# The Naked Scientists Forum

### Author Topic: Lambert's Cosine Law  (Read 51421 times)

#### 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.

#### jeffreyH

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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.

#### jeffreyH

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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.

#### jeffreyH

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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 »

#### 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?

#### jeffreyH

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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.

#### jeffreyH

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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.

#### jeffreyH

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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.

#### jeffreyH

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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 »

#### 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 »

#### 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.

#### jeffreyH

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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]

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.

#### jeffreyH

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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.

#### jeffreyH

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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.

#### jeffreyH

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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 »

#### 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.

#### jeffreyH

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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.

#### jeffreyH

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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.

#### jeffreyH

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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

#### jeffreyH

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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.

#### jeffreyH

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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.

#### jeffreyH

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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 »

#### 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.

#### jeffreyH

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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.

#### jeffreyH

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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:

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 #274 on: 18/04/2015 22:37:00 »