"I wouldn't know a spacetime continuum or a warp core breach if they got into bed with me." -Patrick Stewart
It's the end of the week once again, and so it's time for another Ask Ethan segment! There have been scores of good questions to choose from that were submitted this month alone (and you can submit yours here), but this week's comes from our reader garbulky, who asks:
Why does gravity decrease the further away you are from the object? I've read that it does decrease with distance squared but not why it does this.
This question seems so simple, and yet the answer -- to the limits of our understanding -- is nothing short of profound.
Physics, and science in general, doesn't normally address the question of why when it comes to natural phenomena; it normally sticks to how. You give me an overarching theory, such as a set of laws, and physical objects with specific properties, such as a set of particles, and science tells you how those objects behave according to the predictions of that theory. Gravity is no exception.
For centuries, Newtonian gravitation was the most successful theory describing forces on the largest scales, saying that every object in the Universe that has a mass exerts an attractive force on every other object in the Universe with a mass, and that the magnitude of that force is proportional to the mass of both objects and inversely proportional to the distance between them. That's what Newton's law of universal gravitation says, and what it tells us is -- in principle -- how any system of particles will behave under the influence of gravity.
Can we say something intelligent about why gravity works this way, though? Let's think about our own neighborhood for a minute.
The Sun, the largest mass in our Solar System, is orbited in circles and ellipses by practically every known object, from planets to asteroids and (most) comets. There's something special about circles and ellipses that we don't normally think about as special: they're stable, closed orbits, meaning that these objects return to the same point they started at after what we call a year.
That alone, mathematically, tells you something incredibly interesting. You see, all forces are vectors, meaning they have magnitudes and directions. In the case of our Solar System, the direction of the force on each object is (to an excellent approximation) towards the center of the Sun. Want something to go around the Sun in a closed orbit? Guess what.
You only have two options! One is to have a force that obeys an inverse-square law (like gravity does), and the other is to have a force that increases linearly with distance (like a spring does), and there's a theorem that proves those are the only two possibilities!
So it could have gotten stronger or weaker as the distance increased, but only in one particular way, or we wouldn't have stable, closed orbits.
And since those are the types of orbits required to have stable, moderate temperatures necessary for life, we sure did luck out that these are the laws governing our Universe!
Now there are some forces where the force increases as your distance from the object increases: the strong force is a great example! And there's even an example of a type of force that has no direction and is constant everywhere: that's what dark energy is, permeating all of space equally!
The thing is, though, saying that gravity is an inverse-distance-squared force is an incomplete story. In fact, the very fact that we have an orbit in our Solar System that very clearly isn't closed is how we wound up replacing Newtonian gravity with our modern theory of gravity: General Relativity!
We've come up with a whole host of predictions that have been borne out by experiment and observation, including the gravitational bending of light, the different orbital mechanics of systems with large masses and small distances, gravitational redshift, and many, many others.
But the greatest advance that's related to this question of the strength of gravity is the knowledge that all orbiting bodies do not technically obey an inverse-squared force law.
All orbits under General Relativity come from forces that behave ever so slightly stronger than inverse-square laws, and this means that they will eventually decay over long enough timescales. The innermost planets will have their orbits decay first, followed by progressively outer worlds, because the distance is larger. Eventually, in the absence of all other phenomena, everything would spiral into the gravitational source at the center of all orbital systems.
For an object like Earth that orbited an imaginary, infinitely-long-lived Sun, it would take something like 10150 years for the orbit to decay, but it means that a true stable, closed orbit is a phantasm, something that doesn't really exist in this Universe!
At least, in a Universe governed by General Relativity, which is the best law of nature we have to describe gravity. In the weak-field limit (an approximation) -- when masses are small and distances are large -- this can be shown to reduce to Newtonian gravity, which is where the inverse-square-law-with-distance comes from!
But why do we have General Relativity as the theory that governs gravitation in this Universe, with the particular details that it has? I can't say for certain; no one can.
Which means I have to resort to the standard cop-out answer: the force of gravity is this way because the laws of nature cause it to be. We can imagine a Universe where those laws are different, but this is the one we've got, and we don't fully understand why the laws are this way any deeper than that. We can observe phenomena, infer the laws, test them in new and spectacular ways, and maybe someday we'll understand why the laws are this way. In the meantime, this is the best answer we've got!
Another way to look at it is that it's because our space is three-dimensional. For example, when you make a noise, its volume level falls off with distance according to the inverse-square law, simply because the same amount of energy gets spread out over the surface area of a sphere. As the sphere expands as the noise travels away from its origin, the surface area increases by the square of the radius. In exactly the same way, light dims according to the inverse-square law, as the same energy is spread out over an increasing surface area. So one would expect gravity to behave in the same way.
As Ethan points out, it's not quite that simple, but then General Relativity shows that space isn't really Euclidian....
I've been playing around with a causally connected view of spacetime lately, and there the role of energy density and pressure is to shape the transition from past to future light cones.
So, my personal answer to the stated question would be that when the distance between two gravitating bodies increase, the portion of history they share (the overlap of their past light cones) to their respective total histories decreases as well, which allows for a larger decorrelation of their futures and thus a smaller coupling.
(Why does this site keep giving me "Service unavailable" errors when trying to comment?)
I've been playing around with a causally connected view of spacetime lately, and there the role of energy density and pressure is to shape the transition from past light cones to future ones.
So, my personal answer to the stated question would be that when the distance between two gravitating bodies increase, the portion of history they share (the overlap of their past light cones) to their respective total histories decreases as well, which allows for a larger decorrelation of their futures and thus a smaller observed coupling.
The inverse square law (which even general relativity is using in unchanged way) has its roots in LeSage's shielding mechanism, which has been originally developed by Newton friend, Nicolas Fatio de Duillier, who was genial Swiss mathematician, living in the shadow of Newton. He was much smarter than him from certain perspective - for example he deduced with it, that the gravity must be indirectly proportional to the square of distance, i.e. not linearly, how Newton assumed. The same opinion was occupied with Robert Hooke, who was a competitor and public enemy of Newton. Hooke based his opinion on century old experience of old Arabian astronomers, who were actually first, who deduced the inverse square law. So when it turned out, he was right and Newton wrong in this matter, the otherwise confident Newton got so upset and ashamed with it, he withdrew himself from scientific life and publications for further sixteen years.
Between others Nicolas Fatio correctly deduced, that the shielding must come from flux of corpuscles, which are spreading faster than the speed of light and he called them ultramundanne. Now we know, these corpuscles are actually the gravitational waves and they manifest itself with CMBR noise, which is all around us. The AWT just extends this explanation to composite particle bodies (virtually all fundamental forces can be explained with the same mechanism) and for explanation of cold dark matter (Allais effect), caused with shielding of this shielding with nearby massive objects. The gravitational shielding of longitudinal waves has its supersymmetric counterpart in shielding of photons at short distances, which is known as a Cassimir force
CatMat, there's another site on scienceblogs where they talk about the politics of climate science and deniers are buying time on a spamnet to nuke the site to get at them.
The inverse square law also is the low-energy approximation to a scattering problem, in which two fermions interchange virtual massless bosons. Works out that way for electromagnetism (spin 1 boson = photon), works out for gravity (spin 2 boson = graviton).
If you read the original Einstein you see him saying something like a gravitational is present when a concentration of energy tied up as say a massive star conditions the surrounding space, altering its qualities. The effect of this diminishes with distance. Note that if you had long massive rod, the effect would diminish in a 1/r fashion. But stars are spherical and space is three-dimensional, so the effect diminishes in a 1/r² fashion.
As regards the rubber sheet pictures, imagine you’ve placed a whole lot of parallel-mirror light-clocks in an equatorial slice through and around the Earth. When you plot all the clock rates, your plot resembles the rubber sheet because clocks go slower when they’re lower. The curvature you can "see" relates to Riemann curvature, which relates to curved spacetime. And you measured those clock rates, so it’s a curvature in your metric. It isn't some curvature of space. And it’s important to remember that a clock that's lower doesn't run slower because your plot is curved. It doesn't actually run slower "because spacetime is curved". It runs slower because a concentration of energy conditions the surrounding space, altering it.
Because of the way in which we measure it! only American scientists arrogant enough to "conclude" instead of leaving an open - ended set of observations?
Uncle B, are you arrogant too, or do you go through life without ever coming to any conclusions?
I was also under the idea that it would be similar to a point of light spreading its energy on a sphere, but correct me if I'm wrong, it is implied by the omission of this common explanation that that's not the reason for gravity behaving this way, since (correct me if I'm wrong again) the object is not irradiating energy. That's also why I understand Ethan mentions there's no answer to "why", and Bertrand's theorem.
Quick heads-up to the webmaster - if climate denier freaks are indeed DDOSing the sites, join Cloudflare.
ao9, the particle explanation of the square law is that each virtual particle is massless and can only have the energy that is limited by the uncertainty principle. Since to go further takes more time, the amount of energy the virtual force carrier particle can have by existing drops. And since that internal energy is the force felt between the two points packaged up, the force between those two particles is also less.
Wow: gravity doesn't work because of particles whizzing around. Not does electromagnetism. Virtual particles aren't real particles, hydroigen atoms don't twinkle, and magnets don't shine. We've talked about thisa before, see Ethan's weak force blog and look at teh comments from #25:
John, particles are the QM version of forces.
If you've proven them wrong, where the hell is your prize?
Great work once again! As the old saying goes, science asks and seeks to answer "how" while philosophy asks and seeks to answer "why". Little wonder the highest educational titles merge into "Doctor of Philosophy" regardless of specialty.
John #13: "Wow: gravity doesn’t work because of particles whizzing around. Not does electromagnetism."
Feynman, Schwinger & Tomonaga won the Nobel for showing how electromagnetism works by "particles whizzing around" - or to be more precise, how the exchange of virtual photons creates the force we see as electromagnetism at larger, non-quantum scales. As confirmation that their theory is correct, they (at least, F & S) computed the magnetic moment of the electron. Without a theory of virtual particles, this value should be 2.0; the measured value is 2.0023… The theory agrees with experiment to over 10 decimal digits, and no other theory (without virtual particles) can explain the value correctly.
John, if you think you have a better theory than that, you should show us a prediction that is at least as accurate.
"so it’s a curvature in your metric. It isn’t some curvature of space."
Actually, it is curvature of spacetime since the layout is coded into the metric. Gravitational lensing and bending of light rays is a clear show that yes, spacetime itself curves. It isn't some abstract mathematical curve.
Sometimes you have to give up. I've tried to communicate in a half dozen threads to Wow that virtual particles are different from particles. He does not understand and is not open to understanding the concept. Wow believe particles and virtual particles are the same thing except the virtual ones wink out of existence before they have to be real.
The term 'virtual' is important. Particles and Virtual Particles are not the same thing. A Virtual Particle is a standing wave that links two points. It does not whiz around anywhere. It doesn't move. It can't travel. It exists between two points, then it doesn't exist.
No, you've *communicated* that often enough.
What you've failed to do is to argue the case for it.
David: That's technically true, but there is a difference between the virtual particles of a Perturbation Theory, and the full Quantum Field Theory of electromagnetism.
In Perturbation Theory virtual particles are identical to real particles, you just can't observe them. The picture (specifically the Feynman diagrams) are of little electrons and photons popping in and out of existence and zipping between objects with lifetimes and energies under the uncertainty principle limits.
In QFT, a real particle is a specific type of disturbance in the fields with well-defined momentums, energies, masses, etc, while virtual particles are mathematically different and really have little in common with real ones at all. They are still responsible for transmitting forces between real particles, but the picture of a bunch of photons or electrons, identical to the 'real' ones but unobservable, swimming around, is not a good one. They are still disturbances in the electron/photon fields, but not ones that look like an electron or photon.
What happened is John heard about this and the point that "virtual particle" is a misleading piece of jargon, and ran with it off to la-la land. It's like when he heard about the shear terms in the stress-energy tensor of GR and started saying "well to mean that means spacetime is..." It's semantic-implication-aka-pun-based physics with no real understanding.
Sinisa Lazarek: Ha, thanks for finding that gem, and perfect example of what I was talking about.
In GR "curvature of spacetime" and "metric" are the same thing, as in mathematically equivalent, the most literally "the same" two things can be. The metric *is* the geometry of spacetime.
"Spacetime" as opposed to "space" being another example. Taking "space" *not* to mean the Newtonian concept of space but rather the 3 spacial dimensions of our 3+1 dimensional spacetime, then talking about curvature of space is correct, so long as it is understood that it may also be a curvature in time, or both, to varying degrees depending on relative observer. But space does curve. That's what GR says.
Seems that you are on a personal quest that involves
"The Power of Intention is.. Divine Guidance.. A greater Will that drives you forward on your Life Mission..." from A Cry for Help 2009 by John Duffield
As well you John Duffield are on a scientific quest
"I ponder what might have been (had Einstien lived longer).. Ilkie to think the end product would have dispelled so much mystery that we could not have failed to grasp how the universe works.. If only Einstein had somehow passed on what he knew to Feynman.. then things would have turned out different. So different that by now NASA.. wouldn't be reaching for Mars, they'd be reaching for the stars." fromRelativity plus the Theory of Everything by John Duffield
Now John Duffield, you are entitled to think and believe whatever you wish. I have some unusual thoughts myself. But if you are truelly scientific minded; then it is my opinion that you must be clear about your biases when you speak about science.
Thus, John Duffield, if you were honest; your preface to your comments on this blog would be: I have published at least two books that most scientist think are very speculative. A couple of those unaccepted speculative ideas are thus and thus.
In another place you could clarify, this particular idea is not speculative; it is generally accepted by most scientist.
In my opinion John Duffield, you have not properly introduced yourself; because you have failed to give a sense of your strongly held personal biases.
The point John Duffield is this: a science discussion is not about blindsiding and misleading in order to convince a naive audience of your arguments. No a science minded person must make effort not to mislead; as Feynman says, "The idea is to try to give all the information to help others to judge the value of your contribution; not just the information that leads to judgment in one particular direction or another."
And "give all the information" is exactly John Duffield what you do not do. No, you speak as if you are an expert; and you are not. You speak as if your pronouncements should be accepted; well they have not been.
John Duffield you have published your two or three books. If they speak truth; then trust in the test of time. But if they do not speak truth; well, then I understand why you come out to this blog and try to confuse those who are trying to be part of the honest science discussion.
Be honest that your ideas are seriously speculative or be quiet.
You speak as if your pronouncements should be accepted; well they have not been.
If that troubles you, okthen, then don't ever visit his blogsite: the foolishness will make your head explode.
Tell us something about MOND, Ethan.
Or is this something that history has alredy deured?
You may take it as "Ask Ethan", next episode?
MOND doesn't work. Or it works for galaxy rotation and nothing else, so it doesn't work. It has already been covered here on several occasions. Search through the blog and you'll find topics dealing with it.
Here's a good one for bottom-lining why MOND, while still a neat idea, isn't about to negate the need for Dark Matter:
Thanks. Far as I know, there still are those who follow the idea. How so?
People would often prefer to keep a wrong idea than work out a new one, basically.
There are still those who think Earth stand's still... in the 21st century!! I wouldn't have believed it if I hadn't seen some of them even post on this blog. How so?
.. same as having people still believe all kinds of other things... our nature I guess.
"How so" can't be really answered.
@Sinisa #30, N #28, etc.: Why do good physicists continue to explore MOND? My personal take (and note that I'm *NOT* an astrophysicist, or even a theorist, just an experimental particle physicist!) is that there are two complementary effects.
First, psychology and sociology. A fair community of theorists have developed around Milgrom's model, and have expanded and extended it. It's rather difficult to give up years of research (or to repudiate your adviser's research) if you feel like it's still viable.
Second, good science. The hallmark of a proper _scientific_ theory is that it is falsifiable: it makes concrete predictions for hitherto unobserved phenomena, which can be tested by appropriately designed observations. (Note that does not require _experiment_: observational astronomy is a perfectly valid scientific effort, despite what crackpots and YECs might claim). However, working through the math to actually make those predictions can be exceedingly difficult! Extending MOND to see what effects it could have on cluster/supercluster scales, making it compatible (or at least parametrized) with GR, and so on, are not trivial.
Finally, there's the potential payoff. Suppose we do, at some point, discover that observations actually support MOND (to the exculsion of existing GR/DM/DE predictions). That would be a pretty clear indication of new physics beyond what is already known, something that all _real_ physicists would be extremely excited to find.
About MOND, first I don't disagree with Wow or SL or CB or Ethan.
Rather, I defer to their opinions about MoND.
Yes, yes, it is my turn to be the village idiot. Contradicting even myself.
My personal bias is against MOND; it seems at best to be a useful provocateur theory.
"MoND was proposed by Mordehai Milgrom in 1983" and Milgrom is still publishing papers on it 40 years later http://arxiv.org/pdf/1311.2579v1.pdf. Unfortunately his papers are unreadable to me.
Wikipedia's MoND summary says this, "Within the uncertainties of the data, MoND has remained valid.. the uncertainties on the velocity of galaxies within clusters and larger systems have been too large to conclude in favor of or against MoND. Indeed, conditions for conducting an experiment that could confirm or disprove MoND may only be possible outside the Solar system. . A couple of near-to-Earth tests of MoND have been proposed though.. A test that might disprove MoND would be to discover any of the theorized Dark Matter particles, such as the WIMPs.. Lee Smolin and co-workers have tried unsuccessfully to obtain a theoretical basis for MoND from quantum gravity. His conclusion is "MoND is a tantalizing mystery, but not one that can be resolved now.".. On the other hand, another 2011 study observing the gravity-induced redshift of galactic clusters found results that strongly supported general relativity, but were inconsistent with MoND (Wojtak, Hansen, and Hjorth). A recent work has found mistakes in the work by Wojtak, Hansen, and Hjorth, and confirmed that MoND can fit the determined redshifts only slightly worse than does general relativity with dark halos."
So that's that or what is that?
And what in the world am I as a layman suppose to understand that MOND is proposing?
Scientific American in 2002 gave Milgrom space to describe MOND to us laymen http://www.astro.umd.edu/~ssm/mond/sad0802Milg6p.pdf
Note the first and other pages are blank so scroll down.
Even at this most lucid, Milgrom leaves my eyeballs rolled up and stuck looking at the inside of my skull.
And furthermore maybe quantum gravity will explain things
------- with dark matter or without dark matter (my bias)
------- with MoND or without MoND(my bias)
But hey, I have no, in the detail, reasons for my biases. So until the experts prove otherwise; I defer to the dark matter experts (my bias).
Yes, I notice that I contradict myself in that I am biased against dark matter but I defer to dark matter experts. I'll tell you why!
At least the dark matter hypothesis doesn't leave my eyeballs stuck looking upward in their skull sockets. Rather, just thinking of dark matter hypothesis, for me anyway, brings my eyeballs back to their normal position in their skull sockets.
So I say, let the few experts who fiddle with MoND keep fiddling.
But I warn that it is a very tiresome, on my eyeballs, to even try to follow what MoND experts are arguing. Milgrom's Sci Amer 2002 article leaves me quite unsatisfied; and, as previously noted, the MoND effect, which leaves my eyeballs stuck looking at the inside of their skull sockets, is quite tiring.
Falsifiability isn't really that huge a thing, though it's needed to weed out the patently anti-scientific "Last Thursday Creationism"-type "theories".
The point about falsifiability is more that you have no reason to believe you have it RIGHT if your theory cannot be falsified, since there's nothing consequent from it that would demonstrate it as being valid over any equally compelling theory.
Falsifiability is about weeding out the bad, not accepting the good.
But the existence of special pleading arguments means that in a colloquial sense it carries far more weight than it does for what is more centrally important: predictability.
Falsifiability requires a prediction to test against.
The use of a theory requires prediction to be used for.
Prediction is what the theory is all about and is the prime difference between a theory in science, which gives predictability, and curve fitting, which doesn't.
God, nowadays, gives ZERO predictability. When it used to be "able" to predict stuff (tornadoes, flooding, lightning, etc), it was found that there was no God there.
Then NOMA tried to put a box around science so that predictions in science could not replace predictions in religion. However, that didn't make God-predictions work any better, so the arguments for any god becomes non-prediction.
"Shit happens" is not a scientific theory.
Neither is "Climate has always changed".
"Yes, I notice that I contradict myself in that I am biased against dark matter but I defer to dark matter experts."
I'm "against" Dark Matter too, if it's reified like "cold" or "dark". As a placeholder showing the phenomena's characteristics, I'm 100% fine with it.
Those working on the theories of Dark Matter are, I hope exclusively, working on a theory of what that Dark Matter *is*, and then testing that theory against the rest of science and predicting the results.
Failure then fails that theory of what the stuff is, but the *characteristics* remain.
MOND doesn't display the require characteristics, so in that sense, it is at the very least incomplete in its explanation.
But a non-particle demonstration with the same *characteristics* as Dark Matter would be just as fine as a "matter" demonstration.
Wow #34 #35
For those also wondering, Stephen Jay Gould’s idea of non-overlapping magisteria (NOMA).
So Stephen proposes two areas of human understanding (or misunderstanding) that don't overlap. Really, seems impossible to me. I mean even sense and nonsense seem to overlap everywhere. Oh well, I'm not only the village idiot; I'm a religious chameleon.
Of course, I believe in Santa Claus and a great deal else depending upon where, what and who I am talking to and whether I wish to agree or disagree with them.
@OKThen #35, and Wow #33, #34: Well said, indeed. Good clear statements; I'm not sure I completely agree with your take on falsifiability. The ability of a proper scientific theory or hypothesis to make _concrete_ predictions, and specifically predictions which (at least in principle) can actually be observed, measured, detected, whatever, is critical.
Falsifiability weeds out not just the anti-science YECs, but also the crackpot "just so stories", and the ubertheoretical models which make "predictions" about differences from our current models at scales which are utterly unmeasurable even in principle (string theory, I'm looking at you!).
In any event, I think this is merely a difference in "scale," not a disagreement of principle.
Astronomy is one place where concrete predictions and definite observation is often unavailable.
It's still a science.
Falsifiability is Popper's take, but there was a lot less indirect measurement necessary in his day for science.
Since then the progress of science to inferential propositions means the usefulness of "falsifiability" in determining if something should be considered pseudo or science is not that high.
Still plays a part, but not a central tenet.
Quantum gravity is falsified in the realm of General Relativity.
QG still scientific, but "to an extent" wrong.
My issue with MOND is that in trying to make something work, it makes a whole bunch of other things not work.
If we didn't have relativity, something like MOND could be considered. But GR showed us that ND is already an approximation of GR. So tweaking ND while breaking a more encompassive theory is futile. Imagine a world without GR and only MOND. There would be dozens of phenomena in the world which we would have absolutely no explanation for.
So this brings us to the real issue. Weather or not to accept DM as something that really exists, but we haven't detected it yet, or modifying GR, not Newton. There is also an issue of weather or not we really "know" how GR works on something as large as a galaxy. Have we taken into account all the different components that contribute, have we missed some things.
SL, that's not an issue with MOND, it's an issue with trying to make MOND the only factor at play.
TeVeS tries to correct that deficiency by bringing in relativity, which helps explain a lot of the direct problems with using MOND in a universe that appears to obey GR, like gravitational lensing.
It makes sense, as MOND was developed in the context of the anomalous galactic rotation curves, where the prediction of Newtonian gravity shoulda-woulda been enough and relativity could be safely ignored. But when you want to explain other phenomenon you have to go beyond it.
It still doesn't work to explain all the universe without something like Dark Matter. So as a Dark Matter alternative MOND and its offshoots aren't panning out, but they are still interesting.
I mean, it's not like it's impossible that there are new particles we haven't discovered but out-mass known particles 4:1 *and* our understanding of gravity is not quite right. Even if it just provides another way of looking at things, it could be useful.
This is awesome info.
Is it possible that gravity is caused by dark energy or dark matter? If you imagine a large above ground swimming pool, and you suddenly make the sides of the pool disappear, the water will rush out in all directions. The water furthest from the center moving away from the center much quicker than the water in the middle. Now, think in 3D, and consider outside of the pool as space, and the water dark energy. Anything near the outside would be accelerating faster than an object near the middle. Because of expansion outer objects would be picking up speed as the water pushes them away. The universe expanding?
Think now of the pool on a much larger scale, and the water as dark matter trying to fill any void where it is not. Maybe gravity is not objects being drawn together as much as objects being pushed together by dark energy. Perhaps the reason that the closer two objects are, the more they are attracted to each other (gravitational pull) is because the dark matter between the two accelerates causing a Bernoulli effect between the objects, thereby drawing them closer.
Gravity is not a force. Your view of it is purely Newtonian, which is fine for everyday stuff. But when you start talking about DE, DM and spacetime, you need to understand what current science is saying, before new hypothesis. In short, your model fails on many fronts, mainly because you don't understand how spacetime works in relativity.
I've read that Einstein's calculation of the curvature of light grazing the Sun is exactly twice Isaac Newton's. When anything is exactly twice something else, there must be an explanation, or at least a connection between the two.
What explains this non-coincidence?
"What explains this non-coincidence?"
The maths. Duh! :-)
GR is a geometric change of a 3d space. It's no more weird than the constant of proportionality between the circumference of a circle is 2pi and the area of a sphere is 4pi.