# Can anyone explain Gravity to a biologist?

Dale
Mentor
2020 Award
Has there been any deeper understandings in how gravity acts at a distance
Yes, the non-mathematical summary is that it doesn't act at a distance, it acts purely locally. Esentially, you can write the equations for gravity in a differential form, which means that all of the quantities are defined locally and affect each other only locally. This is analogous to the situation in electrostatics, where you can write Coulomb's law (which appears at first glance to act at a distance) in the mathematically equivalent form of Gauss' law (which is purely local).

Inertia as the ability to move slower than the speed of light ... I have never looked at it that way before. But it seems to me we could could subsitiute mass for inertia in your statement and the meaning wouldn't change. In relativity is mass and inertia different or are they the same thing? Seems to me mass and inertia are always coupled, but I may be wrong.
Since lately I've been really trying to conceptualize mass as a function of energy in a certain state, inertia and gravitation seem like the things that distinguish matter from radiant energy. Mass just seems to be a concept for comparing objects via their relative inertia in a gravity field. Inertia is the ability to resist motion and gravity is the ability to attract other matter. What is mass then? It is not a behavior, is it? Just a descriptive quantity, i.e. a property, right? So my big questions are 1) whether massless radiation can generate gravitation and 2) how could inertia result purely from force-field interactions and 3) how could inertia and gravitation be related in a mechanical sense?

General Relativity is our best theory at the moment.

You forced me to get off my lazy butt and refresh my basic relativity

OK, I think I can get my head around this:

Isaac M. McPhee wrote:

"The moon, to use this example again, is continually orbiting the Earth, and from the perspective of Earth appears to “curve” as it does so. In reality, the moon is moving in a straight line through space which in itself is curved.

Read more at Suite101: Gravity and General Relativity: Einstein's Revolutionary Theory of Curved Spacetime http://www.suite101.com/content/gravity-and-general-relativity-a44580#ixzz11LhEFkqw

Since motion is relative, it doesn't really matter how the objects are moving WRT eachother the curvation and the acceleration will be g. Since the space is curved in an independent and localized manner the space can appear curved and straight depending on your frame of reference.

Neat and tidy but hard to test experimentally.

A thought question:

If you built a steel ring in deep space and measured it to be as close to a perfect circle as possible and then put around a dense massive object (spin matched) how would it behave?

Would it form a straight line of infinate length? If you built a smaller straight edge in deep space as straight as possible and laid it against the ring would they lie flat against eachother or something else?

If you fired electrons around the ring and the massive object had no magnetic field would the electrons emit radiation?

I am guessing the perfect circle would no longer be perfect once it is in the grav field so electrons would do something.

What would be the shape of the ring that would form a perfect straight line in orbit and have electrons do nothing? Probably impossible as no such shape would exist.

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If you built a steel ring in deep space and measured it to be as close to a perfect circle as possible and then put around a dense massive object (spin matched) how would it behave?

Would it form a straight line of infinate length? If you built a smaller straight edge in deep space as straight as possible and laid it against the ring would they lie flat against eachother or something else?

If you fired electrons around the ring and the massive object had no magnetic field would the electrons emit radiation? I am guessing no.

I think the point is not whether a ring would ever appear as anything except a ring but whether appearance or inertial behavior is more important from a physical standpoint. The moon's orbit may be circular or elliptical from a certain point of view, but isn't it more important that it moves through spacetime according to its own inertia without changing trajectory by force?

I think the point is not whether a ring would ever appear as anything except a ring but whether appearance or inertial behavior is more important from a physical standpoint. The moon's orbit may be circular or elliptical from a certain point of view, but isn't it more important that it moves through spacetime according to its own inertia without changing trajectory by force?

Shape probably would depend on your frame of reference. I am interested in more localized phenomena.

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I think the point is not whether a ring would ever appear as anything except a ring but whether appearance or inertial behavior is more important from a physical standpoint. The moon's orbit may be circular or elliptical from a certain point of view, but isn't it more important that it moves through spacetime according to its own inertia without changing trajectory by force?

Doesn't it not change trajectory by force because at any point in that frame space is actually straight? Seems to me part of the elegance of this theory is due to the warping causing straightness.

Ok, now we are getting somewhere. Yes, theories are a conceptual model, just like the statistical models for inheritance.

Going further with biology, the discovery of the function of DNA in terms of genes "solved" the problem of inheritance.

I'm not an expert, but in many ways, GR 'solved' the problem of gravity. There's a lot of people here who can help you understand GR better. It provides a fairly simple conceptual model that predicts and explains measured data.

Just as 'it wasn't until [Newtonian mechanics was developed] that the how questions started getting answered', when GR was developed, *additional* how questions started getting answered. GR also makes some very simple predictions that are extremely difficult to test (and GR is still being tested).

Back to biology- the discovery of DNA explained a lot, but there is still a lot unknown about genes. It's an incomplete model- so in addition to the genome, we have the proteome, metabolome, fillintheblank-ome, because the more accurately we want to model things like the development of an organism, the more factors we have to take into account. We have a quantitative idea of a gene, but not what each one does (the proteome project). Also, we do not understand related things like upstream regulators for expression, post-translational modification and the spliceosome (one gene != 1 protein), modifier genes, etc.

So to be fair, there's still a lot of research regarding genetics.

Does this help?

How very true, turns out genes weren't quite as illuminating as first thought. Otherwise we would be chimpanzees. I pulled out a bunch of different types of antibody gene from the Rhesus monkey and they didn't look much different from our own.

Actually some of the things you mentioned are pretty well understood now:

" but not what each one does (the proteome project). Also, we do not understand related things like upstream regulators for expression, post-translational modification and the spliceosome (one gene != 1 protein), modifier genes, etc."

The genes and gene products, and target signals related to the splicing that occurs in eukaryotes that result in 1 gene making many proteins are fairly well understood. There is good proportion of genes have a "function" associated with them. How many genes are regulated is also fairly well understood at the near atomic level.

But these are like looking at an elephant with a microscope, the big picture of how it all fits together, develops and stays stable is far from clear. Maybe Feyman would say "it just does".

If you had to ask me what question in biology is as puzzling and counterintuitive in Molecular Biology and Evolution as gravity is in physics it would probably be:

How the hell did we ever make it this far anyway? Almost proves the existance of God but if you saw the horrible mismash that our genomes (genetic material) are you would probably think again about grand design. There must be an infinate number of universes.

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The solution I adopted, although not necessarily correct; is as follows:

Imagine that infinity consists of gravitons, now if a group of gravitons are reduced in volume then this causes the remaining gravitons (i.e. infinity) to expand to fill the volume that would otherwise be a volume of absolute nothing (i.e. an impossibility). All the gravitons of infinity are now polarised on radials of the compacted group; that is to say they are acting in opposition to the compacting force and there you have gravity. (Create another compaction and you will see why orbits are needed to prevent collision).

Further compaction of groups within groups creates the other forces of nature, that is why all three primary forces of nature (the weak force is a secondary force) have mathematical theories that are variations of the same (Inverse Square) Law.

Andy Resnick
How very true, turns out genes weren't quite as illuminating as first thought. Otherwise we would be chimpanzees. I pulled out a bunch of different types of antibody gene from the Rhesus monkey and they didn't look much different from our own.

Actually some of the things you mentioned are pretty well understood now:

" but not what each one does (the proteome project). Also, we do not understand related things like upstream regulators for expression, post-translational modification and the spliceosome (one gene != 1 protein), modifier genes, etc."

The genes and gene products, and target signals related to the splicing that occurs in eukaryotes that result in 1 gene making many proteins are fairly well understood. There is good proportion of genes have a "function" associated with them. How many genes are regulated is also fairly well understood at the near atomic level.

But these are like looking at an elephant with a microscope, the big picture of how it all fits together, develops and stays stable is far from clear. Maybe Feyman would say "it just does".

If you had to ask me what question in biology is as puzzling and counterintuitive in Molecular Biology and Evolution as gravity is in physics it would probably be:

How the hell did we ever make it this far anyway? Almost proves the existance of God but if you saw the horrible mismash that our genomes (genetic material) are you would probably think again about grand design. There must be an infinate number of universes.

I'm glad we are starting to speak the same language.

I disagree with how you characterize the state of genetics, but I note that you have come to a conclusion 'stolen' from physics ("it just does").

A theme in your reply makes me wonder if you are confusing 'how?' with 'why?'. Science only can answer 'How'. In biology, it's typically "what is the mechanism", because biology is solely concerned with matter- biochemical reactions and the like. In Physics, the answer can be more abstract (spacetime is curved), but that's just one of the differences between physics and biology.

Evolution, in particular (since you bring it up), cannot be used to answer 'why'. Evolution provides a plausible mechanism and logical foundation to explain things like protein homology. Trying to answer 'why does...' leads to a dead end. Trying to answer 'how does...' leads to a productive research program.

The solution I adopted, although not necessarily correct; is as follows:

Imagine that infinity consists of gravitons, now if a group of gravitons are reduced in volume then this causes the remaining gravitons (i.e. infinity) to expand to fill the volume that would otherwise be a volume of absolute nothing (i.e. an impossibility). All the gravitons of infinity are now polarised on radials of the compacted group; that is to say they are acting in opposition to the compacting force and there you have gravity. (Create another compaction and you will see why orbits are needed to prevent collision).

Further compaction of groups within groups creates the other forces of nature, that is why all three primary forces of nature (the weak force is a secondary force) have mathematical theories that are variations of the same (Inverse Square) Law.

How could gravitons be compacted? If they exist, they don't seem to themselves be affected by gravity, and if they are they can escape a black hole so how could they possibly be compacted?

Hi Sirandar. I too share your interest in gravity. I am just a layman in this field but I think I have come to a good understanding of how gravity works. Not "why" of course, no one knows that. You do not need to be good at math to understand it in Newtonian form. The breakthrough came for me when I understood 1) Newton's universal law of gravitation, 2) frame of reference, and 3) the difference between the concepts of "inertial mass", "active gravitational mass", and "passive gravitational mass". (in that order)

What is the difference between "active gravitational mass" and "passive gravitational mass"? This is the first time I've seen the distinction.

- Curtis

P.S. In another life, I was known as a Turtle master. I even https://www.amazon.com/gp/product/007148664X/?tag=pfamazon01-20.

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I'm glad we are starting to speak the same language.

I disagree with how you characterize the state of genetics, but I note that you have come to a conclusion 'stolen' from physics ("it just does").

Actually, this isn't really meant to be a conclusion ........ perhaps the bio version would be "it just does or we wouldn't be discussing this" ;)

How and why ? two very different questions BUT maybe the same question looked at from a different "frame" of reference. Try that one on a grant proposal ;) Hard to even think about where I could successfully use the word why.

"Why did I supersize that chicken combo at MacDonalds last night?" Because I skipped lunch STOP STOP STOP don't go any further .... too late

To me the key to understanding how gravity works will involve figuring out how inertia works, because inertia is what distinguishes matter from radiant energy. Gravitation seems to be a function of mass, but not as immediately as inertia is, and inertia seems to be responsible for conflicts between objects in motion that result in force-exertion among them. Presumably this is impossible with EM radiation since it always travels at the same speed within any medium, whereas matter can travel at different speeds in the same medium. This is due to inertia, i.e. the ability to move slower than the speed of light. Could inertia be related to gravitation, for example in the way that cellular DNA is related to characteristics of the organism as a whole?

It sure seems like inertial and gravity are related and even have the same underlying root cause/mechanism. The equivalence principle is one of the most interesting observations in all of physics.

Inertia acts like every tiny bit of matter is connected in some 3D harness connected to super tiny springs that are somehow non-interlocking with all the other springs. So when you change velocity there is a slight lag in the springs so that you are stretching one side before the other side catches up to balance the stretching springs. When the mass stops accelerating then the springs can reach an equilibrium.

Gravity acts like those springs are connected between each piece of mattter and they have a nonlinear spring constant such that the force between any two bits of matter vary as $$\frac {1}{r^2}$$.

Very strange circumstances indeed, but it does seem from the equivalence principle that inertia and gravity are related and IMHO probably have the same underlying mechanism.

The core puzzle that needs solving and perhaps the biggest one in all of physics is the question of what mass is, along with the related one of how mass can be converted to energy and vice versa.

Actually, this isn't really meant to be a conclusion ........ perhaps the bio version would be "it just does or we wouldn't be discussing this" ;)

How and why ? two very different questions BUT maybe the same question looked at from a different "frame" of reference. Try that one on a grant proposal ;) Hard to even think about where I could successfully use the word why.

"Why did I supersize that chicken combo at MacDonalds last night?" Because I skipped lunch STOP STOP STOP don't go any further .... too late

Actually contrasting how and why make for facinating discussion in biology but little else because it ends up like GR .... beautiful elegant but hard to test.

A good example would be the related questions "How do we die?" and "why do we die?"

How we die seems pretty well understood except when you dig really deep even though there are actually an almost infinite number of ways to do so. They all share a common theme "a particular order to an unparticualar chaos"

Why we die seems like a completely different question.

I could say that the the only solution that fits all the "hows" of a stable physical existance.

You would say "Prove that that is the only solution when there are a infinate number of other solutions"

I would say "OK, it's the simplest solution that fits all the "hows" of a stable physical existance given what this universe is like." You would say "That's almost the same question, tell me more about these hows even though there are an infinate number of them, can you make a top ten list?

I would say "ability to change is an adaptive trait because things change, as long as it is not too much at once .... living too long makes it hard to change" Maybe that's number one. You would say " Things change but they also stay the same, physical laws don't change so why do we need to."

I would say "physical laws may or may not change but the net interaction of those physical laws can result in change. And you exist in the context of other life which are also changing. If their changes are better than yours you may be excluded from existence"

>>>>>>>>>>>>
Hours later
>>>>>>>>>>>>

I don't think we are getting anywhere, but it was fun

inflector said:
What is the difference between "active gravitational mass" and "passive gravitational mass"? This is the first time I've seen the distinction.
Active gravitational mass specifies a body's role as the source of a gravitational field. Passive gravitational mass specifies a body's susceptibility to being affected by a gravitational field. In Newton's universal law of gravitation active gravitational mass is usually represented as a large M and passive gravitational mass is represented as a small m.

From everything I've read, no one is really sure exactly when and why the distinction first came into usage. However, according to Max Jammer's book "Concepts of Mass in Contemporary Physics and Philosophy" it was not before the 20'th century. I have read that it may have been an attempt to draw an analogy to electric charge. But whatever the reason I think it has caused a lot of confusion.

Gravity has to be the most screwed up force in the universe and the most enigmatic. It pulls on a boulder and a feather to produce the same acceleration. How screwed up is that? It is even more improbable as a moon always showing the same face to the planet it orbits. There is an infinate (or very large finite) number of other states, but moons do and for a relatively understandable reason).

The notion of it is so weird. A large planet sized object without any intermediary particle somehow "knows" I am there and pulls on me at g no matter what my mass is (non relativistic).

Gravity just doesn't make any sense. I remember being outraged at both relativity and that the vacuum of space that transmits electromagnetic radiation even though there was nothing to vibrate. It is a strange counterintuitive universe. I am much better now ;)

Nobody has come up with any theory of gravity that my limited understanding can swallow. Somebody said that gravity is a consequence of mass moving through time. Seems to me that this is as well saying gravity is a result of hedgehogs dancing the rumba.

The explanation of why everything falls at the same rate is that larger objects are pulled more by gravity, but their larger mass also gives them higher inertia, resisting that pull by just as much.

And it would be better to think of gravity relativistically, since it does make more sense that way.

Gravity has to be the most screwed up force in the universe and the most enigmatic. It pulls on a boulder and a feather to produce the same acceleration. How screwed up is that?.....

Gravity isn't screwed up, our understanding of it is. What do we know about gravity? We do not know the mechanism by which it works. A few of its effects have been mathematically described, but we haven't identified the fundamental parameters by which it can be described, as we have with light.

Light once was as enigmatic as gravity, we could see it, it had color and brightness, and it was thought to have an infinite velocity. Once an astronomer determined it had a finite velocity, its mysteries gradually surrendered themselves to intelligent inquiry over a few centuries. Identifying that light could be described by the measurable parameters called wavelength and frequency was a big breakthrough, but it was not known until later that light was an electromagnetic emission. It was found that brightness, a magnitude, was independent of wavelength and frequency.

The force of gravity is independent of the magnitudes of two masses, and as Sirandar stated, "It pulls on a boulder and a feather to produce the same acceleration." The gravitational constant (G) is the constant of proportionality between two masses, regardless of their individual magnitudes. It is my opinion that gravity should be thought of as a type of energy phenomena, and we have not associated the correct parameters to its description. Buried somewhere in the quantum structure of an atom are the energy processes that create gravity. Just as the descriptive parameters wavelength and frequency were already known before they were associated with electromagnetic emissions, we may already know the parameters, measurable factors, that can be used to describe the mechanisms that produce gravity. Once we associate the correct parameters to the phenomena of gravity, its mysteries will be unraveled; hopefully, it won't take centuries.

Gravity is really interesting, what exactly it is, to me is still a mistory, but I feel that finding a 'graviton' might never happen. dont really know why.

I do think that matter, mass and space and time and gravity are tied in ways that make its explination difficult.

If time slows down in the presence of mass, what would be the rate of time in a universe with no mass?

Is it mass and its gravitation that 'moderates' time ?

I wonder, (often wonder about things), if time is standing still on a black hole, there is "NO TIME" available to create any energy, or do any work, but what is radiated from a black hole is gravity. So we are getting a force generated by an object that is not experiencing time.

I would assume if gravitons existed, then that is the only product that would be radiated from a black hole, meaning that a BH is generating energy while experiencing no time.

Time is going so slow, or stoped so that massless photons cannot ascape, but somehow gravity itself is 'radiated' from a black hole in large quantities.

If it was gravitons creating gravity, then under what mechanism would that occur, if a graviton is radiated, would it be at the speed of light ? would it have mass, and would the source of the mass from which it was emmited have less mass ?

That way black holes would eventually evaporate, as it's radiating vast amounts of gravity.

But I expect its just a product of time, gravity effects time, but itself is not affected by time, or at least can still exert a force in the period of NO TIME, that would exist on a BH where time stands still.

Its all quite confusing.

But more so is that gravity is very very very weak, compared to the other forces, electromagnetic, and strong and weak nuclear forces. Gravity is tiny, ultra small.

You can overcome the total gravitational pull of the earth, with a 20c fridge magnet !

So the force coming out of a tiny magnet acting on a key, is able to overcome the gravity of the entire earth acting on that same key.

Gravity is weak, is that because it's leaking away from our universe, into another 2 dimentional 'brane' ?

Or is gravity just the way it is because the universe would simply not work if it was different. Maybe there are many universes, but only the ones with the right amount of gravity turn into universes like ours. who knows ??? I wish I did... :)

Im glad we are asking the questions, and that smarter people than me are thinking about it harder than I could possibly do.

Is it possible to generate light without mass ? would that imply that without mass there would be no light, and without mass or light how 'fast' would time be flowing ?

Does the energy of a photon when its absorbed by an electron increase the mass of the atom of that electron, and therefore increase the mass of the entire object, and if so does that extra mass go away or stay when the source of light is removed.

So when a photon hits the electron, does it permanately increase its electron mass, or the bulk mass of the object.

Does light permanently turn into matter when its absorbed into that matter ?

General relativity is not something you can describe or affiliate with particle physics. It is an understanding of how space time works. As you peel the layers back to its simplest form there is still an infinite amount of questions to answer. Space time is not something that can be itemized like the particles in the LHC, it will be a question that our children will try to answer and there children and so on. Gravity will always remain in limbo because until we know what makes this fabric called space time there is know way to understand everything it contains. Physics is good to educate working knowledge of how the universe works but it does not in any way educate that these things are fact. Physics has always presented itself through teachers that it is fact until proven inconsistent with data. Something cant be theorized and then stated as fact because people cant disprove the theory. All the physics in mainstream are there because they have been observed in the lab and the theory that provoked the experiment will be used until someone can find better descriptions for it. Gravity in its entirety being all that we have to prove how it works is;

Mass+ density=gravity

The mediation of this gravity was produced by Einstein’s general relativity that mass disturbs the fabric of space time. His theory was shown by Eddington when he produced a photo of mercury’s orbit before and after and the stars behind it were in different positions. This shows that light was being affected by gravity. This is the best description for gravity at this time. TM

It sure seems like inertial and gravity are related and even have the same underlying root cause/mechanism. The equivalence principle is one of the most interesting observations in all of physics.

Inertia acts like every tiny bit of matter is connected in some 3D harness connected to super tiny springs that are somehow non-interlocking with all the other springs. So when you change velocity there is a slight lag in the springs so that you are stretching one side before the other side catches up to balance the stretching springs. When the mass stops accelerating then the springs can reach an equilibrium.

Gravity acts like those springs are connected between each piece of mattter and they have a nonlinear spring constant such that the force between any two bits of matter vary as $$\frac {1}{r^2}$$.

Very strange circumstances indeed, but it does seem from the equivalence principle that inertia and gravity are related and IMHO probably have the same underlying mechanism.

The core puzzle that needs solving and perhaps the biggest one in all of physics is the question of what mass is, along with the related one of how mass can be converted to energy and vice versa.

Interesting post. Do you think there could be a connection between the "lag" of inertia and the way that electron orbits shift during photon-generation? Put another way, do you think that the system of force-balancing that occurs within the atom is responsible for inertial resistance to changes in motion? If so, I find this interesting since the atom seems to emit photons as a response to the electron's tendency to return to a previous state. I'm not sure if I'm explaining it well, but this indicates to me that EM emissions are somewhat the result of inertia in the dynamic state of the electrons. Does that make any sense?

Interesting post. Do you think there could be a connection between the "lag" of inertia and the way that electron orbits shift during photon-generation?

First, there's no evidence that inertia itself has a lag that I know of. I was only describing what I thought inertia acted like. The lag applied only to one half of the springs in my analogy, the side in relaxation. The other side, the one in tension acts instantaneously. But again, this is only a picture I have in my mind of what I think it acts like. It is the lag on the one side that would cause inertia because you'd need a force to offset the force of the springs in tension to create the imbalance itself until the springs in relaxation became taught again. But again, this is only an analogy.

brainstorm said:
Put another way, do you think that the system of force-balancing that occurs within the atom is responsible for inertial resistance to changes in motion?

It seems to me that all movement (i.e. the dynamics) is due to a system of force balancing among the various competing forces that affect a mass, strong, weak, electromagnetic, gravity, and their various forms interacting with the mass and its inertia. When you get down to the quantum level this gets tricky to think about.

brainstorm said:
If so, I find this interesting since the atom seems to emit photons as a response to the electron's tendency to return to a previous state. I'm not sure if I'm explaining it well, but this indicates to me that EM emissions are somewhat the result of inertia in the dynamic state of the electrons. Does that make any sense?

Well since we know that the Schrödinger Equation is a wave equation and at the macro level we only get wave equations when there is a restorative force of some sort, there is some tendency for the electron to, as you say: "return to a previous state," as part of its dynamics.

However, in the case of photon emission, this is simply an electron dropping from one quantum level to a lower energy quantum level. I don't think anyone has a good physical model for how this works that maps to macro-level phenomena we see in everyday life at this point.

I think the recently published paper explaining gravity as entropy makes a lot more sense and is intuitively more satisfying than hunting for gravitons or inventing dark energy. You do ask really interesting questions that would seem to point out some "gravitonational" problems with a black hole. The gravity of the situation is a singular sticking point, you might say.

Inertia acts like every tiny bit of matter is connected in some 3D harness connected to super tiny springs that are somehow non-interlocking with all the other springs. So when you change velocity there is a slight lag in the springs so that you are stretching one side before the other side catches up to balance the stretching springs. When the mass stops accelerating then the springs can reach an equilibrium.

Is that not the same as saying that the centre of effort moves forward in proportion to the resistance and stays there as long as the resistance is applied.

Gravity acts like those springs are connected between each piece of mattter and they have a nonlinear spring constant such that the force between any two bits of matter vary as $$\frac {1}{r^2}$$.

Gravity is a linear force, r is a linear measurement, so why is the constant 'non-linear'?

The core puzzle that needs solving and perhaps the biggest one in all of physics is the question of what mass is, along with the related one of how mass can be converted to energy and vice versa.

I cannot quote reference, but amongst the books I read one clearly stated that it is a common error to misunderstand Einstein's equatation in that Einstein stated that his equation proved that mass and energy are two different ways of measuring the same entity (as metres and yards are different ways of measuring distance.

I don't know if this is slightly off topic, but it seems related:

Does anyone know if prior to Newton there was knowledge that gravity decreases with distance from the planet? In other words, was Newton the first person to consider that weight decreased with altitude? If so you would think that that would have been as great a shock as the Earth revolving around the sun, no?

In a documentary, I saw that Halley asked Newton what the shape would be if the force were inverse square. There was speculation and discussion along that line. But only Newton could do the math to prove that it results in an ellipse. He kept his fancy math a secret for the time being, and re-worked a proof that used geometric techniques.