1
New Theories / Different View of Relativity
« on: 02/06/2017 04:44:05 »
Not at all saying the math of Relativity is wrong but below is a different viewpoint of SR via wave mechanics. Please note that I show where all the equations come from but it takes a bit to derive them all because Relativistic effects such as Length Contraction, Time Dilation, "simultaneity shifts," and Relativistic Aberration often occur together. Bare with the format as this isn't a proof read paper but it does contain quite a bit of math.
Classical Doppler Shift (Where v = va + vb):
fmo/f = (1 - v/c) , fms/f = 1/(1 + v/c), and fmso/f = (1 - va/c) / (1 - vb/c)
In general: fmo ≠ fms ≠ fmso
kmo/k = (1 - v/c), kms/k = 1/(1 + v/c), and kmso = (1 - va/c) / (1 - vb/c)
In general: kmo ≠ kms ≠ kmso
(Notice for later the wavenumber forms and frequency forms are the same). The above is the Classical Doppler Shift as found exactly in text books. I choose to use wavenumber with cycles/unit length because the wavenumber forms are symmetrical with the frequency forms with cycles/second.
Classically, the Doppler Shift gives away a preferred frame as something measurable because the moving source equation, moving observer equation, and both source/observer moving equations aren't the same. Frequency and wavenumber shifts leave the signature pointing towards the rest frame.
Set up a light clock perpendicular to the direction of travel. Light of a moving clock goes across at c’ where c² = (c’)² + v². This is just Pythagorean’s Theorem.
c’/c = f’/f = t’/t = (1 – (v/c)²)^½ = ɣ (when perpendicular. Relativistic effects mix at off angles; I can and will show these but it’s not as simple.)
If time slows down by ɣ then the moving source will emit a lower frequency by ɣ and the moving observer will observe a higher frequency by 1/ɣ . Using this idea, we have:
fms = f ɣ / (1 + v/c) = ((1 – v/c)/(1+v/c))^½ (source is moving)
fmo = f (1 – v/c) / ɣ = ((1 – v/c)/(1+v/c))^½ (observer is moving)
fmo = fms even though the mechanism behind them is different. In fact, fmso is also equal (source and observer are moving).
Relativistic Doppler Shift (Where v = (va + vb) / (1+ va vb/c²) I will derive the velocity equation also):
x’/x = k’/k = (1 – v/c)/ ɣ = ɣ /(1+v/c) = (1 – va/c)/(1 + vb/c) ɣb/ ɣa
Note kmo = kms = kmso
t’/t = f’/f = (1 – v/c)/ ɣ = ɣ /(1+v/c) = (1 – va/c)/(1 + vb/c) ɣb/ ɣa
Note fmo = fms = fmso
Where ɣ = (1 - (v/c)^2)^½
f/k = fms/kms = fmo/kmo = fmso/kmso = c The speed of light is constant because of the symmetric treatment of wavenumber and frequency in the Doppler Shift and the relationship x’/x = k’/k and f’/f = t’/t. Notice in the Relativistic form all three k equations and f equations are mechanistically different but yield the same output and have the same input.
The fundamental idea behind all of this is:
1) There is a medium where all fields propagate at c relative to it.
2) ALL particles are excitations of fields and particles with mass (not moving linearly at c) can only occur if two or more fields are interacting causing an ensemble (wave packet) with a rest mass and a non-zero proper time (time passes). An example would be the Higgs Field interacting with excitations of other fields.
3) "Aether Drag" does not need to occur because all particles are excitations AKA waves propagating on the "aether" or fields (whatever you wish to call it). Waves don't need to drag their medium while propagating. A planet moving is not the same as a bowling ball pushing through water. It's more like a wave structure rippling through a super fluid.
4) the ratio of frequencies f'/f equals the ratio of time 't/t. If I transmit a video at 8GHz and someone else receives at 16GHz they will witness my time happening twice as fast. The ratio of cycles received equals the ratio of time observed. Likewise as shown with the Michelson-Morley Interferometer x'/x = k'/k. How I see their length depends on the entire Doppler Shift. In principle we can't break apart our observation and assign chunks to a "classical cause" and other chunks to a "relativistic cause." Instead we see the LT and the DS in their entirety.
Deriving the Lorentz Transform from the Relativistic Doppler Shift (pick one of the Doppler forms; it doesn't matter which one because they are all mathematically identical):
f’/f = t’/t = (1 – v/c)/ ɣ → t’= (t – t v/c)/ ɣ
Use the Einstein Clock Synchronization Convention (sync clocks/rulers with light) c t = x we have:
t’= (t – x v/c²)/ ɣ = Lorentz Time Transformation
x’/x = k’/k = (1 – v/c)/ ɣ => t’= (t – t v/c)/ ɣ
Use the Einstein Clock Synchronization Convention (sync clocks/rulers with light) c t = x we have:
x’= (x – v t)/ ɣ = Lorentz Distance Transformation
The Einstein Clock Synchronization Convention is arbitrary where we assume light is isotropic relative to our frame. I did this because the Lorentz Transform as written this way also makes this assumption. This is only one of an infinite number of possible synchronizations that could have been chosen. Notice the Simultaneity Term v x /c^2 isn't any more amazing here than the Classical v/c term. Creating Planes of Simultaneity to prove 4D spacetime doesn't do much of anything.
Derivation of the velocity addition formula (which can also be explained via wavelike nature):
(1 – v/c)/ ɣ or ɣ /(1+v/c) = (1 – va/c)/(1 + vb/c) ɣb/ ɣa
((1 - v) / (1 + v)).5 = (( 1 - va) / (1 + va))1/2 (( 1 - vb) / (1 + vb))1/2
(1 - v) / (1 + v) = (1 -va) / (1 + va) (1 - vb) / (1+ va)
(1 + va) (1 + vb) (1 - v ) = (1 - va) (1 - vb) (1 + v)
(1 + va) (1 + vb) - v (1+va) (1+vb) = (1 - va) (1 - vb) v + (1 - va) (1 - vb)
(1 + va) (1 + vb) - (1 - va) (1 - vb) = (1 - va) (1 - vb) v + v (1+va) (1+vb)
1 + va + vb + va vb - 1 + va + vb - va vb = (1 -va - vb + va vb) v + (1 +va + vb + va vb) v
2 (va + vb) = (2 + 2 va vb) v
(va + vb) / (1 + va vb) = v This is the Relativistic Velocity Addition Equation.
The full all angle forms (transverse) are:
fmo = f (1 - v/c cos(θmo)) / gamma = fms = f gamma / (1 + v/c cos(θms))
kmo = k (1 - v/c cos(θmo)) / gamma = kms = k gamma / (1 + v/c cos(θms))
Where the angles are related (derived using the velocity addition equation already derived)
cos(θms) = (cos(θmo) - v/c) / (1 - v/c cos(θmo))
Length Contraction:
dx/dt = -v → dx = -v dt →
dΦ/dt -v dΦ/dx =0 → [d/dt -v d/dx][ d/dt +v d/dx] Φ= 0 → d²Φ/dt² = v² d²Φ/dx²
Classical Wave Equation:
0 = d²Φ/dt²- c² d²Φ/dx² - c² d²Φ/dy² - c² d²Φ/dz²
Substitute for the time dependent derivative:
0 = v² d²Φ/dx² - c²d²Φ/dx² - c² d²Φ/dy² - c² d²Φ/dz²
0 = ((v/c)2– 1) c² d²Φ/dx² - c² d²Φ/dy² - c² d²Φ/dz²
0 = ɣ² c² d²Φ/dx² + c² d²Φ/dy² + c² d²Φ/dz²
ɣ dy = dx
This only uses the Classical Wave Equation and an arbitrary shift v in the x direction. The result clearly shows the wave form contracting by ɣ in the x direction. If everything is wavelike in nature with the same fundamental propagation speed c, then Lorentz Contraction occurring on all waveforms (every particle and collection of bound particles) is expected. Classical wave structures also compress as v increases relative to the medium (example sonic booms). Nothing was forced or ad hoc about this result and no time as a fourth dimension was required. Experiments in physics have shown that Relativistic Symmetry naturally arises when the wave sources transform similar or identical to the waves they create. Think of the Couder Walking Drop Experiment and many superfluid experiments. The walking drop deforms in shape a bit (length contracts) when it moves linearly making the Doppler Form closer to Relativistic rather than Classical. Now the Relativistic nature of these mediums does breakdown but no such thing necessarily needs to occur in the "medium of the universe" because this medium could be many tens or thousands orders of magnitude more fine or perhaps it's infinitely fine.
When v is in the x direction. Michealson-Morley Interferometer math to find Lx compared to Ly.
c’ = ɣ c
2Ly / (c’) = Lx / (c - v) + Lx / (c + v)
2Ly / (ɣ c) = Lx / (c - v) + Lx / (c + v)
2Ly / (ɣ c) = Lx (c + v)/ (c² – v²)+ Lx (c - v)/ (c² – v²)
2Ly / (ɣ c) = Lx 2c/ (c² – v²)
Ly / (ɣ c) = Lx c²/ (c² – v²) 1/c
Ly / (ɣ c) = Lx /( ɣ² c)
ɣ Ly = Lx (as we would expect)
GR, SR, QM, and wave equation forms:
The Classical Wave equation below is very much like the equations below it:
0 = d²Φ/dt²- c² d²Φ/dx² - c² d²Φ/dy² - c² d²Φ/dz² (classical wave equation)
0 = c² dt²- dx² - dy² - dz² - c² dtau² (metric of flat spacetime, GR)
0 = d²Φ/dt²- c² d²Φ/dx² - c² d²Φ/dy² - c² d²Φ/dz² + (m c / ħ)² Φ (Klein-Gordon equation, Relativistic QM. Note the trick to find Length Contraction also works here)
0 = guv dxu dxv – c² dtau² (generalized metric, GR [please note I put all terms on the right])
These are all very much similar in form. Local Lorentz Covariance (c = f/k where c is constant at least locally) is the most important law in GR (and SR). Even classically the forms for f and k in the Doppler Shift are identical that this holds true in SR and GR is not surprising. The symmetrical forms would preserve c and the identical Doppler forms for k and f also preserve c. If no backwards time travel is possible there is no need for time to be a true "fourth dimension" but it acts as such in much of the math because of the symmetrical treatment of f and k (f corresponds to measured t and k corresponds to measured x, y, and z). "Curvature of spacetime" would be symmetrical shifts in objects (their size, shape, and "rate through time" relative to other objects in another location).
Please let me know if you have any questions. I could go on for awhile with the formulas but this gets quite tedious to write. I don't think I made any math errors but please let me know if I did. Some math equations didn't copy superscripts/subscripts when I copied them form my notes.
Also from QM E = h f and p = h k so momentum and energy also have the same symmetry as f and k because h is a constant.
I personally find this method of deriving the Relativistic Formulas more intuitive, it bases off of the classical Doppler Shift which is a strong/well-tested concept, and it correctly assumes everything is wavelike. Sure it assumes a medium but Relativity assumes a certain synchronization (I assume none) and Relativity introduces time as a fourth dimension (I'd call that a more complex conclusion). Saying one idea is more "complicated" than another is often subjective. However, I'm glad that the shortcut exists.
Classical Doppler Shift (Where v = va + vb):
fmo/f = (1 - v/c) , fms/f = 1/(1 + v/c), and fmso/f = (1 - va/c) / (1 - vb/c)
In general: fmo ≠ fms ≠ fmso
kmo/k = (1 - v/c), kms/k = 1/(1 + v/c), and kmso = (1 - va/c) / (1 - vb/c)
In general: kmo ≠ kms ≠ kmso
(Notice for later the wavenumber forms and frequency forms are the same). The above is the Classical Doppler Shift as found exactly in text books. I choose to use wavenumber with cycles/unit length because the wavenumber forms are symmetrical with the frequency forms with cycles/second.
Classically, the Doppler Shift gives away a preferred frame as something measurable because the moving source equation, moving observer equation, and both source/observer moving equations aren't the same. Frequency and wavenumber shifts leave the signature pointing towards the rest frame.
Set up a light clock perpendicular to the direction of travel. Light of a moving clock goes across at c’ where c² = (c’)² + v². This is just Pythagorean’s Theorem.
c’/c = f’/f = t’/t = (1 – (v/c)²)^½ = ɣ (when perpendicular. Relativistic effects mix at off angles; I can and will show these but it’s not as simple.)
If time slows down by ɣ then the moving source will emit a lower frequency by ɣ and the moving observer will observe a higher frequency by 1/ɣ . Using this idea, we have:
fms = f ɣ / (1 + v/c) = ((1 – v/c)/(1+v/c))^½ (source is moving)
fmo = f (1 – v/c) / ɣ = ((1 – v/c)/(1+v/c))^½ (observer is moving)
fmo = fms even though the mechanism behind them is different. In fact, fmso is also equal (source and observer are moving).
Relativistic Doppler Shift (Where v = (va + vb) / (1+ va vb/c²) I will derive the velocity equation also):
x’/x = k’/k = (1 – v/c)/ ɣ = ɣ /(1+v/c) = (1 – va/c)/(1 + vb/c) ɣb/ ɣa
Note kmo = kms = kmso
t’/t = f’/f = (1 – v/c)/ ɣ = ɣ /(1+v/c) = (1 – va/c)/(1 + vb/c) ɣb/ ɣa
Note fmo = fms = fmso
Where ɣ = (1 - (v/c)^2)^½
f/k = fms/kms = fmo/kmo = fmso/kmso = c The speed of light is constant because of the symmetric treatment of wavenumber and frequency in the Doppler Shift and the relationship x’/x = k’/k and f’/f = t’/t. Notice in the Relativistic form all three k equations and f equations are mechanistically different but yield the same output and have the same input.
The fundamental idea behind all of this is:
1) There is a medium where all fields propagate at c relative to it.
2) ALL particles are excitations of fields and particles with mass (not moving linearly at c) can only occur if two or more fields are interacting causing an ensemble (wave packet) with a rest mass and a non-zero proper time (time passes). An example would be the Higgs Field interacting with excitations of other fields.
3) "Aether Drag" does not need to occur because all particles are excitations AKA waves propagating on the "aether" or fields (whatever you wish to call it). Waves don't need to drag their medium while propagating. A planet moving is not the same as a bowling ball pushing through water. It's more like a wave structure rippling through a super fluid.
4) the ratio of frequencies f'/f equals the ratio of time 't/t. If I transmit a video at 8GHz and someone else receives at 16GHz they will witness my time happening twice as fast. The ratio of cycles received equals the ratio of time observed. Likewise as shown with the Michelson-Morley Interferometer x'/x = k'/k. How I see their length depends on the entire Doppler Shift. In principle we can't break apart our observation and assign chunks to a "classical cause" and other chunks to a "relativistic cause." Instead we see the LT and the DS in their entirety.
Deriving the Lorentz Transform from the Relativistic Doppler Shift (pick one of the Doppler forms; it doesn't matter which one because they are all mathematically identical):
f’/f = t’/t = (1 – v/c)/ ɣ → t’= (t – t v/c)/ ɣ
Use the Einstein Clock Synchronization Convention (sync clocks/rulers with light) c t = x we have:
t’= (t – x v/c²)/ ɣ = Lorentz Time Transformation
x’/x = k’/k = (1 – v/c)/ ɣ => t’= (t – t v/c)/ ɣ
Use the Einstein Clock Synchronization Convention (sync clocks/rulers with light) c t = x we have:
x’= (x – v t)/ ɣ = Lorentz Distance Transformation
The Einstein Clock Synchronization Convention is arbitrary where we assume light is isotropic relative to our frame. I did this because the Lorentz Transform as written this way also makes this assumption. This is only one of an infinite number of possible synchronizations that could have been chosen. Notice the Simultaneity Term v x /c^2 isn't any more amazing here than the Classical v/c term. Creating Planes of Simultaneity to prove 4D spacetime doesn't do much of anything.
Derivation of the velocity addition formula (which can also be explained via wavelike nature):
(1 – v/c)/ ɣ or ɣ /(1+v/c) = (1 – va/c)/(1 + vb/c) ɣb/ ɣa
((1 - v) / (1 + v)).5 = (( 1 - va) / (1 + va))1/2 (( 1 - vb) / (1 + vb))1/2
(1 - v) / (1 + v) = (1 -va) / (1 + va) (1 - vb) / (1+ va)
(1 + va) (1 + vb) (1 - v ) = (1 - va) (1 - vb) (1 + v)
(1 + va) (1 + vb) - v (1+va) (1+vb) = (1 - va) (1 - vb) v + (1 - va) (1 - vb)
(1 + va) (1 + vb) - (1 - va) (1 - vb) = (1 - va) (1 - vb) v + v (1+va) (1+vb)
1 + va + vb + va vb - 1 + va + vb - va vb = (1 -va - vb + va vb) v + (1 +va + vb + va vb) v
2 (va + vb) = (2 + 2 va vb) v
(va + vb) / (1 + va vb) = v This is the Relativistic Velocity Addition Equation.
The full all angle forms (transverse) are:
fmo = f (1 - v/c cos(θmo)) / gamma = fms = f gamma / (1 + v/c cos(θms))
kmo = k (1 - v/c cos(θmo)) / gamma = kms = k gamma / (1 + v/c cos(θms))
Where the angles are related (derived using the velocity addition equation already derived)
cos(θms) = (cos(θmo) - v/c) / (1 - v/c cos(θmo))
Length Contraction:
dx/dt = -v → dx = -v dt →
dΦ/dt -v dΦ/dx =0 → [d/dt -v d/dx][ d/dt +v d/dx] Φ= 0 → d²Φ/dt² = v² d²Φ/dx²
Classical Wave Equation:
0 = d²Φ/dt²- c² d²Φ/dx² - c² d²Φ/dy² - c² d²Φ/dz²
Substitute for the time dependent derivative:
0 = v² d²Φ/dx² - c²d²Φ/dx² - c² d²Φ/dy² - c² d²Φ/dz²
0 = ((v/c)2– 1) c² d²Φ/dx² - c² d²Φ/dy² - c² d²Φ/dz²
0 = ɣ² c² d²Φ/dx² + c² d²Φ/dy² + c² d²Φ/dz²
ɣ dy = dx
This only uses the Classical Wave Equation and an arbitrary shift v in the x direction. The result clearly shows the wave form contracting by ɣ in the x direction. If everything is wavelike in nature with the same fundamental propagation speed c, then Lorentz Contraction occurring on all waveforms (every particle and collection of bound particles) is expected. Classical wave structures also compress as v increases relative to the medium (example sonic booms). Nothing was forced or ad hoc about this result and no time as a fourth dimension was required. Experiments in physics have shown that Relativistic Symmetry naturally arises when the wave sources transform similar or identical to the waves they create. Think of the Couder Walking Drop Experiment and many superfluid experiments. The walking drop deforms in shape a bit (length contracts) when it moves linearly making the Doppler Form closer to Relativistic rather than Classical. Now the Relativistic nature of these mediums does breakdown but no such thing necessarily needs to occur in the "medium of the universe" because this medium could be many tens or thousands orders of magnitude more fine or perhaps it's infinitely fine.
When v is in the x direction. Michealson-Morley Interferometer math to find Lx compared to Ly.
c’ = ɣ c
2Ly / (c’) = Lx / (c - v) + Lx / (c + v)
2Ly / (ɣ c) = Lx / (c - v) + Lx / (c + v)
2Ly / (ɣ c) = Lx (c + v)/ (c² – v²)+ Lx (c - v)/ (c² – v²)
2Ly / (ɣ c) = Lx 2c/ (c² – v²)
Ly / (ɣ c) = Lx c²/ (c² – v²) 1/c
Ly / (ɣ c) = Lx /( ɣ² c)
ɣ Ly = Lx (as we would expect)
GR, SR, QM, and wave equation forms:
The Classical Wave equation below is very much like the equations below it:
0 = d²Φ/dt²- c² d²Φ/dx² - c² d²Φ/dy² - c² d²Φ/dz² (classical wave equation)
0 = c² dt²- dx² - dy² - dz² - c² dtau² (metric of flat spacetime, GR)
0 = d²Φ/dt²- c² d²Φ/dx² - c² d²Φ/dy² - c² d²Φ/dz² + (m c / ħ)² Φ (Klein-Gordon equation, Relativistic QM. Note the trick to find Length Contraction also works here)
0 = guv dxu dxv – c² dtau² (generalized metric, GR [please note I put all terms on the right])
These are all very much similar in form. Local Lorentz Covariance (c = f/k where c is constant at least locally) is the most important law in GR (and SR). Even classically the forms for f and k in the Doppler Shift are identical that this holds true in SR and GR is not surprising. The symmetrical forms would preserve c and the identical Doppler forms for k and f also preserve c. If no backwards time travel is possible there is no need for time to be a true "fourth dimension" but it acts as such in much of the math because of the symmetrical treatment of f and k (f corresponds to measured t and k corresponds to measured x, y, and z). "Curvature of spacetime" would be symmetrical shifts in objects (their size, shape, and "rate through time" relative to other objects in another location).
Please let me know if you have any questions. I could go on for awhile with the formulas but this gets quite tedious to write. I don't think I made any math errors but please let me know if I did. Some math equations didn't copy superscripts/subscripts when I copied them form my notes.
Also from QM E = h f and p = h k so momentum and energy also have the same symmetry as f and k because h is a constant.
I personally find this method of deriving the Relativistic Formulas more intuitive, it bases off of the classical Doppler Shift which is a strong/well-tested concept, and it correctly assumes everything is wavelike. Sure it assumes a medium but Relativity assumes a certain synchronization (I assume none) and Relativity introduces time as a fourth dimension (I'd call that a more complex conclusion). Saying one idea is more "complicated" than another is often subjective. However, I'm glad that the shortcut exists.