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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 #100 on:
16/10/2014 04:30:18 »
One thing that comes out of this is that the radial distance from a source will be affected by length contraction which is not catered for in gravitational calculations such as that for g or escape velocity. For a distant observer this would be important but not an observer local to the frame that the radius refers to. The factor of 50 must vary as the radius varies so any function to adjust for change in apparent length must operate on this factor as well as the radius. This could be one way of integrating gravity with quantum mechanics.
Note: To achieve the integration the starting point to choose would be the imaginary surface of the event horizon where a fixed value exists.
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Last Edit: 16/10/2014 04:34:06 by jeffreyH
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Re: Lambert's Cosine Law
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Reply #101 on:
16/10/2014 04:58:34 »
I have checked the equation (hbar*M)/(lP^2*c^2*L*r^2) and it was a terrible guesstimate. I will go over it again to find out where I went wrong. [
]
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Re: Lambert's Cosine Law
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Reply #102 on:
16/10/2014 23:01:59 »
Right. L = the distance a photon travels in 1 second. c = the speed of light. lP = the Planck length. hbar is the reduced Planck constant. Our factor of 1 is then [50*L*c^3*lP^2]/hbar. This I have checked and it is near as damn it to 1. So this factor is then applied to M/[50*L*r^2] which is our calculation for g (gravitational acceleration).
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Re: Lambert's Cosine Law
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Reply #103 on:
16/10/2014 23:24:30 »
That then gives us [M*c^3*lP^2]/[hbar*r^2] for our gravitational acceleration. What the units are I have no clue so any help would be appreciated. It may be totally invalid.
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Re: Lambert's Cosine Law
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Reply #104 on:
16/10/2014 23:50:34 »
Since the factor of 50 has canceled what we can say now is there is no energy loss. This means that it is the field density alone that affects its strength. So as in the case of the electromagnetic field the gravitational field must have a force carrier. This will be the spin 2 boson.
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Re: Lambert's Cosine Law
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Reply #105 on:
18/10/2014 09:23:19 »
Taking 2 candidate galaxies and their central supermassive black holes it should be possible to calculate a time dilation gradient between them. The unknowns are the masses of each black hole and the extent of the event horizons of each one. If these were known then our starting point at each of the galaxies would be the imaginary surfaces of the spherical horizons. This has to be imaginary due to any bulging caused by angular rotation. The effects of recessional velocity and any intervening masses would affect this gradient over time. It would be best to start with an idealized model with known variables. This would be made more difficult due to the velocities of the black holes. This causes the point at the horizons surface to be modified away from a fixed universal value. Isn't relativity fun?
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Re: Lambert's Cosine Law
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Reply #106 on:
18/10/2014 09:36:16 »
The solution of course is to have an observer midway between the galaxies and moving relative to both in order to maintain an equal distance from each one. This then becomes our second fixed point and cancels the effects on each event horizon.
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Re: Lambert's Cosine Law
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Reply #107 on:
18/10/2014 21:57:27 »
I am now going to attempt to put together a theory of quantum gravity. Wish me luck.
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Re: Lambert's Cosine Law
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Reply #108 on:
21/10/2014 00:19:57 »
Well I thought nobody was paying attention so I'll come clean. Remember [M*c^3*lP^2]/[hbar*r^2]? Well the c^3 isn't really that. It has L tucked away which is a scalar. What it should be is [M*c^2*L*lP^2]/[hbar*r^2]. So we now have Mc^2 in our gravitational equation. Quantum gravity is easier to deal with at lower energies and includes too many infinite variables at higher energies at the Planck scale. What happens if we have energy implicit in our equation and an intimate link to the Planck dimensions?
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Re: Lambert's Cosine Law
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Reply #109 on:
22/10/2014 00:31:08 »
We can think of c as being L/1. That is 1 light second. We then end up with (L/1)^2 standing in for c^2. This can be viewed as [L^2*lP^2]/1. Square area over time. Multiplying this result again by L gives us a reduced cubic area over time which includes an energy component implicitly by virtue of the mass in the numerator. This resolves to a spatial containment of energy reduced by a proportionality at the Planck scale. This is exactly what we need to start down the path of quantum gravity. By removing the need for G and replacing this with energy we can more easily integrate this with the wave equation and consequently electromagnetism.
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Re: Lambert's Cosine Law
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Reply #110 on:
23/10/2014 23:03:38 »
Reviewing the equation [M*c^2*L*lP^2]/[hbar*r^2] can show us some important points. (L/1)^2 can also be viewed as (dL/dt)^2 where t cannot go below 1 as this would make the speed of light superluminal. So dt >= 1 and then dL <= L giving both time dilation and length contraction implicit within this function. Then M(dL/dt)^2 gives us a changing energy profile as the properties of the system change. Our L*lP^2 gives us a volume of space over which the change of properties occur. Having hbar as the denominator seems to indicate a Planck variant scale in changing frames. Which implies a constant Planck scale over which mass density changes. This is why relativity has not been resolved. The curvature has been viewed in the wrong way. So now we can separate the functions into [M(dL/dt)^2]*[(L*lP^2)/(hbar*r^2)]. This is an energy against volume equation relating energy density to gravitation as both time dilation and length contraction vary. It is also directly connected to the effects of gravity on the photon.
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Re: Lambert's Cosine Law
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Reply #111 on:
24/10/2014 01:09:01 »
Considering the equation [M(dL/dt)^2]*[(L*lP^2)/(hbar*r^2)] it is apparent that r, the radial distance from the source, has to bear a proportionality to dL/dt as this is the distance from the source of gravitation and therefore relates to the strength of the field. The dilation and contraction must fit with experimental observation. In the case of length contraction this is difficult to achieve.
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Re: Lambert's Cosine Law
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Reply #112 on:
24/10/2014 01:11:23 »
Once the proportionality is established this then forms the basis of the time dilation gradient between black hole event horizons. This will include a full model of the behavior of light between these fixed points.
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Re: Lambert's Cosine Law
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Reply #113 on:
24/10/2014 23:03:59 »
There is one more pertinent point to make. To measure in a frame dependent way we would use the form [M(L/t)^2]*[(L*lP^2)/(hbar*r^2)] where L/t does not vary. So that local observers all read the same result. This is equivalent to the form GM/r^2. For observers measuring remote frames then the form is [M(dL/dt)^2]*[(L*lP^2)/(hbar*r^2)].
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Re: Lambert's Cosine Law
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Reply #114 on:
25/10/2014 19:02:21 »
To separate out time dilation or length contraction we can vary time whilst keeping distance constant or vary distance whilst keeping time constant. I have attached two graphs showing this. For the time dilation graph the x axis starts at 1 second which is the local frame. Any point away from this shows the decrease in change over time. At the 2 second point, for instance, it will take 2 seconds for an action to happen that would take 1 second in the local frame. This can also be seen in the length contraction graph. The curves are not equivalent and operate differently. We cannot read off equivalent energy values from both graphs and use the time and distance values taken in our original equation as the energy would differ. The next step will be to develop the dilation/contraction equation.
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Re: Lambert's Cosine Law
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Reply #115 on:
26/10/2014 18:31:28 »
To correct the equation we cannot only vary length or time whilst holding the other constant. We start with E = M(L/t)^2. This can be rearranged to give SQRT(E/M)*t = L. This graph is linear and energy is now constant. This is the time dilation gradient. The graph is attached.
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Re: Lambert's Cosine Law
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Reply #116 on:
26/10/2014 18:33:11 »
Since we already removed the factor of 50 earlier this shows conservation of energy in gravitational interactions. NOTE: This is a graph of energy distribution over time and distance. It relates to the speed of energy flux.
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Last Edit: 26/10/2014 18:52:39 by jeffreyH
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Re: Lambert's Cosine Law
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Reply #117 on:
26/10/2014 21:14:32 »
Here again the local observer is at 1 second and the distance is L.
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Re: Lambert's Cosine Law
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Reply #118 on:
26/10/2014 21:27:21 »
We can see from the graph that if we look at say the 2 second position on the graph and read off the distance we get twice L. This is not length contraction. The speed of light has not changed at all. What does this show us? Well energy MUST vary otherwise gravitation has no effect. The first two graphs separated out time dilation and length contraction and showed a difference in those curves with respect to energy. This means there is a factor missing. This could indicate a negative energy component. This would be our gravitational energy. Without this we cannot validate the effects of the gravitational force. Without it gravity doesn't exist.
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Re: Lambert's Cosine Law
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Reply #119 on:
26/10/2014 21:44:00 »
I asked a question in the the Physics, Astronomy and Cosmology forum as to whether length contraction exists. The next derivation should prove that it doesn't. This is in fact an error that has caused a stall in relativity.If you look at the graph where length is held constant and time changes we see a curve that looks like an inverse square equation. This is absent when we hold time constant and vary length. I will post this another time.
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Re: Lambert's Cosine Law
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Reply #120 on:
27/10/2014 01:15:07 »
We can rearrange the equation for g to get the time factor as SQRT([M*L^2]/g*[L*lP^2]/[hbar*r^2]). This graph is also linear and starts at L/1. This can be thought of as the speed of light at an infinite distance from any gravitational source. Moving horizontally right is equivalent to moving into an intensifying gravitational field where light is effectively slowed down as viewed by an observer at infinity.
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Re: Lambert's Cosine Law
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Reply #121 on:
27/10/2014 01:22:47 »
The time axis in the above graph will reach infinity at the event horizon. Any marked effects represented by the plot will only occur near massive dense objects. The attached plot of earth's g shows no noticeable difference on the time axis as this will be measured in nanoseconds and will be unobservable under normal conditions. This can be considered a linear relationship in a less intense gravitational field.
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Re: Lambert's Cosine Law
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Reply #122 on:
27/10/2014 08:12:00 »
Kinetic energy is given by E = (1/2) mv^2. E is kinetic energy, m is the mass and v is velocity. We can derive momentum as p = √(2Em). The deBroglie equation for wavelength is λ = h/p where here λ is the wavelength, h is Planck's constant and p is momentum. This can be written as λ = h/√(2Em). Since we have already derived t from the gravity equation then we can also derive m. Then time and kinetic energy will be variables in the wave equation. This can be used to show the evolution of the wave under the influence of gravity. In which case m becomes the mass of the particle with r^2 indicating the particle radius squared. The value of g for the particle is most important. This is our way into quantum gravity.
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Re: Lambert's Cosine Law
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Reply #123 on:
27/10/2014 20:16:50 »
The most profitable investigation would be into the relationships between mass-energy, kinetic energy and time.
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Re: Lambert's Cosine Law
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Reply #124 on:
30/10/2014 23:18:18 »
I am now in the position that I need to derive the mass equation. This is to test an hypothesis that gravitation is merely a catalyst and not a force in its own right.
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Re: Lambert's Cosine Law
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Reply #125 on:
31/10/2014 00:34:14 »
So we end up with M =
^{-1}
. Looking at it this way the g force is inherent in the mass with gravity as the catalyst.
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Re: Lambert's Cosine Law
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Reply #126 on:
31/10/2014 01:19:17 »
Finally we can rearrange as M = g
^{-1}
. So now we only need modify g and r to find the mass contained within a radius that will produce a particular g force. We can attempt to apply this to a particle or to a black hole. There is a direct relationship to density inherent in the equation. This neglects time dilation and length contraction but it is straightforward to modify to take these into account. However the complexities rises with four independent variables.
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Re: Lambert's Cosine Law
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Reply #127 on:
31/10/2014 01:51:26 »
Our mass equation can then be substituted into the momentum equation p = √(2Em) where L/t does matter as this has an effect due to both time dilation and length contraction. It can also be a way of showing the effects on kinetic energy. We have to be careful in the application of this formula as it is the mass energy that changes in order to have an effect on the kinetic energy. Which is the wrong way round. Unless we consider it a change in flux rate.
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Re: Lambert's Cosine Law
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Reply #128 on:
31/10/2014 20:59:58 »
I think that gravity as a catalyst is unworkable for a variety of reasons.
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Re: Lambert's Cosine Law
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Reply #129 on:
01/11/2014 01:36:43 »
To get energy Mc^2 we arrive at
^{-1}
. Which we can rearrange as Mc^2 =
^{-1}
. The deBroglie wavelength being h/p we should be able to use these equations to model the effect of gravity on particle waveforms. For momentum we get p = √(2kEg/c^2[{lP^2/r^2}{L/hbar}]^-1).
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Re: Lambert's Cosine Law
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Reply #130 on:
01/11/2014 03:18:47 »
Having the Planck area lP^2 in the equation is of interest.
http://en.wikipedia.org/wiki/Planck_length
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"The Planck area, equal to the square of the Planck length, plays a role in black hole entropy. The value of this entropy, in units of the Boltzmann constant, is known to be given by A/4lP^2, where A is the area of the event horizon. The Planck area is the area by which a spherical black hole increases when the black hole swallows one bit of information, as was proven by Jacob Bekenstein."
The term lP^2/r^2 can therefore link our mass-energy to a density function that relates to the horizon black hole.
Another important point on this page is this.
"In doubly special relativity, the Planck length is observer-invariant."
So is length contraction valid or are space and time separate.
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Re: Lambert's Cosine Law
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Reply #131 on:
02/11/2014 15:11:30 »
Upon reading further there are serious problems with double special relativity so I am going to ignore it for now. The energy equation is important as it relates energy to gravitation rather than mass. As the photon is massless this is the only way we can use the equation with the photon wave equation. I will be looking at Pete's relativistic mass page soon to see how it can all be combined.
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Re: Lambert's Cosine Law
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Reply #132 on:
04/11/2014 23:08:29 »
We can finally rearrange the mass equation from M = g
^{-1}
to M = g
to remove the reciprocal with units of joule second^2 metres^-2.
Correction the units are joule second metres^-2. And of course 1 (joule second) per (square metre) = 1 kg / s
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Re: Lambert's Cosine Law
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Reply #133 on:
05/11/2014 00:06:09 »
So now if we multiply this mass value by 1 second we get our kg value. Since our 1 second value relates to light speed. As time dilation increases our time value increases. Multiplying by the new value gives our increase in mass due to time dilation and increasing velocity.
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Re: Lambert's Cosine Law
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Reply #134 on:
05/11/2014 01:45:09 »
This of course gives us mass flow rate A.K.A mass flux. This is usually used in fluid dynamics although there is no reason not to use it in other ways.
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Re: Lambert's Cosine Law
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Reply #135 on:
06/11/2014 00:30:03 »
The mass equation M = g
can be rearranged in the same way the energy equation was to become M = g
. The term
gives us the number of Planck squares in our radial square area and therefore reduces the magnitude to a scalar Planck multiplier. If the value of r were 2lP this would equal the rs value of the event horizon of a Planck mass black hole. The scalar value then becomes 2. Interestingly this is the factor in the rs equation 2GM/c^2. To derive the Planck mass value from this expression would simply require finding the corresponding value for g.
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Re: Lambert's Cosine Law
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Reply #136 on:
06/11/2014 00:43:24 »
To hold r at 2lP and simply increase g is the same as increasing mass within a set volume. Therefore increasing density. When we get to the point where instantaneous acceleration equals the speed of light we will have found a value of great interest. This will be the point of no return at which a singularity is inevitable.
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Re: Lambert's Cosine Law
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Reply #137 on:
08/11/2014 23:07:00 »
We have kinetic energy e
_{k}
=
and potential energy e
_{p}
= -
. These equations balance as energy is converted from potential to kinetic and visa versa. What is not taken into account here is the effect of time dilation due to a changing gravitational field. The derived equations above can however take this into account. Using energy instead of mass in these equations is the only way to proceed when dealing with massive or massless particles.
To verify length contraction an effect upon kinetic energy must be present. This must be equivalent to a loss of energy when viewed from a remote frame. I intend to show that there is no such effect upon the overall kinetic energy of an object moving through a gravitational field. As the kinetic energy reduces there is an equal amount of increase in the potential energy. This can be thought of as the kinetic energy being negative and the potential energy as positive. Kinetic energy becomes positive only when approaching a mass through its gravitational field. Which is why no force is felt. When accelerating outside of a significant gravitational field, a mass carries with it its own gravitational field and so does feel a force as it cannot be moving through its own field. Its kinetic energy is inherently negative.
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Last Edit: 08/11/2014 23:22:50 by jeffreyH
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Re: Lambert's Cosine Law
«
Reply #138 on:
09/11/2014 00:20:54 »
So we arrive at e
_{p}
=
. We now need the same form for e
_{k}
. Here we have m^2 s^-2 but with hbar having joule second units which signify angular momentum. So how do we square this with a kg unit?
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Last Edit: 09/11/2014 01:14:39 by jeffreyH
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Re: Lambert's Cosine Law
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Reply #139 on:
09/11/2014 01:34:42 »
Well one answer from yahoo is:
https://answers.yahoo.com/question/index?qid=20110820184453AABGGvX
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"Do you really mean m^2/s^2 and not m/s^2, which is just acceleration.
But m^2/s^2 could mean many things. For example, during the recent nuclear disaster in Japan, the radiation dose rate received by workers and citizens was measured in Seiverts/hour. The unit of dose, the Seivert has the dimension m^2/s^2, which is equivalent to Joules (energy) per kg:
Example: Energy(Joules) = force x distance (Nm) = mass x acceleration x distance (kg m^2/s^2)
Energy per mass = Dose (Seivert) = mass x acceleration x distance / mass (m^2/s^2)"
So this could be thought of as energy per mass. What of the hbar in our denominator in joule seconds?
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Re: Lambert's Cosine Law
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Reply #140 on:
09/11/2014 01:50:29 »
Ignoring the hbar for now what of our energy equation e
_{p}
=
. What use is it? We if we consider the g factor to be the gravitational acceleration at the surface indicated by the radius r the we can have an external g
_{e}
value. This value will be the g force from an external mass acting upon the local mass in e
_{p}
=
. We can determine the change in e
_{p}
by using
. If g>g
_{e}
then our potential energy is positive. if g<g
_{e}
then our potential energy is negative. What happens when g=g
_{e}
. This is the situation where the fields are said to cancel. This is equivalent to reducing the potential energy by cancelling g in the equation. So that zero gravity situations will decrease potential energy. Where does it go? Usually we would think of this as becoming kinetic energy. This is one puzzle whose solution will ease our way into a theory of quantum gravity.
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Re: Lambert's Cosine Law
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Reply #141 on:
09/11/2014 02:49:09 »
Because we have a cancellation of g our kinetic energy must be internal. This indicates a low point in time dilation. So that within a hollow cavity at the centre of a mass time dilation will be at its lowest. Since all the forces of the outer mass cancel then the only g force present is that of any particle at the centre of the cavity. This bears out the theory that an event horizon must start at the centre of mass and work its way outwards. This also indicates that composite particles should merge in order for the cancellation to operate outward. If the particles were still individual then a full cancellation would not apply for all particles. Only a unit mass with a single gravitational force can cause such an inward collapse.
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Re: Lambert's Cosine Law
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Reply #142 on:
09/11/2014 04:05:12 »
Having sat and thought about this the equation
is not valid at all. While the situation in a hollow cavity would still apply this subtraction of the external value of g
_{e}
in the above is incorrect.
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Re: Lambert's Cosine Law
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Reply #143 on:
09/11/2014 23:11:38 »
The equation for Planck energy is SQRT([hbar*c^5]/G). This can be derived from the above as [g*h]/[pi*L]. This dispenses with the square root and the gravitational constant and returns the value in joules. Importantly the Schwarzschild radius of the Planck mass is inherent in the equation and so can be rearranged to find g. Our value for g is then 2.77943185E+51. This accelerates mass to superluminal velocities according to this result but does it. Not if we take length contraction into account. The speed of light is never actually violated. Therefore length contraction must exist. This also implies that kinetic energy has a different relationship to gravity than has been thought previously. When moving through a length contracted frame the locally viewed acceleration will appear faster than is apparent to an observer in a remote frame. This will only be noticeable near to a dense massive object with an intense gravitational field. Solving Einstein's field equations in this situation becomes a real challenge. Do we even need to?
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Re: Lambert's Cosine Law
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Reply #144 on:
10/11/2014 00:29:00 »
The g force works out to be very approximately pi*c^6. What significance this has I have no idea. Except that it must relate to length contraction in intense gravitational fields.
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Re: Lambert's Cosine Law
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Reply #145 on:
10/11/2014 01:17:47 »
A while ago I came across the work of Paul Marmet. The significance of his work has only become apparent to me very recently. He was an opponent of general relativity so was largely ignored by the mainstream. I hope to vindicate him not by proving general relativity wrong but by modifying it.
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Re: Lambert's Cosine Law
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Reply #146 on:
14/11/2014 07:54:28 »
I have copied two of my posts from other threads.
1) I don't want to do that. I want an equation of the form
in three dimensions. Here energy is not simple to describe. The components of mass, potential and kinetic energy interact with the gravitational field.
2) If we take the centre of gravity of a perfect sphere and have a plane running through it. We can then define x, y and z axes tilted so each is axis has the same angle to the plane. If we then set a path that when projected onto the intersection of pairs of axes is at 45 degrees all axes that describe the path perpendicular to the plane then change at the same rate. As a baseline for mapping the effects of rates of change this can map a straight line path. This can then be adapted for curved trajectories. Extending this path out to an imaginary spherical surface the mass within the surface can be defined to be of any size with a radius of choice. Comparisons are then easy to make against the baseline straight path. It would be interesting to see what effects we could model on the interchange of energy under various conditions. To include the electric, magnetic and gravitational fields. With an equivalent value for rate of change at equidistant points the effects on energy of multi-mass systems would be fairly straightforward. Just an idea.
Taken together these two ideas can ultimately produce equations of the type derived by Maxwell for the gravitational field. Very early in this thread a gravitational component replaced the permittivity/permeability factor in one of Maxwell's equations.
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Re: Lambert's Cosine Law
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Reply #147 on:
18/11/2014 23:47:00 »
Just as g can be shown to be an intrinsic part of a mass equation can we include a factor describing the de Broglie wave equation itself? If so maybe then we can show exactly how particles behave in a changing gravitational field.
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Re: Lambert's Cosine Law
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Reply #148 on:
28/11/2014 23:24:03 »
E
_{0}
= m
_{0}
c^2 relates rest energy to invariant rest mass. We have derived e
_{p}
=
for potential energy. What if we rearrange like this? e
_{p}
/g =
. Here g is the gravitational acceleration AT THE SURFACE of a mass. It cannot be anywhere else. Here g is an acceleration in square seconds, c^2 is a squared velocity. WE can say that e
_{p}
/g = e
_{x}
where we do not know what x represents. We know it must represent a reduced mass and therefore a reduced energy but cannot be separated from the mass from which it was derived. However a change in the dimension of the radius of the mass will change the inherent value of g at its surface. This also means a change in density. If the radius increases so g decreases and the reduced mass value is larger. If the radius decreases then the value of the reduced mass gets smaller. So therefore the proportion of mass that is involved directly in the gravitational force then increases as is shown by other gravitational equations indirectly. This shows directly how density varies the gravitational force via a redistribution in the balance of mass involved in gravitational interactions.
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Re: Lambert's Cosine Law
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Reply #149 on:
28/11/2014 23:27:55 »
The only 2 values that appear to change in the above equation are mass radius and g at the surface. Here lies the first problem. How do we decide on the radius of a particle? It isn't like a beach ball. In fact how do we really know what it is like or how it is distributed. The only way we can proceed now is via the wave equation that is well defined and experimentally verifiable. Can it be done? At this point I don't know.
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