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Lambert's Cosine Law
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Lambert's Cosine Law
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jeffreyH
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Re: Lambert's Cosine Law
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Reply #195 on:
13/01/2015 22:51:46 »
The reciprocal of momentum 1/p = λ/h. This may relate
somehow??
to phonons or even the reciprocal lattice.
See crystal momentum:
http://en.wikipedia.org/wiki/Phonon
http://en.wikipedia.org/wiki/Reciprocal_lattice
"Simple cubic lattice[edit]
The simple cubic Bravais lattice, with cubic primitive cell of side a, has for its reciprocal a simple cubic lattice with a cubic primitive cell of side 2pi/a (1/a in the crystallographer's definition). The cubic lattice is therefore said to be self-dual, having the same symmetry in reciprocal space as in real space."
How this could be incorporated is beyond me currently. It is simply noted here.
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Re: Lambert's Cosine Law
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Reply #196 on:
15/01/2015 21:16:26 »
We have two interesting components to the unit circle equation. Ke = 1/2mv^2 and (λ/h)^2 = 1/p. Since mv = p we can define Ke as 1/2v(mv). That is 1/2v(p). By combining both we arrive at 1/2v(p)1/p which becomes 1/2v.
So we can now reformulate the equation as 2pi = 2*1/2v*g*r^2*h*(1/c^2)*[1/lP^2*(gamma*t)/L] from which we then get 2pi = v*g*r^2*h*(1/c^2)*[1/lP^2*(gamma*t)/L]. So we now have the particle velocity included in our rearranged formula. This eliminates the kinetic energy altogether. We now have velocity over a time dilated interval. Now 4 components can change and are all related directly. These are the velocity of the particle, the change in time over which the velocity occurs (time dilation as viewed externally), the radius and associated g force at the surface defined by the radius. Do these need to balance to maintain a relationship with the 2pi? That I have not determined yet. However the equation is becoming simpler. It does suggest a tie in with black hole entropy with the 1/lP^2 factor.
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Re: Lambert's Cosine Law
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Reply #197 on:
15/01/2015 23:30:44 »
If we look at the following sequence rearranging for v we can see that velocity is also contained within the gamma factor.
2pi = v*g*r^2*h*(1/c^2)*[1/lP^2*(gamma*t)/L]
2pi/v = g*r^2*h*(1/c^2)*[1/lP^2*(gamma*t)/L]
v/2pi = 1/g*1/r^2*1/h*c^2*[lP^2*L/gamma*t]
v(apparent) = 2pi*([L^3]/[gamma*t^3*g*h]*[lP^2]/[r^2])
Here v(apparent) is not necessarily the same as v in gamma. The values for earth of g and r can be plugged in to see what results are obtained at various velocities inside the gamma factor. This is the next step.
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Re: Lambert's Cosine Law
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Reply #198 on:
16/01/2015 19:37:51 »
Well I think that I can safely say that the above equation is rubbish.
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Re: Lambert's Cosine Law
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Reply #199 on:
22/01/2015 20:40:26 »
Having scrapped the rubbish in post 197 I can now investigate whether or not the earlier wave equation is valid. Firstly I want to point out a conclusion I have reached. If we have two perfect spheres identical in every respect; same radius, surface area, volume and mass, then on a direct line between the two centre's of gravity the force can be said to cancel as it is equivalent and opposite in direction. From this midpoint on the line between the centres we can describe a plane perpendicular to the line on which gravity will cancel at any point as the forces betwwen the masses will cancel. However the vectors will mean that any particle not exactly positioned on the adjoining line will tend to describe a straight line path along this plane until it reaches equilibrium again at the point where the plane meets the line. This is equivalent to an object dropped down a shaft through one perfect sphere that goes from one surface, through the centre of gravity and reverses direction at the opposite surface. Both of these scenarios can be said to be operating in a flat spacetime as long as the only forces present are the masses concerned.
The two sphere scenario raises another point. If we consider the sources to be two black hole an infinitesimal distance apart at the event horizons with enough distance to hold a particle between them without touching either horizon then because the gravitational force operates equivalently and in opposite directions there should be extreme length contraction which also implies extreme time dilation. So in the situation at the centre of the earth we should have a maxima of time dilation for the size of mass of the earth. This also indicates a compression of a central particle in all directions indicating that this is a proof that black holes do in fact form at the centre of a mass and work outwards.
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Last Edit: 22/01/2015 20:43:25 by jeffreyH
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Re: Lambert's Cosine Law
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Reply #200 on:
22/01/2015 23:26:03 »
If we increase the distance between the two black holes and consider the central mass to be emitting a synchronized sphere of photons then it should be possible to calculate the deformation of the photon sphere as it moves outwards. The line connecting the two centres of gravity will form a triangle along the line of the perpendicular plane and this would be an interesting situation to study as the spacetime will experience no curvature. The angle at the apex will form a special relationship between the strength of the gravitational field at a particular point and the effect upon the wavelength and frequency of the photons moving in those directions. The change in the waveform can then be calculated for other directions were the gradient of curvature increases.
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Re: Lambert's Cosine Law
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Reply #201 on:
22/01/2015 23:44:37 »
A note on the two flat spacetimes. The shaft through the sphere is one dimansional being a line through the sphere. Movement away from a straight line path will drift into a curvature in the fabric of spacetime. In the case of the two perfect spheres we have a two dimensional flat spacetime. What is of interest and likely not possible is if we can determine a flat spacetime that is 3 dimensional. If such a spacetime can be determined then we will have either gravity shielding or anti-gravity. Like I just said I don't believe this is possible. We need a third derivative of spacetime that is not at the centre of a mass. This will be a point in spacetime and therefore zero dimensional unless there is some other way of achieving it.
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Re: Lambert's Cosine Law
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Reply #202 on:
24/01/2015 23:24:08 »
In the model with the flat plane between two equivalent perfect spheres we can say that the centre of gravity of the whole system lies outside of either mass. The equilibrium point coincides with the point on the plane that is positioned on the line between the two individual centres of gravity of the perfect spheres. All particles coincident with the plane and initially stationary at points away from equilibrium will be drawn towards this equilibrium point by the combined gravitational forces and their vector directions.
We can then define situations in which particles that are not stationary may be drawn into orbits around the equilibrium point. This flat spacetime is a unique situation to model and removes the complexity of dealing with curvature of the geometry. This is an ideal model with which to examine the change in the wave function due to the influence of gravitation.
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Re: Lambert's Cosine Law
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Reply #203 on:
25/01/2015 19:10:31 »
We can introduce something akin to the uncertainty principle into our two perfect sphere model. If we start with an orbit perpendicular to the plane between the masses that passes directly through the equilibrium point this will be our point of uncertainty. At the point of equilibrium in a perfect orbital path there are now two paths the orbit can take. It can either continue around the original mass or go into a figure of eight orbit around the second mass. As all forces are equal at this point there is a degree of uncertainty here. This is in effect a quantum state and binary in nature.
NOTE: An intriguing third option is that the particle continues on the plane away from both masses on its flat spacetime. This now becomes a three choice situation.
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Last Edit: 25/01/2015 19:13:36 by jeffreyH
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Re: Lambert's Cosine Law
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Reply #204 on:
25/01/2015 19:33:18 »
If we consider the third choice for the orbital path, then any particle following this type of path will act in a similar manner to a jet expelled from the pole of a black hole. The difference is that instead of a directed jet we get a distribution along a plat plane. Does this have anything in common with the relativistic jet?
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Re: Lambert's Cosine Law
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Reply #205 on:
28/01/2015 02:22:40 »
Since a particle that is initially at rest on the plane will tend to move towards equilibrium in a straight line path then we can set two out of 3 dimensions to have zero rate of change. This then produces a scalar value for the gravitational force and is analogous to the path of the particle falling down the shaft through a perfect sphere running from one surface to the opposite surface and passing through the centre of gravity. This one dimensional path still has a direction along the plane and a magnitude. The lack of curvature in the spacetime simplifies the change in wavelength of the particle. The slight complication arises due to the vector directions of the forces of the masses above and below the plane. Since these are equal we can sum them since we already have the direction of the vector of the particle.
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Re: Lambert's Cosine Law
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Reply #206 on:
28/01/2015 02:33:53 »
One reason why it is important to determine how waves may be affected by gravitation is linked to the Penrose Interpretation which is described here:
http://en.wikipedia.org/wiki/Penrose_interpretation
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Re: Lambert's Cosine Law
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Reply #207 on:
30/01/2015 00:32:57 »
A little speculation now. As the density of the gravitational field increases it is my assertion that not only does the remote observer see an object slowing down but it actually does. This is because the increase in density acts against the acceleration due to gravitation. Once inside the ergosphere this density is likely to also trap light. This then prevents acceleration from violating the speed of light as objects approach a black hole. This will also mean that objects disappear upon entering the ergosphere. A similar situation will occur when approaching light speed since unlike the photon tardyons have non zero rest mass that increases relativistically. Also the speed at which the particles will be traveling, being relativistic, will mimic an increase in density of the gravitational fileds of distant objects in the particles vicinity. This will become more pronounced when in the vicinity of a massive object. The attached image, which was posted previously shows the results of earlier calculations of this. At that time I had put this aside but now feel more confident in this assertion.
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Re: Lambert's Cosine Law
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Reply #208 on:
03/02/2015 00:41:50 »
Returning to the flat planar spacetime we can consider the effects of a simple harmonic oscillation about the equilibrium point as a start in determining the change in the wave function. The problem is in finding how this oscillation behaves in such circumstances.
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Last Edit: 12/02/2015 20:11:36 by jeffreyH
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Re: Lambert's Cosine Law
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Reply #209 on:
03/02/2015 21:30:50 »
Two possibilities can be considered for the effect upon the oscillation. It can either be elongated in the direction of both sources or it can be contracted by the effect of the sources. Since length contraction is assumed in a gravitational field it may be best to start with the assumption of a contraction or flattening of the oscillation. This will leave the particle flattened along the plane of the flat spacetime. In which case the effect of the resulting pressure may reduce the energy flux and be the cause of any time dilation. This is worthy of further investigation and may be a fruitful way to proceed in describing wave evolution away from equilibrium.
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