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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
«
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
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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/[2r
s
+L/2
]
However some caution is necessary. The kinematic equation for distance traveled due to free fall is:
d = v
i
t + (1/2)at^2
v
i
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
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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 = v
i
t + (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
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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 Mc
2
G
3
. 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 Mc
2
G
3
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
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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
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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
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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
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Reply #268 on:
28/03/2015 17:11:42 »
To proceed further it would be highly advantageous to read this page through completely.
http://people.bu.edu/gorelik/cGh_FirstSteps92_MPB_36/cGh_FirstSteps92_text.htm
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Re: Lambert's Cosine Law
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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
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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
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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
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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
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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
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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
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Reply #275 on:
26/04/2015 22:06:01 »
If instead we use L instead of
and t instead of
we have
. With t set at 1 second and L at 299792458 m we can simplify as
. To find m we use
. 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
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Reply #276 on:
26/04/2015 22:50:08 »
To really appreciate what this means please read the following.
http://www.google.co.uk/url?url=http://genesismission.jpl.nasa.gov/gm2/mission/pdf/Giantstars.pdf&rct=j&q=&esrc=s&sa=U&ei=u1s9VeKTGcL4atibgMAL&ved=0CBsQFjAB&sig2=FVpc1EFdKKfTZIo8srjslw&usg=AFQjCNHVd9qJvEk-oK6wJRBUlnbXhZ-B2Q
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
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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
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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
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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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