Refractive Field Theory of Gravity

[ammer]

If by preferred frame of reference you mean something like the PPN parameter described that way, education would be greatly appreciated.

The basic idea of Lorentz invariance of the Lagrangian is that it implies the laws of motion to be Lorentz invariant. So the equations are the same in any inertial frame of reference.
If you don't have the same equations in every inertial frame, then most likely there is a frame in which the equations take the simpelest form. That is the preferred frame, in your theory the static frame, I suppose. In terms of PPN parameters: because of the different equations per frame of reference, one or two parameters might deviate outside the preferred frame.

I believe (lazy stance: accept or prove me wrong) the latter is indeed the case with your theory.

I hope this helps,
Regards

[utesfan100]

Okay great, thanks, good post.

How is a negative energy density produced in empty space?

Empty space can be separated into equal and opposite positive and negative energy. Gravitational collapse would do this. The better question is what limits this process? I discuss this more in my paper, though this is essentially speculation. My idea is that the separation of empty space into mass and is strongly connected to the generation of entropy.

The name for the reverse process, of energy cancelling equal and opposite, is called nullification. My speculation would equate this with a violation of the second law of thermodynamics.

I once thought the Hubble red shift could be explained by a tired light model due to interactions with the gravitational field. This would require the Hubble shift to be stronger in dense gravitational fields, such as our galaxy, which confounds the idea that tired light can be applied to this theory.

Thus nullification does not even happen partially between the gravitational field and photons. This also means that the Minkowski metric of the OP would need to become a DeSitter metric on large timescales cosmological distances.

I also suspect the Hubble expansion can eventually be connected to the entropy density of the universe, following an entropy begets entropy exponential increase.

What is negative energy?

I knew this thread would get here eventually!

Negative energy is anything that propagates in a direction opposite the momentum it carries. This complicates our intuitive use of language based on positive mass. For this thread let's agree to use push/pull for forces or momentum changes and attract/repel for accelerations and changes in velocity.

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?

For a source moving with constant velocity, the field would be moving with that same constant velocity (at least between the light cones extending from the points since or until its next impulse is applied.)

At an impulse, or for an accelerating source, your observation is correct. This leads to the issues discussed in post #12. Purcell used a geometric derivation of these effects for electromagnetism that I plan to follow with gravity.

With electromagnetism this effect creates synchrotron radiation. In the refractive field theory of gravity it is presumed that there is no such radiation, which will induce a field equal and opposite this effect, caused by its absence.

A pressure gradient producing gravity sounds a lot like push gravity. Is "refractive field" another way of saying push gravity in your theory?

Well, since it has a negative energy, it is a pull gravity

To answer this question would require a quantum refractive field theory, of which this theory is the continuous limit. So would answering any question regarding the radiating away of the particles composing the field.

I will mention here that in my paper I discussed gravitational shielding in the context of a generic quantum theory, where the mass of the earth-moon-sun system is presumed to come in discrete proton sized chunks of gravitational mass. Shielding on the order of the classical proton radius (where the mass of a proton is accounted for entirely by the weight of its electric field) produces far too much shielding, becoming essentially opaque. Plank length (or better yet, proton gravitational radius) shielding, however, is smaller than our rather tight observed upper bounds for this effect.

[utesfan100]

The basic idea of Lorentz invariance of the Lagrangian is that it implies the laws of motion to be Lorentz invariant. So the equations are the same in any inertial frame of reference.

I believe (lazy stance: accept or prove me wrong) the latter is indeed the case with your theory.

I hope this helps,
Regards

I would argue that every theory has frames where the general equations can be reduced by symmetries of the dynamics being considered, and that the Lagrangian I presented would be an example of this in the general theory.

I know you the current Lagrangian does not satisfy this requirement. My theory asserts that it is Lorentz invariant, but until I provide the transformations, this is a valid open claim.

Let us explore this a bit by considering some inertial frame of reference, within which two equivalent charged masses are in equilibrium, with the electrostatic forces cancelling the gravitational forces. Let us neglect the weights of the fields. Clearly . Oh wait. Their is nothing more of interest to find in this frame.

Now let us examine the dynamics seen by an observer moving with velocity V perpendicular to the line connecting the two objects. Clearly the objects remain at the same distance R relative to each other. Q remains the same, but , with . Further, we have a magnetic force from the moving charges. Thus the gravitational force required must be opposed to the resultant of these three forces.




so


Somehow I was not expecting the fourth powers here. To be honest, I was expecting this force be 0. I guess I was right in the V^2/c^2 terms. It is also a push.

Let us repeat this again from an observer moving parallel to the line connecting the objects.

The magnetic force goes to 0 here, but the distance is reduced by a factor of .



so


Well. That doesn't look promising. It is not clear how this should meld together as the angle varies continuously. That will have to wait until Wednesday night.

At least this should clarify what I mean by a force induced by Lorentz invariance.

[pogono]

If you are interested in contrasting our ideas further, that discussion is best done here.

I wish.
But every time people start to cooperate here, it ends with earning infraction.
ATM is not designed for cooperation. According to ATM rules I have to fight with you and prove you are wrong.
So I recommend email.

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:

If your formulas are not Lorentz invariant and it does not result with Schwarzschild metric (for stationary, symmetric case) you should develop the idea to achieve this aim.
GR is well tested and established monolith. Instead fighting it, I advice you to show rather, that it may be derived other way.

[Copernicus]

I was wondering if you could sum up all the gravitational energy density from the sun, for perhaps 5 billion years and convert that to mass and compare that to the mass of the sun?

[Copernicus]

[QUOTE=utesfan100;1987198]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 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:

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.

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.

Okay so just a rough calculation in my head is that the energy of gravitation from a sun of our sun's mass, over the age of the universe is over 10^42 kg which is 100 billion times the mass of our sun., how would that work? The equivalent energy is way greater than the original mass.

[Copernicus]

[QUOTE=Copernicus;1987488]

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 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:


Okay so just a rough calculation in my head is that the energy of gravitation from a sun of our sun's mass, over the age of the universe is over 10^42 kg which is 100 billion times the mass of our sun., how would that work? The equivalent energy is way greater than the original mass.

I need to correct my statement. The gravitational energy field, of the sun, probably equals the mass of the sun over the life the universe.

Can you explain what you mean by hollow sphere theory?

[utesfan100]

I wish.
But every time people start to cooperate here, it ends with earning infraction.
ATM is not designed for cooperation. According to ATM rules I have to fight with you and prove you are wrong.
So I recommend email.

I did not ask you to cooperate. I asked you to find questions of my theory inspired by your own

If your formulas are not Lorentz invariant and it does not result with Schwarzschild metric (for stationary, symmetric case) you should develop the idea to achieve this aim.

I agree that, especially since my first premise is that space-time is Minkowskian, presenting a Lagrangian that is not Lorentz invariant is a major hole in this theory. My post to Ammer last night is a start in this direction.

I am certain that I don't understand exactly how to do this, and look forward to being educated in those maths by having the mistakes I make along the way corrected.

GR is well tested and established monolith. Instead fighting it, I advice you to show rather, that it may be derived other way.

GR is not as well tested as QED. It has been tested only to the first order PPN limits, and certain higher order phenomenology has been observed in distant high energy situations that make model-invariant numerical determinations difficult.

If my premises predict exactly what GR predicts I don't have a new theory, and am not against the mainstream. I have only a different way of viewing GR.

I show in my OP that the well tested PPN limits of GR are satisfied by this theory, and that my theory disagrees with GR at higher order correction terms. Thus it satisfies the solar system tests of GR, and is a distinct theory. I even reference a proposed mission that can discern between them.

The difference in the parameter that determines time dilation and the bending of light can be determined.

This is just below current measurement errors.

[utesfan100]

[QUOTE=Copernicus;1987506]

I need to correct my statement. The gravitational energy field, of the sun, probably equals the mass of the sun over the life the universe.

I don't under stand the meaning of integrating the energy density with time. To correlate with mass this would need to be an energy flux with time. Since the field is static, their is no energy flux.

This is equivalent to the electric field of a point charge. This has an energy density of:

If a charge is radiating away its electric field, where does this energy come from? (This is why virtual photons are not considered real photons, they can't radiate away any real energy)

The answer is that the field energy was generated by whatever process gave us the charge to start with. The same argument applies to the energy density of the gravitational field.

Can you explain what you mean by hollow sphere theory?

I mean treating the energy inside some radius as if it were an equivalent mass contained entirely on the surface of a sphere at that radius, to express the idea that changes here are unobserved mathematically by making the inside of the sphere 0.

This is opposed to the common modeling of a sphere as a uniform mass of equal density. This is itself an estimate, as density usually has some radial dependence, and a real object is unlikely to be spherically symmetric to begin with.

[utesfan100]

Working towards showing that my theory asserts is Lorentz invariant, by providing the transformations.

Well. That doesn't look promising. It is not clear how this should meld together as the angle varies continuously. That will have to wait until Wednesday night.

At least this should clarify what I mean by a force induced by Lorentz invariance.

It also looks wrong. These equations are not Lorenz invariant because they keep the r from the original frame. The following are the correct equations for any frame, with R, V, gamma and theta measured in that frame.




so


While this works for the test case considered, I am now thinking that I need to find the equations of the field itself, and the equations of the impact on this field on a test mass. This test case will then be a check on my maths.

[utesfan100]

Learn from my mistakes. Don't drink and derive.

Forces are vectors, and their directions need to be considered. Let be along the line connecting the two points, and the normal to this line in the plane of the velocity.




so
[LaTeX ERROR: Image too big 712x30, max 650x600]

The Lorentz corrections to Fg and Fy cancel for massive objects, leaving what could be interpreted as the gravitational force based on rest mass equivalent in form to electromagnetism. The advantage of this representation is that the gravitational of photons can be considered by replacing m*gamma with h*v.

[grav]

Thinking about it some more, there's no reason all of the additional energy density should be concentrated between S and R upon collapse of a body. It may be just wishful thinking, an effort to keep things simple. But really, if the body gains energy upon collapse, then since that's the same as added mass, then the field should be stronger at all r to infinity, but would still be counterbalanced between the positive energy added to the body and the additional negative energy density of the field. Even the potential within the body changes upon collapse in a different way, but if we consider a hollow shell as you have been doing, then we at least know that the negative energy density within the shell is constant, since there is no acceleration of gravity there, so no pressure gradient. I'll keep working on this aspect of it.

[utesfan100]

Thinking about it some more, there's no reason all of the additional energy density should be concentrated between S and R upon collapse of a body. It may be just wishful thinking, an effort to keep things simple. But really, if the body gains energy upon collapse, then since that's the same as added mass, then the field should be stronger at all r to infinity, but would still be counterbalanced between the positive energy added to the body and the additional negative energy density of the field. Even the potential within the body changes upon collapse in a different way, but if we consider a hollow shell as you have been doing, then we at least know that the negative energy density within the shell is constant, since there is no acceleration of gravity there, so no pressure gradient. I'll keep working on this aspect of it.

Let us consider Gauss's Law of gravity. In this theory this is modified to use energy instead of mass as the source.

This law requires any changes observed outside some radius to correspond to a net energy increase inside the radius. The conservation of energy then rules out any observed net change outside the radius from a gravitational collapse within the radius.

Since the negative energy field is generated by the collapse, this requires the positive energy of the collapsing system to increase. The idea of a collapsing system usually evaluates between two static reference states, so we can treat this as a rest mass of the system.

These equations generate a singularity at r=GE/2c^4 that must be an infinite rest mass, surrounded by an infinite mass gravitational field. As long as we don't exceed this density at any r, the binding energy will differ only by the difference in the degree to which mass is concentrated within the outer most sphere the source mass is found, which will be accompanied by an equal and opposite field energy.

[utesfan100]

The basic idea of Lorentz invariance of the Lagrangian is that it implies the laws of motion to be Lorentz invariant. So the equations are the same in any inertial frame of reference.

Actually, it might be better to take a more axiomatic approach. I have gleaned that Lorentz invariance follows from being able to express the theory in differential equations, without time or parity inversion.

I begin with the Minkowski metric and the conservation laws of energy, momentum and angular momentum. These are almost the definition of Lorentz invariance.

On top of this I apply Gauss's Law:

Although not expressed in a trivially Lorentz invariant expression, I see nothing that presents a difficulty in doing so. Specifically, this lacks the problematic time or parity inversions. I assert only that Gauss's Law can be made Lorentz invariant for a static energy distribution.

From these Lorentz invariant principles alone I derive , thus alpha is Lorentz invariant.

I then modify using to introduce an index of refraction. This is equivalent to a metric of the form:
.
For positive real , which we have outside the critical radius, this is clearly Lorentz invariant.

Since these are the principles the Lagrangian were built from, I suggest that it is Lorentz invariant. Or at least a close enough approximation to the actual Lorentz invariant one based on these principles for the PPN limits I calculated.

[Copernicus]

[QUOTE=utesfan100;1987539]

I don't under stand the meaning of integrating the energy density with time. To correlate with mass this would need to be an energy flux with time. Since the field is static, their is no energy flux.

This is equivalent to the electric field of a point charge. This has an energy density of:

If a charge is radiating away its electric field, where does this energy come from? (This is why virtual photons are not considered real photons, they can't radiate away any real energy)

The answer is that the field energy was generated by whatever process gave us the charge to start with. The same argument applies to the energy density of the gravitational field.


I mean treating the energy inside some radius as if it were an equivalent mass contained entirely on the surface of a sphere at that radius, to express the idea that changes here are unobserved mathematically by making the inside of the sphere 0.

This is opposed to the common modeling of a sphere as a uniform mass of equal density. This is itself an estimate, as density usually has some radial dependence, and a real object is unlikely to be spherically symmetric to begin with.

Why not a surface that it self is made of spheres, would it not give the same result?

[utesfan100]

[QUOTE=Copernicus;1987975]

Why not a surface that it self is made of spheres, would it not give the same result?

This would break the spherical symmetry of the model. This requires the energy density to be identical as a given radius for any direction. I don't see how any structure on the surface would be possible at all.

[ammer]

Actually, it might be better to take a more axiomatic approach. I have gleaned that Lorentz invariance follows from being able to express the theory in differential equations, without time or parity inversion.

I'd say it is the just other way around: from Lorentz invariance you obtain the convenience of being able to express the theory independent of the frame of reference. Sort of: comply with the prerequisite and have the math do the work for you.

If you want to claim you theory as being frame independent, just have the yield the same value in every frame of reference. Having said that, coming from here

that is a tall order in my view. And any tweak can turn your theory completely inside out and is likely to invalidate a few results so far.

In my view there are two options: either stick with a theory with a preferred frame, or it's back to the drawing board to further develop your original idea.
Regards

[utesfan100]

If you want to claim you theory as being frame independent, just have the yield the same value in every frame of reference.

Ok. In every frame we have:

=Kinetic energy-Potential energy

Both kinetic energy and potential energy were defined using Lorentz invariant expressions, assuming a Lorentz invariant form of Gauss's Law is used.

If there is a preferred frame, it is defined by the surrounding mass according to Mach's principle, and would amount to frame dragging similar to GR.

In my view there are two options: either stick with a theory with a preferred frame, or it's back to the drawing board to further develop your original idea.
Regards

I am sticking by my idea as it is presented here. This already requires an eventual trip back to the drawing board, as explained in post #12.

[utesfan100]

If there is a preferred frame, it is defined by the surrounding mass according to Mach's principle, and would amount to frame dragging similar to GR.

Actually, I show in my paper that the index of refraction has a second order effect so that for a photon E=Pc becomes E=Pc(1-r_g^2/r^2). This would produce an anisotropy in the CMBR based on the second order moment of the local gravitationally interacting unit relative to earth. In fact, it is within 30 degrees of the Shapely Super-cluster.

This would free the CMBR anisotropy relative to the cosmological principle.

If this difference in energy to momentum of a photon is ruled out to this level by experiment, this would disprove my theory.

[Copernicus]

[QUOTE=utesfan100;1988173]

This would break the spherical symmetry of the model. This requires the energy density to be identical as a given radius for any direction. I don't see how any structure on the surface would be possible at all.

How perfect does your spherical symmetry have to be? To what magnitude?

[utesfan100]

[QUOTE=Copernicus;1988318]

How perfect does your spherical symmetry have to be? To what magnitude?

For the equations to work exactly, the symmetry must be perfect. This is true for any equation in physics.

What we would normally do is use the equation to see how big the effect is, then look for possible side-effects that might alter our results, like the actual distribution of mass. If the determined effects are estimated to be far less than the accuracy needed for our system or experiment they can safely be ignored.

Or, as is the case here, we have a simplistic pet model that is useful for getting a grasp on the theory, not an actual model of any realistic system.

[ammer]

=Kinetic energy-Potential energy

Both kinetic energy and potential energy were defined using Lorentz invariant expressions, assuming a Lorentz invariant form of Gauss's Law is used.

This is not true. Look at the kinetic energy from your first post: , it contains which we know to differ from frame to frame.

I am sticking by my idea as it is presented here. This already requires an eventual trip back to the drawing board, as explained in post #12.

Ok, so please make the trip so you can substantiate your claims. Until then your theory is a moving target.

Regards

[utesfan100]

This is not true. Look at the kinetic energy from your first post: , it contains which we know to differ from frame to frame.

Ok, so please make the trip so you can substantiate your claims. Until then your theory is a moving target.

Regards

I see nothing in your arguments on Lorentz invariance that undermine the integrity of the calculations that I performed from the preferred frame of the central mass.

In reference to Lorentz invariance, I will state that the forum rules specifically state that "I don't know" is an acceptable answer.
Lorentz Invariance/Frame Dragging:
Since my theory agrees with GR in the PPN parameters relative to this preferred frame, I assert that the PPN frame dragging effects of GR must be present in my theory.

I assert that a consistent Lorentz invariant theory of these dynamics can be obtained by determining the required adjustments induced by the Lorentz transformations.

In the event this second assertion fails, I assert that the preferred frame effects follow Mach's principle, meaning the preferred frame is defined by the local masses relative to the location being considered.

The first step in following the analysis of Purcell, first referenced in post #12, will be to determine the equations of the field by the static central force in a moving frame. Until I do the math on this, I can't argue that these assertions must be true, and would be disappointed if this forum did not express skepticism of this claim.

[grav]

Let us consider Gauss's Law of gravity. In this theory this is modified to use energy instead of mass as the source.

This law requires any changes observed outside some radius to correspond to a net energy increase inside the radius. The conservation of energy then rules out any observed net change outside the radius from a gravitational collapse within the radius.

Since the negative energy field is generated by the collapse, this requires the positive energy of the collapsing system to increase. The idea of a collapsing system usually evaluates between two static reference states, so we can treat this as a rest mass of the system.

Hmm, okay, let's try this. I am stationary at a distance r from a body with radius R, where r>R. You are stationary at a distance r from another body with identical mass/energy, but with radius R/2. Would my gravimeter for the body I am at read more, the same, or less than yours?

[utesfan100]

Hmm, okay, let's try this. I am stationary at a distance r from a body with radius R, where r>R. You are stationary at a distance r from another body with identical mass/energy, but with radius R/2. Would my gravimeter for the body I am at read more, the same, or less than yours?

If by equivalent mass/energy you mean the limit of the observed value of mass/energy as R increases without bound, yes. Let us call this value E, in units of energy.

If E is confined within some r, E(r)=E(1-GE/2rc^4)^{-1}. This is what mass/energy we would both derive for the source based on our gravimeter.

You would observe that this is coming from a source of radius R, containing an energy of E(R)=E(1-GE/2Rc^4)^{-1}. I would observe this from a source of radius R/2, containing an energy of E(R/2)=E(1-GE/Rc^4)^{-1}.

This difference in energy, roughly -GE^2/2Rc^4, is exactly the amount of energy I would observe in the field between R/2 and R. Thus I would even see the same E(R) as you do.

If, on the other hand, our spheres are calculated to contained an equal energy, the limits as we go to infinity would differ with my mass being less than your mass at any r>R by the field energy from this source between R/2 and R.

[grav]

If by equivalent mass/energy you mean the limit of the observed value of mass/energy as R increases without bound, yes. Let us call this value E, in units of energy.

If E is confined within some r, E(r)=E(1-GE/2rc^4)^{-1}. This is what mass/energy we would both derive for the source based on our gravimeter.

You would observe that this is coming from a source of radius R, containing an energy of E(R)=E(1-GE/2Rc^4)^{-1}. I would observe this from a source of radius R/2, containing an energy of E(R/2)=E(1-GE/Rc^4)^{-1}.

This difference in energy, roughly -GE^2/2Rc^4, is exactly the amount of energy I would observe in the field between R/2 and R. Thus I would even see the same E(R) as you do.

If, on the other hand, our spheres are calculated to contained an equal energy, the limits as we go to infinity would differ with my mass being less than your mass at any r>R by the field energy from this source between R/2 and R.

If the difference in energy is contained between R/2 and R, then what about within the body? The potential energy within the body increases as well with smaller radius, right? Also, if we both read the same on our gravimeters for bodies with the same mass/energy, then if both bodies originally contained the same mass/energy with the same radius and one collapses to R/2, gaining extra energy in the process so that it now has a greater mass/energy than the one with radius R, then the smaller body would read greater on the gravimeter now in that case, right?

[utesfan100]

If the difference in energy is contained between R/2 and R, then what about within the body? The potential energy within the body increases as well with smaller radius, right?

For a hollow sphere model, the energy and field within the sphere is 0. If we have an estimate of the bodies an energy distribution, we can evaluate the integral with this distribution. This will change the form of the equations above because some of the source will now be outside, but the qualitative observation that the scaling factor increases the further in we go is correct.

Also, if we both read the same on our gravimeters for bodies with the same mass/energy, then if both bodies originally contained the same mass/energy with the same radius and one collapses to R/2, gaining extra energy in the process so that it now has a greater mass/energy than the one with radius R, then the smaller body would read greater on the gravimeter now in that case, right?

Incorrect. Conservation of energy requires that the energy gain within the collapsing sphere be equal and opposite to the field generated. While you are correct in that the attraction from the sphere would increase, the repulsion from the negative mass/energy of the field generated during collapse would cancel this exactly so there is no change in the observations at our gravitimeter.

Now... if the collapse process had a temporary quadrapole moment between the two steady states at R and R/2 it could have emitted gravitational radiation. In this case we would measure an increase in our gravitimeter, as the system would now have gained energy as observed at infinity equal and opposite to the negative energy radiated away.

[ammer]

I see nothing in your arguments on Lorentz invariance that undermine the integrity of the calculations that I performed from the preferred frame of the central mass.

OK. We should be able to get our ideas across. Somehow we fail to do that, both ways imho, and that's not what I'm here for. I'll try to make my assumptions or what I deem obvious explicit and see from there.

Starting from your Lagrangian

I assume that rest mass as well as light speed are the same for every observer in your model. So for any mass represents the same value in every frame of reference.
Consider the Lagrangian of a system with one test particle. If is to be the same value for every observer then so is



Here it stops making sense to me. What if becomes higher than the lhs of the equation?
I haven't recognized anything like this in your OP and therefore my final assumption is that you agree that this misrepresenting your ideas. If so, then we have arrived at a contradiction starting from the premise that your theory is Lorentz invariant.

And unless you come up with a third, I still only see two options: either stick with a theory with a preferred frame, or it's back to the drawing board to further develop your original idea.
Regards

[grav]

For a hollow sphere model, the energy and field within the sphere is 0. If we have an estimate of the bodies an energy distribution, we can evaluate the integral with this distribution. This will change the form of the equations above because some of the source will now be outside, but the qualitative observation that the scaling factor increases the further in we go is correct.

For a hollow sphere, there is no acceleration of gravity within the cavity, so the energy density must be constant within the sphere so that there will be no pressure gradient that produces gravity inside, but there is still a negative energy density there that matches that at the surface of the sphere. If there weren't, then if there were a very small hole in a large hollow sphere that we could freefall through into the cavity, it would be like hitting a wall when jumping from a large negative density at the surface to suddenly zero.

Incorrect. Conservation of energy requires that the energy gain within the collapsing sphere be equal and opposite to the field generated. While you are correct in that the attraction from the sphere would increase, the repulsion from the negative mass/energy of the field generated during collapse would cancel this exactly so there is no change in the observations at our gravitimeter.

It can only be one or the other, right? Either for a body at R/2 with exactly the same mass/energy would we still read the same on our gravimeter at r as for the body with radius R, or we would read the same for a body that collapses to R/2, now with a greater mass/energy due to the energy gained in the collapse. In other words, we wouldn't read the same for two bodies with radius R/2, one with some given mass/energy and one with a greater mass/energy, correct?

[utesfan100]

For a hollow sphere, there is no acceleration of gravity within the cavity, so the energy density must be constant within the sphere so that there will be no pressure gradient that produces gravity inside, but there is still a negative energy density there that matches that at the surface of the sphere. If there weren't, then if there were a very small hole in a large hollow sphere that we could freefall through into the cavity, it would be like hitting a wall when jumping from a large negative density at the surface to suddenly zero.

The infinite energy density at the surface of the hollow sphere model provides the jump discontinuity in the energy density to 0 inside the sphere.

I will provide the solution for a solid sphere with finite energy density later this weekend that does not have this jump discontinuity.

It can only be one or the other, right? Either for a body at R/2 with exactly the same mass/energy would we still read the same on our gravimeter at r as for the body with radius R, or we would read the same for a body that collapses to R/2, now with a greater mass/energy due to the energy gained in the collapse. In other words, we wouldn't read the same for two bodies with radius R/2, one with some given mass/energy and one with a greater mass/energy, correct?

I can not parse this question, nor see how any interpretation of it brings out any point not addresses in posts 55 or 57.