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Author Topic: Lambert's Cosine Law  (Read 1693 times)

jeffreyH

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Re: Lambert's Cosine Law
« Reply #50 on: 24/08/2014 15:13:51 »
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 14e556ac118387864e8a34fbb102e360.gif. The right hand term can be rewritten as e5bfed90d20d5182b5047487241e4b3a.gif 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. e5bfed90d20d5182b5047487241e4b3a.gif should be 957559fc6242b50648307d3958ea52c7.gif.
« Last Edit: 25/08/2014 18:57:25 by jeffreyH »

jeffreyH

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Re: Lambert's Cosine Law
« Reply #51 on: 25/08/2014 19:08:31 »
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.

jeffreyH

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Re: Lambert's Cosine Law
« Reply #52 on: 25/08/2014 22:15:17 »
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.

jeffreyH

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Re: Lambert's Cosine Law
« Reply #53 on: 27/08/2014 01:07:30 »
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.

jeffreyH

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Re: Lambert's Cosine Law
« Reply #54 on: 06/09/2014 13:56:10 »
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.

jeffreyH

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Re: Lambert's Cosine Law
« Reply #55 on: 09/09/2014 21:49:30 »
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.

jeffreyH

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Re: Lambert's Cosine Law
« Reply #56 on: 09/09/2014 23:54:45 »
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 00:03:24 by jeffreyH »

jeffreyH

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Re: Lambert's Cosine Law
« Reply #57 on: 10/09/2014 00:17:34 »
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 00:30:02 by jeffreyH »

jeffreyH

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Re: Lambert's Cosine Law
« Reply #58 on: 11/09/2014 01:09:12 »
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 01:10:49 by jeffreyH »

jeffreyH

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Re: Lambert's Cosine Law
« Reply #59 on: 11/09/2014 01:13:49 »
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 01:17:54 by jeffreyH »

jeffreyH

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Re: Lambert's Cosine Law
« Reply #60 on: 11/09/2014 01:33:21 »
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.

jeffreyH

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Re: Lambert's Cosine Law
« Reply #61 on: 11/09/2014 01:48:34 »
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.

jeffreyH

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Re: Lambert's Cosine Law
« Reply #62 on: 11/09/2014 01:53:01 »
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.

jeffreyH

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Re: Lambert's Cosine Law
« Reply #63 on: 14/09/2014 18:33:08 »
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 18:42:46 by jeffreyH »

jeffreyH

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Re: Lambert's Cosine Law
« Reply #64 on: 14/09/2014 19:22:40 »
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 19:38:50 by jeffreyH »

jeffreyH

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Re: Lambert's Cosine Law
« Reply #65 on: 14/09/2014 20:58:25 »
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 21:02:07 by jeffreyH »

jeffreyH

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Re: Lambert's Cosine Law
« Reply #66 on: 14/09/2014 21:07:20 »
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 21:09:52 by jeffreyH »

jeffreyH

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Re: Lambert's Cosine Law
« Reply #67 on: 16/09/2014 23:43:27 »
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.

jeffreyH

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Re: Lambert's Cosine Law
« Reply #68 on: 17/09/2014 22:38:56 »
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 22:42:48 by jeffreyH »

jeffreyH

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Re: Lambert's Cosine Law
« Reply #69 on: 18/09/2014 00:06:52 »
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.

alancalverd

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Re: Lambert's Cosine Law
« Reply #70 on: 18/09/2014 05:50:47 »
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 3.71017 to 5.91017 kg/m3 (2.61014 to 4.11014 times the density of the Sun), which is comparable to the approximate density of an atomic nucleus of 31017 kg/m3.[7] The neutron star's density varies from below 1109 kg/m3 in the crust - increasing with depth - to above 61017 or 81017 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.

 

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