Thread: Refractive Field Theory of Gravity

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Refractive Field Theory of Gravity

MOTIVATION: 60 years of work to unify general relativity (GR) and quantum electrodynamics (QED) have produced many fanciful ideas, none of which have advanced either theory. (Which is not to say that neither theory has advanced.) While GR has been tested to 4-5 orders of magnitude, QED has had success to over 10 orders of magnitude. This suggests that GR is most likely to be in error.

If gravity could be modeled as a field in a Minkowskian space-time, the unification with the standard model would be greatly simplified. GR has been tested to the parametrized post-Newtonian formalism, which any competing theory should satisfy. The Noether theorem, and the familiar laws concerning conservation of energy and momentum, satisfy all but two of these parameters. This leaves the Eddington–Robertson–Schiff parameters, and .

The first is of these is satisfied if the alternative theory predicts the same gravitational time dilation as GR. This, combined with the precession of nearly circular orbits, shows agreement with the second.

This theory aims to satisfy these requirements, and become an alternative to GR as the continuous limit for a future quantum theory of gravity.

THEORY: The following can be taken as the founding principles of the theory:
1) Space-time is a Minkowskian 3+1 metric.
2) Energy (not just rest mass) IS gravitational mass, including the (negative) energy of the gravitational field.
3) A gravitational field has an index of refraction impacting all dynamics, proportional to the ratio of the energy of an object at that point to its energy at infinity.

The fact that space-time is really an expanding DeSitter space will not impact the solar system tests considered here.

SOUNDNESS: So, how does this theory satisfy the PPN limit? Glad you asked!

Newtonian Limit: We begin by modifying Gauss's law for gravity, where is the energy density and is the acceleration per :

From here we can derive the energy density of the field itself.

Taking a spherically symmetric energy distribution, the source of which is contained within a distance R of the origin, we can use the flat space-time to define the limit of the total energy of the source and its field outside R as we go to infinity. This in tern can be used to express the total energy within some r>R.

For a test object free falling in this field, with energy e(r), we have the following central force law:

Clearly this reverts to Newton's law when the energy is due almost entirely to the rest mass of the objects, using . More interesting is that this can be used to calculate the work needed to take the rest mass to infinity, and find e(r) in terms of this value.

This produces both the energy gain for non-relativistic massive objects, and for photons observed in the Rebka-Pound experiment, to the PPN limit.

Time Dilation: If a gravitational field is composed of gravitons, the interactions with these gravitons may slow processes down as they pass through the field. This theory assumes this is proportional to the ratio of the potential energy to the energy at infinity.

Clearly this is for the spherically symmetric case.

But this last expression is the time dilation from GR.

Before moving on, it should be noted that this alters the special relativistic time dilation as follows:

Also, in cylindrical coordinates (as a reminder):

Precession of Mercury: In classical mechanics the Lagrangian is defined to be the kinetic energy minus the potential energy.

The Euler-Lagrange equation relative to the angle of rotation gives the following constant of motion:

Using the substitution , and evaluating the Euler-Lagrange equation relative to r gives us the following:

For a nearly elliptical orbit, we can find L at the semi-latus recum, , and find the estimated ellipse.

For nearly circular orbits we have:

This is also the value of the precession for GR.

Differences: So, what does this theory predict that is not mainstream? Let me tell you!
The index of refraction is already shown above to differ from GR in the second order term. This could be measurable by the proposed LATOR mission.

The singular surface a gravitationally collapsed object is expected to be 1/4 the size of an event horizon from GR. This could be observed in planned radio imaging of the black hole at the center of the Milky Way. This is confounded by the matter surrounding the singularity, variables in the models of its geometry and the expected enlarging due to the refraction of light.

Real singularities exist. There is no cosmic censorship. All singularities are all naked.

Academic Pedigree: It seems en vogue to stake a claim of my publications and degrees, to support the idea that I be taken seriously.

I graduated Top Waffle from Waffle House University (Norcross, Georgia) in 2005, and used to manage a sports blog dedicated to coverage of the BCS.

2. Welcome to BAUT. Please take some time to read the Advice for ATM Posters and the Rules for Posting. Both are linked at the bottom of this post.

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As I see it the trouble with trying to interpret gravity as a field is that in Newton's theory gravity appeared to act instantaneously. If you make gravity like electromagnetism then it doesn't match the observations - planets would emit much stronger gravitational radiation and spiral into the sun. So I'd like to know:

1:What is the speed of gravity in your theory? Is it instantaneous or does it travel at the speed of light.

2:Can you calculate how rapidly of an orbiting body's energy would be emitted as gravitational waves

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Welcome to BAUT, utesfan100!

May I ask, have you written up this idea, in the form of a paper, and submitted it to an appropriate, peer-reviewed, journal?

If so, would you care to share with us some details (e.g. has it been published? accepted for publication?)? If not, why not?

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I have submitted a more thorough paper on this idea to two peer reviewed journals over the holidays (securing a formal tracking numbers for any future priority disputes), both of which returned it without review.

It is presently being reviewed by Classical and Quantum Gravity. In creating this post, I discovered that section 4 of that paper has severe logical fallacies that were corrected in the presentation here (by taking the corrected result this section aims to demonstrate as a postulate of the theory).

6. It looks to me like you have basically just restated SR and GR as they are currently used, perhaps with a slight twist somewhere that I haven't noticed if your results come out somewhat differently in the extreme limit, but what does this have to do with what you are supposed to be presenting, having something to do with an index of refraction? I can see that a gravitational field can be seen to act as a refractive field as it bends light, for instance, but where in your equations is that demonstrated? If it is different than the ordinary index of refraction, please explain the difference, how it operates, and how it is more useful than just applying ordinary GR.

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Thank you for your observant questions.

Originally Posted by quantropy
As I see it the trouble with trying to interpret gravity as a field is that in Newton's theory gravity appeared to act instantaneously. If you make gravity like electromagnetism then it doesn't match the observations - planets would emit much stronger gravitational radiation and spiral into the sun. So I'd like to know:

1:What is the speed of gravity in your theory? Is it instantaneous or does it travel at the speed of light.
Since my first premise was that space-time is Minkowskian, all information, including gravity, must travel at c. In fact, this induces a gravitational analog to magnetism that causes the Lense-Thirring effect, and reconciles this theory with the observations from Gravity Probe B.

You are setting my up for an argument based on gravitational aberration, that has been shown not to exist in solar system dynamics to O(r_g/r)^4, and does not show up in GR until O(r_g/r)^5.

In fact, you are correct. As presented, the theory should produce a gravitational Larmor radiation at O(r_g/r)^2. Furthermore, since gravitational radiation would have negative energy, this would accelerate the system.

In my detailed paper I suggest that gravity has an additional field equal and opposite to the Larmor field, so this energy is not required. It is expected that this non-EM analog field will also cancel a non-trivial eccentricity term that appears when the eccentricity of an orbit is considered, amounting to 2 arcseconds per year in the case of Mercury.

2:Can you calculate how rapidly of an orbiting body's energy would be emitted as gravitational waves
No. Not until I sit down and follow Purcell's derivation for the Larmor force in the context of the refractive field theory of gravity. Instead I will anticipate the next question.

Orbital Decay Paradox: If the gravitational radiation is negative, why do orbits decay when radiating gravitational waves?

The answer lies in the Lagrangian. The wave radiating away form the system must have originated at one of the objects. This will increase the rest mass of this object. The first order correction to kinetic energy is half that of the potential energy.

Thus both the kinetic and potential energy of the objects increases, but the kinetic energy becomes an increasingly lower percentage of the escape energy needed, causing the observed orbital decay.

The radiated negative energy is a small fraction of the localized negative energy field generated as the orbits decay further.

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Originally Posted by grav
It looks to me like you have basically just restated SR and GR as they are currently used, perhaps with a slight twist somewhere that I haven't noticed if your results come out somewhat differently in the extreme limit, but what does this have to do with what you are supposed to be presenting, having something to do with an index of refraction? I can see that a gravitational field can be seen to act as a refractive field as it bends light, for instance, but where in your equations is that demonstrated? If it is different than the ordinary index of refraction, please explain the difference, how it operates, and how it is more useful than just applying ordinary GR.
This theory assumes SR, but differs for GR in that there is no space-time curvature. The extra impact GR imparts on light-like trajectories is accomplished through an index of refraction.

This is accomplished in the equations by an alteration of the relativistic gamma in a gravitational field, and by adding the field to the Lagrangian, rather than in the metric. The index of refraction is different in solar system tests at one part in 10^8, which would be measurable by the proposed LATOR mission.

The primary perceived benefit of being based on a flat metric is that all of quantum mechanics is premised on a flat space-time. Further, many classical tools of the trade can be brought to bear in a theory based on flat space-time that introduce non-trivial difficulties when applied to a general curved metric. This is why the PPN is often used today when the full GR theory becomes too cumbersome.

9. Originally Posted by utesfan100
This theory assumes SR, but differs for GR in that there is no space-time curvature. The extra impact GR imparts on light-like trajectories is accomplished through an index of refraction.

This is accomplished in the equations by an alteration of the relativistic gamma in a gravitational field, and by adding the field to the Lagrangian, rather than in the metric. The index of refraction is different in solar system tests at one part in 10^8, which would be measurable by the proposed LATOR mission.

The primary perceived benefit of being based on a flat metric is that all of quantum mechanics is premised on a flat space-time. Further, many classical tools of the trade can be brought to bear in a theory based on flat space-time that introduce non-trivial difficulties when applied to a general curved metric. This is why the PPN is often used today when the full GR theory becomes too cumbersome.
But what is being refracted, and in what way? If flat space is SR, then wouldn't (couldn't) a gravitational field still be consider curved space? From what I can see, you are applying the same equations as GR, but just adding the converted mass from the energy of the gravitational field using E = mc^2 to the rest mass, simply giving bodies an overall greater gravitational field according to this greater total mass, is that correct? The results would only differ significantly in the extreme case for a black hole then, approximating Newtonian in the same way GR does for a weak gravitational field. But why should the total mass of the body, rest mass plus that of its gravitational field, be used rather than just the rest mass of the body alone? Furthermore, since the masses of large bodies is determined by the interactions of their gravitational fields to begin with, then according to your theory, wouldn't that calculated mass then be the total mass, rest mass plus gravitational mass, so that to find the rest mass of a body we would have to subtract the gravitational mass from the total mass found from the interactions between bodies?

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Originally Posted by utesfan100
No. Not until I sit down and follow Purcell's derivation for the Larmor force in the context of the refractive field theory of gravity.
It think its important to have calculations for orbital decay. Observations of the binary pulsar PSR B1913+16 agree very well with general relativity, and you would need to check your theory against this.

Roughly I would see things as follows:

Second order terms O(r_g/r)^2 have to be zero, otherwise we fall into the sun. You agree with GR here
Fourth order terms: I think you probably need to agree with GR here to match the binary pulsar observations
Sixth order terms: I would guess that you would disagree with GR here, which would give a prediction that more accurate observations of the binary pulsar might resolve.

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Originally Posted by grav
But what is being refracted, and in what way?
Everything is refracted, as a slowing down of c in a gravitational field. To the PPN limit this is equivalent to GR time dilation.

If flat space is SR, then wouldn't (couldn't) a gravitational field still be consider curved space?
Probably, the field could be expresses in a metric-tensor theory. This would probably be closer to published massive graviton theories than GR.

I would apply Occam's razor to reject curved space-time over a flat field theory, if the two produced the same results. At least until some one produced four points equidistant from each other, with a fifth point equidistant from the other four, at some distance other than the square root of three times the edge length of the tetrahedron.

For optical distances between satellites, the curvature of light due to the refractive field theory must be accounted for.

I completely understand that this use of Occam's razor is subjective. Further, the fact that the (lack of a) Larmor force is an outstanding question of significance would currently favor GR, as it has already explained these observations.

From what I can see, you are applying the same equations as GR, but just adding the converted mass from the energy of the gravitational field using E = mc^2 to the rest mass, simply giving bodies an overall greater gravitational field according to this greater total mass, is that correct?
The field acts much more like the pre-GR theory of Einstein, which he called a Newtonian theory, and that does not predict the time dilation effects.

Rather than abandoning Newton, and accepting space-time curvature, I use the classical idea of an index of refraction.

I am not sure I understand the second half of the question. It is true that as an object collapses, the energy inside the surface horizon increases without bound as the negative energy field is generated equal and opposite around the surface.

The results would only differ significantly in the extreme case for a black hole then, approximating Newtonian in the same way GR does for a weak gravitational field. But why should the total mass of the body, rest mass plus that of its gravitational field, be used rather than just the rest mass of the body alone?
I have already mentioned, twice, an experiment within our solar system that has been proposed to test GR that would distinguish GR from this theory, within our lifetimes. That is certainly not the extreme case of a black hole.

In GR the weight of the gravitational field can be ignored because gravity is not really a field, it is an observed effect of curvature. If gravity is a field, it must have an energy to carry momentum. Why should one form of energy gravitate, while another form does not? Why shouldn't the gravitational field's energy be considered?

Furthermore, since the masses of large bodies is determined by the interactions of their gravitational fields to begin with, then according to your theory, wouldn't that calculated mass then be the total mass, rest mass plus gravitational mass, so that to find the rest mass of a body we would have to subtract the gravitational mass from the total mass found from the interactions between bodies?
To move an object would require moving its gravitational field. The inertial mass would be , except that in the longitudinal direction the changes in must be considered. The gravitational mass observed at a given radius would be increased, altering the inverse square force law and providing a portion of the total precession due to the classical theory of revolving orbits.

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Originally Posted by quantropy
It think its important to have calculations for orbital decay. Observations of the binary pulsar PSR B1913+16 agree very well with general relativity, and you would need to check your theory against this.
I agree that my theory is not mature until this is done.

This will not happen within the next 30 days. GR has a 60 year head start, and has drawn far more research attention

Consider this month's exercise an attempt to determine if bigger holes in the refractive field theory exist, to make such a determination unnecessary.

Roughly I would see things as follows:

Second order terms O(r_g/r)^2 have to be zero, otherwise we fall into the sun. You agree with GR here
First, the 2nd order effect would expel us out of the solar system, not make us fall into the Sun. This is why I mentioned the orbital decay paradox above.

Larmor Field: Any reference I make to a Larmor field will be determined exactly by following Purcell to eliminate the O(r_g/r)^2 term. Essentially, I will assert that their is no radiation due to acceleration, which will induce a new field equal and opposite to the expected Larmor radiation.

I expect that this field will properly address the aberration effects, and eliminate the small eccentricity term a more accurate treatment of eccentricity adds to the precession of Mercury discussed above. Until I do the math on this, I can't argue that this must be true, and would be disappointed if this forum did not express skepticism of this claim.

Fourth order terms: I think you probably need to agree with GR here to match the binary pulsar observations
Sixth order terms: I would guess that you would disagree with GR here, which would give a prediction that more accurate observations of the binary pulsar might resolve.
Once the dipole radiation is eliminated, we will be left with quadrupole radiation. This is what GR produces.

The data shows a parabolic orbital decay. Differences here could be well within the order of magnitude uncertainties of the masses and orbital distances involved. (Tighter calculations can be made, but only once a given model is selected.)

If my theory produces the correct curve, this would suggest that any scaling differences from GR can be attributes to differences in the predicted masses or orbital distances of these objects. Indeed, if my theory requires them to be more massive than GR, that could help explain dark matter. (Wildly speculating, they might even turn out to be less massive-if I even get the right curve.)

13. Originally Posted by utesfan100
In GR the weight of the gravitational field can be ignored because gravity is not really a field, it is an observed effect of curvature. If gravity is a field, it must have an energy to carry momentum. Why should one form of energy gravitate, while another form does not? Why shouldn't the gravitational field's energy be considered?

To move an object would require moving its gravitational field. The inertial mass would be , except that in the longitudinal direction the changes in must be considered. The gravitational mass observed at a given radius would be increased, altering the inverse square force law and providing a portion of the total precession due to the classical theory of revolving orbits.
Okay, so let's say we are stationary in a ship in free space a distance of 'r' from the center of a large body. Our accelerometer reads an acceleration of 'a' toward the body due to its gravity. Normally we would use a = G M / r^2 to find the rest mass of the body, but according to your theory, M should instead be the rest mass of the body plus the mass equivalent of the energy of the gravitational field, correct? The energy of the gravitational field would be G m_rest / R, where R is the radius of the body (or perhaps just that inside the radius r where we are with G m_rest / R - G m_rest / r), so the equivalent mass would be m_g = G m_rest / (R c^2), is that right? And so M = m_rest + m_g.

Now let's say that the radius of the body suddenly collapses from R to a smaller radius S. Our accelerometer would read no difference, right? So using a = G M / r^2, we still calculate the same total mass M = m_rest + m_g. But because the body has collapsed, its gravitational energy increased to G m_rest / S, giving a greater gravitational mass m_g now and leaving a lesser rest mass m_rest of the body since M didn't change. So then, according to this, is your theory saying that as a body collapses, it converts rest mass to gravitational mass? If so, then collapsing to a singularity would require more than infinite initial rest mass, so none could ever form since all of the rest mass would have been "radiated" away long before into its gravitational field. But then, if the rest mass were to dissipate entirely, so too eventually would its gravitational field, right?

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You were correct up to this point:
Originally Posted by grav
But because the body has collapsed, its gravitational energy increased to G m_rest / S, giving a greater gravitational mass m_g now and leaving a lesser rest mass m_rest of the body since M didn't change.
We do get a stronger gravitational field, but the field density is negative. Thus we get more negative gravitational mass in the field and the rest mass of the collapsing, stationary, object must increase. Both increase without bound, equal and opposite, as we approach r_g/2r.

This is the same limit found using an isotropic coordinate chart for the Schwarzschild solution, but, unlike that solution, this represents geometric coordinates.

The rest of your post is dependent on the idea that the gravitational field has a positive energy density, which is not consistent with my theory.

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Hi utesfan100,

This Lagrangian formulation is observer dependent containing both and .

Originally Posted by utesfan100
In classical mechanics the Lagrangian is defined to be the kinetic energy minus the potential energy.
My question: is the Lagrangian invariant under Lorentz transformations?

Regards

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Originally Posted by ammer
Hi utesfan100,

This Lagrangian formulation is observer dependent containing both and .

My question: is the Lagrangian invariant under Lorentz transformations?

Regards
Thanks for the question, Ammer.

One would need to include terms for the magnetism-like force induced by the motion of the source object for the Lagrangian to become Lorentz invariant. In the static case this term goes to 0.

The Lagrangian presented was for the static spherically symmetric case, in order to determine the precession of nearly circular orbits. Indeed, is not properly defined outside this context.

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Originally Posted by utesfan100
One would need to include terms for the magnetism-like force induced by the motion of the source object for the Lagrangian to become Lorentz invariant. In the static case this term goes to 0.

The Lagrangian presented was for the static spherically symmetric case, in order to determine the precession of nearly circular orbits. Indeed, is not properly defined outside this context.
That begs for the question: what is the full version of the Lagrangian?

I'm curious how the concept of potential is woven into the full Lagrangian without introducing some frame bias.

Regards
Last edited by ammer; 2012-Feb-05 at 12:50 PM. Reason: typo

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Originally Posted by ammer
That begs for the question: what is the full version of the Lagrangian?

I'm curious how the concept of potential is woven into the full Lagrangian without introducing some frame bias.

Regards
I also would be interested in seeing the math on the field moving energy induces. I have less faith in my effort to do this than to do the classical math required in the OP. This also seems like a good exercise to hone the presentation of the theory.

The induced field would be derived from the static solution, using Lorentz invariance to derive the required induced field when this is observed from a frame moving relative to the static solution.

There should always be an invariant frame that minimizes the energy of the system, that corresponds to the static frame.

I likely will take a few stabs in this direction in the days ahead.

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Originally Posted by utesfan100
I also would be interested in seeing the math on the field moving energy induces. I have less faith in my effort to do this than to do the classical math required in the OP. This also seems like a good exercise to hone the presentation of the theory.

The induced field would be derived from the static solution, using Lorentz invariance to derive the required induced field when this is observed from a frame moving relative to the static solution.
Of course this will work in getting a consistent theory, but this recipe will very likely lead to a preferred frame of reference. If the Lagrangian in the static frame is not Lorentz invariant, I can't see anything else happen.

Originally Posted by utesfan100
There should always be an invariant frame that minimizes the energy of the system, that corresponds to the static frame.

I likely will take a few stabs in this direction in the days ahead.
I hope you succeed in what you want to achieve. If your goal is to propose a relativistic theory (as opposed to a preferred frame of reference) that might be tough imho.

Regards

20. Originally Posted by utesfan100
You were correct up to this point:

We do get a stronger gravitational field, but the field density is negative. Thus we get more negative gravitational mass in the field and the rest mass of the collapsing, stationary, object must increase. Both increase without bound, equal and opposite, as we approach r_g/2r.

This is the same limit found using an isotropic coordinate chart for the Schwarzschild solution, but, unlike that solution, this represents geometric coordinates.

The rest of your post is dependent on the idea that the gravitational field has a positive energy density, which is not consistent with my theory.
Okay, that's reasonable. So as the body collapses, it gains energy and takes an equal amount of energy from the field, correct? So just the opposite way around from what I stated, the field is converted to rest mass as the body collapses, right? That seems to indicate that the energy of the field has to be taken from something, so are you considering the field to be a refractive medium of some sort, that fills all space with lesser energy density with proximity to a body, and with a refractive index of 1 at infinite distance from the body?

I also stated the gravitational energy to be G M / R, which should have been G M m / R (or - G M m / R when considered negative). This however would be dependent upon the interaction between the fields of two bodies, while we just want the energy of a single body alone. But we can take the gravitational binding energy of the body, starting first with a single particle, then letting another particle fall to it from infinity, then another and so on, building up to a spherical uniformly dense body one particle at a time. This gives a gravitational binding energy of the body of (3/5) G m_rest^2 / R, as found in Wiki here.

Okay, so now, if the body were to collapse from R to a smaller radius S, the gain in energy would be the same as the difference in gravitational binding energies between the body originally forming at a radius R and at radius S, or (3/5) G m_rest^2 / S - (3/5) G m_rest^2 / R. This is the amount of energy that would be gained by the body, so this would also be the energy lost by the field, correct? Assuming that nothing changes outside of R, that our accelerometer at any r>R reads the same before and after the collapse and so forth, then the energy of the field is only taken between R and S. Also, since the field is not static, but moves at c, differences in energy densities must be perpetuated by the body. According to your theory, how is the energy density perpetually lessened with proximity to the body?

Using the gravitational binding energy, we can also find what the negative energy density will be. Let's say the body collapses from R to R-dR. Then the loss of energy within that region becomes (3/5) G m_rest^2 / (R - dR) - (3/5) G m_rest^2 / R = (3/5) G m_rest^2 dr / R^2. The volume of that region is (4 pi / 3) [R^3 - (R - dR)^3] = 4 pi R^2 dr. So dividing the energy by the volume, we get a negative energy density at R of (3/5) G m_rest^2 / (4 pi R^4). At the surface of the sun, then, that would give a negative energy density of 5.34 * 10^13 J / m^3. At the surface of the Earth, it would be 6.86 * 10^10 J / m^3. If this is a negative energy density, taken from the positive energy density of a refractive medium, then the original energy density of the medium must be much, much greater. Do you agree with this? If so, how do you explain these large energy densities. If not, why not?

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Originally Posted by grav
Okay, that's reasonable. So as the body collapses, it gains energy and takes an equal amount of energy from the field, correct? So just the opposite way around from what I stated, the field is converted to rest mass as the body collapses, right? That seems to indicate that the energy of the field has to be taken from something, so are you considering the field to be a refractive medium of some sort, that fills all space with lesser energy density with proximity to a body, and with a refractive index of 1 at infinite distance from the body?
For simplicity, let's consider a hollow sphere of negligible thickness. As the body collapses, new field is generated with a negative density, in the area between the new radius and the old radius. This negative energy is balanced with an equal and opposite positive energy increase to the energy on the surface.

This energy density is thicker the smaller we go, so the interactions of this field impedes the motion of energy more as we enter the field. Making this effect proportional to the energy change relative to an equivalent system at infinity produces the observed time dilation.
I also stated the gravitational energy to be G M / R, which should have been G M m / R (or - G M m / R when considered negative). This however would be dependent upon the interaction between the fields of two bodies, while we just want the energy of a single body alone. But we can take the gravitational binding energy of the body, starting first with a single particle, then letting another particle fall to it from infinity, then another and so on, building up to a spherical uniformly dense body one particle at a time. This gives a gravitational binding energy of the body of (3/5) G m_rest^2 / R, as found in Wiki here.
This is accurate if we ignore the weight of the field. The article at wiki does not include the weight of the field in their calculations. This was not germane to thrust of your previous post, so I did not address it there.

The formula for the energy density can be derived from Gauss's law of gravity alone (well, with the assumption that the curl is 0). The binding energy for the hollow sphere model can be found to be:

The rest of your post follows, from a uniform binding energy, neglecting the energy of the field, until we get to here:
[QUOTE]Also, since the field is not static, but moves at c, differences in energy densities must be perpetuated by the body. According to your theory, how is the energy density perpetually lessened with proximity to the body?[\QUOTE]
For a static source the field is static. This theory aims to be an alternative continuous limit for a future quantum theory of gravity.

When that theory arises, gravitons will be radiated out, and an explanation for their regeneration will then be needed. Unless gravitons are like virtual-photons that exist, but don't really exist because the energy does not add up, but our math predicts exactly the right results when we use them.

Finally, you examine the energy density of the field.
At the surface of the Earth, it would be 6.86 * 10^10 J / m^3. If this is a negative energy density, taken from the positive energy density of a refractive medium, then the original energy density of the medium must be much, much greater. Do you agree with this? If so, how do you explain these large energy densities. If not, why not?
The binding energy is not calculated by integrating potential energy over a volume. One must integrate the work done to produce that potential to start with.

The correct form for the energy density of the field is given in the OP.

A derivation of this can be found in a web search for the energy density of an electric field.

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Originally Posted by ammer
Of course this will work in getting a consistent theory, but this recipe will very likely lead to a preferred frame of reference. If the Lagrangian in the static frame is not Lorentz invariant, I can't see anything else happen.
If by preferred frame of reference you mean something like the PPN parameter described that way, education would be greatly appreciated.

I expect that a mathematical effect parallel to the difficulties called "non-localizability" of the field in GR may appear at some point.

I hope you succeed in what you want to achieve. If your goal is to propose a relativistic theory (as opposed to a preferred frame of reference) that might be tough imho.

Regards
Me too, if it were easy it would have been done already, and thanks!

23. Nice defense.

Are you sure orbital decay is required by the equations? The Moon's mean distance from the Earth is increasing, likewise, we are not certain that 1au is not increasing.

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Originally Posted by Jerry
Nice defense.
Thanks!

Are you sure orbital decay is required by the equations? The Moon's mean distance from the Earth is increasing, likewise, we are not certain that 1au is not increasing.
I will not be sure about what is required by the equations until I complete the analysis outlined in post #12.

As I understand it, the change in distance to the Moon is due almost entirely to classical tidal effects from the tug of the Sun. This suggests that the orbit of the Earth-Moon system should actually be decaying slightly as a result of the classical energy transfer. These effects are much larger than the relativistic residue.

State of the art ephemeris use armies of data processed by teams of computers to determine hundreds of independent parameters simultaneously. This amplifies the precision of our measurements to where we can rule out non-classical relativistic perturbations to a maximum of O(r_g/r)^4. Any theory with a perturbation of lower order would not match observation.

The orbital decay of binary pulsars requires this to eventually break down, requiring a relativistic decay, as there is not a large mass in proximity to produce the required tidal effects. Thus if my analysis results in no perturbation, this also would not match observation.

25. Originally Posted by utesfan100
For simplicity, let's consider a hollow sphere of negligible thickness. As the body collapses, new field is generated with a negative density, in the area between the new radius and the old radius. This negative energy is balanced with an equal and opposite positive energy increase to the energy on the surface.

This energy density is thicker the smaller we go, so the interactions of this field impedes the motion of energy more as we enter the field. Making this effect proportional to the energy change relative to an equivalent system at infinity produces the observed time dilation.

This is accurate if we ignore the weight of the field. The article at wiki does not include the weight of the field in their calculations. This was not germane to thrust of your previous post, so I did not address it there.
The field is included as negative energy. If it were neglected, we would only have the original rest mass of the body plus the gravitational binding energy for the particles that make it up as they fall to the body. Since that extra gravitational energy is added to the body upon collapse, it gains mass, so it would gain stronger gravity and we would have a greater reading on our accelerometer. But if we figure that the positive energy the body gains is counterbalanced by the negative energy of the field, then the accelerometer reads the same. But you didn't answer my question. Are you considering that the negative energy of the field is taken from an overall refractive medium that fills otherwise empty space? Also, if the field is continually regenerated as it is not static as a solid would be, but travels outward from the body at c, then the field would not move with the body when accelerated, right? Otherwise, how would different parts of different fields associated with many different bodies distinguish themselves so that only those parts of the intermingling fields will move with a body when that particular body is accelerated?

The binding energy is not calculated by integrating potential energy over a volume. One must integrate the work done to produce that potential to start with.

The correct form for the energy density of the field is given in the OP.

A derivation of this can be found in a web search for the energy density of an electric field.
What I used is the gravitational binding energy, that amount of energy required to form gravitationally particle by particle into a spherical uniformly dense body, although that is holding the body stationary while particles fall to it rather than the particles and body falling toward each other, but I would think that has been taken into account in the Wiki article. It is the work done, according to the amount of energy gained by each particle as it falls to the body. The article doesn't add the energy gained by each particle to the total mass of the body as it builds up, which would supply greater gravity, but that works for our purpose anyway if we are considering that the positive energy gained by the body is counterbalanced by the negative energy of the field. If the body collapses from R to R-dR, then the difference in energy divided by the difference in infinitesimal volume is the negative energy density at R if the positive energy gained by the body is counterbalanced by the negative energy taken from the field and our accelerometer reads the same at any r>R, thereby taking the "weight" of the field into account. So it should be accurate. Where did you get your calculation specifically?

26. Hi utesfan100,
Originally Posted by utesfan100
If gravity could be modeled as a field in a Minkowskian space-time, the unification with the standard model would be greatly simplified. GR has been tested to the parametrized post-Newtonian formalism, which any competing theory should satisfy.
I wrote an article describing gravity in the way you propose.
My publication was accepted in peer review journal and is expected to be published in February.

Originally Posted by utesfan100
Time Dilation: If a gravitational field is composed of gravitons, the interactions with these gravitons may slow processes down as they pass through the field. This theory assumes this is proportional to the ratio of the potential energy to the energy at infinity.
Clearly this is for the spherically symmetric case.

But this last expression is the time dilation from GR.

Before moving on, it should be noted that this alters the special relativistic time dilation as follows:
Explanation for GR based on Minkowski: http://www.bautforum.com/showthread....49#post1971349

Originally Posted by utesfan100
Precession of Mercury: In classical mechanics the Lagrangian is defined to be the kinetic energy minus the potential energy.
Lagrangian, the same as yours: http://www.bautforum.com/showthread....43#post1928243

And also field equations and derived GR main equation: http://www.bautforum.com/showthread....ield-equations

I propose some cooperation to develop this common concept.
Please, reach me by my email.

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Originally Posted by grav
But you didn't answer my question. Are you considering that the negative energy of the field is taken from an overall refractive medium that fills otherwise empty space?
No. I am considering that the gravitational field IS a refractive medium that fills an otherwise empty space. It happens to have a negative energy density.

Also, if the field is continually regenerated as it is not static as a solid would be, but travels outward from the body at c, then the field would not move with the body when accelerated, right? Otherwise, how would different parts of different fields associated with many different bodies distinguish themselves so that only those parts of the intermingling fields will move with a body when that particular body is accelerated?
I reject the idea that the equivalence of inertial and gravitational mass implies a phenomenological equivalence between acceleration and gravity. They can be distinguished by observing the relative motion of distant stars.

So far you have only considered the static spherical solution, with a static accelerometer (which now makes it a gravitometer). Since the mass, its field and the graviometer are static, I don't see what you think is moving. You have considered a gravitational collapse withing the radius if the gravitometer, that we agree is not observed by this device.

Where did you get your calculation [on binding energy] specifically?
I got them from my paper on this idea that is currently under review. The derivations are shown in the appendix to that paper. Since this is not yet published, that does not help you much

The late Edmond S. Miksch had derived this, excluding the energy of the field itself at his now expired domain www.negative-mass.com.

Your formulas are for a uniform density, which presents difficulties in my theory. For a hollow shell I will accept the limit -1/2GM^2/R. Any structure within this radius, and its field, would not be observed outside this radius anyways. Here is a back of the envelope sketch of the average energy density of the solar gravitational field between the surface of the sun and Earth.

Using SI units I get, G=6.67E-11, Msun=2E30, 1AU = 1.5E8 and Rsun=7E5. This gives a binding energy for a solar mass with a radius of 1AU of -1.8E42. At a radius of Rsun we have a binding energy of -3.8E44. Thus 99.5% of the binding energy is between the Earth and the solar surface.

The volume of the larger sphere is 1.41E25. The volume of the inner sphere is a negligible 1.4E18. This gives an average energy density of -2.7E19. Most of this is located near the surface of the Sun.

Converted to mass we have an average density of -300, for a total mass of -4E27. The energy density at Earth is -1.3E17, which amounts to a mass density of -1.4. I must confess, having never run these numbers before, I was struck by their magnitude. This suggests a density at Earth similar to the air, which should be clearly observable.

Indeed, we can observe the gravitational bending of light by this field. This energy density can be viewed as a pressure, so it would tend to cancel in most analysis. When it does not, its effect would be the gravitational forces of the Sun.

Including this mass changes the inverse square power law, causing an effect due to the classical theorem of revolving orbits that produces a significant portion of the relativistic precession of elliptical orbits. The time dilation from the index of refraction this field produces provides the remaining relativistic precession. These are solved together by using a Lagrangian including both effects.

What convinced me go public with this theory is that I completed the math showing that these corrections agree with GR for nearly circular orbits.
Last edited by utesfan100; 2012-Feb-07 at 12:25 AM.

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Using SI units I get, G=6.67E-11, Msun=2E30, 1AU = 1.5E8 and Rsun=7E5. This gives a binding energy for a solar mass with a radius of 1AU of -1.8E42. At a radius of Rsun we have a binding energy of -3.8E44. Thus 99.5% of the binding energy is between the Earth and the solar surface.

The volume of the larger sphere is 1.41E25. The volume of the inner sphere is a negligible 1.4E18. This gives an average energy density of -2.7E19. Most of this is located near the surface of the Sun.

Converted to mass we have an average density of -300, for a total mass of -4E27. The energy density at Earth is -1.3E17, which amounts to a mass density of -1.4. I must confess, having never run these numbers before, I was struck by their magnitude. This suggests a density at Earth similar to the air, which should be clearly observable.
Ahem ... Yeah. The solar system length values were given in km, while G and c used m. This is how you crash land a Mars rover into the polar cap.

The correct binding energy estimates, really using SI units, are now -2E45 at Earth, and -4E47 at the surface of the Sun. The volume of an AU sphere is 1.4E34, resulting in an average energy density of -3E13, corresponding to a mass density of -3E-4. At the Earth the density is -1.3E11, with a corresponding mass density of -1.4E-6. This is far less dense than the air.

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Originally Posted by pogono
Hi utesfan100,

I wrote an article describing gravity in the way you propose.
My publication was accepted in peer review journal and is expected to be published in February.
I look forward to reading it!
Lagrangian, the same as yours: [...] And also field equations and derived GR main equation:
I have briefly examined the forum you linked to, and see several key differences. While we share the idea that the bending of light due to gravity is in large part due to an optical property of gravity, I don't see the medium in your theory that is causing this optical effect. This is a key flaw in previous optical theories of gravity.

Our Lagrangians are different also in the sign of the gravity component, with mine becomming:

Further, your theory is still metrical, in that the spatial coordinates are not geometric coordinates.

The differential equations of an index of refraction are very similar in appearance to a metric. One could argue that what I call an index of refraction is a disguised metric:

My theory does not aim to satisfy the EFE. Indeed, the solutions I present do not. You appear to have aimed to derive the EFE from your staring point, while I aimed only to approach its results in the PPN limit. The negative energy density of the gravitational field provides the attractive dynamics the EFE aims to achieve.

I propose some cooperation to develop this common concept.
Please, reach me by my email.
If you are interested in contrasting our ideas further, that discussion is best done here. If you want to pm me a link to an arVix copy of the pre-print of your article (or similar online copy), I can asses whether collaboration on a paper idea (resolving the issues addressed in post #12) this summer may be in order.

30. Okay great, thanks, good post.

Originally Posted by utesfan100
No. I am considering that the gravitational field IS a refractive medium that fills an otherwise empty space. It happens to have a negative energy density.
How is a negative energy density produced in empty space? What is negative energy?

Originally Posted by utesfan100
I reject the idea that the equivalence of inertial and gravitational mass implies a phenomenological equivalence between acceleration and gravity. They can be distinguished by observing the relative motion of distant stars.

So far you have only considered the static spherical solution, with a static accelerometer (which now makes it a gravitometer). Since the mass, its field and the graviometer are static, I don't see what you think is moving.
I mean that the field is continually travelling outward from the center of the body at c, regenerating. It does not act instantaneously. If we move the body, the field is now generated from the new position of the body while that which was already emitted before does not move with the body, but continues on as it did before, away from the old position, correct? In other words, the entire field is not like a solid that moves as the body moves, but only begins generating outward at c again from the new position, right?

Originally Posted by utesfan100
Indeed, we can observe the gravitational bending of light by this field. This energy density can be viewed as a pressure, so it would tend to cancel in most analysis. When it does not, its effect would be the gravitational forces of the Sun.
A pressure gradient producing gravity sounds a lot like push gravity. Is "refractive field" another way of saying push gravity in your theory?

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