# Lambert's Cosine Law

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#### 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.
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#### 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.
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#### 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.
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#### 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 »
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#### 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?
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#### 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.
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#### 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
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#### 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.
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#### 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 »
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#### 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.
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#### 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.
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#### 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.
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#### 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.
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#### 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.
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#### 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.
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#### 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.
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#### 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 »
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#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #217 on: 12/02/2015 23:58:59 »
The coulomb force constant 1/$$4\pi\epsilon0$$ is similar to the factor in the original derivation of the Maxwell equation in the early posts of this thread. In that derivation 1/$$\epsilon0\mu0$$ 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 »
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#### 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.
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#### 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.
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#### 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.
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#### 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.
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#### 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?
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#### 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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#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #225 on: 13/02/2015 23:29:27 »
If we rewrite v = SQRT(Gm/r) as s/t = SQRT(Gm/r) then we can eliminate the square root (s/t)^2 = Gm/r. This can be restated as (s/t)(s/t) = Gm/r. If we move one of our distances (s) to the other side it becomes (1/t)(s/t) = Gm/r * 1/s. The value of 1/s then relates to the tangent to the orbit that moves us through an angle to a higher radial position. So we have s/t^2 = Gm/r * 1/s. By multiplying both sides by 1/t we then have an equation for jerk s/t^3 = Gm/r * 1/s * 1/t. Of course we really need to be looking at ds, d^3t and dt in these in this case.
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#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #226 on: 14/02/2015 00:07:28 »
Of course if we did not want to maintain our perfectly circular orbit we could simply remove the accelerating force and allow an elliptical orbit to develop. This leads to the thought that maybe electron orbitals have a lot in common with elliptical orbits.
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#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #227 on: 14/02/2015 01:23:20 »
If we consider we have an acceleration (a) and a jerk (j) our final equation forms become:

a = Gm/dr * 1/ds

j = Gm/dr * 1/ds * 1/dt

With our final velocity after deceleration :

v = SQRT(Gm/r)

We also have an adjusted mass equation:

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

This differs from what is usually understood as mass as the electric field and charge components have been added in. So this is a composite mass equation. An investigation of substituting this into the velocity equation would be interesting.
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#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #228 on: 14/02/2015 02:01:52 »
If we consider a large central mass with a smaller mass in orbit and both perfect spheres with a unifomr mass density we can say something about our velocity equation. Perpendicular to the field of the larger central mass we have a gradient of velocities that describe concentric circular orbit. Throughout the volume of the smaller mass these orbital profiles have differing effects from the nearest to the furthest away. At the point in the smaller mass that is nearest to the central body the velocity is at its peak speed and dies away at the surface furthest away. If the centre of gravity of this smaller mass passes exactly along the orbit at the correct velocity then those portions of the mass further away are traveling at a speed that would take a smaller mass out of the orbit. At the nearest point of the smaller mass to the larger body the speed is lower than needed to maintain an orbit at that altidude and a smaller mass at that point would tend to fall out of orbit towards the larger body. Therefore an induced angular momentum of rotation could be induced by such an imbalance. The point nearest the larger mass would be retarded in motion whilst the furthest point will advance in its motion. This is without taking into account any time dilation. Even in elliptical orbits this would be true.
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#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #229 on: 14/02/2015 02:26:47 »
It is interesting to note that there are already equilibrium points known as the Lagrange points. At these points in the 3 body situation objects will maintain an orbit and even a small deviation from equilibrium will be corrected to bring a mass back to equilibrium.

Details can be found here:

http://hyperphysics.phy-astr.gsu.edu/hbase/mechanics/lagpt.html

EDIT: Equivalent points should be present throughout the universe due to the interaction of the gravitational fields of galaxies.
« Last Edit: 14/02/2015 02:30:23 by jeffreyH »
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#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #230 on: 14/02/2015 22:25:21 »
In moving away from a stable circular obit we need an acceleration. However, acceleration tells us nothing about the mass that is moving. It tells us nothing about the force that needs to be applied to accelerate the mass. Different sized masses can not be moved away from equilibrium at exactly the same rate of acceleration by applying the same force. Different forces are needed. This says something about the propagation of force through mass. When considering free falling objects of differing masses in the same gravitational field this is not the case. It is always the same force. This indicates an action at the particle level right through a mass and acting simultaneously.
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#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #231 on: 17/02/2015 01:00:15 »
New thought.

x = [kQ/(Er^2) * G/(gr^2)] - [kQ/(Er^2) + G/(gr^2)]

Don't know where this leads yet. I need to look at some particle data. Not sure what x will actually mean.

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#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #232 on: 18/02/2015 01:26:19 »
The two fundamental equations for the de Broglie wave are:

$$\lambda$$= h/SQRT(2*Ke*m)

$$\upsilon$$= Ke/h

In both equations the energy is kinetic represented here by Ke. This is energy added to the system. Kinetic energy itself has the formula Ke = (1/2)mv^2. We saw earlier that momentum p = mv and that $$\lambda$$= h/p. The momentum is the motion of mass though space and the kinetic energy is a result of this motion. If we consider time dilation and its effects upon motion over time we should be able to derive equations that describe the changes in wavelength, frequency and kinetic energy in any gravitational field. What we also need is the effect on the particles electromagnetic field.

As I continue this thread I will be introducing relativistic mass.
« Last Edit: 18/02/2015 02:12:33 by jeffreyH »
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#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #233 on: 19/02/2015 01:19:10 »
A short digression before I progress this thread. Hawking radiation is due to pairs of particles, a particle and it's anti particle. One of these falls through the horizon whilst the other escapes. In the case of an anti particle falling through the horizon it should, if possible, be able to annihilate with a particle in the interior. If the particle falls in, however, then the universe loses mass. The particle will simply become part of the singularity. This may not be true, I don't know. However, whether a black hole grows or shrinks has profound consequences. I will be following up on this next.
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#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #234 on: 19/02/2015 01:30:35 »
When plotting change of wavelength of the photon that escapes just outside a black hole horizon there may be a linear relationship between frequency and the speed of the photon as viewed from infinity. The graph of this situation is attached.

With the frequency held constant and the speed of the photon plotted as viewed from infinity we have a linear plot. If we were to add a gamma factor, which I believe is incorrect, the photon will lose all its energy at infinity.

This plot brings to mind the Hubble data. If we consider that the black holes at the centres of galaxies are constantly changing, either shrinking or growing, then the shift that they cause in photon wavelength may also be changing in as yet undetected way. This may be regular across all such black holes and may follow a law of its own. In which case the expansion of the universe may be biased away from its true rate.

It all depends upon how these black holes behave. The black hole sag A* may be able to help answer such questions. In the end the red shift may be only very marginally affected by such black holes. Only time will tell.
« Last Edit: 19/02/2015 01:33:10 by jeffreyH »
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#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #235 on: 19/02/2015 01:38:15 »
For comparison I have attached the plot with the gamma factor.
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#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #236 on: 21/02/2015 01:15:50 »
If we re-examine the factors kQ/(Er^2) + G/(gr^2) these can be re-arranged as:

kQg/(Egr^2) + GE/(Egr^2)

This is an interesting set of relationships. The charge is paired with gravitational acceleration and the Electric field is related to the gravitational constant. Since the charge is involved in the attraction and repulsion forces and g is a consequence of the attractive force of gravitation this is worth further investigation. It is also of note that the denominator has the electric field and gravitational acceleration as factors. The units still need to be sorted for validity and this is what shall be undertaken next. This is not a mass equation by any means. It is something entirely different. If this is valid the value should be determined by unit cancellation. What we have left will guide the rest of the investigation.
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#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #237 on: 21/02/2015 04:01:33 »
In the equation x = [kQ/(Er^2) * G/(gr^2)] - [kQ/(Er^2) + G/(gr^2)] when we remove the gravitational and electric components we may be revealing the magnitude of the quark energy hidden by confinement. This should be taken as pure speculation because it is in no way validated. The equation E = mc^2 defines the energy we can detect. It is the confined energy that is hidden from immediate view mathematically. The jury hasn't even deliberated on this one.
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#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #238 on: 21/02/2015 17:03:20 »
If we consider an individual particle interacting with a perfect sphere we can then say that at the surface of the sphere the particle will experience a force represented by Sg. For a perfect sphere this will apply at any point on its surface. We can then say that the total potential around that surface is Sg*4*pi*r^2. We could then calculate the surface area the sphere would have if contained within its Schwarzschild radius rs. We can then define this as As. The surface of then sphere at normal density is A = 4*pi*r^2. If we consider this as a ratio we can say As/A is a starting point with A/As representing the magnitude of A with respect to As. Can we then say that Sg*A/As represents the true increase in force?
« Last Edit: 21/02/2015 17:12:59 by jeffreyH »
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#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #239 on: 23/02/2015 02:18:54 »
Looking again at kQg/(Egr^2) + GE/(Egr^2) what does the numerator kQg + GE tell us? We have Coulomb's constant incorporating the speed of light, magnetic permeability and electric permittivity. This produces a product with the charge and the gravitational acceleration. Added to this is the electrostatic field strength as a product with the gravitational constant. For a particle such as the proton, what values do we get? Do these balance in any way? That is the next test.
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#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #240 on: 27/02/2015 15:49:25 »
Enough outrageous speculation. Let's get down to business. If we consider the wave as circular rather than sinusoidal we can describe the total velocity by (2*pi*1/4*w*f)/t. Here w represents wavelength and f represents frequency. This is an arbitrary choice of variables. So now we have an angular velocity over time. However if we look at the Lagrangian $$L = \frac {1}{2}m(\dot{x}^2 + \dot{y}^2 + \dot{z}^2) - mgz$$ this is unsuitable as the velocity is not angular. As the wave moves through space it is stretched and so there needs to be a combination of both angular and forward velocity to produce a combined velocity whose momentum can be used in a modified Lagrangian. Tying the two velocities together in this manner and with the frequency held constant we can show how the system will evolve when viewed by a remote observer. This also ties the time dilation directly to the internal actions of the particle. We can thus show the evolution of of the change in the rate of change of a system moving into a gravitational field.

NOTE: This is why a flat spacetime is crucial to the development of these equations.
« Last Edit: 27/02/2015 15:52:19 by jeffreyH »
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#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #241 on: 27/02/2015 16:29:46 »
It should not be assumed that the forward velocity and angular velocity change in unison. Maybe for the photon this can be true but this exercise is not concerned with photons. We cannot examine the effects of gravitation if these velocities are exactly proportional.
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#### PmbPhy

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##### Re: Lambert's Cosine Law
« Reply #242 on: 27/02/2015 16:52:21 »
Jeff! What in the world are you doing in this thread? It's as if you're having a very long drawn out conversation with yourself. What gives?

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #243 on: 27/02/2015 18:37:33 »
I am just using it to put ideas down. Don't worry I haven't gone bonkers.
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#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #244 on: 28/02/2015 00:39:01 »
If z is the axis of the path of the particle we can derive the equation:

$$L = \frac {1}{2}(m \dot{z}^2 + mr^2\dot{\phi}) - mgz$$
« Last Edit: 28/02/2015 01:17:36 by jeffreyH »
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#### PmbPhy

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##### Re: Lambert's Cosine Law
« Reply #245 on: 28/02/2015 01:26:23 »
Okay. You remind me or yor_on when you do this.  [^]

#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #246 on: 28/02/2015 01:37:52 »
So I'm in good company then  []
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#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #247 on: 28/02/2015 04:45:02 »
In the equation x = [kQ/(Er^2) * G/(gr^2)] - [kQ/(Er^2) + G/(gr^2)] when we remove the gravitational and electric components we may be revealing the magnitude of the quark energy hidden by confinement. This should be taken as pure speculation because it is in no way validated. The equation E = mc^2 defines the energy we can detect. It is the confined energy that is hidden from immediate view mathematically. The jury hasn't even deliberated on this one.

Interestingly a reduced mass formulation would be better for this equation. Especially when considering the effects of gravitation.

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

The equation then becomes x = [kQ/(Er^2) * G/(gr^2)] / [kQ/(Er^2) + G/(gr^2)]
« Last Edit: 28/02/2015 04:46:51 by jeffreyH »
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#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #248 on: 28/02/2015 20:02:31 »
There is something interesting we can do with the gravitational acceleration equation. This equation is:

g = Gm/r^2

This is like using a reduced mass term as we have neglected the second mass. If we rearrange this equation we can arrive at:

r^2/G = m/g

We know that force is F = ma and as g is an acceleration we can form F = mg. If we rearrange to relect this we get:

r^2G = mg

But r^2G does not equal mg as the form r^2/G = m/g represented a proportionality.

This can be illustrated via

1/2 = 2/4

1*2 <> 2*4

1*2*x = 2*4

In this case it is easy to determine that the missing factor is 2^2

However if we investigate the units of both sides of the gravitational acceleration equation something interesting can be found.

G units = m^3 kg^-1 s^-2 cubic metres per kilogram second squared

left side units = m^5 kg^-1 s^-2

mg units = m kg s^2 metres per second squared

missing units kg^2 m^-4 is surface density squared

So we can make the equations equal with the right value for surface density squared as:

r^2Gd^2 = mg

Where d is the surface density squared. The value of d here is undetermined. However, calculating its value is very important. This indicates that it is only the surface density and not the mass as a whole that generates the force of gravity. This also explains the event horizon surface area relationship to entropy. The remaining gravitational energy is internal to the mass.
« Last Edit: 28/02/2015 20:06:05 by jeffreyH »
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#### jeffreyH

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##### Re: Lambert's Cosine Law
« Reply #249 on: 01/03/2015 17:06:49 »
So if the force is related to surface density, then as we increase the radius away from the surface of the gravitating object what density are we then relating to? It must still be the surface density of the gravitating mass but as it dies away in an inverse-square manner can this tell us anything about the density of the gravitational field at points away from the source? In other words, can we quantize gravitation?
Fixation on the Einstein papers is a good definition of OCD.