.png, total energy of the atom is

energy_before.png

However just at the moment of  gamma_photon2.png  emission, E_atom_p.e.png loses an amount of energy hf_o.png such that

atom_p.e._energy_loss.png

where the moving atom experiences a recoil and acquires a new velocity velocity_v_prime.png and f_o.png is the atom's natural oscillating frequency when observed at rest and the observer staying right at the gamma_photon2.png  reference frame of the atom source.

Furthermore,

energy_after.png

§ The Law of Energy Conservation Dictates:

energy_before_after2.png

§ The Components of Linear Momentum:

From the above gamma_photon2.png  emission diagram, we can now deconstruct the following x, y - components of linear momentum in this following traditional manner:

compression_factor.png

Hence, the difference between f_prime.png and f_o.png gamma_photon2.png  frequencies is wholly determined by the compression factor

which for  beta_equals_zero.png  gives  cos_beta_equals_1.png  and for which this compression factor for straight - line observation easily reduces down to

note: relative velocity velocity_v.png is taken to be negative when the atom source is relatively moving away from the observer in stationary system ; otherwise, relative velocity velocity_v.png 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 gamma_photon2.png  increasingly moves away from the stationary observer in system at greater and greater amounts of velocity_v.png, the observed f_prime.png frequency decreases with a concomitant increase in observed wavelength for gamma_photon2.png. That is,

red_shifted.png

Conversely, for an oscillating atom source for gamma_photon2.png  moving towards a stationary observer in system

blue_shifted.png

The ratio

doppler_factor.png

is what produces the Doppler Effect for our source atom emitting gamma_photon2.png !

In the non - relativistic limit where v << c,

red-shift.png

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 f_prime.png to an observer in system becomes

relativistic_analysis.png

See: special relativity - time dilation

Notice here, again, that for a receding gamma_photon2.png  atom source where receding velocity velocity_v.png is taken as negative_v.png,

red-shift2.png.

Finally, for

transversal_doppler_effect.png

and in terms of energy

transversal_doppler_effect_1.png.

§ Straight - Line Observing Oscillating Photon - Emitting Atom:

That is, take up another perspective in stationary system so that the atom source for gamma_photon2.png lies strictly along the x - axis and hence beta_equals_zero.png implying cos_beta_equals_1.png .

This then simplifies our equation for observed frequency

simplifies_derived_equation.png.

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 positive_v.png as being taken as negative_v.png.

Notice also that for

simplified.png

if velocity velocity_v.png is taken as positive_v.png where the object is approaching, therefore

blue-shifted2.png.

§ Some Further Mathematical Manipulation for Straight - Line Observation of Atom Emitting Photon:

receding_approaching_source.png

Also,

wavelength_receding_approac.png

§ Atomic Pair Production:

We know from an earlier Lorentz transformation that momentum, mass and energy are related as follows:

momentum_mass_energy.png.

That is, momentum momentum.png and mass mass.png transform from one system positive_v.png into another system according to the Lorentz transformation equations and the relativistic addition of velocities. Hence,

energy_doppler.pngLorentz transformation of the energy - momentum vector

In the extreme case of v_equals_c.png, the atom source for gamma_photon2.png no longer exists - i.e., it goes to zero and disappears!

For the other extreme situation where velocity_v.png is extremely large but not at c.pngE_prime.png becomes infinitely small as can also be seen from the inverse relation

E_prime_inverse_v.png

but yet the numerator

numerator.png

never quite goes to zero.

Of course we know that

E_equals_hf.png

so we should not be surprised that

frequency_equivalent_energy.png.

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 E_prime.png becomes extremely small whenever velocity_v.png is extremely large but not at c.png, however there surely must be a lower limit for E_prime.png when gamma_photon2.png  hits an atomic nucleus and thereby produces an elementary particle and its anti - particle!

That is, atomic pair production occurs whenever a gamma_photon2.png 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 gamma_photon2.png, E_o.png, be greater than the combined energies of the electron and positron. Or,

pair_production.png.

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 gamma_photon2.png 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 gamma_photon2.png  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 gamma_photon2.png ; 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,

whistle_pitch.png

is the Doppler factor  giving a mathematical expression to the Doppler Effect.

§ Earth as a Platform for Astronomical Observations:

aberration_starlight.png"On The Electrodynamics of Moving Bodies", by A. Einstein, 1905

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 velocity_v.png 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 velocity_v.png; 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

relativistic_aberration.png

We now need to re - visit the equations for the relativistic addition of velocities as follows:

Abstract #2

world_lines.png

From the perspective of [ relatively ] stationary system , sun, we thus have

sun_perspective.png

On the other hand, from the perspective of [ relatively ] moving system , earth, we have

perspective_earth.png

since we are observing the path of starlight just within earth's x - y plane.

§ Aberration Constant:

aberration_constant.png

That is, the angle of deflection beta.png 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:


angle_deflection.png

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

path_starlight.png

However with respect to the moving frame of reference of earth, we must employ Lorentz inverse transformation equations as follows:

relativistic_aberration2.pngrelativistic aberration formula, Einstein, 1905

§ Angle of Deflection:

In fact

angle_deflection2.png

where angle_deflection3.png is the angle beta.png 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

aberration_constant2.png

§ Relativistic Transversal Aberration on Platform Earth:

relativistic aberration equation, Einstein, 1905

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

relativisitic_aberration3.pngrelativistic aberration equation, Einstein, 1905

Hence,

Relativistic beaming occurs since

§ Relativistic Red - Shift:

Relativistic red - shift z can now be determined as follows:

relativistic_red_shift.png

For "line of sight" relative motion, line_of_sight.png and hence

line_of_sight2.png

And, simplified

simplify.png

§ 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 O2 7593.70 759.370 L Fe 3820.44 382.044
B band O2 6867.19 686.719 N Fe 3581.21 358.121
C (H-alpha) H 6562.81 656.281 P Ti+ 3361.12 336.112
a band O2 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
E2 Fe 5270.39 527.039 e Fe 4383.55 438.355
F (H-beta) H 4861.34 486.134 b1 Mg 5183.62 518.362
G Fe 4307.90 430.790 b2 Mg 5172.70 517.270
G Ca 4307.74 430.774 b3 Fe 5168.91 516.891
H Ca+ 3968.47 396.847 b4 Fe 5168.91 516.891
K Ca+ 3933.68 393.368 b4 Mg 5167.33 516.733
a 1 Angstrom = 1x10-10 meter = 0.1 nanometer
b 1 nanometer = 1x10-9 meter = 10.0 Angstrom

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:

transversal_doppler_energy.png

Analogous to Lorentz inverse transformation equations

Lorentz_inverse.png

is

energy_momentum_equivalent.png

Now,

aberration_constant3.png

Also, since

transversal_doppler_energy2.png

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:

lorentz, redshift, 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 Extra-Galactic 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:

Riemann flat space metricWilkinson Microwave Anistropy Probe - WMAP

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!!

electromagnetic_spectrum.gif

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 gamma_photon2.png frequency depending upon the velocity of the approaching or separating star or galaxy whereas the speed of light c.png forever remains unalterably constant!

§ References:

  1. "The Special Theory of Relativity", by David Bohm, A close Einstein colleague at Princeton University's Institute for Advanced Study, copyright © 1965, 1966 by Sarah Bohm

  2. "A Relation Between Distance And Radial Velocity Among Extra-Galactic Nebulae", by Edwin Hubble, Mount Wilson Observatory, Carnegie Institution of Washington, January 17, 1929

  3. "Super - Eddington Accreting Massive Black Holes as Long-Lived Cosmological Standards", 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, publication date: 02/2013, reference: 2013PhRvL.110h1301W, Physical Review Letters, vol. 110, Issue 8, id. 081301



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