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### Author Topic: Lambert's Cosine Law  (Read 51563 times)

#### jeffreyH

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##### Re: Lambert's Cosine Law
« 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.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« 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.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« 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.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« 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.
« Last Edit: 25/01/2015 19:13:36 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« 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?

#### jeffreyH

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##### Re: Lambert's Cosine Law
« 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.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« 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

#### jeffreyH

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##### Re: Lambert's Cosine Law
« 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.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« 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.
« Last Edit: 12/02/2015 20:11:36 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« 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.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #210 on: 04/02/2015 02:39:13 »
If the oscillation is directed mostly in the direction of the plane this is equivalent to the freedom of movement perpendicular to the direction of a gravitational field. The constraint in this case is created by field cancellation and not due to proximity to the surface of a gravitational mass. This then becomes a case of special interest.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #211 on: 05/02/2015 00:22:51 »
Before proceeding with an investigation of gravity's effect on the wave it would be useful to examine the attached graph. This shows the change in wavelength over speed for the electron. The values have not been rigorously checked but this gives a good starting point. Relativistic mass has been taken into account but no gamma factor has been applied. The wavelength shortens with velocity and therefore the frequency/energy increases exponentially indicating the increase in mass.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #212 on: 05/02/2015 01:31:35 »
If we were to include not only the gamma function, but also the derived mass equation with an inherent value for g, then as the relativistic mass increases the internal gravitation will also increase and have an effect on the internal functioning of the particle. The rate of change of energy flux will be effectively slowed down within the particle radius.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #213 on: 08/02/2015 23:33:54 »
If we return to our two perfect sphere model and consider a particle on the plane of our flat spacetime then we can show something interesting. No matter what starting position we take away from equilibrium there is a second plane that is perpendicular to the flat spacetime plane that runs through the centres of gravity of both masses. On this plane lie the vectors from the centres of gravity to the starting position that indicate the strength of the gravitational acceleration of each mass. If we label these g1 and g2 respectively then we can say that g1 = g2 as all forces are equivalent at every point in the flat spacetime. We can use this perpendicular plane mathematically to determine the vector magnitude of the combined forces at the starting point. If we consider this perpendicular plane to be in the z direction then we can consider the motion of the particle to be moving on this plane as well as that of the flatspacetime plane. Knowing this we can ignore the flat plane to simplify the mathematics, the x and y values being zero. This can now be considered a two dimensional problem in the z plane.

This may also simplify the integration of the wave function.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #214 on: 08/02/2015 23:39:00 »
It is said that the spin of a particle can neither be speeded up or slowed down. This is like saying that light always travels at c. Yet we know that light slows down in a gravitational field due to time dilation. Taking this into consideration we can say that spin must also be modified by gravitation otherwise dilated observers would see an increase in particle spin. This is how we must view the wave function when it is affected by gravitation.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #215 on: 11/02/2015 23:28:01 »
As well as the path through the flat spacetime we can consider another path. This is described by the equation v = SQRT(Gm/r) and would normally represent the velocity required to maintain a circular orbit at a particular altitude. In the scenario with two masses this is not the case simply because of the interactions of the fields and how they will affect each other. This is of particular interest in relation to the behavior of the wave function and can be investigated later.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #216 on: 12/02/2015 19:47:12 »
In cases where we have well defined constraints, such as a circular orbital around a single perfect spherical mass or a particle traveling on a plane in flat spacetime, we can work with virtual displacements. The method can be studied here:

http://en.wikipedia.org/wiki/Virtual_displacement

Time is removed from the equation due to the constraints involved, so long as there is no violation of those constraints. It is not so straight forward in the case of the flat spacetime plane sitting between the identical masses as all points on the plane are not equivalent. However the case of the single circular orbit is a good starting point in the use of virtual displacements and a way to derive equations of displacements in time for circular orbits around one or both of our two masses perpendicular to the flat plane and passing through the equilibrium point.

EDIT: This is useful because there will be no change in the wave function in this type of balanced system. When a modification introduces change we can then describe the effects on the wave.
« Last Edit: 12/02/2015 19:49:44 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #217 on: 12/02/2015 23:58:59 »
The coulomb force constant 1/ is similar to the factor in the original derivation of the Maxwell equation in the early posts of this thread. In that derivation 1/ was the value used. If we want the electric field incorporated in our mass we must derive a new equation.

E = kQ/r^2

g = Gm/r^2

Here k is the Coulomb force constant and Q is the charge. So we now have electric and gravitational equation. What can we do with them. If we consider the mass to charge ratio m/Q and the charge to mass ration Q/m we can now derive a new equation.

E/Q = k/r^2

g/m = G/r^2

These can be further re-arranged:

1/Q = k/(Er^2)

1/m = G/(gr^2)

If we want charge to mass this then becomes:

Q = (Er^2)/k

Giving:

Q/m = [(Er^2)/k] / [G/(gr^2)]

Q/m = [(Er^2)/k] * [(gr^2)/G]

If we now want a mass equation we first re-arrange as:

m/Q = k/(Er^2) * G/(gr^2)

And finally our mass equation is:

m = kQ/(Er^2) * G/(gr^2)

Now we have two components, the product of which gives our mass. The Electric field and its charge and the gravitational field and the acceleration at the surface. We now have united the electric and gravitational fields within the mass equation from which we can derive the magnetic component. With a means to derive the electromagnetic field we can now unite this with quantum mechanics. My knowledge of quantum mechanics is currently sorely lacking so I will be investigating simple effects on the wave equation until I can fill in the gaps in my knowledge.

EDIT: I made a correction to the value for k
« Last Edit: 13/02/2015 00:20:55 by jeffreyH »

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #218 on: 13/02/2015 00:11:21 »
Now this shows exactly why the wavelength affects energy and therefore mass, the electric and gravitational fields being intrinsic to the wave equation.

#### PmbPhy

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##### Re: Lambert's Cosine Law
« Reply #219 on: 13/02/2015 04:28:34 »
In cases where we have well defined constraints, such as a circular orbital around a single perfect spherical mass or a particle traveling on a plane in flat spacetime, we can work with virtual displacements. The method can be studied here:

http://en.wikipedia.org/wiki/Virtual_displacement
I don't understand why you're talking about virtual displacements here. The only place I know of where they're used is in analytical mechanics, i.e. Lagrangian mechanics.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #220 on: 13/02/2015 17:01:35 »
In cases where we have well defined constraints, such as a circular orbital around a single perfect spherical mass or a particle traveling on a plane in flat spacetime, we can work with virtual displacements. The method can be studied here:

http://en.wikipedia.org/wiki/Virtual_displacement
I don't understand why you're talking about virtual displacements here. The only place I know of where they're used is in analytical mechanics, i.e. Lagrangian mechanics.

All will become as clear as mud eventually. I am just trying something out. If it fails I will come back and eat humble pie m'lud.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #221 on: 13/02/2015 20:43:21 »
Particles have spin. Not strictly the same as angular momentum but they are considered to be in motion internally of an undetermined kind. (Physics has models for this but no definitive answer.) If we consider the two components of the electric and gravitational fields in the above equation with respect to this internal motion what does it tell us? Are the fields also in motion and if not why not? If they move are they oriented in the same or an opposing way? If they are static is this because of opposing forces? The strengths of these two fields differ enormously. It is hard to reconcile a static nature due to opposition when they are unequal in strength. I do not intend to provide any definitive answers to these questions, it would likely be impossible. However, these kinds of consideration need to be kept in mind.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #222 on: 13/02/2015 22:12:29 »
We can re re-examine the equation v = SQRT(Gm/r) for a circular orbit around a single perfect sphere containing a uniform mass density. If instead of velocity we start in a circular orbit and then start applying an infinitesimal acceleration our orbit must expand with this increase in kinetic energy. Mush like an increase in energy of the electron moves it into a higher orbital. If we continue increasing the acceleration we will describe a spiral trajectory away from the central mass. This can be said to be similar to the vortex described by Einstein. This first infinitesimal increase in velocity, while different to a virtual displacement, can be useful in a number of ways. As the electron moves further out from the nucleus its wavelength changes. This will be the same situation with our orbit around our single mass. At any point we can determine how the wave equation is affected. We require less effort in this regard than in escaping in a straight line path as we start with all points in the orbit perpendicular to the gravitational field making it easier to move. Therefore any exchanges of energy will be trivial.

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #223 on: 13/02/2015 22:45:48 »
Now here we have a conundrum. We are increasing velocity and yet to maintain a higher orbit we need a decrease in velocity due to the inverse square nature of the gravitational force. Otherwise we continue moving away from the source. In order to rest in a higher orbit we will need to manage a controlled and precise deceleration. This throws up some issues about the electron as the field around a particle SHOULD also be inverse square. So what puts the breaks on the electron?

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #224 on: 13/02/2015 22:53:10 »
Well the electron must undergo a very short range and short lived acceleration of just the right amount of energy to shift to the correct orbital. This has to be precise and so does not involve uncertainty although this in no way invalidates the uncertainty principle. The position and momentum can still not be determine as it is a scalar determinacy.

EDIT: This scenario is at the heart of the issue of determining change in the wave equation.
« Last Edit: 13/02/2015 22:55:15 by jeffreyH »

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##### Re: Lambert's Cosine Law
« Reply #224 on: 13/02/2015 22:53:10 »