# Lambert's Cosine Law

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#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #300 on: 08/08/2015 14:33:18 »
If we take $$\frac{GM}{v^2} + r_s = r$$ and modify this to be $$\frac{GM}{c^2} + r_s = r$$ we can remove the addition of rs and we arrive at $$\frac{GM}{c^2}$$. The Schwarzschild radius calculation itself is $$\frac{2GM}{c^2}$$ so that with our equation we ar dealing with half the Schwarzschild radius. That is 1 Planck length. So adding this to rs gives us an orbital velocity of c at 1.5rs. This means that, for a Planck mass black hole, particles with rest mass can only maintain a steady orbit at twice the radius at a velocity of (1/2)c. This also means that photons can ONLY orbit at 1.5rs. Of course the scales are so small that particles would actually have to be point sources. It reality this would be an impossible situation. As a model though it may yield some useful results.
« Last Edit: 08/08/2015 14:34:58 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #301 on: 08/08/2015 15:44:55 »
If we consider the photon's wave it is best viewed as a geodesic with an oscillation about the direction of its path. This is because moving to a higher or lower radial distance would destroy the orbit. This is an idealized situation. The length of the orbit along its straight line geodesic path is $$2\pi r$$ where r is 3 Planck lengths or 1.5rs. The peak of the wave occurs over an extended period of time and will take an astronomical number of orbital periods to occur and so for all intents and purposes we can view the photon as traveling a perfectly straight line geodesic. This APPEARS to imply that at Planck scales the photon can be viewed as having zero energy. This is a strange concept as it also implies no wave function. At these scales it is as if the photon is only a particle. This makes sense however as the photon is not an infinitesimal point.

When viewed in this light the weakness of the gravitational field becomes less important at the Planck scale. It all depends upon how small the unit vectors for the extent of the instantaneous action of the force carrier of gravitation. If the vectors are at a scale where the energy of the photon is infinitesimally small then the action of gravity will have a high magnitude compared to the photon. This will also be a function of force carrier density.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #302 on: 08/08/2015 16:06:09 »
Now we can see the crux of the problem. Since the strength of gravity is so much smaller than that of the electromagnetic field the wavelength should be longer. This implies an astronomically high density of gravitons to affect 1 photon. Since at the Planck scale the photon appears to have zero energy then this has to be true of the graviton. We could state that $$Gx - \gamma x > 0$$ is the relationship that needs to be satisfied. Where G stands for the graviton and gamma for the photon. At the Planck scale x cannot be a stand in for energy. So what is x?

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #303 on: 08/08/2015 16:10:08 »
This could mean a different probability density profile to either the photon or any particle with rest mass. That is that the graviton is smeared over a larger area than other particles.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #304 on: 08/08/2015 21:05:57 »
It could be worthwhile to investigate Causal Fermion Systems with regard to this model. Especially the states of the Dirac sea.

https://en.wikipedia.org/wiki/Causal_fermion_system

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #305 on: 10/08/2015 00:45:12 »
To map a set of equal incremental increases in orbital magnitude we find a sequence of 1/2rs, rs, 2rs, 4rs, 8rs, 16rs etc. Each of these increments show an equal increase in orbital velocity. This profile is much different to that of the force in the direction of the gravitational field which is inverse square.
« Last Edit: 10/08/2015 00:49:16 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #306 on: 10/08/2015 20:05:42 »
There is something I need to correct here. After having run calculations I have found that it is a radial distance of 3rs that has an orbital velocity of 1/2c. At 2rs the velocity is 70% the speed of light. Light speed is still at 1.5rs.

I am currently looking for a well defined equation for relativistic escape velocity. Once I have this then I will be trying to determine the relationship between the two. If any.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #307 on: 10/08/2015 20:25:32 »
For the orbital equation the relativistic mass terms were removed. It would be useful to apply this to escape velocity. Then the correlation between the two can be found.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #308 on: 15/08/2015 13:59:58 »
I have attached a graph of orbital velocity and escape speed at set radial distances from a black hole. Escape speed is so named as it is not a vector with direction. Any direction that does not intercept the horizon will do. We can simply choose an arbitrary direction that is normal to the surface of the horizon.
« Last Edit: 15/08/2015 14:06:57 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #309 on: 15/08/2015 14:26:45 »
The attached graph shows the difference between the magnitude of the two velocities and therefore the kinetic energy required in the perpendicular and normal directions. At a point 2 radial distances from the source these are equal. This is a radial distance of some significance.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #310 on: 15/08/2015 16:04:26 »
This difference in kinetic energy is of the form $$\frac{1}{2}m\frac{GM2r-GMr+GMr_s}{2r^2-2rr_s}$$
« Last Edit: 15/08/2015 16:12:13 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #311 on: 15/08/2015 20:17:16 »
Quantization should be attempted at the point of convergence of kinetic energies. This is not possible at the Planck scale so the next step is determining the best mass size to use in this process.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #312 on: 16/08/2015 14:56:43 »
Since the Planck scale of a Planck mass black hole allows no stable orbit for light the mass size chosen should be to a scale that does allow a light speed orbit. This should be the smallest mass for which this occurs so that force carrier quanta can be determined. This will be a function of spatial separation of force carriers due to the inverse square nature of the gravitational field. It will also be a function of force carrier density.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #313 on: 16/08/2015 23:20:54 »
Could this zero kinetic energy cross over relate to the UV Fixed Point.

https://en.wikipedia.org/wiki/Ultraviolet_fixed_point

#### PmbPhy

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##### Re: Lambert's Cosine Law
« Reply #314 on: 17/08/2015 00:27:50 »
Could this zero kinetic energy cross over relate to the UV Fixed Point.

https://en.wikipedia.org/wiki/Ultraviolet_fixed_point
I can't believe it. You're still talking to yourself, Jeff? I didn't know that you were still doing this after all this time.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #315 on: 23/08/2015 11:31:05 »
I have the best arguments with myself  []

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #316 on: 23/08/2015 11:36:43 »
To correct the final form of the equation we have $$\frac{1}{2}\gamma m\frac{GM2r-(GMr+GMr_s)}{2r^2-2rr_s}\,=\,0$$

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #317 on: 23/08/2015 12:42:39 »
Now for some speculation. One problem has always been the loss of gravitational energy over time. How can a mass generate gravitational energy and not have an accumulating loss of energy over time? What if we had graviton pairs appearing out of the vacuum where one graviton returns to the vacuum whilst the other interacts with mass and takes some of its kinetic energy? Ultimately, there should be a mechanism for this virtual graviton to return to the vacuum. This way we are only dealing with the loss of kinetic energy and gravitation is no longer inherent to mass. Just like the photon is not inherent to mass, but a separate and distinct particle. This graviton pair production mechanism then echoes the process of hawking radiation.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #318 on: 23/08/2015 13:35:50 »
If we consider the inflationary period as a type of runaway expansion we can then propose something else. That gravitation only became aparent during the radiation-dominated era.

The interaction of vacuum graviton pairs with photons and neutrinos then acted as a brake on inflation. The intensity of the interaction would then have partly contributed to the production of the CMBR via extraction of kinetic energy from highly energetic photons.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #319 on: 23/08/2015 13:50:47 »
To sink gravitons back to the vacuum we have only two choices. Infinity or the centre of gravity of an object. The most realistic option is the centre of gravity. This implies that black holes, as well as being entropy sinks are also gravity sinks. The proposed fossil gravity field outside the event horizon of a black hole may be a consequence of this process.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #320 on: 23/08/2015 13:54:52 »
This sinking of gravity inside a black hole, over cosmological timescales, may be the trigger to converting a black hole to an equivalent white hole. Thus a big bang.

#### PmbPhy

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##### Re: Lambert's Cosine Law
« Reply #321 on: 23/08/2015 19:13:38 »
I have the best arguments with myself  []

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #322 on: 23/08/2015 19:53:18 »
I have the best arguments with myself  []

Lol. I was joking Pete.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #323 on: 06/09/2015 12:59:19 »
If it is considered that gravitation originates in the vacuum then what happens when two gravitons interact? The attached image shows this possible interaction as a Feynman diagram using gluon symbols as placeholders for the graviton. the two original intersecting gravitons swap paths at the intersection and draw one of a pair of vacuum gravitons out and in the process both donat kinetic energy to the new particle. At the center of a dense enough mass this will result in an initial singularity which then propagates outwards.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #324 on: 06/09/2015 13:07:19 »
There should be an uncertainty involved in this process as to whether a graviton will appear from the vacuum or return to the vacuum. Otherwise we have an infinite increase in the strength of gravity. This would be apparent at Lagrangian points which it obviously is not.