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### Author Topic: Lambert's Cosine Law  (Read 51532 times)

#### jeffreyH

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##### Lambert's Cosine Law
« on: 17/07/2014 01:17:45 »
I am looking to adapt Lambert's Cosine Law for the gravitational field. What do others think? Is this already done? Can it be done?

#### JP

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##### Re: Lambert's Cosine Law
« Reply #1 on: 17/07/2014 02:43:12 »
It has to do with incoherent light coming off a perfectly reflecting surface.  What is the motivation for applying it to gravity?

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #2 on: 17/07/2014 21:23:45 »
It has to do with incoherent light coming off a perfectly reflecting surface.  What is the motivation for applying it to gravity?

It is the way you can relate area to particle emissions per second. See the section "Details of equal brightness effect" at wikipedia.

http://en.wikipedia.org/wiki/Lambert's_cosine_law

If gravity has force carrying particles there may be something to be gained by investigating a relationship. While it is nonsense to talk of luminosity for gravity field strength over an area is relevant. Field strength comes down to particle density. I have considered an equilibrium point between electromagnetism and gravitation with increased mass sizes when considering collapsing systems.
« Last Edit: 17/07/2014 21:28:23 by jeffreyH »

#### JP

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##### Re: Lambert's Cosine Law
« Reply #3 on: 18/07/2014 04:06:17 »
...which is fine so long as you're modeling particles that follow Maxwell's equations and add incoherently (they don't interfere with each other).

But that's a pretty bold assumption unless you're dealing with incoherent light.  And as a bold assumption, it needs strong evidence to support it, so why do so for gravity?

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #4 on: 19/07/2014 19:34:58 »
...which is fine so long as you're modeling particles that follow Maxwell's equations and add incoherently (they don't interfere with each other).

But that's a pretty bold assumption unless you're dealing with incoherent light.  And as a bold assumption, it needs strong evidence to support it, so why do so for gravity?

I have absolutely no answer to that. I have thoughts on the subject and I am working on theories but that is speculative. Self-interaction is a big issue here and is a theory killer in many respects.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #5 on: 20/07/2014 12:55:30 »
If we go back to Maxwell we find c = 1/. This can then be modified to represent C^2 = 1/ or ^-1. For rs we could then produce 2GM/^-1. This may seem like an insignificant substitution. Excuse the very bad latex. What if we make rs a unit radius by setting it to a value of 1? Then the only unknown is the mass which can be calculated by re-arrangement.
« Last Edit: 20/07/2014 13:11:20 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #6 on: 20/07/2014 13:17:07 »
We may now be able to link this to the curl equations of the electromagnetic field. It may be more productive to set the mass equal to the Planck mass.
« Last Edit: 20/07/2014 13:23:50 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #7 on: 20/07/2014 15:01:54 »
This then results in the relationship:

(epsilon0*mu0)^-1 = 2*G*Pm/2*Pl

epsilon0mu0 = Pl/G*Pm

NOTE: To substitute in the curl equations we have basically halved the gravitational component by canceling both factors of two. This then balances the dual equation and could indicate two separate gravitational components to the interaction.
« Last Edit: 21/07/2014 20:31:41 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #8 on: 20/07/2014 16:00:10 »
I haven't really checked any of this and it may be hogwash. If right it may already be known anyway and have no significance. At this point I do not have the information to draw conclusions.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #9 on: 20/07/2014 16:18:09 »
Now back to Lambert's Cosine Law. This would only be valid if the rates of emission between photons and gravitons differed in a distinct proportion yet to be determined. This of course relates back to the work on black body radiation and has a strict set of boundary conditions.
« Last Edit: 20/07/2014 16:21:10 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #10 on: 20/07/2014 17:59:50 »
We have two curl functions

-
-

Which can be substituted with

-
-

Here we can consider Pl to be a fixed constant and then the only variable quantity in the first term is the mass represented by Pm. This can not be increase otherwise we violate the Schwarzschild metric. This should then run from some lower bound up to the Planck Mass. What this lower bound should be is also undetermined. This then introduces a density variation into the curl equations. Is this valid? I don't know.

NOTE: This does however tie us to the Planck scale. This is where we need to be to produce any kind of theory of quantum gravity.
« Last Edit: 21/07/2014 20:34:48 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #11 on: 20/07/2014 23:06:30 »
The reduction in mass should coincide with a reduction in escape velocity. What effect would a reduction in mass mean for epsilon and mu (see image). Actually this is the wrong way round. This should actually model how light slows down in an intensifying gravitational field. It should model redshift. I need to delve deeper to make sure this is true.
« Last Edit: 21/07/2014 20:53:28 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #12 on: 22/07/2014 00:56:29 »
In our mu epsilon equivalence our Planck length is the numerator and appears in the place of time. As light speed is 1 Planck length in one Planck time this appears reasonable. We are also relating our mass to the Coulomb field. The epsilon mu value will increase with decreasing mass and decrease with increasing mass. As was stated previously the mass value can only decrease as increasing this will violate the speed of light limit. We would be calculating using values greater than c which is prohibited.
« Last Edit: 22/07/2014 01:02:50 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #13 on: 22/07/2014 01:13:38 »
It may be only the permittivity that is affected by changes in mass. This would explain the reduction in reactivity expressed as time dilation. The substitution in the curl equation is still incorrect as this does not express the limit at an event horizon so more work needs to be done to correct this. What the missing factor is remains unclear.
« Last Edit: 22/07/2014 01:42:17 by jeffreyH »

#### JP

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##### Re: Lambert's Cosine Law
« Reply #14 on: 22/07/2014 03:26:27 »
I've moved this topic as its veered well into New Theory territory.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #15 on: 22/07/2014 20:42:45 »
To revisit this it can be shown that f(z-ct) and f(z+ct) will satify the differential equation. In - Pl and should be at the same scale so to convert this we need to assume t to equal 1 Planck time. Then needs to be adjusted accordingly. This then sets our wave equation in the environment of the black hole event horizon. Or at least I think it does. I will be investigating this. Both E and H the electric and magnetic components have three separate parts in x, y and z. This is where the difficulties may arise.

NOTE: Treating this as a scalar wave equation is not very useful. Having the wave expressed in only the z direction and holding x and y at zero just won't describe reality in extreme conditions. While this is how Maxwell linked c to light this should be discarded. The resulting formula is more complex but defining the electromagnetic wave and its path near an event horizon should be a good starting point for a path into quantum gravity.
« Last Edit: 22/07/2014 22:40:54 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #16 on: 23/07/2014 21:46:54 »
I have a new method to add to this thread that plots in 3 dimensions with an implicit time function that will describe the progression of a wave with shifting of the waveform and time dilation incorporated. That will have to wait until the weekend. A scalar time function can also be applied to the x, y and z dimensions to show the shifts from blue to red and visa versa. This is then expressed in terms of the gravitational field. This is modeled at the quantum scale using an implicit unit vector for time. It can then be applied macroscopically to planetary motion.
« Last Edit: 23/07/2014 22:13:39 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #17 on: 26/07/2014 11:06:26 »
Before trying to link the gravitational field to Lambert's Cosine Law I need to take a detour. This starts with the unit sphere and the unit circle. Using the unit sphere and circle shows some interesting relationships and can be scaled up. This can then be used to describe both subatomic and macroscopic domains.

The circumference of the unit circle is 2*pi. To determine the angle of an arc around the circle whose arc length is equal to the radius we can use (1/2*pi)*360 which can be simplified to 7/44*360. This proportionality will become important when viewing interactions at differing scales and relates to wave frequency, length contraction and time dilation effects. The angle we have determined can be converted to radians to use in calculations.

It is interesting to note that the period of sin x is 2*pi. This can be utilized by considering forward motion and angular rotation as it relates to the unit sphere. The relationship between these two properties can describe the evolution of a wave and can be related directly to the gravitational field. When used it can be shown to show the underlying mechanism of the Pauli Exclusion Principle and the difference in energy levels required between electrons.

There are 3 directions of motion under consideration within this model. One motion is forward direction and is considered to be aligned with the poles of the sphere. The two other directions are angular. The first is around the equator and the second follows a longitudinal path intersecting both poles. The maximum unit of motion in unit time in the polar direction is equal to the unit sphere radius. The maximum unit of motion of the angular paths is 7/44*360 as stated above. If viewed at the Planck scale the angular components cannot reach this speed or none of us would be here. Therefore we can deduce that this dampening in angular momentum must be due to gravity which is what the current physical theories state.

If we follow this line of thinking through to its conclusion we can show that when considering the universe as a whole system light might get infinitesimally near to c but will never actually reach it as long as any gravitational field remains. I will demonstrate the reasons for this conclusion as I proceed.

NOTE: In case this limit on light speed seems absurd consider that if even light cannot reach its own maximum speed then certainly nothing else will. Also all mass is affected in the same proportional way as the photon within an equivalent field and the speed of any one component of the system is relative to everything else. One other consequence of this model may be the discarding of Lorentz transformations and the use of frames of reference. I say may because I have yet to properly determine this.
« Last Edit: 03/08/2014 21:37:40 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #18 on: 26/07/2014 12:37:36 »
Before proceeding further it would be useful to consider a thought experiment. When considering planetary motion we encounter the elliptical orbit as found by Kepler and later explained by Newton. From the relationship that equal areas of the orbit based at one focal point are equal for equal time periods we can imagine a scenario with planets. Consider a planet tipped on its side so that both poles align with the path of the orbit. The planet is then spinning around the orbital path as it moves. If we imagine a point on the spinning equator that we follow as it describes a path centered on the orbital path we see that frequency of the wave increases with distance from the focal point. We can then show that we have a redshift described nearer the object and a blue shift described as the planet moves further away. Light does not behave this way as shown by the receding galaxies indicating an expansion of the universe. The model being described here will eventually explain this.

NOTE: This thought experiment disregards the time component of the wave in case anyone was going to object. That was on purpose and will be explained later.
« Last Edit: 26/07/2014 12:57:00 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #19 on: 26/07/2014 14:33:53 »
This model will ultimately describe two types of photon. One with diverging electric and magnetic fields and the other with converging electric and magnetic fields. The type with diverging fields will propagate away at c while type with the converging fields has lower velocity, is imbalanced and will converge with a positive field. The converging photon is more severely affected by gravitation and interacts with it differently and may explain levitation in superconductors.

NOTE: The photon with converging fields may be equivalent to the hypothesized magneton. The period of this convergence is pi.
« Last Edit: 26/07/2014 14:37:37 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #20 on: 27/07/2014 10:23:50 »
We have a very interesting explanation of how DE Broglie derived his wave equation here:

http://chemwiki.ucdavis.edu/Physical_Chemistry/Quantum_Mechanics/Quantum_Theory/De_Broglie_Wavelength

His starting point was to assume E whilst represented by mc^2 could also equal hv which is Planck's constant times velocity. This term operates for light only so examining equation 5 we see that he ended up with essentially h/mv. If we take angular momentum as a stand in for energy and because we have 2 fields in motion we should be able to derive an equation at the Planck scale that equals De Broglie's. This is dependent upon the radius of a particle.

It has to be understood that we are no longer describing the wavelength here. If we are working in natural units the angular momentum will range between 0 and 1. The velocity is forward momentum and so describes the stretch of the wave with time. Hench this formula describes the rate of change in the wavelength.

Now if we go back to we also have a mass component. If we have Va as our angular momentum we have h/Va*v as the velocity of a wavelength segment which can then be related to the time over which it evolves. As describes our wave and relates this to mass and gravity we should be able to combine these. Remember that the mass was the only true variable in and in the De Broglie equation it is mass we have modified by replacing it with angular momentum.

Since we are only modifying mass all our calculations are done at a set radials distance as mass dereases. This is because Pl never changes. If we take the earth as an example we need to correlate a change between the surface and twice the radius to our fixed radius. We need to see how the mass should decrease to give us the right gravitational field strength. This factor is important to determine. Because we hold the radius fixed we can then model the wave evolution over a spherical surface as described above. This is not meant to reflect the reality of particles but is a device to determine the relationships.

In our left hand term the mass factor relates to the radial distance the wave travels away from the source. This has to be a density change. It is not only related to the density of the source mass but also the gravitational field density at any radial distance. Our factor will be p with p= 1/r1^2 where r1 is the distance away from the source hence we now have as our first term. The use of radial distance now connects the permittivity/permeability term directly to the wave evolution via the vector r1. The starting point will have r0 (our static radius) equal to r1. As r1 moves away the density is reduced by our term p. This will produce a gradual blue shift and can be related to time dilation via an implicit time on the z-axis. This does NOT describe the evolution of a light wave. Other factors come into play for light.

From now on the model uses linear algebra.
« Last Edit: 03/08/2014 17:33:53 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #21 on: 03/08/2014 17:50:20 »
At this point I need to state one conclusion I have already reached. The magnetic field force carrier should be a symmetry broken photon. It needs a partner particle which I firmly believe is a symmetry broken graviton. We should be thinking in terms of a gravitomagnetic field as well as an electromagnetic one. Now all I have to do is prove it.  [8D]
« Last Edit: 03/08/2014 17:53:34 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #22 on: 03/08/2014 20:36:37 »
The next posts will be concentrating on examining the effects on the partial differentials for E and H in the Maxwell equations.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #23 on: 03/08/2014 21:39:59 »
I have changed the first sentence of paragraph 2 of reply #17 from "The diameter of the unit circle is 2*pi" to "The circumference of the unit circle is 2*pi" as the former was in error and was confusing.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #24 on: 03/08/2014 22:15:12 »
At this point 3 things about the De Broglie equations should be understood.

Frequency and Energy are directly proportional.
Wavelength and Energy are inversely proportional.
Wavelength and frequency are inversely proportional.

Angular momentum was substituted for energy above so for our model evolving a wave over a spherically rotating and forward moving surface we can say:

Frequency and Angular Momentum are directly proportional.
Wavelength and Angular Momentum are inversely proportional.
Wavelength and frequency are inversely proportional.

« Last Edit: 03/08/2014 22:21:05 by jeffreyH »

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##### Re: Lambert's Cosine Law
« Reply #24 on: 03/08/2014 22:15:12 »