# Thread: Time, Matter, and Gravity

1. ## Time, Matter, and Gravity

Morris Anderson has posted his "the universe in wave equations" book on the web.

If you are seriously debunking, or looking for alternatives to the BB model, this is a very promising one. (If you do not have broadband, you may only want to view his presentation, the paper is 132 pages of HTML so, click on the link and start another pot of coffee...)

There are a lot of things I like about Morris's paper:

It is based upon solitons - matter as standing waves, just like Jacque Moret-Bailly's.

He is able to correctly model virtually all gravitational effects using wave equations, including those addressed by general relativity.

It is unbelievably simple - making GR predictions has never been so easy.

Not all of his predictions agree with GR, which is also a plus in my book, because he is able to increase the mass and density of objects without creating a black hole: If this is correct, it explains how Quasars could be intrinsically redshifted by gravity to very high redshifts - this is not possible with GR.

The only real heartburn I have with it is his treatment of inertial energy - he does not agree with the equivalence principle, and isolates gravity as a force independent of inertial fields. (I think it is likely inertial energy is not independent of gravitational forces.) This leads to a different definition of space, although when the dust has settled, we may find the concepts are equivalent.

Morris and I both presented in the same 'Gravity' session of the APS Four Corners chapter conference. My presentation was on "Twisting Einstein to Assign Causality", a concept we have been kicking around on the "Gravity Behaving Badly" thread. We were both quite stunned, because even though we started from opposite ends of the universe, we are within one or two pages of each other.

And we all wonder what happens next.

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HI Jerry,

Thanks for the link to Anderson’s work. The prediction of a red shift from the sun, derivation of E = mcc and the rather straight forward prediction of the advance of the perihelion of the planets indicates his work should get more attention than it has received so far.

Anderson defines time as “ the position where something is”

After describing time “Future, present and past”, he puts forth the following postulate.

Postulate 1.

For any TIME (past, present and future), the position of any object with respect to matter is unique.

For me, Anderson’s use of this measure of time is a historical measure of time. Each point in spacetime is found at a unique location in historical time. It is my opinion that this historical measure of time, (also called Cosmic time or by me “absolute time”) is a separate unique dimensional measure of time. One dimension of time is the time interval between points as expressed by the speed of light. This measure of time is not complete in any philosophical description of nature. Besides the time interval separating points, each point has a unique location in history. For the most part, Anderson’s derivations appear to be based upon a historical use of the measure of time.

Also, on page 82 Anderson should have placed the predicted deflection for starlight along the side of the sun to be ˝ that observed based upon a Newtonian model, just a clerical error of no real significance.

Snowflake

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The view expressed in this book can be compared to the one described in the new topic (presenting my paper): "The Four Space-times Model of Reality". Readers must take into account that there are experimental proofs showing that space-time is a single entity. In Special Relativity and General Relativity there are four dimensions. Relativity says that Time is a dimension, not a parameter.
Few sentences from my paper:
Treating space and time the same way is the scope of Euclidean Special Relativity (ESR). The metric of flat space-time in ESR has the signature ++++, therefore our 3+1 space-time is described embedded in the 4-space.
In ESR there is a single 4-space in which all particles, for which Lorentz transformation is applicable, travel at the speed of light.
Time t is an integral local function of changes of the four space coordinates, dimensionally aligned by c. Time exists if there is motion in the 4-space.
ESR has been derived from SR, therefore it inherits a partial experimental validation from SR. 4DO, that is a theory of gravity in 4-space, inherits a partial experimental validation from GR. In fact it is possible to convert the metrics of GR into refractive indices and both theories can give equivalent or comparable prediction (Almeida, 2004b;
Montanus, 2001). Obviously, the interpretation of the results is different.

IN EUCLIDEAN RELATIVITY THE AMBIGUOUS NATURE OF TIME IS SOLVED. EVERYBODY CAN UNDERSTAND WHAT "TIME t" REALLY IS.

Best Regards

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Very interesting development by Morris G. Anderson in his "Time, Matter, and Gravity", though I've only gone through parts of it, the definitions and illustrations, and his postulates. Also read his notes on "Wave Propagation in a Gravitational Field".

It seems that Anderson redefined the usual "time, length, and force" all in terms of light "c", which makes a lot of sense, since everything we see and feel and live within is made of light. It's all Energy! I have not yet delved into it sufficiently to see what his redefinitions will do to Gravity, though I sort of understand how he redefines it with c. I also am puzzled that he does not treat magnetism in his treatise, though I might just have missed it, or not gotten there yet. Of course, Maxwell's equation automatically incorporates magnetism in his Em/ Bm = c.

I think the real test of this new way of measuring using light c is how it works in its predictive capacity. If it can do this, then we have an easier alternative, in my opinion, to traditional Relativity. Also, does Anderson predict any variability for Gravity? Since he says in his list of postulates that G is a Universal constant, I guess probably not. Still, very impressive.

5. Originally Posted by Lunatik
...I think the real test of this new way of measuring using light c is how it works in its predictive capacity. If it can do this, then we have an easier alternative, in my opinion, to traditional Relativity. Also, does Anderson predict any variability for Gravity?
Anderson's equation's modify gravity, predicting that large bodies
Originally Posted by Anderson
lose the ability to absorb matter. This is because as it captures a particle it converts most of the matter to light and emits this back into space as radiation, much of which could be in the form of long wavelength electromagnetic energy.
mmmm...John Kierein has been telling us for decades there is oodles of very low frequency radiation out there....

Is this where gravity waves are? Particle physicists tell me I am up in the night when I question the Rayleigh-Jean and Wien limits. And I am. This is not a comfortable plank to be walking on...

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Somebody please put this mathese in words that the rest of us can understand.

So correct me if I'm wrong but his equations appear to treat photons and particles identically in a gravity field. But photons and particles do not act identically in a gravity field. Just as a basic example; particles accelerate toward the field where as photons do not. Simple enough to understand, though vastly more complicated to explain. Do his equations show this disparogy?

We can say with conviction that both matter and energy show a certain wave/particle duality both mathmatically and intuitively (to which only the latter I can attest). Matter behaves predominately as particles. Energy behaves predominatly as waves. Both can be coaxed into showing their other sides. Although the universe does tend to favor particle physics if we were to weigh one against the other. We cannot say with conviction that both matter and energy always behave in exactly the same way in the midst of a gravitational field.

Thought experiment: Create an extremely massive long thin plane. So massive that it bends light as it travels by. Shine a beam of light down the length of the plane parallel to its surface. The light must bend to the object's gravity. Will the light "fall" and and strike the surface of the object? Or will the light find its way around the object due to space/time distortion?

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## Re: Time, Matter, and Gravity

Originally Posted by Jerry
Morris Anderson has posted his "the universe in wave equations" book on the web.
I've looked at a few of the first pages of the book.

Chapter 1)

Morris makes the claim that the wave nature of matter is related to gravity. This is not correct. While the de Broglie wavelength of matter does tell you information about the relative velocity of your particle, this is completely independent of gravity. He has simply used the fact that the object is traveling inertially in the x-direction to allow for the reference velocities to be related to the sine of the angle made between the vertical and the velocity. The relationship to Snell's law is actually coincidental. Snell's Law works because light actually slows down crossing into a new medium. This acts like an instantaneous "constant acceleration" of sorts on the light that allows for the relationship to be exactly correct for the two straight light beams. However, there is no analogous effect of a constant acceleration of a light beam in the atmosphere. It seems to me that the author has conveniently parametrized his motion in terms of two instants in time and has found a coincidental arrangement simply based on looking at optics and seeing that the change in momentum of a particle -- in this case light -- is equal to a constant force.

However, this doesn't work only for gravity. Any constant, conservative force acting in one direction will do.

In his second chapter he defines a derivative incorrectly.

I haven't gotten any further.

8. ## Re: Time, Matter, and Gravity

Originally Posted by Astronomy
In his second chapter he defines a derivative incorrectly.

I haven't gotten any further.
Are you talking about equation 2-1?

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## Re: Time, Matter, and Gravity

Originally Posted by A Thousand Pardons
Originally Posted by Astronomy
In his second chapter he defines a derivative incorrectly.

I haven't gotten any further.
Are you talking about equation 2-1?
Yes.

10. ## Re: Time, Matter, and Gravity

Originally Posted by Astronomy
Originally Posted by A Thousand Pardons
Originally Posted by Astronomy
In his second chapter he defines a derivative incorrectly.

I haven't gotten any further.
Are you talking about equation 2-1?
Yes.
Well, he's implying that the instantaneous change is already contained in the notation, as in x=delta(x) I think. I haven't been able to read much of the paper but that part seems okay to me. In principle anyway.

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## Re: Time, Matter, and Gravity

Originally Posted by Normandy6644
Well, he's implying that the instantaneous change is already contained in the notation, as in x=delta(x) I think. I haven't been able to read much of the paper but that part seems okay to me. In principle anyway.
The correct definition should be that both delta(x) and delta(t) go to zero. If one goes to zero, the other has to as well for the derivative to exist.

12. ## Re: Time, Matter, and Gravity

Originally Posted by Astronomy
Originally Posted by Normandy6644
Well, he's implying that the instantaneous change is already contained in the notation, as in x=delta(x) I think. I haven't been able to read much of the paper but that part seems okay to me. In principle anyway.
The correct definition should be that both delta(x) and delta(t) go to zero. If one goes to zero, the other has to as well for the derivative to exist.
Well, that little nitpick doesn't have much to do with the rest of the paper, though I do agree it's a bit cavalier for a scientific paper (why does he really even need to say that velocity is the derivative of position, shouldn't we all know that by now?). I'm more critical of his whole section below that on momentum and energy. There are some things in there that strike me as very wrong, but I want to find a way to formalize it before I post my criticism of just that bit for now.

13. I've been studying the paper and here are a few preliminary comments on some of the definitions and postulates.

Two of the definitions are practically meaningless. They are:
Originally Posted by In his paper Morris Anderson
Definition 3: Position is where something is.

Definition 16: Matter is the substance of existence.
That last one is a lot like definining cotton by (cue the Richie Havens impersonator!) "Cotton is the fabric of our lives".

On the other hand, it is very difficult to offer good definitions of these concepts and I don't know what I would offer in their place that would be any better. But this I know, I would not be using a definition with as little content as Definition 16 as a justification in a mathematical proof as Mr. Anderson does later on.

His definitions of momentum and energy (Definitions 18 and 20 respectively) contain the ratio of the local speed of light to some reference value. Is there any observational justification for this?

Several of his postulates have problems.
Originally Posted by In his paper Morris Anderson
Postulate 2: It is not possible to create or destroy matter.
We do this all the time at particle accelerators. Protons and anti-protons annihilate one another and produce many other particles and anti-particles. However, energy, momentum, angular momentum and many other quantum numbers are conserved.
Originally Posted by In his paper Morris Anderson
Postulate 5: A body of matter distorts the medium of space causing the speed of light to be a function of position. Gravity is the influence of this gradient on matter.

Postulate 9: The charge of an object distorts the medium of space. The interaction of these distortions causes the force of charge attraction or repulsion.
There is a lot of distortion of space going on, but no calculation of how this space is distorted. This is a little strange, since this is a theory that wants to dispense with general relativity and yet claims that the mechanism by which it acts is the distortion of space.
Originally Posted by In his paper Morris Anderson
Postulate 7: the ratio of the permittivity of space to the permeability of space in a vacuum is contant and independent of position.

Postulate 8: The ratio of the rest matter to charge of an electron is contant or independent of location. The ratio of a proton is also constant.
Have these things really been observed? What experiments could we perform to verify these postulates?

Also, there is more to the universe than protons and electrons. There are neutrons and neutrinos, muons and tauons, quarks and gluons, weak force bosons, the Higgs boson, etc. Presumably the ratios of the various charges of these particles to their rest masses is also constant. Maybe.
Originally Posted by In his paper Morris Anderson
Postulate 10: The force of interaction between an atomic nucleus and a single electron can be calculated as: F=Z*e^2/(4*pi*epsilon*r^2).

Postulate 11: An electron-atomic nucleus system is stable when its structure corresponds to integer values of the wavelenth of interaction.
Postulates 10 and 11 relate mainly to the Bohr model for single-electron atoms and ions (H, He+, Li++, etc.) and has severely limited accuracy for this special case. Unfortunately Mr. Anderson then tries to use this to justify the power of (c/c_s) appearing in his mass formula.

Finally, I point to:
Originally Posted by In his paper Morris Anderson
Postulate 12: Space and matter are mutually dependent upon each other. Where there is space there is matter, and where there is matter, there is space. One does not exist without the other.

Definition 11: A vacuum is a volume of space where there are no particles or electormagnetic waves.
If Postulate 12 is true, then there can be no vacuum because matter and electromagnetic waves are everywhere.

I will have more to say on section 2 in a later post.

14. Originally Posted by Astronomy
Morris makes the claim that the wave nature of matter is related to gravity. This is not correct. While the de Broglie wavelength of matter does tell you information about the relative velocity of your particle, this is completely independent of gravity. He has simply used the fact that the object is traveling inertially in the x-direction to allow for the reference velocities to be related to the sine of the angle made between the vertical and the velocity. The relationship to Snell's law is actually coincidental.
Thanks for jumping in on this. Although you have stated the correct textbook answer, I don't think the textbook is right. Roy Keys steered me to this Consoli, &amp; Costanzo morsel. C&amp;C have shuffled through the original Michelson and Morley data and concluded a reduction in the speed of light in an gravitational or electromagnetic field is consistent with an “ether flow” if and only if a Lorentz transform is also performed. In this case, the 'ether' is the electromagnetic radiation, and the slowing of the speed of light proportional to both mass and electromagnetic density.

No, this is not far fetched. It is mathematically equivalent to Einstein's interpretation with one exception: Einstein used the transfer to vary time, the vacuum M&amp;M lasar interferometer reslults make it clear varying the speed of light in the same proportions is just as valid of an interpretation of the data: In fact more so, because it allows assignment of causality.

This means Anderson’s assumption the de Broglie wavelength is dependent upon gravity may be valid. (This is the wiggle room I was using to form similar hypotheses in the Gravity Behaving Badly thread.)

Originally Posted by Astronomy
However, this doesn't work only for gravity. Any constant, conservative force acting in one direction will do.
This is true, but doesn’t this also demonstrate the equivalence principle, in that there is no deference between gravity and other (electromagnetic) causes of acceleration?

Originally Posted by Quartermain
So correct me if I'm wrong but his equations appear to treat photons and particles identically in a gravity field. But photons and particles do not act identically in a gravity field. Just as a basic example; particles accelerate toward the field where as photons do not. Simple enough to understand, though vastly more complicated to explain. Do his equations show this disparogy?…
Again, if you look closely at Consoli, &amp; Costanzo, it is a very straightforward interpretation of Michelson &amp; Morley to say photons are slowed by a Lorentz transfer in a gravitational field, rather than that space and time are bending.

Originally Posted by Quartermain
Will the light "fall" and and strike the surface of the object? Or will the light find its way around the object due to space/time distortion?
Excellent paradoxical thought experiment. I will guess that the interaction with the gravimetric field will slow the photon, increasing it’s wavelength until it interacts with the plane, where it is most likely reflected.

15. ## Re: Time, Matter, and Gravity

Originally Posted by Normandy6644
Originally Posted by Astronomy
Originally Posted by Normandy6644
Well, he's implying that the instantaneous change is already contained in the notation, as in x=delta(x) I think. I haven't been able to read much of the paper but that part seems okay to me. In principle anyway.
The correct definition should be that both delta(x) and delta(t) go to zero. If one goes to zero, the other has to as well for the derivative to exist.
Well, that little nitpick doesn't have much to do with the rest of the paper, though I do agree it's a bit cavalier for a scientific paper (why does he really even need to say that velocity is the derivative of position, shouldn't we all know that by now?).
I don't think that the comment applies to equation 2-1. It just expresses a limit--if the limit doesn't exist, the derivative doesn't exist.

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## Excell spreadsheet for plotting the precession of an orbit

I have created an Excel spreadsheet based on the equations of wave propagation - see Path Program.xls http://anderson.morris.home.att.net/. Comparing results from this program with data demonstrates that the wave propagation method outlined in the book "Time, Matter, and Gravity" provides accurate results. This spreadsheet includes several examples including setups for calculating the bending of light past the Sun and the precession of Mercury's orbit.

I have been assigned to teach a class on this method at my place of employment. As part of this effort, I would like to present a comparison of this method with General Relativity. Can anyone create a similar spreadsheet based on GTR for me to use?

Morris

17. ## Re: Excell spreadsheet for plotting the precession of an orb

Jerry, last November,[/url]]Morris Anderson has posted his "the universe in wave equations" book on the web.
There are a lot of things I like about Morris's paper:

It is based upon solitons - matter as standing waves, just like Jacque Moret-Bailly's.
Solitons are cool.
Originally Posted by Morris
Morris
Morris? Morris Anderson? Welcome to the BABB. I suppose followups should go to the thread Can anyone show me how to do this with General Relativity?

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A Thousand Pardons wrote:

Solitons are cool.
Morris wrote:

Morris

Morris? Morris Anderson? Welcome to the BABB. I suppose followups should go to the thread Can anyone show me how to do this with General Relativity?
Thanks, I put a lot of work into being able to obtain these solutions and yes, My name is Morris Anderson and I really am interested if anyone can show me how to plot the path of motion in space using the equations of general relativity.

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I have a question.
Our Observable Universe is filled by a gravitationally field and electromagnetic field. An empty space does not exist.

The space in our Observable Universe is discrete because of the oscillation of this field. There is minimal uncertainty Planck length 10^-35 m close to Black Hole because high energy frequency oscillation. In Cosmic Voids is this minimal length longer because of the distances between the matter and low energy of the gravitationally field.

A kinetic energy joint with the potential energy of the gravitationally field gives us an inertial frame. In one inertial frame the maximal speed of the information is equal constant speed of light. In other inertial frames time, space changes according to Lorentz transformations.

Some people call this single element of the gravity+energy field with a minimal length - Zero Point Energy, Elemental Space Deformation, Elemental Vortex, Ether…
Is it truth ?

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In general the wavelength of interaction between objects decreases near a gravitational body. You will find the equations for this on slide 3 of the presentation Wave Propagation In A Gravitational Field.pdf. The physical size of objects (such as atoms) also decrease. You can find the full derivation for this in my book available at the following web sight: Time, Matter, and Gravity. You may also find it interesting to note that based on my approach General Relativity provides only a 1st order approximation of the gravitational redshift. The errors associated with this approximation have led to the concept of a black hole. In reality we have something much different in the form of a quasar as predicted in my book on Page 83.

21. Originally Posted by Morris
You may also find it interesting to note that based on my approach General Relativity provides only a 1st order approximation of the gravitational redshift.
So, you've already done the calculations for General Relativity, for this part of the problem?

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The redshift for GTR is a published equation that is readily available. What I am not able to do is plot the path of motion for an object based on the equations of General Relativity. I have never found an example of this. If anyone has an example, I would like to know about it.

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Originally Posted by Morris
In general the wavelength of interaction between objects decreases near a gravitational body. You will find the equations for this on slide 3 of the presentation Wave Propagation In A Gravitational Field.pdf. The physical size of objects (such as atoms) also decrease. You can find the full derivation for this in my book available at the following web sight: Time, Matter, and Gravity. You may also find it interesting to note that based on my approach General Relativity provides only a 1st order approximation of the gravitational redshift. The errors associated with this approximation have led to the concept of a black hole. In reality we have something much different in the form of a quasar as predicted in my book on Page 83.
Does this wavelength decrease near a gravitational body have a relationship to atomic clocks slowing when traveling through this gravitational field? If so, then clocks slowing may not indicate time slowing, only atomic oscillations slowing. Is this a reasonable hypothesis, per your work?

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Hi Morris
If our Observable Universe expands – some billions years ago was more dense (2 times less radius = 8 times higher density) . Is the light of that time gravitationally intrinsic redshifted because of the average density change ?
Chesslaw

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Originally Posted by Lunatik wrote
Does this wavelength decrease near a gravitational body have a relationship to atomic clocks slowing when traveling through this gravitational field? If so, then clocks slowing may not indicate time slowing, only atomic oscillations slowing. Is this a reasonable hypothesis, per your work?
[/quote]

The short answer is yes. The longer explanation to this question depends on how we define time. As far as I can tell Einstein never defined what time is. If I am wrong please correct me. In section 2 of my book I define the characteristics of time in terms of how we use the word in our language. Based on this approach time is not a variable. This eliminates a lot of confusion and allows us to calculate the influence of motion or the influence of a governing body on the natural frequency of matter. By doing this we end up with the lorentz transformations of Special Relativity. But we do not end up with the same gravitational redshift as given by General Relativity.

Originally Posted by czeslaw wrote
Hi Morris
If our Observable Universe expands – some billions years ago was more dense (2 times less radius = 8 times higher density) . Is the light of that time gravitationally intrinsic redshifted because of the average density change ?
[/quote]

The natural frequency of matter is a function of the properties of space. Therefore, if the universe was denser in the past, then the natural frequency of the matter as it existed in the past would be readshifted compared with matter of today. I discuss the equations related to this in detail in my book.

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Hi Morris,

When your work came up the first time, I noticed that there was similarity between your expression for kinetic energy (Eq. 2-15) and the one used in Autodynamics . Since you're here now, I though I'd point this out. Most physicists prefer the normal SR expression, but I find the AD expression compelling. For one thing, it avoids the problem of an object having infinite energy as it approaches c. Carezani published several papers and also a book. I'm not really endorsing AD in all respects, because there is much to be skeptical of there, and they also have an unfortunate tendency to fashion AD as a sort of science cult. But I thought you might find this interesting for your own work.

One question I would have is this. If you are breaking away from SR, does this not automatically take you into new GR ground as well? Or are you trying to keep GR?

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Originally Posted by ExpErdMann Wrote
Hi Morris,

When your work came up the first time, I noticed that there was similarity between your expression for kinetic energy (Eq. 2-15) and the one used in Autodynamics . Since you're here now, I though I'd point this out. Most physicists prefer the normal SR expression, but I find the AD expression compelling. For one thing, it avoids the problem of an object having infinite energy as it approaches c. Carezani published several papers and also a book. I'm not really endorsing AD in all respects, because there is much to be skeptical of there, and they also have an unfortunate tendency to fashion AD as a sort of science cult. But I thought you might find this interesting for your own work.

One question I would have is this. If you are breaking away from SR, does this not automatically take you into new GR ground as well? Or are you trying to keep GR?
Thanks for the link. This is interesting. I have also seen the equation I use for kinetic energy in college text books. So I don't think there is anything unique about it.

My intent in this effort is not to disprove Einstein’s work. It is only to figure out how things work. The results of my efforts have provided me with a different solution than Einstein came up with. In the end, we should be able to judge which on is right based on the data. I am not an expert on GR. However, it is clear that my approach provides an explanation for the existence of quasars that is becoming more and more difficult to do with relativity.

The bottom line is that I do not agree with the theories of Special or General Relativity.

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That seems like a pretty good approach to me. It's counterproductive to get too bogged down in trying to disprove SR or GR (especially if those theories are making correct predictions in many cases). A good alternative theory in the end will make the same correct predictions that SR and GR did, but also make some new, interesting ones that those theories didn't.

I haven't read through your whole book - will if I get a chance - but would you say that your main results hinge on Eqn. 2-15? It's a pretty major break from SR.

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Originally Posted by Morris
Originally Posted by ExpErdMann Wrote
Hi Morris,

When your work came up the first time, I noticed that there was similarity between your expression for kinetic energy (Eq. 2-15) and the one used in Autodynamics . Since you're here now, I though I'd point this out. Most physicists prefer the normal SR expression, but I find the AD expression compelling. For one thing, it avoids the problem of an object having infinite energy as it approaches c. Carezani published several papers and also a book. I'm not really endorsing AD in all respects, because there is much to be skeptical of there, and they also have an unfortunate tendency to fashion AD as a sort of science cult. But I thought you might find this interesting for your own work.

One question I would have is this. If you are breaking away from SR, does this not automatically take you into new GR ground as well? Or are you trying to keep GR?
Thanks for the link. This is interesting. I have also seen the equation I use for kinetic energy in college text books. So I don't think there is anything unique about it.

My intent in this effort is not to disprove Einstein’s work. It is only to figure out how things work. The results of my efforts have provided me with a different solution than Einstein came up with. In the end, we should be able to judge which on is right based on the data. I am not an expert on GR. However, it is clear that my approach provides an explanation for the existence of quasars that is becoming more and more difficult to do with relativity.

The bottom line is that I do not agree with the theories of Special or General Relativity.

Wow! Hey! Welcome to the board!

30. Originally Posted by Morris
The bottom line is that I do not agree with the theories of Special or General Relativity.
Just glancing at your webpage (I haven't opened the pdfs yet), I'm wondering why you wouldn't write equation (1) as z = e^phi' - 1, with phi' = ( MG/c^2 )( 1/r2 - 1/r1 ), or is that where it came from?

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