Lambert's Cosine Law

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Online jeffreyH

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Re: Lambert's Cosine Law
« Reply #50 on: 24/08/2014 16:13:51 »
The first derivative of g = GM/r^2 is 2GM/r^3 which describes the change in acceleration or jerk. The Schwarzschild metric has 2GM/c^2 and the right hand factor of our wave function has [tex]\frac{\partial ^2 E}{pc^3}[/tex]. The right hand term can be rewritten as [tex]\frac{\partial ^2 E}{r^2c^3}[/tex] where r is distance of the wave from the source of gravitation. Before getting back to Lambert's Cosine Law it will be necessary to investigate Coulomb's Law with respect to the previous equations. This may seem like a very strange path to take, which it in fact is, but I want to determine certain relationships before proceeding.

A correction to the above. [tex]\frac{\partial ^2 E}{r^2c^3}[/tex] should be [tex]\frac{r^2 \partial ^2 E}{c^3}[/tex].
« Last Edit: 25/08/2014 19:57:25 by jeffreyH »

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Re: Lambert's Cosine Law
« Reply #51 on: 25/08/2014 20:08:31 »
Light is affected by gravity which experiment shows is true. A photon leaving a gravitational field is slowed. Where then does the energy come from to restore the loss of kinetic energy whilst leaving the gravitational field? The photon will gain momentum due to the fall off of gravitational effect due to the inverse square nature. Can it be only the fall in gravitational strength that is the cause? Or is it the relationship between forward and angular momentum which is redistributed according to a law of nature? We say this is the conservation of energy. Then energy IS momentum. Conservation of energy thus means conservation of momentum which again suggests quantization of momentum.

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Online jeffreyH

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Re: Lambert's Cosine Law
« Reply #52 on: 25/08/2014 23:15:17 »
Attached is a graph of the gravitomagnetic wave of the graviton. Both positive and negative energies are presented. Maxwell's concern about negative energies can be overcome if momentum and energy are considered equivalent. The positive and negative energies are then due to opposing spin in a two component particle. The positive and negative elements cause attraction with both positively and negatively charged particles. This double opposing spin also indicates that the graviton cannot be polarised, unlike the photon. This plot is slightly reminiscent of the amplituhedron. I say slightly as it cannot be said that they are equivalent in any way.

The positive portion of the wave exhibits the loop back inherent in the magnetic field. The positive portion has an altogether different profile.

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Online jeffreyH

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Re: Lambert's Cosine Law
« Reply #53 on: 27/08/2014 02:07:30 »
I have come to a conclusion driven by the mathematics that I didn't expect. I had assumed that for any black hole the point at which escape velocity reaches c would be at a greater radius than the point where g equals c. This was not the case. Perpendicular motion, if my math is correct, is overcome before the event horizon is reached. I do need to check this but if correct then any photon is doomed before it actually reaches the event horizon. I am still skeptical about this result as it means black holes should consume mass more aggressively than expected. This is not what happened to the G2 gas cloud.

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Online jeffreyH

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Re: Lambert's Cosine Law
« Reply #54 on: 06/09/2014 14:56:10 »
One point to note for anyone even remotely interested in reading this thread. I am breaking mathematical notation in ways that physicists would consider invalid. Don't try this at home because you can't determine why.

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Online jeffreyH

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Re: Lambert's Cosine Law
« Reply #55 on: 09/09/2014 22:49:30 »
The reason why there may be a region at a point outside the event horizon that seals a particles fate may be because magnetic field lines first fall beyond the event horizon making it a sink for the field. To maintain the circulation of the field the particle would be doomed to enter the horizon once the field line is trapped.

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Online jeffreyH

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Re: Lambert's Cosine Law
« Reply #56 on: 10/09/2014 00:54:45 »
For the first time I have been able to plot the profile of acceleration toward light speed taking gravitational self interaction into account. The plot is attached and shows the view from a frame external to the acceleration. This SOLVES the infinite mass problem due to the time dilation effects. The external observer would initially see the object accelerate and then appear to slow down and become fainter. This is similar to the effects when approaching an event horizon.

NOTE: This profile suggests an intensifying gravity well around the accelerating object.
« Last Edit: 10/09/2014 01:03:24 by jeffreyH »

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Re: Lambert's Cosine Law
« Reply #57 on: 10/09/2014 01:17:34 »
This could mean that during the early stages of the universe some black holes were formed during the inflationary period when velocities were much higher and not simply due to collapse. These would be the so called primordial black holes. This would also explain the apparent slow down after the inflationary period. The other strange thing when viewing the plot is that it suggests that things appear to be accelerating away precisely because we are slowing down and the time dilation and length contraction are now reversing.
« Last Edit: 10/09/2014 01:30:02 by jeffreyH »

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Re: Lambert's Cosine Law
« Reply #58 on: 11/09/2014 02:09:12 »
Counter to what I stated previously the universe has not actually left its inflationary phase. This is a consequence of gravitation. The gravitational field evolves away from the source into an expanding spacetime. The energy of the field occupies a compressed space near to the source. Gravitation therefore cannot itself be affected by dilation or contraction. The consequence has to be that its propagation is superluminal until it reaches infinity where it will equal the speed of the photon. A consequence of this is that the gravitational field can in fact radiate out of a black hole.
« Last Edit: 11/09/2014 02:10:49 by jeffreyH »

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Re: Lambert's Cosine Law
« Reply #59 on: 11/09/2014 02:13:49 »
Another consequence of this is that it must be mass itself that compresses the spacetime and not gravitation. The operation of gravity merely accumulates the mass in the first place. I can currently think of no way that gravitational force can be carried by a boson with such properties. It may be a consequence of the spin 2 nature but how I don't know.
« Last Edit: 11/09/2014 02:17:54 by jeffreyH »

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Re: Lambert's Cosine Law
« Reply #60 on: 11/09/2014 02:33:21 »
The only way a force carrier can appear to operate under these circumstances is by losing mass with distance from source. This is puzzling. However it does suggest that gravitational waves would only be detectable near to a strong source. This also implies that at infinity the particles ceases to exist. If there is a way of focusing gravitation using light then this could be a mechanism for detection.

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Re: Lambert's Cosine Law
« Reply #61 on: 11/09/2014 02:48:34 »
The loss of mass of the graviton could be the source of dark matter as more mass would be lost nearer the source. In galaxies with central massive black holes this loss of mass would explain the galactic halos. The loss to dark matter/energy at source would also prevent orbital decay due to massive bosons near to the source. This loss would allow the boson to maintain a constant speed free of time dilation.

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Re: Lambert's Cosine Law
« Reply #62 on: 11/09/2014 02:53:01 »
Thinking about the G2 gas cloud encounter brings up another possibility. When a star has lost enough gravitational energy and the amount of dark matter around the object reaches a critical point then collapse could be initiated by pressure from this excess of dark matter. Dense object could then radiate a lower strength gravitational field than currently thought. Since G2 did not exhibit the expected behavior this could be a feasible explanation. The pressure from dark matter/energy would aid in keeping the cloud intact.

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Re: Lambert's Cosine Law
« Reply #63 on: 14/09/2014 19:33:08 »
The graph of acceleration to light speed can also be applied to black holes. As the time tends to infinity the motion through space tends toward zero. At an infinite distance light will have a velocity of 1 Planck length in 1 Planck time. The knee occurs before half light speed and this point should correspond to a turning point in the tidal forces. These forces then increase exponentially. Calculating the point at which the escape velocity of a black hole equals this value will give the exterior zone of no return. Particles are still free to circulate around the black hole but only light can ever escape this region. This then confines the accretion disk.
« Last Edit: 14/09/2014 19:42:46 by jeffreyH »

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Re: Lambert's Cosine Law
« Reply #64 on: 14/09/2014 20:22:40 »
When using the graph to calculate the extension out from the event horizon we end up with the equation hc = Rs + 2*pi*Rs. Where hc is the radial distance to the boundary of the confinement zone. Therefore the extension of the radius out to the accretion confinement zone is equal to the circumference of a great circle of the sphere describing the hypothetical perfectly spherical event horizon of a black hole. This is derived directly from the knee of the curve in proportion to the fraction of light speed represented by that point. The fact that this is equivalent to 2 * pi * Rs appears to validate prior assumptions.

NOTE: This hypothesis should be easy to falsify through astronomical observation of dense objects and candidates black holes. This all depends upon being able to reliably determine the extent of the accretion zone.
« Last Edit: 14/09/2014 20:38:50 by jeffreyH »

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Re: Lambert's Cosine Law
« Reply #65 on: 14/09/2014 21:58:25 »
There is a second point of inflection on the graph nearer to the event horizon which should indicate the start of the ergosphere. This I have yet to calculate. When studying the curve there appears to be no indication of the position of the event horizon itself. This is simply the termination of the curve at infinite time.
« Last Edit: 14/09/2014 22:02:07 by jeffreyH »

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Re: Lambert's Cosine Law
« Reply #66 on: 14/09/2014 22:07:20 »
The portion of the curve following the start of the ergosphere describes a complex geometry. This indicates that the frame dragging induced in this region is within a complex space. The plot terminates at this point as it involves complex numbers.
« Last Edit: 14/09/2014 22:09:52 by jeffreyH »

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Re: Lambert's Cosine Law
« Reply #67 on: 17/09/2014 00:43:27 »
There is an error in my method which can be illustrated in the section "2. THE INNER DISK RADIUS" here: http://iopscience.iop.org/0004-637X/476/1/278/fulltext/35151.text.html

The 2*pi*Rs is incorrect. This calculation should instead refer to the spin of the object related to torque. This is due to the scales of distance and time on the graph. The value for confinement radius is therefore incorrect.

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Re: Lambert's Cosine Law
« Reply #68 on: 17/09/2014 23:38:56 »
The above graph has no torque component at all because it has no relation to angular momentum. There is however some connection to angular momentum in a relativistic manner which at present is undetermined. The axes on the graph should really have been labelled as time dilation over length contraction but I just couldn't be bothered to change them.
« Last Edit: 17/09/2014 23:42:48 by jeffreyH »

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Re: Lambert's Cosine Law
« Reply #69 on: 18/09/2014 01:06:52 »
A note on stellar collapse. Due to extremes of internal heat the mean density within a star will be lower than other celestial bodies such as planets. As the fuel is exhausted and this mass cools the density increases. This can then cause a vacuum in internal cavities. I have no idea at this point if this would be a valid reason for black hole formation or how it would work. Without a density profile for the interior of a star it is impossible to determine. A combination of internal gravitation within the cavities and the vacuum could be the initial cause of collapse.

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Offline alancalverd

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Re: Lambert's Cosine Law
« Reply #70 on: 18/09/2014 06:50:47 »
A note on stellar collapse. Due to extremes of internal heat the mean density within a star will be lower than other celestial bodies such as planets.
not true. Consider a neutron star

Quote
A typical neutron star has a mass between ~1.4 and about 2 solar masses with a surface temperature of ~6 x 105 Kelvin [3][4][5] (see Chandrasekhar limit).[6][a] Neutron stars have overall densities of 3.7◊1017 to 5.9◊1017 kg/m3 (2.6◊1014 to 4.1◊1014 times the density of the Sun), which is comparable to the approximate density of an atomic nucleus of 3◊1017 kg/m3.[7] The neutron star's density varies from below 1◊109 kg/m3 in the crust - increasing with depth - to above 6◊1017 or 8◊1017 kg/m3 deeper inside (denser than an atomic nucleus).[8] This density is approximately equivalent to the mass of a Boeing 747 compressed to the size of a small grain of sand. A normal-sized matchbox containing neutron star material would have a mass of approximately 5 billion tonnes or ~1 km≥ of Earth rock.
helping to stem the tide of ignorance

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Re: Lambert's Cosine Law
« Reply #71 on: 19/09/2014 18:08:47 »
A note on stellar collapse. Due to extremes of internal heat the mean density within a star will be lower than other celestial bodies such as planets.
not true. Consider a neutron star

Quote
A typical neutron star has a mass between ~1.4 and about 2 solar masses with a surface temperature of ~6 x 105 Kelvin [3][4][5] (see Chandrasekhar limit).[6][a] Neutron stars have overall densities of 3.7◊1017 to 5.9◊1017 kg/m3 (2.6◊1014 to 4.1◊1014 times the density of the Sun), which is comparable to the approximate density of an atomic nucleus of 3◊1017 kg/m3.[7] The neutron star's density varies from below 1◊109 kg/m3 in the crust - increasing with depth - to above 6◊1017 or 8◊1017 kg/m3 deeper inside (denser than an atomic nucleus).[8] This density is approximately equivalent to the mass of a Boeing 747 compressed to the size of a small grain of sand. A normal-sized matchbox containing neutron star material would have a mass of approximately 5 billion tonnes or ~1 km≥ of Earth rock.

Thanks for the info. It was a bit of cock-eyed speculation really.

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Re: Lambert's Cosine Law
« Reply #72 on: 20/09/2014 00:25:51 »
Related to the time dilation and length contraction plot is the spacetime density over momentum plot which is attached. As particles reach relativistic velocities the spacetime density increases exponentially.

NOTE: This is linear and not volumetric density. It can be centred at a point in spacetime. The implications of this are that every black hole has an equivalent density due to spacetime density. More massive black holes have the same density due to spacetime contraction as smaller black holes. It is this equivalent density at the event horizon that traps light. This indicates that the event horizon and the singularity are exactly the same thing.
« Last Edit: 20/09/2014 00:58:11 by jeffreyH »

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Re: Lambert's Cosine Law
« Reply #73 on: 20/09/2014 01:23:28 »
All this leads to the conclusion that recession of galaxies is related to a decreasing spacetime density as a natural consequence of the big bang.

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Re: Lambert's Cosine Law
« Reply #74 on: 21/09/2014 05:49:36 »
Trapped in a prism, in a prism of light
 Alone in the darkness, darkness of white
 We fell in love, alone on a stage
 In the reflective age

Entre la nuit, la nuit et l'aurore.
 Entre les royaumes, des vivants et des morts.
 If this is heaven
 I don't know what it's for
 If I can't find you there
 I don't care

I thought, I found a way to enter
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Now, the signals we send, are deflected again
 We're still connected, but are we even friends?
 We fell in love when I was nineteen
 And I was staring at a screen

Entre la nuit, la nuit et l'aurore.
 Entre les voyants, les vivants et les morts.
 If this is heaven
 I need something more
 Just a place to be alone
 'Cause you're my home

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Re: Lambert's Cosine Law
« Reply #75 on: 21/09/2014 05:51:36 »
I thought, I found a way to enter
 It's just a reflektor (It's just a reflektor)
 I thought, I found the connector
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 Will I see you on the other side? (Just a reflektor)
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 We all got things to hide (Just a reflektor)

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 Our song escapes, on little silver discs
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I thought, I found a way to enter
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 I thought, I found the connector
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It's just a reflektor

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 Will I see you on the other side? (reflektor)
 We all got things to hide (reflektor)
 Just a reflektor
 Will I see you on the other side?

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Online jeffreyH

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Re: Lambert's Cosine Law
« Reply #76 on: 21/09/2014 15:01:53 »
For light the geometry at the event horizon no longer has a recognizable 3 dimensions. The outward path normal to the horizon now equals zero and this dimension is inaccessible. This does not mean that we only have a two dimensional environment across the surface of the horizon as gravity will still act on the photon. This leads to the conclusion that beyond the horizon we have a negative 3 dimensional space. If the spacetime is flattened at the horizon this can be the only conclusion. This ultimately leads to a breakdown in the physics. To pass through the horizon light would have to be two dimensional with no depth to the energy. Internally this then translates to negative energy. This was one of the reasons that Maxwell could not proceed from electromagnetism on to gravity. However his consideration of negative energy was nothing to do with black holes.

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Re: Lambert's Cosine Law
« Reply #77 on: 28/09/2014 22:31:29 »
I am currently looking into this and whether it can relate at all to symmetry broken photons.

http://physics.aps.org/articles/v5/44

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Re: Lambert's Cosine Law
« Reply #78 on: 29/09/2014 00:50:20 »
If we consider gravity losing energy and mass over time and distance from source to dark matter and dark energy then at an infinite distance it will no longer exist as gravitation. The dark energy/matter can no longer operate as a repulsive force at infinity so must be a mainly localized phenomena and follow the same inverse square law as gravitation. The majority of this dark matter and energy would reside in the vicinity of the source of the gravitation. That is the galaxies that originated the gravitation. The gas cloud G2 would therefore be prevented from serious interaction with sag a* simply because of the higher concentration of dark matter/energy coincident with the accretion disk and surrounding environment.

NOTE: The amount of this mass loss is directly related to the spacetime density surrounding the mass generating the gravitational field so this would not even be noticeable in a terrestrial environment.
« Last Edit: 29/09/2014 00:58:41 by jeffreyH »

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Re: Lambert's Cosine Law
« Reply #79 on: 04/10/2014 16:36:36 »
I am now of the view that mass loss may not be the answer. Gravity probe B results show the vortex around the earth as predicted by Einstein. This runs counter to the direction of angular momentum as if, like an induced magnetic field, gravitation is the interaction of an external field with the moving mass. The Higg's field imparts some of the mass to a particle. This relationship may be part of the answer. Energy may still be lost by gravitation but not sufficient to explain dark matter and dark energy. The voids between the galaxies are always being affected by gravitation however infinitesimally small the effect in those regions. This could over time lower the general mass density which in turn lowers spacetime density. I am looking at the falling slinky effect in order to determine a method of describing the density change in spacetime.

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Re: Lambert's Cosine Law
« Reply #80 on: 05/10/2014 02:52:53 »
I now have an equation to determine the radius of the start of the accretion zone around dense objects, including black holes. The conclusions explain the failure of G2 to be consumed by sag a*. It also brings up the possibility that once a black hole forms nothing will ever fall into it. While this sounds like it must be wrong there are reasons for this conclusion. These involve non-violation of light speed and were calculated indirectly from the gravitational binding energy.

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Re: Lambert's Cosine Law
« Reply #81 on: 06/10/2014 22:41:29 »
Gravitational binding energy U is given by U = (3*G*M^2)/(5r). Where G is the gravitational constant, M is the mass and r is the radius to the surface. THe mass must be considered of uniform density. Gravitational acceleration g can then be determined by g = (5*U)/(3*M*r). If we extend the radial distance and consider a drop in density within our new surface boundary then U decreases proportionally. This appears to indicate that density is a critical factor in gravitational calculations. This density can only be expressed through a variation in G which makes it frame dependent. This relates G not only to mass density but to spacetime density.

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Re: Lambert's Cosine Law
« Reply #82 on: 07/10/2014 00:03:36 »
This all indicates that we are chasing a ghost by trying to find a constant value for G.

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Re: Lambert's Cosine Law
« Reply #83 on: 10/10/2014 00:25:26 »
I am now starting work on a universal time dilation gradient with its reference point as the event horizon of a black hole. I will be able to use this gradient in further equations and possibly then link gravitation directly to maxwell's equations.

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Re: Lambert's Cosine Law
« Reply #84 on: 10/10/2014 21:39:49 »
If we take GM as our starting point and find the fractional portion of the mass used in gravitational calculations we can then find the depth of distribution this would be around the surface. This then shows a relationship between the depth of this mass and the radius of the event horizon. The concentration of the mass then reaches a critical point at which the intensity of gravitation produces the energy required to overcome the escaping photons. This surface depth then relates directly to the radius of the horizon. It also shows just how weak the gravitational field is.

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Re: Lambert's Cosine Law
« Reply #85 on: 10/10/2014 21:50:29 »
It could be that most of the gravitational energy in a mass is confined and only a small portion of the flux leaves the mass at its surface. This cannot be determined without an investigation of this surface depth at increasing densities in relation to gravitation forces. If this depth equals the radius at the horizon this would answer some questions. It may also confirm the frozen star hypothesis.

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Re: Lambert's Cosine Law
« Reply #86 on: 10/10/2014 23:43:40 »
The speed of light can be determined by the equation 1/(G*50). This equation can be rearranged to find the exact value for G in the local frame. The factor of 50 relates to the fraction of mass across the surface found by using GM.

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Re: Lambert's Cosine Law
« Reply #87 on: 11/10/2014 00:00:24 »
Therefore 1/50c = G.

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Re: Lambert's Cosine Law
« Reply #88 on: 11/10/2014 10:24:58 »
A while ago whilst investigating the 2GM in Shwarszcchild's calculation I came across the factor of 0.02 which I knew was important but didn't know why. Well 0.02 = 1/50 and now I know why.  [8D]

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Re: Lambert's Cosine Law
« Reply #89 on: 11/10/2014 16:05:54 »
Our G value is in seconds per meter so if we take the reciprocal 1/G this gives 50*c which may well be the speed of gravity. The speed is then 14,983,877.348 km/s. This would be why gravity cannot be detected easily. You would need to take 50 equally spaced measurements and look for a repeating pattern of vibrations that could not be explained any other way.
« Last Edit: 11/10/2014 16:15:05 by jeffreyH »

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Re: Lambert's Cosine Law
« Reply #90 on: 11/10/2014 17:08:06 »
The electromagnetic spectrum could them be linked to gravitation by interference of the wavelength of the graviton with that of the photon. That is if the graviton exists.

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Re: Lambert's Cosine Law
« Reply #91 on: 13/10/2014 00:06:47 »
Just for good measure I have attached a plot of a composite wave interaction with a standing wave.

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Re: Lambert's Cosine Law
« Reply #92 on: 13/10/2014 21:33:16 »
Attached is a model representing the removal of the gravitational component from a compressed mass. The length contraction decreases to zero through the progression. The gravitational constant has been replaced by the 1/50 factor. This plot starts with a mass compressed within its event horizon radius. The length contraction dies off gradually until during the last 20% the rate of change increases dramatically.
« Last Edit: 13/10/2014 21:35:55 by jeffreyH »

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Re: Lambert's Cosine Law
« Reply #93 on: 13/10/2014 22:18:49 »
The equation for our starting point is [tex]\frac{M}{50c^3}[/tex] whose units are cubic seconds per cubic meter. In other words 3 dimensional time in 3 dimensional space. This is a much better solution than using G.

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Re: Lambert's Cosine Law
« Reply #94 on: 14/10/2014 23:49:28 »
Attached are two plots. One calculating g using the gravitational constant and the other using a factor of 1/50c. The question is this; is the factor of 50 actually a constant. We view gravity from an isolated bubble and have limited data on differing strength gravitational fields. This is one thing I am looking into. The other is gravitational interaction at the particle level.

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Re: Lambert's Cosine Law
« Reply #95 on: 15/10/2014 01:33:09 »
Although the results match there is a real problem in using c in the denominator. Instead this should be L which then represents simply the distance traveled by light in 1 second. G is used as a unit conversion and can be calculated approximately by using hbar/(lP^2C^3). However we need a unit conversion that instead of producing G will produce the unit value of 1. So hbar/(lP^2C^3) requires such a factor before being added to our equation. The reciprocal of this equation will give us this value as a constant. Then it needs to be determined what this constant means.
« Last Edit: 15/10/2014 01:59:00 by jeffreyH »

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Re: Lambert's Cosine Law
« Reply #96 on: 15/10/2014 02:21:04 »
So fingers crossed our equation for g should be (hbar*M)/(lP^2*c^2*L*r^2). Of course the Planck length would need to be recalculated using 1/50c in place of G to get an adjusted value but ignoring units.

Note: Changed g to G related to the 1/50c factor.
« Last Edit: 16/10/2014 02:16:56 by jeffreyH »

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Re: Lambert's Cosine Law
« Reply #97 on: 16/10/2014 02:20:24 »
I have not checked the results for the above equation for consistency yet. The point of interest is that the factor of 50 has disappeared. This suggests that there can be no energy loss in gravitation. So no link with dark energy/matter. Dark energy is entirely separate from gravitation.

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Re: Lambert's Cosine Law
« Reply #98 on: 16/10/2014 03:52:13 »
Well now like Joyce's Finnegan's wake we go back to the start with:

-[tex]\frac{Pl}{GPm} \frac{\partial ^2 E}{\partial t ^2}[/tex]
-[tex]\frac{Pl}{GPm} \frac{\partial ^2 H}{\partial t ^2}[/tex]

What about replacing G with 1/50c in this equation? The other point of note here is that we have 3 constants in the left hand term. That is G, Pl and Pm. It was shown that the factor of 50 can cancel out but in this case it would still be present, I think. This is the next project. To incorporate gravitation into the electromagnetic wave equation. In this way a formula for the affect on the wave over time by gravitation can be developed. Hopefully!

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Re: Lambert's Cosine Law
« Reply #99 on: 16/10/2014 03:53:15 »
Then it will be back to the standing wave interaction.  [:0]