A Systems Theory of Relativity and Gravity, #2.

(Mattson, 2015)

TABLE OF CONTENTS

1. Introduction: Small Worlds

2. Every System Is Incomplete

3. The Speed of Gravity

4. The Three-Body Problem

5. The Singularity Isn’t Near

6. Conclusion: The Great Awakening

The essay that follows will be published in three installments; this, the second installment, contains sections 3 and 4.

But you can also download and read or share a .pdf of the complete text of the essay, including the REFERENCES, by scrolling down to the bottom of this post and clicking on the Download tab.

3. The Speed of Gravity

A picture held us captive. And we could not get outside it. (Wittgenstein, 1953: §115, p. 48^e)

One of the most confounding mental pictures in modern physics – as pernicious, perhaps, as the chimera of a closed system – is our notion of massive bodies “exuding” or “radiating” gravity. We are accustomed, because of Isaac Newton’s formulae for measuring gravity, to perceive gravity as a function of an object’s mass. It would probably help us, in a variety of ways, to think of that relationship in reversed terms: an object’s mass is the result of its gravity. The more an object interacts with things around it, the more its gravity increases. In this sense, gravity seems to create “closed systems”—like our solar system. Objects gain or lose mass in direct proportion to the local amplitude of their relations with other matter in the universe.

The clearest demonstration of this interrelationship was described by Albert Einstein as part of his theory of relativity (Einstein et al., 1923/1952). Einstein showed that the mass of an object (that is, of an object with nonzero initial mass) increases, and time dilates, as the object’s velocity approaches c, the speed of light. This has never been a particularly comfortable formulation for other physicists, however, and there is currently a vogue for insisting that the mass of objects remains constant at all velocities. What “increases,” according to this bastardized version of Einstein’s original insight, is an object’s energy and momentum. This makes relativity appear more consistent with the First Law of Thermodynamics, concerning the conservation of energy and matter in the universe.

Unfortunately for this “updated” version of relativity, the equations that describe the change in an object’s mass as its velocity grows are not linear, which makes them incompatible with the formula (m*v) that enables us to calculate momentum. Furthermore, to claim that an object’s “energy” increases with its velocity, in some fashion distinct from its momentum, is a nonsensical hedge that imputes “energy” to an object in an abstract, harmless way, as if the object is storing it up in an offshore bank account. This abstract energy then must re-emerge, and become very consequential, when the object suddenly cannot reach or exceed c. We do not have any physical way of representing all this because it is nothing but a convenient fiction. Converting relativistic mass increase to an unspecified energy increase amounts to offloading all the difficult implications of special relativity, which produced a lot of novel, refractory mysteries en route to a complete explanation of the relationship between energy and matter. We have to go back to Einstein, armed with quantum physics, to perceive how velocity and gravity truly interact.

Earlier I claimed that mass is an effect of gravity. In order to square this with classical thermodynamics, it is crucial to understand that objects do not literally “grow,” like some inflatable pool toy, as their velocity increases. Rather, what we measure as their apparent mass increases because they “encounter,” in the same instant, more of the other matter (and, therefore, more of the gravity) in the universe. Another way of putting this is that an object’s systemic boundaries become more porous in proportion to increases in its positional uncertainty, and this enables the object to maintain localized gravitational relationships across a wider swath of space as it becomes increasingly “smeared” over a larger surface area. This is consistent with both Heisenberg’s uncertainty equations and special relativity. No object with mass can move at or above c because doing so would mean engaging with an infinite totality of other gravitational fields. A single hydrogen atom, moving at the speed of light, would drag the entire universe along in its train.

Inevitably, this raises another fundamental question: what limits the speed of light, and why do gravitational waves (like those produced by a supernova) propagate at precisely that speed? The answer is that photons in our universe do not, as it were, choose to confine themselves to 3 x 10⁸ m/s. The phrase “speed of light” (and measuring distance in “lightyears”) is almost a misnomer; what we are talking about, in reality, is the speed of gravity. That’s why gravity, a force that is not particulate (or even, like light, a wave and a particle) travels at a perfectly consistent, finite speed across the cosmos. Why is it so shockingly slow? Gravity is not a form of energy transfer, and although it is an effect of massive bodies, it has no mass of its own. It is, according to general relativity, simply a factor that curves the spacetime around it. This curvature is not just an effect produced by objects with mass. It is also an indispensable condition of possibility mass-bearing objects require in order to exist in the first place. Rather than conceiving of gravity as a kind of exceptional, distortive “warp” in the fabric of spacetime, it is more accurate to think of spacetime itself as a consequence of gravity. Gravity produces the extensible space that surrounds us; objects “make room” for themselves in proportion to their mass.

That’s why gravity travels through space at a finite, consistent speed, when it “should”—considered logically, but superficially—affect spacetime instantaneously. Gravity “should” propagate, therefore, instantaneously: what hinders spacetime from assuming whatever shape it pleases? It is not an object; it is the medium through which objects push forward, blind as sturgeons. What delays it? Nothing other than, once again, gravity interfering with itself. Just as there is no theoretical limit on quantum entanglement, the gravitational interaction between different massive objects never zeroes out, no matter how far apart they are. This presents three important corollaries:

The mass of an object represents the minimum “degrees of freedom” matter needs in order to subsist at all, in any discrete form (Verlinde, 2010).

Neither matter nor pure energy (light) can escape the total gravity that gives space its volume.

The local gravitational effect of mass-bearing objects is increasingly superseded, at macroscopic scales, by a homogeneous interference pattern created by the total gravity of all matter, extending in all directions. This is why the gravity of a body with mass “declines” in proportion to its distance from the observer measuring it. It arrives at our position as quickly as it can, braving a soup of interfering fields.

This last observation has actually been anticipated for millennia by diligent human observers of cosmic phenomena. In his lecture “Lightness,” the Italian writer Italo Calvino said that

what strikes the literary imagination about Newton’s theories is not the subjection of all things and people to their own inescapable weight but rather the balance of forces that allows celestial bodies to float in space. (Calvino, 2016: p. 28, italics added)

This moves us to propose the following Normalization Principle (NP) : the entirety of space suspends the mass and energy of the universe in a homogeneous, non-local gravitational field that arises from the distributed interrelation of all matter, due to gravity. This field resists the propagation of both local gravitational disturbances and pure energy. This inherent resistance also delimits (i.e., localizes) the gravitational effects produced by discrete massive bodies.

4. The Three-Body Problem

I don’t want to serve as the instrument of some… triang[le]…between you two. (Hurlyburly, 1998: line spoken by Mickey, aka Kevin Spacey)

This brings us, at last, to another hitherto insoluble problem I mentioned at the beginning of this essay: the so-called “three-body problem.” As we usually understand it, the “three-body problem” articulates the chaotic behavior of three objects co-located within a closed gravitational system. The reciprocal interaction of three overlapping gravitational fields produces chaotic outcomes that cannot be adequately described or predicted by classical, Newtonian physics. As it happens, the problem has already been solved (to the extent that a solution is possible) by two Israeli physicists, Yonadav Ginat and Hagai Perets, who mapped the branching probability of different trajectories resulting from three interacting gravity-bearing objects (Ginat and Perets, 2021).

These trajectories do not produce infinite positional variations, since many identical positions recur across different “branches” (possible sequences) of states.[i] They are not infinitely prolonged, either, since eventually the highly identified gravity of two such objects, close enough to become a loosely concentrated “inner binary,” slingshots the third one out of range (“eject[ing it] into infinity,” as the authors phrase it). This is known as a “dissipative encounter,” and is an inevitable, entropic result of such closed gravitational systems. (Of course, as I have already asserted, the inverse relationship between gravity and distance is itself a product of “external” influences baked into any gravitational calculation. There is no such thing as a closed system.) The method Ginat and Perets used to calculate these sequential possibilities is known, colloquially and quite vividly, as “the drunkard’s walk.” Their own terms for their solution—they call it “approximate,” “analytical,” and “statistical”—remind us that their calculations are, at best, probabilistic distributions, rather than exact predictions.

But, of course, even this pioneering work isolates three gravitational fields from the rest of space as long as all three objects stay within nondissipative bounds. In real space, gravity is not precisely homogeneous for anything other than maximally entangled phenomena. Why isn’t the Earth subject to the same chaotic perturbations as the inebriated planets modeled by Ginat and Perets? This is really no different than asking why there are no “naked” singularities—black holes large enough to have infinite event horizons. This is why the NP is so essential to a rational understanding of gravitational effects in real space: without it, every event horizon could become infinite, as long as the black hole producing it was able to assimilate mass faster than it was dissipating via Hawking radiation.

Another implication of the NP is that the fundamental gravitational field that produces c as a universal constant, and constrains the velocity of objects with mass, is produced by self-interfering simultaneous entanglements. These entanglements mostly remain below the threshold of observable forces while making spacetime fundamentally (albeit weakly) resistant to the propagation of discrete phenomena across its entire area. This is consistent with Richard Feynman’s path integral equations for collapsing all possible trajectories of a quantum particle moving through space: his equations set the stage for “destructive interference” between different extreme trajectories that help us convert uncertain (waveform) probabilities into the classical, linear behavior of particles, without invalidating either kind of propagation. (On the contrary: Feynman makes them complementary and mutually interdependent.) Similarly, the NP makes the weak impact on the Earth of, say, the Crab Nebula, mutually interdependent with the Earth’s classical gravitational trajectory in orbit around the sun.

The NP is also upheld by the gravitational theories of John Wheeler, including his promising, essentially unfinished work on the possible topologies of overlapping gravitational fields. Specifically, Wheeler proposed—after an initial, failed attempt to explain cosmic phenomena via geons, hypothetical structures that combined electromagnetic and gravitational effects—that the laws of the universe might emerge from gravitational waves that became stable due to interference from other gravitational waves. He abandoned his research into these “revised and updated” geons after finding himself unable to account for their emergence or their stability. His hypothesis wasn’t wrong, however. It was just incomplete.

The research I cited earlier, on the three-body problem, showed how non-hierarchical (and “non-dissipative”) arrangements of all three bodies in a closed gravitational system can phase-shift into a 2v1 arrangement, composed of a semi-stable “inner binary” and a smaller orbital satellite. The combined gravity of two planets (constellated into a “binary”) sometimes expels the third “towards infinity,” sending it beyond the threshold necessary for the tripartite system to continue operating in the same chaotic, interdependent fashion as before. The binary “remainder,” with only two planets, then becomes entirely stable and predictable. I can speculate, with reasonable confidence, that the chaotic interaction of mass-bearing particles creates similarly opportune conditions for isolated pockets of emergent order where the mutual, destructive interference of other gravitational fields in the surrounding environment becomes unusually symmetrical. In other words, Wheeler was right: gravity, by itself, is adequate to explain how objects with nonzero mass and “classical” properties can, in fact, emerge ex nihilo from a seething vacuum. They are born, like Aphrodite, from permutations of the quantum foam. Furthermore, once they do emerge, they persist on a scale that makes them impervious to chaotic quantum fluctuations.

NOTE

[i] These transpositions are not precisely equivalent, however, because the information unique to each separate sequence is preserved in the form of each object’s orbital angular momentum.

In the next installment, we’ll look at quantifying the amount of information in the universe. Prepare yourselves for “The Singularity Isn’t Near” and “The Great Awakening”!

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