Please comment on the accuracy of the perspectives on long-range gravitational attraction between distant particles as discussed herein. Thanks in advance!

In the following two sections, I will state my contention/observation from a classical, and then a GR perspective:

Classical: From a classical perspective, the gravitational attraction between two distant objects such as between a star and an orbiting planet, or between two galaxies, is equal to the sum of the gravitational forces between each of the separate particles with a non-zero mass in the first object, and each of the separate particles with a non-zero mass in the second object. I say non-zero mass to exclude particles such as photons. This simplification ignores the "mass defect" of bound nucleons in atoms of roughly 0.8%, and the lesser mass defect due to the chemical binding energy in molecules, etc. Stated mathematically, the total gravitational force $F$ between the two distant very massive objects, such as between two galaxies, stated in terms of gravitational attraction between individual particle pairs, is given by:

$$ F \approx \sum_{i=1}^{n1}\sum_{j=1}^{n2}G\frac {m_i m_j} {r_{ij}^2} $$

where n1 is the number of particles in the first galaxy, n2 is the number of particles in the second galaxy, and where each particle in the first galaxy has mass $m_i$, each particle in the second galaxy has mass $m_j$, where the distance between each particle pair in the two galaxies is $r_{ij}$, and where $G$ is the gravitational constant. "$\approx$" is used instead of "=" because mass defect is not considered in this simplification. This simplification also assumes that the vectors between each particle pair are coincident (which is not strictly true) though different in magnitude. Further, the word particles as used above may be of nucleons and electrons, or nucleons may be divided into quarks, as long as the same standard is used for both galaxies.

The point of this consideration is to emphasize that while the gravitational force between say two nucleons or electrons at the distances between a star and one of its planets possibly billions of kilometers apart, or between two nucleons or electrons at the distances between two galaxies millions or billions of light-years apart is vanishingly small, it is in fact the sum of those vanishingly small attractive forces that combine to yield the total gravitational attraction between star and planet, or between two galaxies in the above examples. This perspective may yield a new appreciation for many, that without those "vanishingly small" forces between individual particles across distances of even billions of light-years, the composite force between those distant massive bodies of planets, stars, and galaxies, would not exist.

GR: From a general relativity perspective, where the "force" of gravity is replaced with spacetime curvature induced by the very existence of a non-zero mass particle, or a system of such particles such as in an atom, planet, star or galaxy, we may consider that the gravitational "attraction" between the two very massive distant systems such as star-planet, or galaxy-galaxy, is merely a result of the combined spacetime curvatures associated with each individual particle, acting on each particle in the distant object, such as between one electron or nucleon in one galaxy, and another electron or nucleon in the distant galaxy.

The point of this exercise is to reflect upon the idea that while we usually think of spacetime curvature as being induced by very massive bodies such as planets, stars, and galaxies, the vanishingly small spacetime curvature induced by even a single electron, at distances of even billions of light-years, is entirely significant. While this observation may be mundane and obvious from the perspective of a trivial mathematical summation, it struck me as profound and thought provoking that the spacetime curvature of for example a single electron, billions of light-years distant from that particle, is not only present, but essential for the functioning of cosmology on a macro scale.

For simplicity, I do not mention here the contribution of other non-zero-mass particles beyond electrons, nucleons, and quarks.

I invite comment on both: 1) The accuracy of my statements about the gravitational attraction between very distant particles and their contribution to the total gravitational attraction between large and distant celestial objects, as well as 2) My reflection on the significance though vanishingly small spacetime curvature associated with each individual electron and nucleon at distances of even billions of light-years.

If this second point is correct, then it strikes me as profound that the spacetime curvature of a single electron or single neutron at billion light-year distances is not only present, but critically important for the structure and functioning of the entire universe.

Thus every single non-zero mass particle makes its presence known to the entire universe, and I found that idea to be astonishing and awe inspiring in its reality and implications.

To state this more prosaically and playfully, the gravitational influence of an electron in the tip of my nose, influences an electron (and the hierarchical systems to which it is bound - i.e. atom, molecule, cell, organism) in an electron in the tip of the nose of a purported being in the Andromeda galaxy, contributing meaningfully to the total gravitational attraction between our two galaxies, and by extension, even to gravitational interactions with far distant galaxies such as those revealed in the Hubble Deep Field, about 13 billion light years away.

And thus in conclusion, the spacetime influence of individual particles are present even billions of light-years distant from each individual particle, and are critically important in the structure and function of the entire universe.

Thanks in advance for your interest in my question/comment, and your responses!

  • $\begingroup$ Interesting to think about dimensions. Why is the square of distance in the formula? There seems much more potency if you consider differences between celestial masses and rest masses of particles. - Imagine the universe being attracted by surrounding "distant eyes", sort of black holes, and the universe located in a center where you never know who's turn it is to "lock on you" (whole picture: expanding). Then "direction" in my opinion - angles - unifies gravitation and wave particels in their being forced under more differing angles the more they are distant from each other. "Imagine." $\endgroup$ Commented Nov 9, 2022 at 16:01

2 Answers 2


There are two ways of thinking about theories of physics:

The first, and popular for theoretically inclined physicists is that mathematics defines reality. The old Pythagorean and Platonic view: mathematics exists and nature dances to its tune. The second is the pragmatists, who say "data and observations can be fitted with mathematics formulas that will also be predictive of future behavior'.

The history of physics up to now tells us that what the previous generation thought was a complete description of nature ( only engineering is left) , the next generation experiments and theories showed to be an approximate.

General relativity completes Newtonian mechanics for large masses and energies because it was found necessary to be used for planetary and cosmic observations, and has been validated up to now.

Validated means that there are no contradictory data or observations to say that GR is falsified in a region of variables where it is expected to hold.

Falsification needs comparison of measurement numbers with predictions of a theory to be discordant. But measurements come with errors, in the case of cosmology large errors compared to centimeters and seconds. Thus, any prediction of GR for situations as the one you describe, of an electron light years away to affect an electron here, is way within possible errors of measurement. This means that the statistical effect of matter light years away can only be rationally considered, as only gross effects can be observed and measured. At the moment that is the assumption, just the masses and spins of planets are enough for the GPS accuracy we need for driving around.

The measurement errors overwhelm any effect of individual particles, and the statistical randomness ( temperature of star) of particles making up a star also remove any theoretical attribution of the curvature from a specific particle to influence the curvature of a specific particle light years away. There is no way to see any measurable effect of your proposition, anyway.


I thought your question was very elegantly put. Yes, the enormous forces that govern the structure of the universe are the cumulative effect of a vast number of extremely small forces that are individually utterly negligible.

  • $\begingroup$ Long delayed response!... Thank you Marco for your comment and interest! I am glad you share the profound implications of my reflections. We live in an amazing universe! $\endgroup$
    – Roger
    Commented Mar 6, 2023 at 5:29

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