Lambert's Cosine Law

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jeffreyH

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Re: Lambert's Cosine Law
« Reply #250 on: 01/03/2015 23:29:35 »
It may then be shown that surface density is:

d = SQRT([mg]/[r^2G])

When we expand the radius this can describe two situations. Firstly the whole mass can assume a lower density within the volume described by the new radius. Secondly we describe a point away from the surface of the mass that leaves density unaffected. The mass retains its original radius value. Both situations are equivalent. Therefore this equation should also describe the field density at any point away from the surface of the source. This can only be true for a perfectly spherical object with uniform density. This is suggestive of the quatization of the gravitational field.

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Re: Lambert's Cosine Law
« Reply #251 on: 02/03/2015 07:33:39 »
While it would be useful to determine if d is a constant in this situation it would not help with determining anything relating to a black hole. It is escape velocity that we need to relate to density to determine if quantization can explain the effects on light. There would need to be a reformulation of the escape velocity equation to relate it to the same surface density. Both situations could then be examined. Along with the relationship between gravitational acceleration and escape velocity. The next step is the reformulation of the escape velocity equation.

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Re: Lambert's Cosine Law
« Reply #252 on: 03/03/2015 11:34:56 »
Before attempting to reformulate the equation for escape velocity it should be noted that these equations have not been independently verified for validity. It may well be wrong to include surface density in this manner.

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Re: Lambert's Cosine Law
« Reply #253 on: 04/03/2015 01:57:37 »
WE have escape velocity as Ve = SQRT([2GM]/r). We can first remove the square root as in Ve^2 = [2GM]/r. Kinetic energy Ke = 1/2mv^2 and we already have Ve^2. Ir rearranged we have (1/2)Ve^2 = [GM]/r. If we multiply both sides by M we get (1/2)MVe^2 = [GM^2]/r. Force F = M/a so to get the force we need an acceleration to be able to proceed. This now becomes more complex. For now we can simply view this as a derivation of the kinetic energy of our escape velocity. The square of the mass on the right hand side echoes the square of the surface density to some extent.

EDIT: This is a strange equation in reality because we are not considering a mass moving away from a source but the source itself. Which can't be moving away from itself. So what exactly does this formulation of kinetic energy actually represent? Well if both masses were equivalent it could be viewed as the kinetic energy required to separate them at the required escape velocity. I am very unsure about this one.
« Last Edit: 04/03/2015 02:10:20 by jeffreyH »

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Re: Lambert's Cosine Law
« Reply #254 on: 04/03/2015 02:14:39 »
Can we then say that the kinetic energy required to separate any two masses is GMm/r?

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Re: Lambert's Cosine Law
« Reply #255 on: 04/03/2015 17:40:14 »
WE can view this in a different way and similar to the way the last equation was derived. From (1/2)Ve^2 = [GM]/r we can rearrange as [(1/2)Ve^2]/M = G/r. Then taking the same step as in [(1/2)Ve^2]*M = G*r the units necessary on the right hand side are now kg^2 m^-1. The units kg m^-1 actually represent linear mass density but without the square of the kg unit. I have no idea what kg^2 m^-1 represents.

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Re: Lambert's Cosine Law
« Reply #256 on: 05/03/2015 01:09:15 »
We can eliminate the mass from the left hand side and include linear mass density as:
(1/2)Ve^2 = G*D*r

Where the d in the previous equation represents surface mass density and here D represents linear mass density. So rather than an energy equation we now again have a velocity equation. We can then restore a more original form.

Ve = SQRT(G*D*r)

The other equation

d = SQRT([mg]/[r^2G])

can now be reformulated.

d^2 = [mg]/[r^2G]

1/g = m/[r^2d^2G]

So we have

g = [r^2d^2G]/m

And

Ve = SQRT(G*D*r)

Here the main point of interest is the combination of the gravitational constant with the radius and density in both equations. Both the value for g and Ve will be with respect to the mass surface. These may be only applicable to the perfect spherical mass.

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Re: Lambert's Cosine Law
« Reply #257 on: 05/03/2015 01:50:16 »
Working through the equation for surface mass density we find the following. At any point on a perfect sphere with roughly the mass of the earth only 0.36 % of the total mass can be considered to be having any gravitational effect.

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Re: Lambert's Cosine Law
« Reply #258 on: 05/03/2015 02:18:09 »
Of interest in regard to all this are the items on this list of astronomical anomalies.

http://www.technologyreview.com/view/414539/the-puzzle-of-astronomys-unexplained-anomalies/

EDIT: Could this be due to an incoherence in the gravitation fields of masses? This brings us back to Lambert's Cosine Law strangely enough.
« Last Edit: 05/03/2015 02:21:02 by jeffreyH »

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Re: Lambert's Cosine Law
« Reply #259 on: 05/03/2015 21:23:57 »
A correction to the escape velocity equarion.

Ve = SQRT(G*D*r)

Should be

Ve = SQRT([D*G*r]/[(1/2)M])

And the density equation becomes

D = [(1/2)Ve^2M]/[GR]

EDIT: Here the units are kg^2 m^-2. The density relationship here is not straight forward. Here kinetic energy is related to the density and makes sense when considering this is derived from escape velocity.
« Last Edit: 06/03/2015 00:59:32 by jeffreyH »

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Re: Lambert's Cosine Law
« Reply #260 on: 11/03/2015 20:15:22 »
Having gone back and thought about the shift in wavelength then length contraction has to be proportional to the length of the wave as calculated from a remote frame. This proportionality is possibly direct but maybe indirect due to a difference in the gradient of the change of each. Time dilation is then related to an inherent twist in spacetime due to the gravitational field. This will only be observable in the vicinity of extremely dense objects and when strong enough will result in frame dragging.

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Re: Lambert's Cosine Law
« Reply #261 on: 15/03/2015 22:32:24 »
The following equation has been derived to try to determine the minimum mass for a stable black hole.

g = c^2/[2rs+L/2$$\sqrt{1-\frac{r_s}{r_s+L/2}}$$]

However some caution is necessary. The kinematic equation for distance traveled due to free fall is:

d = vit + (1/2)at^2

vi is the initial velocity, t is elapsed time, a is acceleration and d is the displacement. Any object having an initial velocity when far from a black hole may ultimately acquire superluminal velocity before reached the event horizon due to its initial velocity. If instead of the Chandrasekhar limit we use a value of 3 solar masses we find that the acceleration falls below c. In order for this to be the minimum black hole mass it needs a proof of non-superluminal speed before the event horizon. Otherwise physics breaks down exactly where it shouldn't.

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Re: Lambert's Cosine Law
« Reply #262 on: 15/03/2015 23:48:25 »
A paper on maximum netron star mass can be found here:

http://arxiv.org/abs/1307.3995

This is pertinent to the above equation.

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Re: Lambert's Cosine Law
« Reply #263 on: 16/03/2015 06:01:32 »
The equation

d = vit + (1/2)at^2

will function perfectly well in a gravitational field such as the earth's. When it comes to the region near to a black hole things are radically different. The amount of change in the displacement increases more rapidly. This results in the elongation or spaghettification of matter as it approaches ever closer to the horizon. Smaller and smaller increments of time are then required to determine the actual displacement along the path of the in-falling matter. It is therefore more sensible to assume a point particle along this path. The time increments are still necessary but no consideration need be taken of the effects along a mass made up of multiple particles.

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Re: Lambert's Cosine Law
« Reply #264 on: 28/03/2015 04:30:34 »
A note on something I concluded whilst considering the speed of gravity. In order for gravity to operate at c the equation for the energy of the gravitational field should be Mc2G3. Where M is the mass, c is the speed of light and G is the gravitational constant. The energy is then 30 orders of magnitude less than the overall energy of the mass. This is in line with the difference in strength between the electromagnetic force and the gravitational force.

EDIT: Note that Mc2G3 is NOT an energy equation. It is an example of the magnitude of difference between the forces. With this equation you end up with 11 spatial dimensions.
« Last Edit: 28/03/2015 14:22:22 by jeffreyH »

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Re: Lambert's Cosine Law
« Reply #265 on: 28/03/2015 14:58:58 »
If we consider a slightly different equation GMt where t is time then G has the units m^3, kg^-1 and s^-2. We can cancel the units of kg^1 with the mass and s^-2 becomes s when canceled with the time parameter. We now have units of cubic metres per second which is flow rate through a volume of space. The GM would normally be converted to an acceleration via division by r^2. For a mass the size of the earth this gives 3.98574405E+14 m^3/s. The question is how do we interpret this flow rate? It is not due to the rotation of the earth as angular momentum is not included and neither is an angular velocity. What is always there is particle spin.

What we have here is a reduced mass term via G. So we are not considering a 100% flow rate. At the particle level this may indicate is that a proportion of spin angular momentum is responsible for generating the gravitational field. This can be considered as a twisting field at the lowest level source.

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Re: Lambert's Cosine Law
« Reply #266 on: 28/03/2015 15:06:05 »
This proportionality raises other questions. Why is there such a proportionality at the particle level? Can we determine a relationship between G and the elementary particles involved? Does this indicate that not all particles are involved in generating the field? I do not have the knowledge of the standard model to carry this further. If any can find this useful and wishes to pursue this they have permission to use the ideas as long as credit to the source is given. Then again no one may find these ideas valid in which case nothing is lost.

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Re: Lambert's Cosine Law
« Reply #267 on: 28/03/2015 15:17:01 »
A final point on this is that if spin angular momentum is slowed by time dilation then the force of gravity generated by a smaller mass in the field of a much larger mass is reduced. The larger mass will be almost unaffected. For the speed of gravity to vary in proportion to the speed of light in a gravitational field them this should be true.

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Re: Lambert's Cosine Law
« Reply #268 on: 28/03/2015 17:11:42 »

http://people.bu.edu/gorelik/cGh_FirstSteps92_MPB_36/cGh_FirstSteps92_text.htm

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Re: Lambert's Cosine Law
« Reply #269 on: 03/04/2015 17:46:48 »
On the gravitational constant. A constant value of between 48 and 50 should allow the calculation of the constant without the circularity involved in using the Planck values. This is because the gravitational constant is itself used in determining these values. An equation of the form [r^2/(NMct)]V^2 is therefore required. Where N takes the place of the constant (48-50). If we introduce relativistic gamma into this equation we can then determine the coordinate change in the gravitational constant that will relate to the effects of time dilation and length contraction. The value of r used as a the numerator is the unit normal vector. To use gamm we first reformulate the equation as [r^2/(NMc)](s^2/t^3) and then gamma is applied to (s^2/t^3). The derivation of this equation will require some further explanation. Its form was initially derived to preserve the units of the calculated value of G. The equation itself leads to some interesting consequences.

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Re: Lambert's Cosine Law
« Reply #270 on: 04/04/2015 21:00:12 »
Considering [r^2/(NMc)](s^2/t^3) s = 1 metre, t = 1 second and M = 1 kilogram. The gamma function normally has v^2/c^2 but here we take the escape velocity Ve which makes this functional become Ve^2/c^2. The escape velocity Ve will only ever reach c at the event horizon and this is regardless of the size of mass. Using this form of the function we can calculate the coordinate value of G at any point away from the event horizon. To apply this to a black hole we first must find the radial distance of the black hole that gives the required value of Ve and plug this back into the equation to find the coordinate value of G. This in turn can gives us a coordinate value for the gravitational acceleration g at points outside the horizon. So in this way we can calculate how time dilation affects this coordinate acceleration. This explains why outside observers should see objects slow down when approaching an event horizon in strictly mathematical terms.

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Re: Lambert's Cosine Law
« Reply #271 on: 05/04/2015 03:55:44 »
It should be noted that these coordinate equations are only valid for the Schwarzschild metric, That is a non-rotating and uncharged black hole.
« Last Edit: 06/04/2015 12:59:56 by jeffreyH »

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Re: Lambert's Cosine Law
« Reply #272 on: 08/04/2015 21:31:50 »
The attached graph is a tentative attempt to plot the coordinate value of G, the gravitational constant, from infinity to the event horizon of a black hole. This is not a verified equation by any means but is interesting none the less.

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Re: Lambert's Cosine Law
« Reply #273 on: 16/04/2015 01:21:22 »
I have seen it stated succinctly elsewhere that "Energy is the timelike component of the four momentum". So can we derive coordinate Lagrangians from a point near to the value at infinity to a point very close to the event horizon. This will require a coordinate kinetic energy and a coordinate potential energy. The potential energy we can derive using the coordinate value of G. Now it is time to look at coordinate kinetic energy.

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Re: Lambert's Cosine Law
« Reply #274 on: 18/04/2015 22:37:00 »
The relationship between G and the Planck values can be expressed by:

$$G=\frac{\ell_P^3}{m_P\cdot t_P^2}$$

This correlates with the expected entropy of a black hole. That is the cube of the Planck length divided by the Planck mass times the Planck time will give a value for G. However this is circular because you need G to derive the Planck units themselves. What came first the chicken or the egg?

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Re: Lambert's Cosine Law
« Reply #275 on: 26/04/2015 22:06:01 »
If instead we use L instead of $$\ell_p$$ and t instead of $$t_p$$
we have $$G = \frac {L^3}{mt^2}$$. With t set at 1 second and L at 299792458 m we can simplify as $$G =\frac {L^3}{m}$$. To find m we use $$m = \frac {L^3}{G}$$. The attached graph shows the coordinate acceleration towards the event horizon viewed from infinity for a mass of m. The value of m is 4.03726E+35 kg.

The pink line shows a standard calculation of acceleration and the blue line the coordinate value. The turning point in the blue line shows the point at which the observed object falling into the black hole appears to slow down.

The y axis shows acceleration m/s-1 and the x axis radial distance from the event horizon. [Correction]The y axis shows acceleration m/s-2 and the x axis radial distance from the centre of gravity.
« Last Edit: 30/04/2015 01:19:55 by jeffreyH »

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Re: Lambert's Cosine Law
« Reply #276 on: 26/04/2015 22:50:08 »

If true then galactic sized black holes are the only ones that can exist and must have been formed during the early stages of the universe and were responsible for the formation of galaxies. The amounts of mass involved can not exist as ordinary stars and must be a consequence of the slowdown of expansion following the inflationary period.

EDIT: A better source is:

http://www.space.com/858-study-stars-size-limit.html
« Last Edit: 26/04/2015 23:00:16 by jeffreyH »

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Re: Lambert's Cosine Law
« Reply #277 on: 30/04/2015 01:08:49 »
The derivative of GM/r^2 is 2GM/r^3 which will show the rate at which g is changing at varying radial distances. This derivative is also used in Malcolm S Longair's book Galaxy Formation Second Edition. I am about to buy this book to pursue this further. I have attached a graph of 2GM/r^3 for the proposed minimum mass black hole.

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Re: Lambert's Cosine Law
« Reply #278 on: 02/05/2015 05:43:37 »
The compressibility of matter is the crucial point in this investigation. This relates to gravitational collapse.

http://en.wikipedia.org/wiki/Gravitational_collapse
[Gravitational collapse is the inward fall of an astronomical object due to the influence of its own gravity which tends to draw the object toward its center of mass. In any stable body, this gravitational force is counterbalanced by the internal pressure of the body acting in the opposite direction. If the gravitational force is stronger than the forces acting outward, the equilibrium becomes unstable and a collapse occurs until the internal pressure increases sufficiently that equilibrium is once again attained (the exception being a black hole).]

The crucial sentence is "If the gravitational force is stronger than the forces acting outward, the equilibrium becomes unstable and a collapse occurs until the internal pressure increases sufficiently that equilibrium is once again attained".

The key thing is to plot all potential stages of equilibrium for a variety of mass sizes.

Another crucial point is this.

"According to Einstein's theory, for even larger stars, above the Landau-Oppenheimer-Volkoff limit, also known as the Tolman–Oppenheimer–Volkoff limit (roughly double the mass of our Sun) no known form of cold matter can provide the force needed to oppose gravity in a new dynamical equilibrium. Hence, the collapse continues with nothing to stop it."

The Tolman–Oppenheimer–Volkoff limit is then the key to determining if black holes of 3 solar masses can actually form.

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Re: Lambert's Cosine Law
« Reply #279 on: 02/05/2015 06:07:32 »
In discussion of the Tolman–Oppenheimer–Volkoff limit it is instructivbe to note this:

http://en.wikipedia.org/wiki/Tolman–Oppenheimer–Volkoff_limit

[In the case of neutron stars this limit was first worked out by J. Robert Oppenheimer and George Volkoff in 1939, using the work of Richard Chace Tolman. Oppenheimer and Volkoff assumed that the neutrons in a neutron star formed a degenerate cold Fermi gas. They thereby obtained a limiting mass of approximately 0.7 solar masses, [2][3] which was less than the Chandrasekhar limit for white dwarfs. Taking account of the strong nuclear repulsion forces between neutrons, modern work leads to considerably higher estimates, in the range from approximately 1.5 to 3.0 solar masses.[1] The uncertainty in the value reflects the fact that the equations of state for extremely dense matter are not well known. The mass of PSR J0348+0432, 2.01±0.04 solar masses puts a lower bound on TOV limit.]

where "the equations of state for extremely dense matter are not well known".

Discussion of the uncertainty can be found here:

http://en.wikipedia.org/wiki/QCD_matter#Phase_diagram

"The phase diagram of quark matter is not well known, either experimentally or theoretically."

This then opens the debate on lower black hole mass limit.

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Re: Lambert's Cosine Law
« Reply #280 on: 03/05/2015 10:11:32 »
After having read a post in another forum I am going to assume something rather odd. That all particles travel at the speed of light. This is a mathematical device only as NO they don't all travel that fast. However what it does do is make all things equal. The speed of light is also considered to be the speed of gravity. So why not just use the photon? Well the photon is massless. If this assumption is also combined with massive particles then we may learn something from a mathematical derivation. It can always be corrected later to show realistic values.

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Re: Lambert's Cosine Law
« Reply #281 on: 03/05/2015 12:42:57 »
For the purposes of this investigation relativistic mass has to be ignored. The object generating the gravitational field will be a Planck mass Schwarzschild metric black hole. So that we can disregard angular momentum.

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Re: Lambert's Cosine Law
« Reply #282 on: 03/05/2015 12:54:02 »
If we take c=G=1 then a photon orbital becomes 2M which is at the horizon of the black hole. If we were to direct a constant directed light source so that the photons in the orbital increased over time there would come a point where the gravitation generated by the light would start to cancel with the gravitation immediately inside the horizon such that Ve would fall below c. This will become important later.
« Last Edit: 03/05/2015 12:55:45 by jeffreyH »

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Re: Lambert's Cosine Law
« Reply #283 on: 13/05/2015 20:58:43 »
One thing that it would be useful to do would be to determine the forces active in the internal cavity of a mass. I don't think this has been attempted experimentally underground. What it would be nice to be able to do is this.

$$G\sum_{i=1}^{n} \frac{M_i}{r_i^2}\hat{r}$$

However, since these are vectors , at the centre of gravity they are said to cancel so a summation is not possible. A method is needed to determine the action of opposing forces that does not simply assume cancellation. The effect on particles will not be zero. Something must happen due to the outward attractive force that applies in all directions. This is not a trivial exercise nor is it a worthless one. It has implications for the examination of extreme gravitational sources.

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Re: Lambert's Cosine Law
« Reply #284 on: 29/05/2015 23:14:29 »
Going back to the coordinate value of G we can first investigate the relationship between surface area and entropy. If we take the equation $$\frac {4\pi GM}{g} = 4\pi r_s^2$$ then we can find a value of g that will give the area of the surface of the event horizon. We can then substitute the value of coordinate G that would apply at the horizon. In calculating coordinate G it was found that G does not tend to zero as the radius approaches rs. This indicates that the size of a black hole indicated by the range of the apparent horizon when viewed by a remote observer will be smaller than expected. This could be one explanation of why Sag A* did not consume the G2 gas cloud. If the black hole is actually larger than we think but compressed into a smaller area due to spacetime compression then the tidal forces will be lower than expected.

EDIT: Of course as well as the coordinate value of G we also need to substitute the coordinate value of g.
« Last Edit: 29/05/2015 23:21:40 by jeffreyH »

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Re: Lambert's Cosine Law
« Reply #285 on: 13/06/2015 15:17:33 »
Going back to the idea of all masses moving at light speed we then have an equation for kinetic energy of the form $$\frac{1}{2}Mc^2$$. This is obviously neglecting relativistic mass. How is this useful? That is the next step in the analysis.
« Last Edit: 13/06/2015 15:19:06 by jeffreyH »

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Re: Lambert's Cosine Law
« Reply #286 on: 28/06/2015 11:27:42 »
During the time that the electromagnetic field was being quantized Planck came up with $$E = h\upsilon$$ where h is Planck's constant and $$\upsilon$$ is wave frequency. This was verfied experimentally via Compton scattering and the equation $$\lambda^{\prime} = \lambda + \lambda_c (1 - cos \theta)$$. Here $$\lambda$$ is the incident wavelength $$\lambda^{\prime}$$ is the scattered wavelength and $$\theta$$ is the scattering angle. Now $$\lambda_c = \frac{h}{mc}$$ where m is the particle mass and c is the speed of light. What was shown above was (1/2)mc^2. How does this relate? That will be answered next.

EDIT: $$\lambda_c$$ is the Compton wavelength of the target particle.
« Last Edit: 28/06/2015 11:47:17 by jeffreyH »

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Re: Lambert's Cosine Law
« Reply #287 on: 28/06/2015 12:45:55 »
Quote from: jeffreyH
During the time that the electromagnetic field was being quantized Planck came up with $$E = h\upsilon$$ where h is Planck's constant and $$\upsilon$$ is wave frequency.
You have to be careful with this. It wasn't Planck who quantized the EM field. That happened when quantum field theory was created much later on. $$E = h\nu$$ was postulated by Planck as the quantization condition for harmonic oscillators in a black body. It was Einstein who quantized the electromagnetic wave, i.e. light
Quote from: jeffreyH
This was verfied experimentally via Compton scattering ...
Compton scattering confirmed that light was made of particles.

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Re: Lambert's Cosine Law
« Reply #288 on: 28/06/2015 15:17:11 »
Quote from: jeffreyH
During the time that the electromagnetic field was being quantized Planck came up with $$E = h\upsilon$$ where h is Planck's constant and $$\upsilon$$ is wave frequency.
You have to be careful with this. It wasn't Planck who quantized the EM field. That happened when quantum field theory was created much later on. $$E = h\nu$$ was postulated by Planck as the quantization condition for harmonic oscillators in a black body. It was Einstein who quantized the electromagnetic wave, i.e. light
Quote from: jeffreyH
This was verfied experimentally via Compton scattering ...
Compton scattering confirmed that light was made of particles.

Thanks for that clarification Pete. It makes things clearer for the reader.

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Re: Lambert's Cosine Law
« Reply #289 on: 25/07/2015 12:28:41 »
I was going to progress this thread on the theme of light speed particles. However I wish to post an equation without any derivation. This has come from various conclusions which I will get to later. The equation is this.

$$F^2 = \frac{GmPm_e^2}{r-rs}$$

Here F is force, mP is the Planck mass, me is the mass of the electron, rs is the Schwarzschild radius of the Plank mass and r is the radial distance of the electron from the event horizon.

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Re: Lambert's Cosine Law
« Reply #290 on: 25/07/2015 12:46:56 »
Please note: Before anyone does dimensional analysis and complains that this is not a force equation, there is a parameter missing.

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Re: Lambert's Cosine Law
« Reply #291 on: 25/07/2015 13:02:36 »
With the missing parameter (tP) Planck time we can derive the following definite integral.

$$\int_{r_s+lP}^{r_s+2lP} \frac{GmPm_e^2}{tP^2(r - r_s))} dr$$

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Re: Lambert's Cosine Law
« Reply #292 on: 26/07/2015 20:44:58 »
The equation for F2 is not the same as for gravitational potential energy which is of the form:

$$\int_r^\infty \frac{-GMm}{r^2}dr = \Big[\frac{GMm}{r}\Big]_r^\infty = \frac{GMm}{\infty} - \frac{GMm}{r}$$

The reason for this is the derivation used to arrive at the F2 form. The major difference is in the power of the radius r.

jeffreyH

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Re: Lambert's Cosine Law
« Reply #293 on: 26/07/2015 23:57:13 »
If we take the equation for a circular orbit

$$v = \sqrt{\frac{G(M+m)}{r}}$$

we can derive the following

$$v^2 = \frac{G(M+m)}{r}$$

multiplying both sides by m gives

$$mv^2 = \frac{Gm(M+m)}{r}$$

dividing both sides by 2

$$\frac{1}{2}mv^2 = \frac{Gm(M+m)}{2r}$$

simplifying gives

$$\frac{1}{2}mv^2 = \frac{GMm+Gm^2}{2r}$$

This results in the total combined kinetic energy of the orbit. Under the right conditions and with the right modifications could this be a way to quantize the gravitational field?

jeffreyH

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Re: Lambert's Cosine Law
« Reply #294 on: 29/07/2015 22:30:28 »
Some of the subtleties and difficulties involved in the quantization process can be seen on the following page.

http://edition-open-access.de/sources/5/25/

jeffreyH

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Re: Lambert's Cosine Law
« Reply #295 on: 03/08/2015 21:59:26 »
In partial relativistic terms we can start with $$\frac{1}{2}mv^2 = \frac{GMm}{2\left( r - r_s \right)}$$. However we need to modify this to be $$\frac{1}{2}\gamma mv^2 = \frac{GM\gamma m}{2\left( r - r_s \right)}$$ where M is much larger than m.

jeffreyH

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Re: Lambert's Cosine Law
« Reply #296 on: 03/08/2015 22:10:15 »
From there we can reformulate first as $$\frac{GM}{v^2} = r - r_s$$ and then as $$\frac{GM}{v^2} + r_s = r$$.

NOTE: This only applies to the Schwarzschild metric and circular orbits.
« Last Edit: 03/08/2015 22:13:10 by jeffreyH »

jeffreyH

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Re: Lambert's Cosine Law
« Reply #297 on: 04/08/2015 00:04:26 »
When the orbit is coincident with the horizon of the black hole v can take on only two values. Zero and infinite. Whilst infinite is the expected value, zero should also be given due consideration. If we look at the situation where maximum kinetic energy is given by $$\frac{1}{2}mc^2$$ and then take v to be half light speed in the orbital equation this then equates to a momentum of $$\frac{1}{2}mc$$. If considering the Planck mass a velocity of 1/2c would give as twice the Schwarzschild radius. Therefore in a distance of 2 Planck lengths the velocity has to either change to infinite or zero. In my personal opinion zero is a better bet. Since light speed is 1 Planck length in 1 Planck time. This gives a particles 2 Planck lengths in which to come to a stop. Since it would have to be traveling 1/2 a Planck length in 1 Planck time this seems more reasonable than achieving an infinite number of Planck lengths.

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Re: Lambert's Cosine Law
« Reply #298 on: 04/08/2015 00:12:18 »
The crucial point to consider is the transition between r at 2rs and r at 1.5rs.

jeffreyH

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Re: Lambert's Cosine Law
« Reply #299 on: 08/08/2015 01:15:08 »
There are two types of kinetic energy that can be considered in this model. The energy required to maintain a circular orbit at a particular distance and the energy required to achieve escape velocity at a particular distance. Looking at escape velocity it is interesting to look at the magnitude of change in the field perpendicular to the surface but working towards the surface from a remote radial distance. This way discrete differences can be shown by the relationship of (r+1)^2/r^2 where the units are in integer increments of the objects radius. The graph of this function, which applies to any size of mass, is attached. It starts at 20 radial distances and ends at the objects surface.