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Lambert's Cosine Law
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Lambert's Cosine Law
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Re: Lambert's Cosine Law
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Reply #150 on:
29/11/2014 00:01:10 »
What happens when the value of g = c^2? Then e
_{0}
/g can be said to be similar to e
_{0}
/c^2. However it is not an equality. It is an artificial modification. The gravitational acceleration becomes superluminal for a start. The points of interest are the resulting reduced mass term and the mass radius.
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Re: Lambert's Cosine Law
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Reply #151 on:
29/11/2014 19:27:13 »
The basic problem with the mass and energy equations derived in this thread are their application. They can tell us nothing significant about the particle since the radius is uncertain and variations of mass/energy at that scale are too small to investigate. In the case of a macroscopic mass the equations neglect the nature of the mass as individual particles within molecules which combined together have individual interactions that would invalidate results. This is why the derivation of an equation with an intrinsic wave component is one of the only ways to proceed.
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Re: Lambert's Cosine Law
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Reply #152 on:
30/11/2014 13:36:30 »
Can we incorporate the wave into the mass equation? Let's start with two equations. The first for kinetic energy and the second for the wavelength itself.
Here KE is kinetic energy, m is the mass and v its velocity.
KE = (1/2)mv^2
For the wave equation we have:
f = h/p
Where f is the frequency, h is Planck's constant and p is momentum. To incorporate kinetic energy into the equation the following steps are required.
KE = (1/2) mv^2
2KE = mv^2
2KEm = m^2v^2
2KEm = (mv)^2
Since momentum equals mv we can derive momentum to include kinetic energy using SQRT(2KEm). Then for the wavelength we have:
f = h/SQRT(2KEm)
We can never have zero kinetic energy because we always have zero point energy. Therefore KE has to be intrinsic to mass which means mass always has momentum. Only for purposes of mathematical derivation can we use rest mass. Since mv would require velocity to be a numerator we would be multiplying velocity with hbar so no we cannot incorporate the wave equation into a mass equation. The same can be said for the energy equation. This indicates that the wave is merely an effect of the motion of the particle through space. Either in a straight line path or via angular momentum. From this we can reach the conclusion that because the wave is not intrinsic it can be directly affected by the gravitational field. Since the gravitational field will affect trajectory.
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Re: Lambert's Cosine Law
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Reply #153 on:
30/11/2014 18:03:13 »
Now in a previous post we did see how a wave can be affected by gravitation. The plot is shown again here. This was arrived at by examining Maxwell's equations. It is not a verified equation by any means. What it does attempt to show is the shift in wavelength as a particle moves outward from gravitational field source. As a starting point this needs to be re-examined rigourously.
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Re: Lambert's Cosine Law
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Reply #154 on:
30/11/2014 18:10:25 »
What the above plot does bring to mind are the discrete energy levels and integer wavelengths of the electron orbitals.
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Re: Lambert's Cosine Law
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Reply #155 on:
30/11/2014 18:25:39 »
We can develop a 3 particle model to show gravitational interactions as vectors. We have to use 3 since we can describe a plane that all the particles line up with at any point in the evolution of the interactions. Any more particles cannot be assumed to sit on this moving plane. The inherent values of g for each particle can then be described as vectors in the system as it changes over time. Determining how each wave evolves during the interactions will be of interest. Exactly how do they behave?
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Re: Lambert's Cosine Law
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Reply #156 on:
03/12/2014 03:03:02 »
A correction to one of the posts above. In the equation f = h/SQRT(2KEm) of course f is wrong as it is the frequency and not the wavelength. In case I confused everybody. It should be λ = h/SQRT(2KEm). If we hold mass as invariant then the kinetic energy determines a change in the wavelength. Since gravitation alters the kinetic energy this can be used to describe the effects of gravitation on the wave. More on this later.
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Re: Lambert's Cosine Law
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Reply #157 on:
04/12/2014 03:12:00 »
The Kaluza-Klein theory is described here:
http://en.wikipedia.org/wiki/Kaluza
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–Klein_theory
This is a scalar theory of gravitation. Interestingly from here:
http://en.wikipedia.org/wiki/Scalar_theories_of_gravitation
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We find that:
"Kaluza–Klein theory involves the use of a scalar gravitational field in addition to the electromagnetic field potential in an attempt to create a five-dimensional unification of gravity and electromagnetism. Its generalization with a 5th variable component of the metric that leads to a variable gravitational constant was first given by Pascual Jordan."
The fact that this leads to a variable gravitational constant is of interest. Only one of the papers appears to have a translation. This theory has some interesting consequences and should be pursued vigorously in my opinion. I will be investigating this in conjunction with the work shown in this thread.
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Re: Lambert's Cosine Law
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Reply #158 on:
04/12/2014 03:28:37 »
For reasons I will explain when I have worked out the proof, mass can exceed light speed but only when approaching an event horizon. Within a defined region surrounding the horizon nothing will be visible. It is not that things disappear once the horizon is crossed. They will vanish BEFORE the horizon is reached. The innermost point of an accretion disk will mark the extent of the outer visible area. X-ray sources must then emanate from this region as they would not escape the region beyond this.
NOTE: It MAY be possible to achieve superluminal interstellar velocities but only if it is possible to shield against gravitation. This has to be considered with caution since it relies on propositions that are entirely without proof.
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Last Edit: 04/12/2014 03:37:18 by jeffreyH
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Re: Lambert's Cosine Law
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Reply #159 on:
04/12/2014 22:17:12 »
I have discovered a relationship in the gravitational field density. This is shown in the attached graph. I will not be showing how this was derived at the moment as this has far reaching consequences if correct. I will be developing a set of equations around this initial equation.
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Re: Lambert's Cosine Law
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Reply #160 on:
04/12/2014 22:59:21 »
Contrary to what I may have posted earlier in this thread the gravitational field in fact does lose energy but the equation describing the rate of change is not a simple relationship. More energy is lost nearer the source than further away. In fact the field later regains some of the lost energy from somewhere. This is puzzling. Gravity well is an understatement of the situation. I can now derive the density variations outside the event horizon and the 'no light zone' before the start of the accretion disk..
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Re: Lambert's Cosine Law
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Reply #161 on:
04/12/2014 23:01:57 »
If we consider the energy of the gravitational field as negative then this implies the field is becoming more positive as it moves away from the source. This does not mean that it will ever become a repulsive field but may explain the accumulation of dark energy due to energy conservation.
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Re: Lambert's Cosine Law
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Reply #162 on:
04/12/2014 23:39:28 »
The possibility of gravity giving rise to the dark matter/energy halos comes from the profile of energy loss. Starting low then reaching a peak and dying away again. This would produce such a halo effect with most of the dark material concentrated at the peak of energy loss. The dark matter/energy produced would be even weaker than the gravitational energy and would accumulate over time. A galaxy that has had the material stripped by an encounter with another galaxy could over time re-acquire its halo due to future energy loss from the gravitational field of the central black hole. These anomalies may be detectable and it may be fruitful to find such galaxies.
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Re: Lambert's Cosine Law
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Reply #163 on:
05/12/2014 22:51:30 »
One consequence of the above hypothesis is that the electromagnetic field will lose energy in the same proportions as the gravitational field. This does not apply to the magnetic field itself which loses no energy. This makes sense as it circulates and would be unable to sustain circulation if energy was lost. So the electric portion of the field and the
possibly
photon itself lose this energy the further from the source the position of the field or particle is. I say possibly in the case of the photon as I just don't know for sure.
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Re: Lambert's Cosine Law
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Reply #164 on:
05/12/2014 23:48:31 »
Gravitational lensing should be more pronounced at a set radial distance from the mass generating the gravitational field. This will be within the halo region around the mass at a set radial distance from its surface.
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Re: Lambert's Cosine Law
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Reply #165 on:
06/12/2014 00:16:31 »
The other thing all this finally establishes in my opinion is that gravity does in fact travel at exactly light speed and itself undergoes dilation due to its interactions with the electromagnetic field. They affect each other proportionally. Like the charge of the proton and electron being the same while the mass differs the electromagnetic and gravitational field affect each other proportionally even though their energies are not equivalent. I have no idea how this works.
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Re: Lambert's Cosine Law
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Reply #166 on:
09/12/2014 02:00:45 »
Finally we come to our equation for the wave. If we rearrange the mass equation to be M =
we now have a time dilation component implicit to the function. This includes the velocity of the mass. Applying
to the inverse light speed t/L also relates to the dilation of the photon in a gravitational field. However this form is concerned with velocity alone. Interestingly the square of the radius and the surface value of g also increase. This echoes the thinking of Paul Marmet that the Bohr radius must increase with velocity. The next step is to test the equation for the results it will produce to see if in fact it does reflect the real world experimental data.
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Last Edit: 09/12/2014 02:10:30 by jeffreyH
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Re: Lambert's Cosine Law
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Reply #167 on:
10/12/2014 17:40:25 »
The mass dilation equation has some of the fundamental constants incorporated into it. It would be interesting, although maybe not very informative in its present state, to try values at the Planck scale for radius and g. This can be done using the variation derived for the Planck mass black hole. Using this a fixed reference point can be set at the event horizon.
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Re: Lambert's Cosine Law
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Reply #168 on:
13/12/2014 15:37:07 »
I have found an interesting papaer which may be of interest. I haven't read it yet. It is from December 2011 so is only 3 years old. The title is "Where is hbar Hiding in Entropic Gravity?". This is to do with the proposal of entropic gravity by Erik Verlinde which I also haven't reviewed. This is apparently a classical Newtonian gravity theory with origins in quantum mechanics. The link is:
http://arxiv.org/abs/1112.3078
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Just a note from me. Is hbar hiding in the definition of mass as above?
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Re: Lambert's Cosine Law
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Reply #169 on:
20/12/2014 16:59:19 »
One concrete proposal has come out of these investigations which I will need to provide the equations for eventually. For a black hole with the mass of the earth with rs set at around 1cm there is a special region which extends outward radially to a distance of 815.4 metres (Approx.).I am not entirely sure if light would be trapped within this zone, probably not. However any other tardyon mass WILL be trapped within this zone. So therefore the event horizon is not the danger zone for the possibility of escape. This already occurs further out. This may in effect prevent the measurement of the mass of black holes with any accuracy.
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Re: Lambert's Cosine Law
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Reply #170 on:
21/12/2014 15:41:45 »
The results above had led onto an equation that includes one function for angular momentum and another for specific area. I am not sure what the specific area indicates as it is only indirectly related to specific volume. These equations when viewed with respect to the earlier equations in this thread should provide some new insights on the interaction of mass with gravity. I am working on this now and will post the details when the equations are complete.
EDIT: If a function of angular momentum can be used in the mass equation then this will allow the evolution of the wave due to gravitation to be an inherent property of mass. This may also become a way of describing the action of time dilation.
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Last Edit: 21/12/2014 15:48:29 by jeffreyH
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Re: Lambert's Cosine Law
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Reply #171 on:
22/12/2014 18:22:34 »
In the equation below we can determine the distance traveled during an amount of time t when the gravitational acceleration equals g.
Δy = 1/2*g*t^2
If we then set g to equal
g = 2L/t^2
we can show that in one second due to cancellation of the following
1/2*2L/t^2*t^2
That we will have traveled L distance in one second. Since L is equal to 1 light second of distance this means that we will have reached light speed during this acceleration. Applying this to the parameter for earth gives us an exclusion zone around an earth sized black hole. This is the first step in deriving our new mass equation.
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Re: Lambert's Cosine Law
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Reply #172 on:
23/12/2014 00:04:27 »
If we set g = 2L/t*dt we have
2L/t^2 = Gm/r^2
Re-arranging for r
r = SQRT(t*dtGm/2L)
If r = L then
L^2 = t*dtGm/2L
Re-arranging for m
m = 2*L^3/t*dt*G
Setting G equal to the approximation 1/50c
m = 2*L^3/t*dt*50*L/t
Restoring c
m = 100c*L^3/t*dt
And finally re-arranging
Volumetric acceleration
m/100c = L^3/t*dt
Cumec for volumetric flow is in the units m^3/s so here we have the potential volumetric acceleration of gravitation for the whole mass. We need to reconcile this with g at the surface that will be the next step.
The above is equivalent to the following simply formula.
Gm/2
Here the factor of 2 appears again. So if we let a equal this volumetric acceleration we arrive at:
a = Gm/2
EDIT: To cater for length contraction in 1 spatial dimension we would modify the equation thus:
m/100c = L^2*dL/t*dt
Now we can cater for the effects of both relativistic changes with respect to the potential volumetric acceleration.
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Re: Lambert's Cosine Law
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Reply #173 on:
23/12/2014 01:09:56 »
The difficulty of reconciling gravitational acceleration with volumetric acceleration lies in the fact that while the first follows a linear geodesic the second represents an infinite number of radial directions in 3 dimensional space. These all emanate from the centre of gravity.
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Re: Lambert's Cosine Law
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Reply #174 on:
23/12/2014 16:46:22 »
So how to reconcile this. Well we saw that y = 1/2gt^2. If we re-express a = Gm/2 as a = Gm/(2r^2) we find the equivalent acceleration for the length y during a 1 second interval. What needs to be determined now is how an expression for angular momentum can be achieved.
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Re: Lambert's Cosine Law
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Reply #175 on:
27/12/2014 02:03:59 »
For a general wave equation we can show the following progression:
m = g*r^2*h*1/(2pi)*(1/c^2)*[1/lP^2*(gamma*t)/L]
P^2 = 2*Ke*g*r^2*h*1/(2pi)*(1/c^2)*[1/lP^2*(gamma*t)/L]
λ = h/SQRT(Ke*g*r^2*h*1/pi*(1/c^2)*[1/lP^2*(gamma*t)/L])
In the final wave equation the velocity v should be considered the only variable. It is a component of both Ke and gamma. Since we have components such as c^2, h and the Planck length squared it would be interesting to review equations containing these combinations to see if this gives us an insight into gravitation, mass and quantum mechanics.
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Re: Lambert's Cosine Law
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Reply #176 on:
27/12/2014 03:17:29 »
As a starting point for an angular momentum formula we need to review the following.
http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/qangm.html
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Another useful reference is on wikipedia.
http://en.wikipedia.org/wiki/Azimuthal_quantum_number
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Re: Lambert's Cosine Law
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Reply #177 on:
28/12/2014 18:15:30 »
To develop our equation of angular momentum we must first review an earlier post.
"Before trying to link the gravitational field to Lambert's Cosine Law I need to take a detour. This starts with the unit sphere and the unit circle. Using the unit sphere and circle shows some interesting relationships and can be scaled up. This can then be used to describe both subatomic and macroscopic domains.
The circumference of the unit circle is 2*pi. To determine the angle of an arc around the circle whose arc length is equal to the radius we can use (1/2*pi)*360 which can be simplified to 7/44*360. This proportionality will become important when viewing interactions at differing scales and relates to wave frequency, length contraction and time dilation effects. The angle we have determined can be converted to radians to use in calculations.
It is interesting to note that the period of sin x is 2*pi. This can be utilized by considering forward motion and angular rotation as it relates to the unit sphere. The relationship between these two properties can describe the evolution of a wave and can be related directly to the gravitational field. When used it can be shown to show the underlying mechanism of the Pauli Exclusion Principle and the difference in energy levels required between electrons.
There are 3 directions of motion under consideration within this model. One motion is forward direction and is considered to be aligned with the poles of the sphere. The two other directions are angular. The first is around the equator and the second follows a longitudinal path intersecting both poles. The maximum unit of motion in unit time in the polar direction is equal to the unit sphere radius. The maximum unit of motion of the angular paths is 7/44*360 as stated above. If viewed at the Planck scale the angular components cannot reach this speed or none of us would be here. Therefore we can deduce that this dampening in angular momentum must be due to gravity which is what the current physical theories state.
If we follow this line of thinking through to its conclusion we can show that when considering the universe as a whole system light might get infinitesimally near to c but will never actually reach it as long as any gravitational field remains. I will demonstrate the reasons for this conclusion as I proceed."
The angle derived above becomes important only when the radius of an object is 1 Planck length. As this will only apply in some string theories we can disregard it. Not only because it relates to string theory but it relates to light speed which angular momentum should never reach. This will be discussed later.
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Re: Lambert's Cosine Law
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Reply #178 on:
29/12/2014 15:48:42 »
In examining angular momentum it is not sensible to have Planck units. The best units to select would be nanometres and nanoseconds. Thus the speed of light can be represented by L*1 nanometre divided by 1 nanosecond. We can then apply a factor to L to determine a non-relativistic speed. Now we may have a problem as the angular momentum of a particle is L = r x mv. The two values for L, 1 light second versus L for angular momentum mus NOT be confused. L = r x mv is the cross product mv the linear momentum and r which is the point of rotation. This is best illustrated in the animation on wikipedia.
http://en.wikipedia.org/wiki/Angular_momentum
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Re: Lambert's Cosine Law
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Reply #179 on:
01/01/2015 00:12:17 »
I have decided to temporarily suspend posts to this thread until I have reviewed the following.
http://phys.org/news/2013-11-proton-radius-puzzle-quantum-gravity.html
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http://arxiv.org/abs/1412.4515
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Re: Lambert's Cosine Law
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Reply #180 on:
02/01/2015 00:14:46 »
What if gravity or any other such force which is considered inverse square is not that at all but appears that way the further from the source and at macroscopic scales. If instead the law was 1/r^(1+1/n) and n started at an as yet undetermined value > 1 and approached 1 but only got there at infinity then we would have an entirely different situation microscopically where gravity starts as a 1/r law and then changes over time to become nearer and nearer to a 1/r^2 law. What if we started the value of n at 50 as an arbitrary assignment?
EDIT: This may validate the dark matter halo hypothesis.
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Re: Lambert's Cosine Law
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Reply #181 on:
03/01/2015 02:42:28 »
In one of the papers by Roberto Onofrio the Schwarzschild radius of the elementary particles becomes boosted by 33 orders of magnitude. As light should still be trapped then the energy of the gravitational field should become boosted by an equivalent factor. So we could replace the value of 50 by 33. If we stay with 50 for now then the attached graph shows the change in gravitation with respect to an earth sized mass. The radius and mass are held constant while the power of the radial term moves from 1/r^(1+1/50) to 1/r^(1+1/1). At the left of the x axis the situation mimics a Planck mass situation where the whole mass can be treated as a single particle. At the right of the x axis we return to our 9.81 value of g where the mass is made of separate particles. The x axis has been graded so that the intervals between successive points are more closely spaced at the source and separate with movement to the right. This is the only way to achieve a sensible scale and illustrates the change from 1/r to 1/r^2 as a gradient.
EDIT: The x axis is derived by starting with a value of 50/50 at the left and ending at 50/1 at the right. This does not mean that this is valid in this scenario. It was a quick experiment with the values.
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Re: Lambert's Cosine Law
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Reply #182 on:
03/01/2015 03:17:52 »
This would of course imply a decrease in energy of the gravitational field the further from the source it is. As stated above the intervals on the x axis were derived. If we remove the derivation then we get the attached plot which does show a very different profile altogether.
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Re: Lambert's Cosine Law
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Reply #183 on:
04/01/2015 23:38:56 »
We can carry out a very rough guestimate on our first value by using the mass of the hydrogen atom and the mass of the earth. We then get a factor of Mh/Me where Mh is the mass of the hydrogen atom and Me is the mass of the earth. Our first value for g was 38913462.45 so we end up with (6.7e-28/5.97219e24)*5.97219e+24. Our result is 4.36557e-45 for g. However this is a 1/r relationship and does not describe an acceleration in this form so the actual acceleration will be much less. If we apply the same proportionality to the radius of the earth as we did to the mass we can use this in our next guestimate.
EDIT: We would need to obtain the square root value of the reduced radius as a correction. If this value does not match a reasonable radius for the hydrogen atom then this invalidates this method entirely. This will need to match the Bohr radius within a certain range of precision.
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Re: Lambert's Cosine Law
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Reply #184 on:
05/01/2015 00:07:05 »
What we get is 2.67495e-23 metres which is 31 orders of magnitude larger than the Schwarzschild radius of the proton. Since Roberto Onofrio is suggesting a particle's Schwarzschild radius is boosted by 33 orders of magnitude this is a very surprising result and only 1 order of magnitude out.
CORRECTION: This should state a 2 order of magnitude difference.
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Re: Lambert's Cosine Law
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Reply #185 on:
05/01/2015 00:19:30 »
The attached graph is a log/log plot of the evolution of the variation in the profile of the gravitational field in terms of induced acceleration. The values for mass and radius are still those of an earth sized mass and no scaling down has been attempted.
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Re: Lambert's Cosine Law
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Reply #186 on:
06/01/2015 21:43:37 »
The scale at which the forces would be acting is the attometre. This is 10^-18 metres which is 5 orders of magnitude greater than the boosted proton Schwarzschild radius of the proton. Our scale could be defined from the boosted radius up to the currently accepted proton radius which is in the femtometre range 10^-15. The proton radius is around 0.84–0.87 fm in size. Can an equation be derived that shows a morphing of gravitation between the microscopic and macroscopic domains? This is the key question. It will not be like the previous guestimate.
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Re: Lambert's Cosine Law
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Reply #187 on:
07/01/2015 03:38:51 »
The next step is the investigation of the gravitational coupling constant. Details can be found on wikipedia.
http://en.wikipedia.org/wiki/Gravitational_coupling_constant
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Re: Lambert's Cosine Law
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Reply #188 on:
09/01/2015 02:18:39 »
A short detour is worth taking before continuing. Consider a point P which is at an infinitesimal distance outside the event horizon of a black hole. At point P a photon is emitted in a direction directly away from the gravitational source. Since gravitation will slow light in its field and the the escape velocity is infinitesimally near c at point P then the emitted photon will not have the energy to achieve a velocity equaling c. Since it cannot reach the escape velocity it should therefore lose kinetic energy as it moves away from the source. The question that arise is does this mean that the photon is therefore trapped and will eventually stop and reverse direction. Since, without any other forces acting on the photon, it will only experience a constant velocity it cannot accelerate to escape the gravitational field. This is an important consideration when investigating the ergosphere surrounding a black hole. This is suggestive of a zone outside the horizon that can still potentially trap photons. Since it has been shown experimentally that light can be slowed substantially when moving through certain mediums it may be that the intensity of a gravitational field can mimic such a medium. Infalling photons that are not directed at the source may well fall into near-horizon orbits. Being perpendicular to the direction of the gravitational field may allow photons to reach a velocity infinitesimally close to c. However this may not be the case at all.
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Re: Lambert's Cosine Law
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Reply #189 on:
09/01/2015 02:41:51 »
It is useful to read the following article.
http://news.harvard.edu/gazette/1999/02.18/light.html
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The extremely low temperature of the medium will effectively transfer kinetic energy in the form of heat. This implies that the ergosphere around a black hole is acting like a bose-einstein condensate. This can come about via infalling matter achieving this state with low vibrational energy. This implies a uniform change in acceleration on the particulate matter within the ergosphere so that no heat is generated and no force is felt. Since kinetic energy is equivalent to a potential heat transfer the field, or something, must be taking up this heat. Either that or x-rays with enough energy are effectively escaping the ergosphere and taking this heat away. However, what if there is no matter close enough to the black hole to be falling into the ergosphere? In this case there will be no medium through which light can be effectively slowed down. In the case of the encounter of gas cloud G2 with sag A* it may simply be the case that the black hole was starved of matter with which to generate a sufficiently dense medium with which the gas would react.
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Re: Lambert's Cosine Law
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Reply #190 on:
10/01/2015 18:25:25 »
Consider the whole surface of the horizon of a rotating black hole. Because of this rotation frame dragging will carry matter around the equatorial disk. Velocities will be high. However at the poles this velocity is absent and therefore makes it easier for matter to enter the black hole's event horizon at those regions. This could result in energy release in the form of photons in all directions. Some will fall into the black hole at the pole. Some will initially form eccentric orbits across the horizon while others will be propelled away from the poles. The jets away from the poles being intense will carry some matter with them in the form of maybe gas molecules or nuclei and separate electrons.
NOTE: Due to entanglement a proportion of the photons expelled via the jets could provide information on entangled photons falling into the black hole.
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Re: Lambert's Cosine Law
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Reply #191 on:
10/01/2015 19:11:40 »
If we investigate path profiles around a rotating sphere it could be possible to construct a model of this scenario. This can then be related to smaller scales. Ultimately it can be applied to the particle itself. The following is a good point in the history of particle physics to start.
http://en.wikipedia.org/wiki/Spin_(physics)#History
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"Spin was first discovered in the context of the emission spectrum of alkali metals. In 1924 Wolfgang Pauli introduced what he called a "two-valued quantum degree of freedom" associated with the electron in the outermost shell. This allowed him to formulate the Pauli exclusion principle, stating that no two electrons can share the same quantum state at the same time.
The physical interpretation of Pauli's "degree of freedom" was initially unknown. Ralph Kronig, one of Landé's assistants, suggested in early 1925 that it was produced by the self-rotation of the electron. When Pauli heard about the idea, he criticized it severely, noting that the electron's hypothetical surface would have to be moving faster than the speed of light in order for it to rotate quickly enough to produce the necessary angular momentum. This would violate the theory of relativity. Largely due to Pauli's criticism, Kronig decided not to publish his idea.
In the autumn of 1925, the same thought came to two Dutch physicists, George Uhlenbeck and Samuel Goudsmit at Leiden University. Under the advice of Paul Ehrenfest, they published their results. It met a favorable response, especially after Llewellyn Thomas managed to resolve a factor-of-two discrepancy between experimental results and Uhlenbeck and Goudsmit's calculations (and Kronig's unpublished ones). This discrepancy was due to the orientation of the electron's tangent frame, in addition to its position.
Mathematically speaking, a fiber bundle description is needed. The tangent bundle effect is additive and relativistic; that is, it vanishes if c goes to infinity. It is one half of the value obtained without regard for the tangent space orientation, but with opposite sign. Thus the combined effect differs from the latter by a factor two (Thomas precession).
Despite his initial objections, Pauli formalized the theory of spin in 1927, using the modern theory of quantum mechanics invented by Schrödinger and Heisenberg. He pioneered the use of Pauli matrices as a representation of the spin operators, and introduced a two-component spinor wave-function."
Pauli's objections were overcome but should they have been? This will be investigated on the way to our wave interaction equation.
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Re: Lambert's Cosine Law
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Reply #192 on:
11/01/2015 00:32:02 »
Another useful piece of history to review is this:
http://en.wikipedia.org/wiki/Llewellyn_Thomas
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"Llewellyn Hilleth Thomas (21 October 1903 – 20 April 1992) was a British physicist and applied mathematician.[1] He is best known for his contributions to atomic physics, in particular:
Thomas precession, a correction to the spin-orbit interaction in quantum mechanics, which takes into account the relativistic time dilation between the electron and the nucleus of an atom.
The Thomas–Fermi model, a statistical model of the atom subsequently developed by Dirac and Weizsäcker, which later formed the basis of density functional theory.
Thomas collapse - effect in few-body physics, which corresponds to infinite value of the three body binding energy for zero-range potentials."
Particularly this:
"While on a Traveling Fellowship for the academic year 1925–1926 at Bohr's Institute in Copenhagen, he proposed Thomas precession in 1926, to explain the difference between predictions made by spin-orbit coupling theory and experimental observations."
A section on the Thomas interaction energy can be found here:
http://en.wikipedia.org/wiki/Spin
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–orbit_interaction
This includes a Lorentz factor for a moving particle. This is an important consideration for interacting wave equations, especially with regard to changes in a gravitational field. Time dilation can be related directly to this factor.
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Re: Lambert's Cosine Law
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Reply #193 on:
11/01/2015 23:33:48 »
If we return to the poles of the black hole it should be true that a higher proportion of Hawking radiation will be generated here than elsewhere along the surface of the horizon. The interaction of particle wave ensembles away from the poles should be a good area to investigate for particle interactions with gravitation within an intense field. If entanglement is prominent here we may also be able to determine some aspects of the behavior of particles having fallen through the horizon in these polar regions. A study of the profiles of many polar jets should give valuable data in this regard.
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Re: Lambert's Cosine Law
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Reply #194 on:
12/01/2015 06:01:45 »
Now if we want a balanced wave equation we first restore the factors of 2.
This:
λ = h/SQRT(Ke*g*r^2*h*1/pi*(1/c^2)*[1/lP^2*(gamma*t)/L])
Becomes:
λ = h/SQRT(2Ke*g*r^2*h*1/2pi*(1/c^2)*[1/lP^2*(gamma*t)/L])
Then we need h on the right hand side so:
λ/h = 1/SQRT(2Ke*g*r^2*h*1/2pi*(1/c^2)*[1/lP^2*(gamma*t)/L])
Inverting we get:
h/λ = SQRT(2Ke*g*r^2*h*1/2pi*(1/c^2)*[1/lP^2*(gamma*t)/L])
Squaring:
(h/λ)^2 = 2Ke*g*r^2*h*1/2pi*(1/c^2)*[1/lP^2*(gamma*t)/L]
Multiplying by 2pi:
2pi(h/λ)^2 = 2Ke*g*r^2*h*(1/c^2)*[1/lP^2*(gamma*t)/L]
Finally:
2pi = (λ/h)^2*2Ke*g*r^2*h*(1/c^2)*[1/lP^2*(gamma*t)/L]
We can now adjust g, r, wavelength or gamma*t but they must all balance in order to conserve the equality. This involves conservation laws somehow. Which ones, maybe all, I don't know yet. I may easily have made a mistake here so beware. This does not take into account the effects of an external g force but since 2pi is the circumference of the unit circle a multiplication be ge (external g) will result in either an expansion or contraction of radius depending upon where it is less than or greater than 1. Being based upon metre units this is the correct pivot point.
NOTE: Since an increase in the unit radius would imply a slower spin rate then as the gravitational field intensifies this will correlate with time dilation. In this form the equation says nothing about a change in radius with increasing velocity.
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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
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http://en.wikipedia.org/wiki/Reciprocal_lattice
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"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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