# A doubt in gravitation

Can anyone give me a situation when the gravitational potential due is 0 but gravitational field is non-zero.

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If any of the good physicists whom I greatly admire on this forum see this I'll probably take a beating, a real bad beating for this. But then maybe I deserve it

Maybe the center of the Earth?

I think of gravitational potential much like I think of other potentials - their values are arbitrary! That is, you can set your "zero" wherever you like. What matters for the theory's ability to describe our observations is not the value of the potential, but rather is differences in the value of the potential, or the change in the potential.

Two common places to set the "zero" of gravitational potential energy are at the surface of your system or at a distance infinity far from your system. Each location of zero will produce the same results, but you can place the zero wherever you like.

I dont know if you have had calculus... The gravitational field is the negative derivative of the gravitational potential. Recall that a function can have a zero value and a non-zero derivative at the same time.

A.T.
Maybe the center of the Earth?
OP asked about a place where the gravitational field strength is non-zero.

hi, everybody .. as said by modus, i think he is correct. It should be the center of earth as gravitational field is the negative derivative of gravitational potential wrt position so it field exists whereas potential doesnt..hope that makes sense..

The field in the centre of the earth is zero as well as the potential. Think of Newton's apple.

yes..ur right..

BruceW
Homework Helper
The gravitational field in the centre of the earth is zero for sure. And as modus said, the gravitational potential is arbitrary. Only changes in the gravitational potential are important. But usually the gravitational potential is defined to be zero at infinitely far away.

So, using this definition, what is the gravitational potential at the centre of the earth? Not zero, buddy. In fact, it is
$$- 2 \pi G \rho R^2$$

If any of the good physicists whom I greatly admire on this forum see this I'll probably take a beating, a real bad beating for this. But then maybe I deserve it

Maybe the center of the Earth?
Other way around.
At the center of the earth, the gravitational field is zero but the gravitational potential is at a local minimum.

The gravitational field is the gradient of the gravitational potential. Because of the spheroid symmetry of the earth's mass, the gradient of potential at the center of the earth is zero. In principle, one would be "weightless" at the center of the earth.

The gravitational potential at the center of the earth is much larger than the limit of the gravitational potential at an infinite distance from the earth. The limit of the gravitational potential at an infinite distance from the earth is by the usual convention zero. Because gravity is always attractive, the gravitational potential has to be non positive.

So the gravitational potential at the center of the earth is by the same convention much lower than zero. It is a negative number with a large magnitude.

It might be easier to explain in terms of the magnetic field, and then we can take the analogy back to gravity. Take two bar magnets. Position them such that they are fairly close to each other, both south poles face each other, and they are both aligned on an axis (say X axis), symmetrical and equidistant from some origin. I think you get the picture. If we place a piece of magnetically sensitive object exactly between them, say an iron BB, it will stay in the center as long as it is perfectly balanced.

Yes I know that in practice this is extremely difficult to do. Any variation in the parameters will offset the balance and the bb would immediately slam to one magnet or the other. Even approaching absolute zero temperature there remains some slight motion. But at least for some short period of time, at some exact center location, the bb can remain balanced.

The Superposition Principle tells us that the net magnetic field strength felt at the center point is zero. It doesn’t say that there is no magnetic field there at all. There is a big difference!

To continue with our thought experiment with magnets, we now place two more magnets symmetrical from the origin along the Y axis, and two as such along the Z axis, and then more and more at many locations equidistant from the center such that we have an approximate sphere with the south pole point in the center and the north pole pointing out.

Magnets have another difficulty in that they never come in monopoles, and in reality it is impossible to create our perfect magnetic sphere as we would like, but gravity doesn’t have this problem. So to complete the analogy we replace both ends of all the magnetic dipoles with a gravitational monopole (a piece of mass will do) and we state what we already know which is that is that unlike magnetism, gravity is an attractive force (like graviational charges attract instead of repel).

Photons mediate the magnetic force. I interpret that to mean that the magnetic field is composed of photons, albeit virtual photons. If the gravitational force works in a similar manner, then I maintain my original argument that the center of the earth has a non-zero gravitational field.

As for the potential, ModusPnwd is correct. It’s arbitrary and we can set zero to whatever we choose.

BruceW
Homework Helper
Photons mediate the magnetic force. I interpret that to mean that the magnetic field is composed of photons, albeit virtual photons. If the gravitational force works in a similar manner, then I maintain my original argument that the center of the earth has a non-zero gravitational field.
I don't think that it is necessarily true that the gravitational field does work in a similar manner. Even if it does, then you're saying that there is a gravitational field in the centre of the earth in the same sense that there is a gravitational field at any arbitrary point in space, due to quantum fluctuations.

I don't think that it is necessarily true that the gravitational field does work in a similar manner.
That’s o.k., there are many different views on how gravitation works as it is still an open question in physics. Your input on this is appreciated if you wish to elaborate a little, but if it has to do with QLG that is covered to a great extent in other sections on this forum.

there is a gravitational field at any arbitrary point in space, due to quantum fluctuations.
Yes, I agree completely, but I was actually trying to make a different point. I don’t want to beat a dead horse, but I thought of a better (and shorter) analogy. Your boss tells you to go up to 50 kW transmitting antennae to make a repair. Some idiot keys the mic while you are up there. You’re going to be in a world of hurt.

Now if you are at the center of two such antennae’s and they both get keyed, you will receive large doses of radiation at the center point. The radiation is coming from both directions at you, even if the divergence of the field there is zero.

That’s it for now. If anyone denies that, then I’ll admit defeat for now.

It might be easier to explain in terms of the magnetic field, and then we can take the analogy back to gravity. Take two bar magnets. Position them such that they are fairly close to each other, both south poles face each other, and they are both aligned on an axis (say X axis), symmetrical and equidistant from some origin. I think you get the picture. If we place a piece of magnetically sensitive object exactly between them, say an iron BB, it will stay in the center as long as it is perfectly balanced.

Yes I know that in practice this is extremely difficult to do. Any variation in the parameters will offset the balance and the bb would immediately slam to one magnet or the other. Even approaching absolute zero temperature there remains some slight motion. But at least for some short period of time, at some exact center location, the bb can remain balanced.
This is very much unlike the situation with gravitational orbits. Objects in gravitational orbit will not immediately slam into another object in gravitational orbit. This is because the effect of gravitational dipoles on orbit are much smaller that the effect of gravitational monopoles in orbit.

The Superposition Principle tells us that the net magnetic field strength felt at the center point is zero.
The Superposition principle says nothing of the kind. The magnetic field strength inside a bar magnet is supposed to be strong according to mainstream physics.

It doesn’t say that there is no magnetic field there at all.
You got that right.
There is a big difference!
Which you got precisely backwards.

The Superposition Principle does not say that the magnetic field of a magnet is zero at the center. This is because magnets have a positive magnetic dipole moment and a zero magnetic monopole moment. Ever well educated physicist would claim that the gravitational field in the center of the earth is effectively zero. This is because the earth has a large gravitational monopole moment and a gravitational dipole moment which is negligible.

To continue with our thought experiment with magnets,
This is an invalid thought experiment since it starts with hypotheses that have been experimentally refuted time after time. A thought experiment is a short cut in the mathematics, not in experimental physics.

Relative to a coordinate system at the center of the earth, the earth has a very small gravitational dipole moment and a very big gravitational monopole moment. Therefore, the gravitational field in the center of the earth is going to be very small, if not zero.

I will disagree with those who say that the earth has a zero gravitational dipole with respect to the center of the earth. Therefore, there may be a very small gravitational field at the center of the earth associated with asymmetry of the mass density distribution. However, I don’t think anyone has ever measured this dipole moment.

I'll bet you that it will be measured some day. When a gravitational dipole moment is detected relative to the center of the earth, you can claim there is a small but nonzero gravitational field at the center of the earth. However, there will be a point a few inches north or south of the geometric center where the gravitational field strength is zero. I will then redefine the center of the earth to be at the point where the gravitational field strength is zero.

Although not ruled out by theory, the theory does show that the effect of the dipole on the gravitational field has to be negligibly small. Since no one has ever measured such a dipole by experiment, and since the effects of such a dipole would be very large, one can not assume that the gravitational dipole is significant.

we now place two more magnets symmetrical from the origin along the Y axis, and two as such along the Z axis, and then more and more at many locations equidistant from the center such that we have an approximate sphere with the south pole point in the center and the north pole pointing out.
The motion of bar magnets is completely irrelevant to the motion of gravitational masses.

Two monopoles of equal magnitude but opposite sign placed a small distance apart are equivalent to a pure dipole. Each magnet is equivalent to a North and South pole set a small distance apart. However, a gravitational dipole would have to be equivalent to a positive and negative mass located a small distance apart. Since there is no such thing as a negative mass, you can’t make a pure gravitational dipole. Scientists have looked for such things, but never found them. So your thought experiment has no significance.

Magnets have another difficulty in that they never come in monopoles, and in reality it is impossible to create our perfect magnetic sphere as we would like, but gravity doesn’t have this problem. So to complete the analogy we replace both ends of all the magnetic dipoles with a gravitational monopole (a piece of mass will do) and we state what we already know which is that is that unlike magnetism, gravity is an attractive force (like graviational charges attract instead of repel).
Your analogy is not complete because there is only one type of gravitational mass. Magnets always have two magnetic charges, North and South.
Two monopoles of equal magnitude but opposite sign placed a small distance apart are equivalent to a dipole. Magnetism has two “charges”, North and South. You can arbitrarily chose a sign for each charge. However, gravitational mass is always positive.
Gravity is always attractive. However, magnetism can be attractive or repulsive. So right there, you have a fundamental difference between gravity and magnetism.

Photons mediate the magnetic force. I interpret that to mean that the magnetic field is composed of photons, albeit virtual photons. If the gravitational force works in a similar manner, then I maintain my original argument that the center of the earth has a non-zero gravitational field.
It doesn’t matter if gravity is mediated by gravitons, space-time, or leprechauns. The earth may not have a gravitational field at it center. The mediating particle, or even the existence of a mediating particle, has nothing to do with whether a field is zero or not zero.
As for the potential, ModusPnwd is correct. It’s arbitrary and we can set zero to whatever we choose.
This is correct. Therefore, the value of the absolute potential has no physical significance. The choice of zero potential is arbitrary with respect to experiments.
Therefore, you can set the gravitational potential anywhere you like. This includes your navel. I admit that the choice of ones navel is an interesting gauge.
The choice of zero is part of what one calls the gauge of a calculation. All the potentials, vector and scalar, have arbitrary constants associated with them. The choice of constants is called a gauge.
The gauge does not affect the predicted measurements. You should be advised that the mathematics of orbital calculations will become much more difficult if you chose your navel. However, the approach should be novel enough for a theoretical thesis. Most professionals chose the zero potential to be at a point in the limit of infinite distance from earth.

Different Volts for different Folks.

Here is an article that says explicitly that you can’t form a pure gravitational dipole because there is no such thing as negative mass.

http://www.lightandmatter.com/html_books/genrel/ch09/ch09.html [Broken]
“Since our universe doesn't seem to have particles with negative mass, we can't form a gravitational dipole by putting positive and negative masses on opposite ends of a stick --- and furthermore, such a stick will not spin freely about its center, because its center of mass does not lie at its center!”

Note that they are only talking about an "isolated" gravitational dipole. The earth may have "bubbles" that that cause a small but nonzero dipole moment relative to the center of the earth. The continents of the world may form such "bubbles", since they are made of rock which is much less dense than ocean bottom. Or there could be dense nodes of material that cause a gravitational dipole.

What the article doesn’t say is that an impure dipole can be made by place a large mass at one end of the stick and no mass on the other end of the stick. The stick/large mass composite will have a gravitational dipole.

If this scenario is what you are talking about then please let us know. However, the earth to a very high approximation has a spheroidal density distribution. Therefore, it doesn’t have a gravitational dipole to a very high degree of approximation.

The important thing is that no one has been able to measure this gravitational dipole. If it exists, it is negligible. Unless there is a nonzero gravitational dipole relative to the center of the earth, there can be no gravitational field in the center of the earth.

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First of all, wow, what a mean and negative post Darwin. I’m truly sorry that I pissed you off so badly. Let me try to be civil as I defend myself.

A thought experiment is a short cut in the mathematics, not in experimental physics.
Really? You’re making me wrong about using the term thought experiment? Here is what Wikipedia says about it…
“A thought experiment or Gedankenexperiment (from German) considers some hypothesis, theory,[1] or principle for the purpose of thinking through its consequences.”

This is very much unlike the situation with gravitational orbits. Objects in gravitational orbit will not immediately slam into another object in gravitational orbit.
I never mentioned anything regarding gravitational orbits in any of my postings in this thread. Use the control ‘f’ key and search this page.

Regarding superposition, here is a quote from the Wikipedia article…

“the net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually.”

I take that to mean that the net response of a field (any of the force fields) can be zero at some location and time, while at the same time having a non-zero response from each individual stimulus. That seems to me like mediators of a force occupying (or moving through) a region of zero response.

the earth has a very small gravitational dipole moment
That is pure conjecture on your part! Now I think I see where you are coming from. My posts here are an honest attempt at understanding physics. If you wish to express new ideas, honestly I am all for that. Just please find a more appropriate way. When I’m wrong I say I’m wrong. Bust me on the facts as understood by the current state of the art in physics, not on the semantics of what a thought experiment is, or how my understanding doesn’t fit your personal theory.

BruceW
Homework Helper
That’s o.k., there are many different views on how gravitation works as it is still an open question in physics. Your input on this is appreciated if you wish to elaborate a little, but if it has to do with QLG that is covered to a great extent in other sections on this forum.
I really know very little about the research into possible quantum gravity theories unfortunately. Just out of interest, what does the L stand for in QLG ?

MikeGomez said:
Now if you are at the center of two such antennae’s and they both get keyed, you will receive large doses of radiation at the center point. The radiation is coming from both directions at you, even if the divergence of the field there is zero.
I don't understand :( sorry. The divergence of what field? The electric field has zero divergence in any region which contains no charges. So I'm guessing you don't mean that. Do you mean the change in the electric field as you move from one antennae to the other? In other words, if we denote a coordinate which goes from one antennae to the other in a straight line as 's' then the partial derivative of the electric field with respect to 's', keeping other coordinates constant? So then it does make sense to me that this quantity (call it $\partial \vec{E}$ ) is zero at the middle point between the two antennae (just from symmetry).

I don't really see the analogy. In this case, you're going to be in a world of pain even if the electric field was constant throughout space, right? So why does it matter if $\partial \vec{E}$ is zero? And in the situation of the gravity at the centre of the earth, the field is zero but it changes when you move away from the centre.

Just out of interest, what does the L stand for in QLG ?
Sorry, I meant LQG, Loop Quantum Gravity.

So then it does make sense to me that this quantity (call it $\partial \vec{E}$ ) is zero at the middle point between the two antennae (just from symmetry)
Yes, thank you. We agree that the partial derivative of the electric field at the middle point is zero. My use of the term divergence is probably incorrect.

My point is simply that we will get fried by radiation if we are at the middle point (zero point) when someone keys the two mic's. Are we in agreement on that?

Nugatory
Mentor
I don’t want to beat a dead horse, but I thought of a better (and shorter) analogy. Your boss tells you to go up to 50 kW transmitting antennae to make a repair. Some idiot keys the mic while you are up there. You’re going to be in a world of hurt.

Now if you are at the center of two such antennae’s and they both get keyed, you will receive large doses of radiation at the center point.
Right, because there are very powerful time-varying electric currents in the towers. If we just placed powerful charges on each tower and let the charges sit, the resulting fields would be constant everywhere, and zero at the midpoint.

You can use stationary electric charges to make good analogies with gravitation; they're both 1/r2 static forces. Moving charges and time-varying fields, not so much.

BruceW
Homework Helper
My point is simply that we will get fried by radiation if we are at the middle point (zero point) when someone keys the two mic's. Are we in agreement on that?
yes, I understand that bit. But I don't get where it makes an analogy to the situation of gravity at the centre of the earth.

First of all, wow, what a mean and negative post Darwin. I’m truly sorry that I pissed you off so badly. Let me try to be civil as I defend myself.

Really? You’re making me wrong about using the term thought experiment? Here is what Wikipedia says about it…
“A thought experiment or Gedankenexperiment (from German) considers some hypothesis, theory,[1] or principle for the purpose of thinking through its consequences.”

I never mentioned anything regarding gravitational orbits in any of my postings in this thread. Use the control ‘f’ key and search this page.

Regarding superposition, here is a quote from the Wikipedia article…

“the net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually.”

I take that to mean that the net response of a field (any of the force fields) can be zero at some location and time, while at the same time having a non-zero response from each individual stimulus. That seems to me like mediators of a force occupying (or moving through) a region of zero response.

That is pure conjecture on your part! Now I think I see where you are coming from. My posts here are an honest attempt at understanding physics. If you wish to express new ideas, honestly I am all for that. Just please find a more appropriate way. When I’m wrong I say I’m wrong. Bust me on the facts as understood by the current state of the art in physics, not on the semantics of what a thought experiment is, or how my understanding doesn’t fit your personal theory.
Any physicist, including me, can construct a hypothetical distribution of mass that has a nonzero gravitational monopole and a nonzero dipole. The nonzero dipole term exists even though mass can not be negative. However, a nonzero dipole has to be superimposed on a nonzero monopole contribution. The famous dictum "gravitational dipoles do not exist" only applies to isolated gravitational dipoles.

Multipole expansions can be done with any distribution of mass. This is not speculation. This is mainstream physics which drops right out of classical mechanics. Any perturbation in density that the inversion operator changes can generate a gravitational dipole. However, one needs an asymmetric distribution of mass.

There was a question on some physics talent test concerning a lead sphere with a bubble of air in it that was off center. The problem was to calculate the gravitational potential outside the sphere. The potential when expanded had a large monopole contribution, a dipole contribution, and even higher order dipoles.

I am surprised that some posters here responded by saying there is no such thing as a gravitational dipole. The existence of a dipole term in an asymmetric density distribution is rather obvious. Yet, I keep on seeing it appear in newspaper articles, popular science texts, and so forth. It is rather frustrating. I know the authors are trying to simplify physics for people who are not professional. However, there is such a thing as over simplifying physics. I dislike seeing oversimplified statements with a fraction of truth used like they were unquestionable fact.

The statement "there is no such thing as a gravitational dipole" isn't precisely true. However, there is some truth to it. A proper understanding requires some qualifiers.

The earth is close to symmetric but not quite. The earth is not invariant to an inversion operation. Merely looking at a globe of the earth shows that there are more continents on the North hemisphere then in the South hemisphere. Continental crust is not as dense as oceanic crust. Therefore, the asymmetry in continents shows that there may be a small gravitational dipole in the earth. However, it would have to be small since the crust is only a very small fraction of the mass of the rest of the earth. So far, the gravitational dipole of the earth has not been measured.

I was giving you the benefit of the doubt. Maybe you are intuiting the existence of a gravitational dipole but don't know the words for it. Maybe you are being confused by all those people who say that a gravitational dipole "can't" exist. So I was addressing the scenario where maybe the earth has a small gravitational dipole. You could call this the "Darwin123 thought experiment". Assume that the earth does have a nonzero but small dipole moment that hasn't been discovered yet.

Even if such a gravitational dipole were to be discovered, all it could do is shift the point of zero gravitational field a very small distance. The gravitational field in the center of the earth would still be effectively zero. There would still be a point of zero gravitational field strength close to the center of the earth. However, the gravity at the center of the earth would still be insignificant in the sense that it would be unmeasurable.

I was mentioning obits because the existence of a curved orbit implies the presence of a nonzero gravitational field. The motion of your magnets implies, as you pointed out, the existence of a nonzero magnetic field. In fact, the motion of the magnets implies a nonzero gradient in the gravitational field. However, the two fields don't resemble each other in the least.

There was no more meanness in my statement then in yours. You said, with emphasis and sincerity, that one can eliminate the differences between the earth and the magnets by replacing the ends of the magnets with monopoles. You did not ask whether it could be done. You said that it could be done. I merely pointed out that you can't. You interpreted this as meanness.

Read your own post carefully. Read it like someone else unfamiliar with you would read it. You were making emphatic statements of "fact" that were wrong. I perceive emphatic statements of fact as a type of condescension. However, my response was merely to explain why you were wrong in great detail.

If I was irritated, then it was my own business. I did not call you anything. I merely said that you were wrong. Then I told you why. Maybe you were embarrassed by the fact that your emphatic statements of fact were wrong. You have the right to explain in more detail why I am wrong.

You were implying that your analogy with magnets somehow disproved the fact there there is no gravitational field near the center of the earth. You were wrong. I not only said you were wrong, but I told you in great detail. I don't recall saying anything explicitly "mean".

I may have been irritated. However, I channeled the irritation by explaining the situation in detail. I hope the moderators don't think that explaining things in detail is improper. The guidelines don't say that "detail" is synonymous with "rude".

Can't someone be irritated without being rude? Can't someone answer questions in detail without being called rude?

yes, I understand that bit. But I don't get where it makes an analogy to the situation of gravity at the centre of the earth.
If we were not speaking of radio antennae’s, but rather strong gravitational sources, would we not have a similar situation? I am assuming the gravitational equivalent of radiation. Perhaps that is the folly of my thinking.

Darwin says "it doesn’t matter if gravity is mediated by gravitons, space-time, or leprechauns. The earth may not have a gravitational field at it’s center." I choose leprechauns (in good humor Darwin, I promise). The net effect of the leprechauns at the center of the earth is zero. But that is because the leprechauns at the center of the earth balance each other out. The leprechauns themselves still exist at the center of the earth, do they not?

Any perturbation in density that the inversion operator changes can generate a gravitational dipole. However, one needs an asymmetric distribution of mass.
Darwin, I really don’t know much about inversion operators. With regards to gravitational dipoles, does this imply the existence of positive and negative gravitational charges at some elementary level? Or is it more of an effect such as you mentioned with the perturbation in density?

Regarding the aversion that some people have regarding gravitational dipoles, perhaps it is because they think it implies the possibility of negative mass. I can’t speak for anyone else, but I know that I personally have been guilty of having prejudices against the idea of negative mass.

You can use stationary electric charges to make good analogies with gravitation; they're both 1/r2 static forces. Moving charges and time-varying fields, not so much.
Yes. I was thinking exactly the same thing right before your post came up. Using the magnetic field was a mistake. I am aware that the magnetic field is based on a time varying charge, that the magnetic flux lines will never be perfectly aligned and perfectly still, that even a bar magnet is only a magnet due the motion of it’s atomic components, etc. I was hoping to get a pass on all that, because (so I thought) that at a large scale those factors would be negligible.

Yes. I was thinking exactly the same thing right before your post came up. Using the magnetic field was a mistake. I am aware that the magnetic field is based on a time varying charge, that the magnetic flux lines will never be perfectly aligned and perfectly still, that even a bar magnet is only a magnet due the motion of it’s atomic components, etc. I was hoping to get a pass on all that, because (so I thought) that at a large scale those factors would be negligible.
Dipoles are short range compared to monopoles. Therefore, dipoles have no significant effect on the "large scale". Monopoles have a large effect on the "large scale".

If your magnets were placed in orbit 100 miles apart, the magnetic dipoles would not affect each others orbit. If they were placed 10000 miles apart, the magnetic dipoles would certainly not affect each others orbit.

The field strength of monopole decreases as a square law with the distance. The field strength of a dipole decreases as the cubic law with distance. The monopole has far more range than a dipole. So when you talk about "long range", you are generally talking about monopole effects.

Their gravitational monopoles of these same magnets may have a small effect over a long time. This is why shattered asteroids and comets often stay together in similar orbits, and even merge back together.

The total scattering cross section in scattering of a Coulomb Law point charge (monopole) is infinite. The total scattering cross section of a point dipole is finite. The infinite cross section is a limit which breaks down at some point due to screening effects. Therefore, it isn't totally physical. However, the infinite range of a the Coulomb potential should be read as being "much, much larger than the range of a dipole."

Darwin, I really don’t know much about inversion operators. With regards to gravitational dipoles, does this imply the existence of positive and negative gravitational charges at some elementary level? Or is it more of an effect such as you mentioned with the perturbation in density?

Regarding the aversion that some people have regarding gravitational dipoles, perhaps it is because they think it implies the possibility of negative mass. I can’t speak for anyone else, but I know that I personally have been guilty of having prejudices against the idea of negative mass.
I don't know what your mathematics level is. Mathematically, the inverse of a series of points is a series of points coordinates equal to the negative of the original coordinates.

"Inversion operation of a system" means turning the whole system inside out but keeping the relative distances. Or equivalently: turn a system upside down and make a mirror reflection.

If new image is identical to the original unmoved object, then the original system is considered invariant to inversion.

If you make a copy of a globe of the earth which is inside out with respect to the original, the copy will look not look like the original seen in a mirror and upside down. If the copy of the original globe had the continents concentrated near the top, then the continents on the copy will be concentrated near the bottom of the globe.

The lack of inversion symmetry implies that there is a small gravitational dipole of the earth. However, I don't think that it has been measured.

My thought experiment was imagine that the earth really had a gravitational dipole that was nonzero but very small. The gravitational field near the center of the earth would be nonzero but negligible for all practical purposes.

I don't think that you were talking about the effect of very small gravitational dipoles. You could correct me if I am wrong. In order to understand your question, I had to be sure that you weren't talking about the effect of very small gravitational dipoles.

The existence of small gravitational dipoles embedded in a large monopole would come down to Euclidean geometry. Geometrically, the earth is slightly asymmetric. Given the asymmetry and Newton's Law of gravity, the earth may to have a very small dipole at "the center of the earth". However, the rest of your post indicates that you have something else in mind. What remains to be seen.

If we were not speaking of radio antennae’s, but rather strong gravitational sources, would we not have a similar situation? I am assuming the gravitational equivalent of radiation. Perhaps that is the folly of my thinking.

Darwin says "it doesn’t matter if gravity is mediated by gravitons, space-time, or leprechauns. The earth may not have a gravitational field at it’s center." I choose leprechauns (in good humor Darwin, I promise). The net effect of the leprechauns at the center of the earth is zero. But that is because the leprechauns at the center of the earth balance each other out. The leprechauns themselves still exist at the center of the earth, do they not?
There are leprechauns (i.e., virtual particles) jumping around everywhere. Even in a vacuum, in intergalactic space, there are virtual photons balancing each other out. Some names for virtual particles in an area with no measurable field is "vacuum state", "zero point energy", "quantum fluctuations", etc. They aren't a "field".

What an experimenter calls a field is not everywhere. There would not be any measurable gravitational field in intergalactic space, any more than there would be a measurable gravitational field at the center of the earth. However, there would be the same density of "virtual particles" jumping around.

In any case, what other people are trying to tell you is this. The gravitational field can not reach a peak near the center of the earth. In fact, There is no measurable gravitational force on the atoms near the center of the earth. The gravitational field at the center of the earth adds up very close to zero. In fact, there will be a point very close to the center of the earth where it will add up to zero.

Of course, no one can yet reach the center of the earth yet. Therefore, it can't be directly tested. However, the theories regarding gravity and quantum mechanics have been tested in many other environments. So far as these theories predict, the gravitational field strength is at a minimum near the center of the earth.

The theories predict that the gravitational field strength is effectively zero near the center of the earth. Your arguments contradict both the theory and experiments done everywhere other than the center of the earth.

If the virtual particles cancel each other out with respect to force, then by definition the field is zero. The gravitons at the center of the earth supposedly cancel each other out.

Some measurable quantities will be very big at the center of the earth. The pressure at the center of the earth will reach a maximum and be very large. The pressure is indirectly related to gravity. However, pressure is not the same as gravitational field strength. The pressure can be extremely large even when the gravitational field strength is zero.

The difference in potential between the center of the earth and a point infinitely far away from the center is also supposed to reach a relative maximum which is very large. However, this relative potential difference is not a gravitational field strength, either.