Philosophy, Physics

Abstract:  In this article I recommend that substantivalism change its focus from what it has been traditionally—a thinly veiled effort to explain the current standard models of physics—and instead embrace the proper focus of philosophy—explaining reality itself. Exploiting that new perspective, I define a rigorous and testable model of spacetime as a substance, and propose a hypothesis that explains the curious rotational characteristics of spiral galaxies, a phenomenon currently explained by dark matter.

This article is designed to take the substantivalism/relationalism debate in a different direction. If you are new to this topic, the competing theories are:

Relationalism: The proposition that spacetime is an abstract, geometric coordinate system, a heuristic, that helps physicists understand and quantify the relationships and interactions between objects. Only the objects themselves are real, and talk of spacetime curvature, for example, is essentially metaphorical.

Substantivalism: The proposition that spacetime, over and above its role as mere coordinate system, is also and primarily the one and only ontologically real substance of the cosmos. This bold hypothesis argues that all of physical reality—from the gossamer vacuum of “empty” space to the dense nuclei of atoms and black holes—is nothing but spacetime, a single substance, manifesting in a variety of different forms.

A great deal has been argued on both sides of this debate over the past century and a half. In the middle to late nineteenth century, substantivalism was dominant, culminating with Hendrik Lorentz’ work on the luminiferous ether. At that time it struck most scientists as common sense that light, like other waves, required a medium through which to be transmitted. Then suddenly, with the release by Einstein of his Special and General Theories of Relativity, relationalism gained a seemingly insurmountable advantage. Relativity Theory does not depend on a physical medium pervading all of space and it outperforms Lorentz’ notions of an absolute, or stationary, spacetime. Relativity also seemed to explain the failure of the Michelson-Morley experiment, which was designed to detect the stationary ether postulated by Lorentz. This triumph of Einstein continues to inform the conventional wisdom of almost all modern physicists.

While no one seriously questions the power of modern physics to make accurate predictions about a vast range of phenomena, philosophers are burdened by an additional criterion of truth beyond the mere utility of a scientific theory. They must also ask whether or not the theory in question provides a satisfying picture of reality. For many if not most scientific theories (e.g., the germ theory of disease), the utility and reality of the theory in question overlap to such an extent that there is no noteworthy distinction between the two. Our understanding of microorganisms (e.g., the varicella zoster virus) as real ontological things in the world corresponds very closely with our theory of their effects (e.g., chicken pox) on other things. Modern genetics even enables scientists to sequence a virus’ genome and identify the precise proteins and cellular mechanisms responsible for its associated disease. Microbiologists are never expected to acknowledge an empirical effect by a microbial cause that can be neither observed nor even conceived. Any such chasm between physical cause and effect would be regarded as a fatal weakness of the hypothesis, and yet modern theoretical physics is loaded from top to bottom with exactly these sorts of chasms. Biology is entirely realistic while modern physics is largely positivistic.

Given the popularity of substantivalism, it is clear that many philosophers are made very uneasy by the anti-realism of physics. Put simply, positivism argues that if the math works it does not matter why; there is nothing to be gained by trying to visualize the underlying reality, the objects, the things-in-themselves that cause the effects we observe. This position might be regarded as simply a common-sense nod to the wisdom of Kant. But it might instead be seen as a premature surrender in the face of a difficult problem. The jury is still out.

Though it can hardly yet be credited as a hypothesis, substantivalism hopes to inject realism into modern physics, to insert spacetime into the vacant ontological slot at the foundations of reality. Its goal is to put a comprehensible, physical cause behind the empirical effects. For a substantivalist, spacetime is the thing-in-itself.

A New Direction for Substantivalism

If we were to rank the two theories on a scale of 1 to 10, relationalism (understood as the current array of standard models) earns a good solid 7 while substantivalism barely registers an anemic 1. The standard models (e.g., Lambda-CDM Model, Standard Model of Particle Physics, Standard Solar Model) while admittedly incomplete and often incompatible are, nonetheless, astonishing monuments to the human mind. Substantivalism, by contrast, has no comparable record of accomplishments and cannot even boast a widely accepted formulation that might one day form the basis of a testable hypothesis. Perhaps it is no surprise then that the history of substantivalism has been marked by a series of futile efforts, most notably John Wheeler’s geometrodynamics, to recreate at least some of the success of its relationalist big brothers.

Understandable as it is that substantivalists feel compelled to pay homage to the standard models by attempting to reimagine them with spacetime at the core, that project is self-defeating and doomed to fail. When scientists overlay reality with a particular spacetime-as-coordinate-system, that structure comprises all of the mathematical, scientific, and philosophical underpinnings of the latest standard model. It is emphatically not simply a piece of blank graph paper onto which, or clear lens through which, reality is directly and objectively recorded. Indeed, it functions as nothing less than the horizon, the current limitations, and the conditions under which reality itself can appear at all. For example, there is currently no accepted explanation for the curious rotational characteristics of spiral galaxies, though dark matter is the leading candidate. But because dark matter is not understood, there is no aspect of the latest spacetime-as-coordinate-system, when overlaid onto reality, that will allow it to appear. Among other things, this failure announces to us that the current spacetime-as-coordinate-system, fortified with all the best ideas to-date, is incomplete in at least one major respect. Dark energy is another such example, and between them—dark energy and dark matter—we have two gaping holes right at the heart of the standard model, and therefore two gaping holes in the latest spacetime-as-coordinate-system.

Substantivalists believe that the acceptance of spacetime-as-substance and its subsequent elaboration will fundamentally alter our understanding of ontology in general and physics in particular, potentially filling the two holes mentioned above. It follows directly from that assumption that they also believe that the current standard models that are built into the latest spacetime-as-coordinate-system are wrong at the most basic level, and may be very far from the truth in most detailed respects as well. Yet the debate between substantivalists and relationalists usually unfolds from a curious effort on the part of substantivalists to conceive of a spacetime substance that faithfully explains the models embedded in the latest spacetime-as-coordinate-system. Essentially, the standard of proof for a theory of spacetime-as-substance seems to be that it is able to replicate whatever success is accorded to the current spacetime-as-coordinate-system—even though substantivalists ought to recognize that any such standard would require a theory that they themselves have already at least implicitly rejected, and which demonstrably fails to explain both dark matter and dark energy, among many other things (e.g., black holes, cosmogony).

Treating spacetime as a substance rather than as a coordinate system does not simply change the complexion or terminology of the standard models while leaving them otherwise largely intact. It completely transforms, at their theoretical foundations, the models themselves. Therefore, since recreating the standard models using spacetime-as-substance is both impossible and self-defeating, then there is really only one other option. Substantivalists need to formulate a robust model of spacetime-as-substance and then advance a testable hypothesis based on that model. This shift in focus is little more than an acknowledgement, however unnerving, of what this debate has been all along: a genuine choice between two different theories of physics, not merely between two slightly different flavors of the existing standard models.

Assumptions

Building on the observations above, the goal of the substantivalist ought not to be the recreation of the current standard models using spacetime-as-substance. Instead, the substantivalist is called upon to advance a compelling model of spacetime as a genuine ontological substance, and then embark on the difficult project of demonstrating that this substance is superior to the relationalist model that denies it any such existence. For the purposes of that demonstration, therefore, I will make the following [provisional] assumptions about spacetime as a fully extant substance:

  1. Spacetime is the one true substance of the cosmos, a proposition often referred to as radical super-substantivalism.
  2. Spacetime is neither created nor destroyed, it only changes form. Specifically, it exhibits a range of pressure values.
  3. Spacetime has an equilibrium pressure very close to the vacuum pressure.
  4. A gravitational field is a spacetime pressure gradient that manifests according to the inverse square law.

Assumption 1 is nothing more than the polemical commitment made by substantivalists that is required to kick off the debate and is, in the end, the central claim in need of validation. Assumption 2 is a reasonable extrapolation from the well-understood physical principle that mass/energy is neither created nor destroyed. Assumption 3 follows from the fact that the vacuum has a nearly constant pressure (~2.7 on the Kelvin scale); if spacetime is indeed the fundamental substance, then it is the obvious source of that equilibrium pressure. Assumption 4 reflects the fact that, when the two-dimensional analogy of spacetime-as-a-coordinate system is translated into the three-dimensional reality of spacetime-as-a-substance, the two-dimensional curvature becomes a three-dimensional pressure gradient (Fig. 1). Failure to make this final assumption leaves spacetime hopelessly mired in the geometrical abstraction of relationalism and irrevocably denies it any genuine ontological reality.

Figure 1: Spacetime Curvature Versus Pressure Gradient
Figure 1. A two-dimensional spacetime curve describes a geometric abstraction with no ontological reality, while a three-dimensional spacetime pressure gradient is fully consistent with the requirements of a physically extant substance.

I make no claim at this point in the discussion that these assumptions are either obvious or already proven. Instead, they constitute, collectively, an adequately robust version of spacetime-as-substance to function as a testable hypothesis when suitably explicated. The rest of this article is designed to demonstrate the validity of that hypothesis. If successful, the descriptions of galactic rotation that follow will be sufficiently compelling that the four assumptions can be regarded as describing a legitimate candidate for an ontologically real spacetime-as-substance.

Spiral Galaxy Rotation (“Dark Matter”)

Back in the 1930s, a physicist named Fritz Zwicky discovered that objects in spiral galaxies do not seem to observe Newton’s laws. In the 1960s, physicist Vera Rubin took up his work and discovered that objects far away from the center, unlike those in our solar system, do not slow down in proportion to their distance from the core (Fig. 2).

If our solar system behaved in this way, Neptune’s orbital velocity (5.4 km/sec) would be at least as high as Mercury’s (48 km/sec), nearly nine times its true value. So far as Newton is concerned, that would exceed the escape velocity of an object at that distance from the sun and send it flying off into interstellar space. Nevertheless, that is exactly how it appears to be with our galaxy as a whole. Objects near the rim somehow manage to orbit as rapidly as those near the core without flying off into intergalactic space. This has prompted at least one physicist to speculate that Newton’s laws might change in relation to the velocity of the object under consideration.

Newtonian Predictions on Velocity
Figure 2: If galaxies behaved according to Newtonian predictions (b), the velocity of objects, relative to the galactic plane, would steadily decrease the farther they are from the core. Instead, it has been observed that objects maintain a nearly constant velocity (a) regardless of their distance.

Most physicists, to their credit, are hesitant to discard Newton’s laws without a very good reason, but the alternative they have conjured up is no less peculiar. In order to keep the objects at the rim of the Milky Way moving as fast as they do, there must be a very powerful gravitational field that pulls in the opposite direction to the galactic core. Indeed, to account for the motion of our galaxy, the calculated field strength implies a quantity of matter roughly ten times as great as the whole visible galaxy. Unfortunately, when astronomers look they find nothing out there to generate such a field. Therefore, physicists reason, there must be a species of particle that is virtually undetectable, invisible, and only weakly interacting with others of its kind, but which nonetheless exerts a strong gravitational pull. This dark matter—hypothesized to be made of WIMPs (weakly interacting massive particles)—the theory goes, exists as a gigantic halo surrounding the galaxy that exerts exactly the right gravitational counterforce to keep the galaxy rotating the way it does.

No direct evidence exists for these WIMPs. Their predicted properties are just whatever is necessary to explain this unexplained phenomenon—not necessarily a bad idea in itself. Unexplained phenomena are great for science; they present an opportunity for discovery. The problem here is that these particular particles are way too convenient. To hold a galaxy in place without tearing it apart, this halo of matter must mold itself into an impossibly unlikely geometric configuration. Also, because galaxies rotate at different velocities and come in different sizes, the quantity of dark matter must exactly balance a specific galaxy’s angular velocity and total mass. That means the quantity of dark matter must be intimately related somehow to the galaxy for which it is responsible. On top of all that, it has to be virtually impossible to detect since, when we point our telescopes at the place where this matter should be, there is nothing to be found.

Thankfully, there is a perfectly reasonable explanation for galactic rotation that does not require us to either banish Newton or accept the existence of dark matter.

Assumption 2 states that spacetime is neither created nor destroyed. Therefore, all stellar phenomena (stars, black holes, neutron stars, quasars), by converting mass into energy, are actually transforming spacetime from one form into another. For the purposes of this paper, the exact mechanisms of this transformation are of secondary importance, though nuclear fusion, Hawking radiation, and neutron decay are among the currently accepted theories. What matters here is that spacetime is transformed from an extremely dense (atomic) form into an extremely decompressed (vacuum pressure) form. Indeed, from a substantivalist perspective, all mass/energy transformations must be of this nature: physical phenomena that cause spacetime to decompress. The energy associated with this decompression, described by Einstein’s famous equation E=mc2, is the vigorous expansion of spacetime as it strives to achieve its equilibrium pressure (Assumption 3).

All of this spacetime is liberated by objects that are themselves orbiting the core of the Milky Way at a virtually identical velocity. That is, unlike our solar system, the orbital velocity of all objects in the galaxy is roughly the same and so angular momentum increases as the distance from the core increases. That, in turn, means the centripetal force required to hold the object in its orbit must also increase. The farther an object is from the galactic core, the harder it pulls against whatever force  is holding it in its orbit. It is easy to see why this is such a confounding problem. The farther an object is from the galactic center, the harder it pulls against the core, despite the fact that gravity ought to decrease the farther one gets from the core. As far as galaxies are concerned, Newton’s laws seem to have been turned upside down. So, what’s the answer?

The centripetal acceleration of any object in the galaxy is related to its orbital radius. By mass, most of these objects are stellar objects, busily churning out spacetime by way of one or another mass/energy transformation mechanism. That spacetime, in turn, is liberated into the galaxy with a velocity proportionate to the centripetal acceleration of the object that liberated it. Therefore, spacetime liberated by any stellar object will tend to flow toward the rim of the galaxy at a speed proportionate to the centripetal acceleration of its source. That, in turn, means that the velocity of the spacetime flow at any point in the galaxy is directly proportionate to the distance of that point from the core. Finally, in accordance with Assumption 3, spacetime resists compression above its equilibrium value. Consequently, as spacetime is accelerated off the edge of the galactic disk, it presses against the cosmos as a whole and the cosmos pushes right back.

Figure 3 - Pressure Gradients
Figure 3. Spacetime accelerated off the rim of a spiral galaxy is compressed as it attempts to move into the intergalactic medium—spacetime at its equilibrium pressure. The resulting pressure gradient creates gravitational fields both inside and outside the ring of maximum compression. This phenomenon, not dark matter, is responsible for galactic rotation as well as gravitational lensing.

This counterforce exerted by the cosmos against the flow of spacetime just beyond the outer edge of the disk is a consequence of Assumption 3, the resistance of spacetime to compression above its equilibrium value. The intergalactic medium, like all “empty” space, is already full of spacetime at or very near the vacuum pressure. The galaxy, by contrast, is a major source of “new” spacetime, that is, spacetime that has been recently liberated from atomic matter in the galaxy’s vast reservoir of active stellar phenomena. Inasmuch as the entire cosmos is already replete with spacetime at the vacuum pressure, this new spacetime has nowhere to go.

As a result, spacetime, flowing off the disk, is bottled up and compressed as it attempts to move into intergalactic space. That compression is most intense where the spacetime flow has the highest velocity, namely, near the rim where its source object had the highest centripetal acceleration. This compression creates a spacetime pressure gradient (Fig. 3). It is strongest just outside the rim and gradually diminishes in the direction of the core, as well as way out beyond the rim. And according to Assumption 4, a pressure gradient is a gravitational field. It is this phenomenon, not dark matter, that holds the galaxy together.

The evidence for this model is significant. Notice that no matter what the rotational velocity of a given galaxy, no matter how rapidly or slowly it spins, and no matter how massive it is, the strength of the resulting gravitational field will always be exactly calibrated to hold it together. This is because the centripetal accelerations of the sources of spacetime are proportionate to their angular velocities, and the intensity of the spacetime gradient is, in turn, proportionate to centripetal accelerations of its various stellar sources. These variables, whatever they happen to be, always find a stable equilibrium. The spacetime flowing off the edge of the disk pushes against the cosmos in proportion to the angular velocity of the galaxy, and that is what determines the intensity of the gravitational field (the spacetime pressure gradient). Notice also that the intensity of the gravitational field at any point in the galaxy is proportionate to the radius at that point, meaning that the field strength decreases smoothly toward the core. A giant mass of dark matter would tend to pull the galaxy into a ring or donut shape unless its physical distribution and consequent gravitational pull just happened to correlate exactly with the disk shape we actually observe, and that would be very hard to explain. Neither do we have to explain how exactly the right quantity of dark matter happens to be present to generate the proper gravitational field for any particular galaxy.

Another piece of evidence: Draw a line from the galactic core through the earth and on to the rim (Fig. 4). If we measure the Doppler shift of starlight anywhere along this segment, we find that it is directly proportionate to the difference between the velocity of the spacetime flow at that point and its velocity here at earth. Starlight is red-shifted anywhere along the segment between the earth and the core and blue-shifted between the earth and the rim. Moreover, the extent of the blue shift increases as the star in question approaches the rim. This happens because the light from blue-shifted stars must flow upstream against the spacetime current and is compressed in the process, shortening its wavelength. Meanwhile, light from red-shifted stars is stretched. These Doppler shifts have little to do with the relative motions of the stars. Stars near the rim are not moving much, if at all, toward the earth, despite their blue-shifted light.

Figure 4: Red-Blue Shift
Figure 4. Starlight reaching the earth is red-shifted if its source is between the earth and the galactic core, and blue-shifted if it comes from a star between the earth and the rim. The flow of spacetime toward the rim of the galaxy, not the relative motions of the stars, is primarily responsible for this Doppler effect.

Finally, this galactic gravitational field is shaped like a large flattened out donut with the thickest part just beyond the rim. In essence, it is a huge circular convex lens that, not coincidentally, is exactly the necessary shape to account for the gravitational lensing that astronomers have observed (Figs. 3 & 5).

Figure 5 -Hubble Image of Horseshoee Lens
Figure 5. On rare occasions a distant (lensed) galaxy, a foreground (lensing) galaxy, and the earth are lined up perfectly. If the lensing galaxy rotates in a plane perpendicular to our line of sight, the exact shape of the gravitational (spacetime) gradient is easily observable. Lensshoe Hubble Courtesy ESA/Hubble & NASA.

Conclusion

The ultimate aim of substantivalism, like all philosophy, is to explain reality. Its goal is not to explain the existing standard models of physics. The standard models are competing theories, not independent ontological entities in their own right that demand philosophical explanations. Further, substantivalism was proposed in order to solve problems, like dark matter, with which the standard models have struggled. Hence, even if a substantivalist achieved some measure of success demonstrating the efficacy of spacetime-as-substance to recreate certain aspects of the standard models, that success would likely come at the cost of transforming spacetime into a substance that is no longer capable of meeting its primary objective. That is, if spacetime is made compatible with the standard models, it will thereby adopt the very limitations it was originally proposed to overcome, rendering the entire project pointless.

The model of spacetime-as-substance presented in this article largely ignores the standard models and focuses instead on physical reality itself. I made no effort, in the spirit of geometrodynamics, to define dark matter particles that are made out of spacetime. That approach would have been a clear capitulation to the existing theory, and would have, even if successful, done little more than change the terminology of the standard model. Dark matter particles are necessary only for relationalism; in that theory, gravity demands a gravitational source, an object with mass. By contrast, the model I have proposed recognizes spacetime as a genuine substance in its own right. It is capable of forming a pressure gradient (gravitational field) without any need of additional particles (WIMPs) that, it should be obvious by now, do not exist and are never going to be discovered. In general, this new approach to substantivalism—judiciously ignoring certain aspects the standard models while embracing the relatively simple model of spacetime expressed in the four assumptions above—provides a powerful explanatory tool for not only dark matter, but for many other physical phenomena as well.

Philosophy, Physics

Abstract: This paper highlights and examines a critical assumption embedded within the framework of the substantivalism/relationalism debate. This assumption is the unacknowledged priority given to the traditional properties of spacetime-as-coordinate-system that have been handed down over the course of the twentieth century. In this paper, I attempt to substantiate my claim that such an unreflective assumption exists and is decisive, and evaluate the propriety and consequences of accepting that assumption. Finally, I will offer an alternative framework within which to move the debate forward on terms that prioritize spacetime-as-substance over spacetime-as-coordinate-system.

Traditional Spacetime

Even before the ancients, such as Aristotle and Euclid, started pondering the nature of space and time, before they created elaborate systems and defined the relationships between points in three dimensions, there was already a far older and more sophisticated neurological mapping system built-in to all human brains. The perception of spacetime as a matrix wherein objects and events are located, relative to one another and to one’s body, was already instinctive and familiar long before it was standardized by cartographers and mathematicians. The notion of spacetime as a coordinate system is old, instinctive, and unreflective.

What is new and radical is the idea that, over and above its role as a matrix that contains objects, spacetime is also a substance, a thing in its own right. And far from being instinctive and unreflective, only the most nuanced reasoning (Relativity Theory) and counterintuitive scenarios (Newton’s bucket argument) can bring to light even the mere possibility that spacetime might exist independently. With this stark contrast in mind I believe it is fair to say that:

  • Spacetime as non-existent coordinate system is intuitive, natural, ancient, genetically encoded, obvious
  • Spacetime as existing substance is radical, innovative, new, counterintuitive, contrary to experience, obscure.

Participants in the substantivalism/relationalism debate might well be nodding their heads in agreement, but also skeptically questioning my claim that there is anything hidden or surreptitious about this distinction. Of course the idea that spacetime is a substance is radical, counterintuitive, and unproven; that’s the whole crux of the debate. Fair enough, but I have something slightly more elusive in mind.

The Epistemological Priority of Spacetime as a Coordinate System

The first thing we need to note about the priority of spacetime-as-coordinate-system over spacetime-as-substance is that it is entirely epistemological. It is the ease with which we arrive at the respective conclusions that dictates their priority, not anything to do with their ontological natures. Spacetime-as-coordinate-system is an epistemological heuristic; it helps explain things. Spacetime-as-substance is ontological and remains to be explained. Indeed, spacetime-as-coordinate-system does not have any obvious ontological nature at all. By claiming it is non-existent, nothing but a geometric abstraction, we more or less explicitly admit that it is not so much a thing to be examined, as it is a mental overlay used for navigation, or a heuristic employed to simplify and categorize the events that are actually taking place. Spacetime-as-substance, by contrast, is proposed as the fundamental stuff of reality, the ultimate answer to the most basic ontological question: What is?

Herein lies the issue. Spacetime-as-coordinate-system is simple and has epistemological priority (exactly because of its historical precedence and great explanatory power) over spacetime-as-substance. On the other hand, spacetime-as-substance has ontological priority over spacetime-as-coordinate-system because, if it really exists, any coordinate system that results from spacetime’s genuine structure will be dictated entirely by that structure. That is, if spacetime exists, its role as coordinate system is entirely derivative of its ontological character. It is not the case that spacetime-as-substance must bend to accommodate whatever notions of reality are embedded within our currently accepted notions of spacetime-as-coordinate-system. And there are a huge number of such notions embedded there.

The priority of spacetime-as-coordinate-system is, fairly obviously, based on the long history and undeniable success of Western science, and specifically the great sophistication and accuracy (within many domains) of modern physics. That entire history of success is built into the latest epistemological spacetime-as-coordinate-system. Meanwhile, current notions of spacetime-as-substance are new, immature, enjoy no comparable history of success, and can boast no great explanatory power. It is entirely understandable, perhaps even justified, that we would prioritize spacetime-as-coordinate-system and unreflectively assume that it ought to form the starting point of our investigations even if we have, formally, made the shift to substantivalism. In effect, it appears reckless, disrespectful, and a bit daft to replace, wholesale and outright, the highly refined coordinate system we already have with an unproven substance that we cannot even define. Certainly at first, and perhaps for the foreseeable future, this newly postulated substance will lack anything approaching the explanatory power of our highly developed spacetime-as-coordinate-system. Trailblazers who venture into this uncharted territory should expect more than a little ridicule as they build their case. But if there is any merit to substantivalism, and a key success or two are forthcoming, it will eventually begin attracting converts.

Contrast Between a Substance and the Coordinate System

When scientists overlay reality with a particular spacetime-as-coordinate-system, that structure comprises all of the mathematical, scientific, and philosophical underpinnings of the latest cosmological thinking. It is emphatically not simply a piece of blank graph paper onto which, or clear lens through which, reality is directly and objectively recorded. Indeed, it functions as nothing less than the horizon, the current limitations, and the conditions under which reality itself can appear at all. For example, there is currently no accepted explanation for the curious rotational characteristics of spiral galaxies, though dark matter is the leading candidate. But because dark matter is not understood, there is no aspect of the latest spacetime-as-coordinate-system, when overlaid onto reality, that will allow it to appear. Among other things, this failure announces to us that the current spacetime-as-coordinate-system, fortified with all the best ideas to-date, is incomplete in at least one major respect. Dark energy is another such example, and between them—dark energy and dark matter—we have two gaping holes right at the heart of the current theory, and therefore two gaping holes in the latest spacetime-as-coordinate-system.

Substantivalists believe that the acceptance of spacetime-as-substance and its subsequent elaboration will fundamentally alter our understanding of ontology in general and physics in particular, potentially filling the two holes mentioned above. It follows directly from that assumption that they also believe that the current state of the scientific art that is built into the latest spacetime-as-coordinate-system is wrong at the most basic level, and may be very far from the truth in most detailed respects as well. Yet the debate between substantivalists and relationalists usually unfolds from a curious effort on the part of substantivalists to conceive of a spacetime substance that faithfully explains the theories embedded in the latest spacetime-as-coordinate-system. Essentially, the standard of proof for a theory of spacetime-as-substance seems to be that it is able to replicate whatever success is accorded to the current spacetime-as-coordinate-system—even though substantivalists ought to recognize that any such standard would require a theory that they themselves have already at least implicitly rejected, and which demonstrably fails to explain both dark matter and dark energy, among many other things (e.g., black holes, cosmogony).

The Ontological Priority of Spacetime as a Substance

For substantivalists, this observation ought to be liberating and unnerving at the same time. Liberating, because there is no requirement to bend one’s notions of spacetime-as-substance to fit into the latest spacetime-as-coordinate-system. In fact, any such attempt will invariably lead to a characterization of spacetime-as-substance that is beset by all of the inadequacies of the current theories. So, for example, it ought not to be accepted as axiomatic that any coherent formulation of spacetime-as-substance must fully replicate quantum theory, with little or no alteration. And it is by reference to that radical degree of freedom that the trepidation ought to set in. Presumably, a fully developed and defended theory of spacetime-as-substance will be radically different from our current set of theories. And at least two things follow from that stark acknowledgement.

One, the relatively straightforward (not to say easy) project of shoehorning spacetime-as-substance into the latest spacetime-as-coordinate-system is almost certainly of very limited utility and may be wholly counterproductive. Nevertheless, and to be fair, it is unlikely that absolutely everything that is currently propounded by modern cosmologists is complete nonsense, and for that reason it will be, in most cases, extremely difficult to determine which elements need to be respected and which questioned or discarded—this is a perennial challenge of all scientific progress. But however it ultimately unfolds, a healthy skepticism of especially those aspects of the standard models that bear directly on spacetime, matter, and ontology will be indispensable.

Two, substantivalists need to fully embrace the ontological priority of spacetime-as-substance over spacetime-as-coordinate-system. As mentioned, this paradigm shift feels somehow unnatural, disrespectful, and premature, but in the end the latter must entirely conform to the former. On a more positive note, such a shift utterly transforms the landscape and horizon of discovery. Whereas previously the main focus of substantivalists has been the futile shoehorning exercise discussed below, going forward their investigations need to focus less on what it would take to make a physically real coordinate system, and more on what sort of substance is sufficient to explain everything in the physical universe. Whatever it turns out to be, the appropriate spacetime-as-coordinate-system will follow naturally and be entirely derivative thereof.

Geometrodynamics—Folding the Coordinate System

The most comprehensive and ambitious effort to-date to shoehorn spacetime-as-substance into a spacetime-as-coordinate-system was John Wheeler’s geometrodynamics. Wheeler’s program was to treat matter, in all its various manifestations, as so many complex geometric contortions of spacetime. He reasoned that if we can imagine a large, simple spacetime curve as a gravitational field, we can also imagine a very small and tightly curved spacetime region as a subatomic particle, what Wheeler called a geon. To slightly oversimplify it, he conceived all of reality as a kind of spacetime origami—surface-like spaces twisted and knotted such as to give rise to all the fundamental particles at one extreme of scale, and all of the large-scale astrophysical phenomena at the other extreme.

It is widely agreed that Wheeler’s efforts were not entirely adequate, but it is worth taking some time to appreciate his attempt and glean what knowledge we can from it. In particular, geometrodynamics points to just how confounding the relationship is between spacetime-as-substance and spacetime-as-coordinate-system. Wheeler, a classically trained physicist, naturally accepted the priority of spacetime-as-coordinate-system and attempted to fold and twist that coordinate system in such a way as to create spacetime-as-substance. The project hinged on the dubious notion that it is possible to contort a massless, geometric abstraction in such a way as to bring ontologically genuine matter into existence. It also points to a fundamental issue related to the distinction between properties and substances.

With reference to the ontological argument for the existence of God, Kant pointed out that existence is not a predicate. The exact same list of properties can be ascribed to the idea of God as can be ascribed to God Himself. All that distinguishes the two is existence. Kant explained, correctly I believe, that properties are predicated of a substance while existence is the substance itself. It is meaningless to predicate existence of a substance because, in effect, existence and substance are synonymous. Substance is nothing other than whatever we mean by existence. To say that substance exists is like saying heat is hot—the predicate adds nothing to the concept because it is the concept.

In a strangely analogous way, Wheeler was attempting to create existence by merely altering the description of the underlying coordinate system. In effect, he was trying to add the property of existence to an otherwise purely abstract concept, thereby bringing the previously non-existent idea into full reality.

Black Holesblackhole

All of these ideas come to a head in the most enigmatic entity in the universe—a black hole. The current (relationalist) theory postulates that there is a real thing (an infinitely dense singularity) in the center of a very intensely curved, but formally non-existent, spacetime region. In this model, the gravitational relationship between a black hole and objects within its influence is exhaustively determined by the singularity and the external object. The spacetime-as-coordinate-system surrounding the black hole is merely a mathematical overlay, fortified with the latest theories (including, for example, gravitons, Higgs fields, etc.), that functions entirely as a heuristic. But what is a substantivalist committed to in this case?

According to a substantivalist, spacetime is not merely a geometric abstraction but is the one true substance of the cosmos. So, here’s the big question: What, exactly, exists in the region one inch from the center of a black hole? If we were to demand of a relationalist that he discard temporarily his helpful heuristic, the spacetime-as-coordinate-system, and instead say what is really there, he might hypothesize that there is likely a very high graviton particle density and a Higgs field with an array of very extreme values, but no genuine “matter” in any standard sense of the term. All of the actual mass is concentrated into an infinitely dense, infinitesimally small, object in the center of the black hole. A substantivalist would say something completely different.

If we take seriously for a moment the central tenet of substantivalism, then the spacetime surrounding the center of a black hole is real, not a mere abstraction, not simply a coordinate system. From that single assumption, we are suddenly faced with a fascinating question: Is the real mass—viz. the spacetime surrounding the center—different from or the same as the mass traditionally attributed to the singularity in the center itself? In other words, are there two types of substance, one type in the center and another type surrounding it? Or, on the contrary, is it spacetime all the way down? If spacetime is the one and only substance, then it follows necessarily that this substance (regardless of how we ultimately characterize it) becomes increasing dense as we approach the center of a black hole, and the center itself is nothing more than the densest location, but it is not qualitatively different from any other point in the entity. And, most importantly, there is nothing special at the core; there is no singularity.

The Singularity as Absurdum

What does any of this have to do with the thesis of this paper, that spacetime-as-coordinate-system is mistakenly prioritized over spacetime-as-a-substance? The singularity hypothesized to exist at the center of a black hole is exactly where the rubber hits the road in this whole debate. And in nearly every respect, this is great news for the substantivalist. In the first place, a singularity is only necessary because spacetime is not currently thought to be a real substance. If spacetime is a substance, then there is no difficulty explaining the extreme mass of a black hole. Whatever else this implies, it means that the current (relationalist) theory has been forced to fabricate, ex nihilo, an utterly preposterous entity in order to maintain its stance that spacetime is a non-existent coordinate system. The singularity exists entirely outside of all other accepted science, has no coherent physical (or logical) description, and will most likely never enjoy a single shred of empirical evidence. It very clearly runs afoul of Karl Popper’s falsifiability test of scientific coherence. One could easily imagine the singularity being used by an opponent of the existing theory rather than by one of its proponents. “You say, my good man, that spacetime does not exist. But you realize, don’t you, that you must then assume the existence of a pure fantasy, a hole in reality, at the core of black hole? Are you sure you wouldn’t like to reconsider?”

Cosmologists know that the singularity, whether at the core of a black hole or at the origin of the universe, is an ominous threat to the standard models. The singularity is no less disconcerting than quantum theory’s abysmally inaccurate prediction of zero-point energy—the vacuum pressure—120 orders of magnitude greater than its measured value. It also draws uncomfortable attention to the intractable incompatibility of quantum and relativity theories, invalidating at least one if not both of them. Indeed, if we were to recast the entire standard model as an elaborate reductio ad absurdum proof, the singularity fits very nicely into the role of absurdum. Immediately upon demonstrating that this absurdity follows necessarily from the theory, we can say with perfect justification: therefore, the opposite must be true, spacetime is a substance. However comfortable we have become talking about singularities, until such time as they are given a physical description that is not utterly incoherent, their necessary existence ought to be regarded as strong counterevidence to the standard models.

My rationale for bringing this to light here is to cast doubt on the wisdom of starting from spacetime-as-coordinate-system and attempting to build spacetime-as-substance up from there. That traditional prioritization is entirely epistemological, built partly on our innate neurobiological mapping skills and partly on the notions of space and time (absolute, non-relativistic) used throughout most of scientific history. But as black holes make clear, spacetime-as-substance has ontological priority and should be assumed as the starting point for the purposes of understanding substance itself. Among many other things, this reprioritization means that we are under no obligation to demonstrate how mass can be magically created, via some form of spacetime origami (e.g., geometrodynamics), out of an inherently massless and theoretically inadequate spacetime-as-coordinate-system.

Over and above the manner in which black holes expose the dubious singularity notion, is the fascinating way in which they reveal spacetime’s mass-like characteristics at extremely high densities. Nothing new or special or mathematically complex (geometrodynamic) had to be done to spacetime to make it into matter. All we had to do was compress it. Indeed, if we examine—armed with our new assumption that it is nothing but spacetime all the way down—the entire gravitational (spacetime) field of a black hole, we will find a smooth continuum of spacetime, distributed according to the inverse square law, approximating the vacuum pressure at the most distant extent and the density of atomic matter at the core, and nothing else. That is, once we discard the singularity, a black hole illustrates all by itself the manner in which spacetime behaves as both matter and gravity. And that is not a coincidence.

Rethinking the Third Dimension

Finally, another very questionable assumption sneaks into the debate when we prioritize spacetime-as-coordinate-system over spacetime-as-substance; we accept, without even realizing it, that it makes sense to talk about a three-dimensional substance that does not exist. That is, we embrace the bizarre notion that nothing fundamentally new results from adding a third dimension to the two-dimensional surface-like analogy that undergirds spacetime-as-coordinate-system. It is easy enough to imagine a three-dimensional coordinate system that is nothing but a heuristic, a geometric abstraction. But is not possible to imagine a three-dimensional substance that lacks any form of existence whatsoever. In a nutshell, the lack of depth, and therefore lack of ontological reality, of a two-dimensional surface cannot be translated into a three-dimensional system. Coupled with the assumption that it is a substance, the addition of the third dimension is enough to make spacetime exist. To suppose that some additional form of folding or knotting or twisting is also required is, at bottom, an incoherent idea that follows from taking the two-dimensional analogy too far—from embracing uncritically the priority of spacetime-as-coordinate-system. Consider that, if the non-existence, the pure geometrical abstractness, of the coordinate system persisted into its conception as a genuine substance, then no amount of folding or twisting would do any good. If we start folding or stacking two-dimensional planes, one atop the other, each one possessing exactly zero depth, we will never end up with a 3-D object with mass.

We can see this clearly if we go back to our new conception of a black hole. Whether we use spacetime-as-coordinate-system or spacetime-as-substance, the curvature is very simple and follows the inverse square law from the center of the black hole all the way out to the farthest reaches of its gravitational influence. What made the spacetime into a substance was not any elaborate folding or twisting but simply our elimination of the singularity at the center. Once we grant ontological priority to spacetime, the third dimension is enough to make it a real substance all by itself. Indeed, it is not even possible to imagine a massless, but nevertheless 3-dimensional, substance. If a substantivalist assumes that spacetime is a three-dimensional substance, he has already assumed it has mass, otherwise he has not really assumed it exists at all. There is no additional work to be done on that score. What needs to be done to prove that the substantivalist theory is correct is not to demonstrate how spacetime-as-coordinate-system can be creatively folded to create matter—that would be to, once again, prioritize the coordinate system—but rather to demonstrate the explanatory power of a fully developed spacetime-as-substance with mass as an already integral component. If such a demonstration is successful, as it appears to be with a black hole, then the burden of proof will fall onto the relationalist, and it will be abundantly clear that the ontological priority of spacetime-as-substance trumps the epistemological priority of spacetime-as-coordinate-system.

Summary

In summary, spacetime-as-coordinate-system and spacetime-as-substance serve two different purposes. The former is an epistemological heuristic, embedded with the sum total of our current knowledge of physics. The latter is immature, but is proposed as the fundamental ontological substance of the cosmos, and it is hoped that its full development will result in radical changes to our picture of reality—in other words, that it will supplant the current coordinate system. Any notion of substance that is derived from the current spacetime-as-coordinate-system will inevitably be plagued by exactly the same problems that substantivalism is chartered with solving. Therefore, if only as a formal matter, spacetime-as-substance must be recognized as ontologically primary, and all conflicting aspects of spacetime-as-coordinate-system should be regarded as highly suspect and subject to change in accordance with new ontological discoveries.

Physics

What does a proton look like? The simple answer: Don’t ask. As with other subatomic particles, a proton is thought to be a complex of quantum wave functions—one for each of its quarks—distributed probabilistically through space. There is no “physical thing,” understood in realistic, everyday terms to which one could point and say, “There’s a proton!” At least that’s the theory.

Nevertheless, it would certainly be interesting, if only as a thought experiment, if we could build a simple, physical model of proton spin, using only standard Newtonian principles. To be convincing, it must generate a series of wavelengths analogous in its structure to hydrogen’s electromagnetic emission spectrum (Figure 1).

Balmer Series Wavelength
Figure 1. The Balmer Series is one of several series of spectral line emissions from the hydrogen atom. It includes four bright lines within the visible spectrum and a large number of increasingly compressed and dim lines in the ultraviolet spectrum. The other hydrogen series (e.g., Lyman, Paschen, Brackett) exhibit analogous distributions of lines—dim and compressed in the shorter wavelengths, bright and spread out in the longer wavelengths. The Rydberg formula, introduced by Johannes Rydberg in 1888, predicts the spacing between these lines with great accuracy, but does not attempt to explain the physical mechanism behind them

As it happens, such a model is possible and is surprisingly easy to explain. The dynamics involved are sufficiently straightforward and transparent that no rigorous methodology is required to illustrate the concept. A reasonably discerning non-scientist should be able to quickly grasp the basic ideas. The model is easy to visualize and simulate on a computer, and can be adjusted along a number parameters that will allow it, after some trial and error, to generate any of the various series of lines in hydrogen’s electromagnetic emission spectrum.

The “Proton”

The primary component of this model is a dense spheroidal object composed of a smooth, cohesive gel, analogous in its properties to a firm but fluid variant of silicone. Flexibility is critical because this object spins in a very particular way. We start by drawing in a standard axis, but instead of setting the object spinning in the usual way, around the axis, this object is driven by a toroidal convection current that is rotated 360° about the axis (Figure 2). In effect, the object turns itself inside out as it circulates. For the purposes of this discussion, the cause of this convection is not important, only that it is very fast and energetic.

Spheroid
Figure 2. The spheroid is created by convection cells rotated about a central axis, resulting in a toroidal configuration.

Assume further that this object is suspended, weightless, within an atmosphere composed of a gaseous version of the very same gel that, as a fluid, makes up the spheroid. And finally, we will assume that the entire system is located within a large, sealed chamber (with the walls well beyond the phenomenon itself) that maintains the gel-gas at a constant atmospheric pressure (Figure 3). For simplicity, we will refer to the point at which the surface is pulled into the spheroid as the north pole, and the opposite end, where the core of the spheroid emerges back onto the surface, as the south pole.

Spheroid withinSealed Container
Figure 3. We will assume the spheroid is located within a sealed container, maintaining a constant average pressure inside the vessel.

This simple system is all we need to generate the electromagnetic spectrum of hydrogen. Everything hinges on the fascinating properties that result from the convective circulation of the spheroid.

Angular Momentum

The vigorous spinning motion of a gyroscope yields a high angular momentum vector, perpendicular to its plane of rotation, stabilizing the object into which it is integrated (Figure 4). Because of their capacity to self-stabilize, gyroscopes are used in rockets and missiles to keep them upright and aimed in the right direction. The angular momentum vector is lined up with the long axis of the rocket, preventing it from rotating on a short axis and flying off course. At first glance, the vigorous convective motion of the spheroid would seem to have similar properties. But only at first glance.

To understand the angular momentum characteristics of this spheroid, we need to examine any one of an infinite number of possible sets of radially symmetrical cross sections (Figure 5(a)). Each cross section intersects two opposing convective cells. We can draw in the angular momentum vectors for each cell (Figure 5(b)), and then use simple vector addition to calculate the net value for the entire object. Incredibly, no matter what set of cross sections we select, the net angular momentum for the spheroid as a whole always has a value of exactly zero (Figure 5(c)).

Angular Momentum Vector
Figure 4. Gyroscopes are used in rockets and missiles to stabilize them along their long axes and prevent them from rotating on a short axis and flying off course.
Radial Cross Sections of a Spheroid
Figure 5. Various radial cross sections (a) of the spheroid reveal angular circulations (b) that might seem to contribute to its spin and angular momentum. However, when we use the head-to-tail method to add the angular momentum vectors of all possible circulations (c) they sum to zero. Hence, the spheroid has no intrinsic spin.

This zero angular momentum value is fascinating because it means the whole object does not, like a gyroscope, resist any effort to push it out of a particular orientation. No matter how vigorous the convection, any applied force will cause the object to move or rotate just as if it were a solid, stationary ball. It also means that a tremendous amount of kinetic energy can be stored inside the spheroid without it having the slightest effect on anything outside of the object. And, correlatively, no matter how energetic the spheroid is internally, even the slightest external force can have an effect on it. In fact, the spheroid, because it possesses no net angular momentum, will only move in response to externally applied forces.

Gas Jet

As the spheroid circulates, it draws the ambient gas into its north pole (Figure 6). At that location, gas particles are in contact with the surface of the object. And because they are composed of the same gel substance as the object, they have a tendency to stick to the surface. When that happens, the gas particles are pulled down into the object. Once inside, the gas is compressed as it is funneled down into the narrow channel that forms the spheroid’s core. It is also greatly accelerated by the vigorous movement of the gel through the core, along its major axis. When the gas emerges from the south pole it is propelled outward as a highly focused jet, into the ambient gas.

Naturally, this gas jet, as it exits the south pole, is at a much higher pressure than the ambient atmosphere. At the same time, the pressure near the opposite side of the spheroid is dramatically lowered by the vigorous suction exerted by the north pole. And because the spheroid has no net angular momentum, the force exerted on the object by the pushing and pulling of the poles causes it to move.

The propulsion generated by the south polar jet is far more significant than the suction generated by the north pole, so for the purposes of explaining the motion of the spheroid, we can ignore the north pole altogether. That is not to imply this force does not exist, only that the limited goal of this demonstration can be achieved without it. Indeed, if there is, in fact, a physical analog of this model, all forces, however subtle, are well worth investigating.

Gas Jet
Figure 6. Gas is pulled in through the north pole and expelled from the south pole, resulting in low pressure around the north and high pressure at the south.

Spin

Because it has a zero net angular momentum value, this spheroid is not at all like a rocket equipped with a gyroscope. A propulsive force will tend to spin the object rather than send it flying off in a straight line, similar to a bottle rocket without the stick. We can start by examining the simplest component of this spinning (Figure 7). Now that the entire object is spinning in a more standard way—rather than exhibiting only its internal convection—we might expect to be confronted with a new candidate for angular momentum. But even this straightforward rotational motion does not confer angular momentum upon the object.

If we take a cross-section of a standard solid sphere, parallel to its rotation and perpendicular to its axis of rotation, we can examine the behavior of the physical material that, together with the matter in all similar cross-sections, lends the object its angular momentum. In a solid sphere all of these cross-sections are composed of matter that is spinning in parallel planes, and so we would expect the calculation of the net angular momentum to be a simple matter of adding up the contributions of all such cross-sections. But that is not what happens when we try to create similar cross-sections of our spinning convective spheroid. Though the entire object, abstracted from its internal dynamics, seems to behave much as a solid sphere, in fact none of the matter of which the spheroid is composed rotates in a plane parallel to the entire object’s rotational plane. If, then, we were to add up the contributions of each cross-section to determine the angular momentum of the entire object, we end up, once again, with an answer of zero.

Spin
Figure 7. Gas is drawn in through the north pole and expelled from the south pole (a), resulting in a disequilibrium state—high pressure around the south pole and low pressure around the north pole. As the poles push and pull on the gas, the spheroid rotates (b) in order to put the north pole into a region of higher pressure and the south pole into a region of lower pressure.

This curious conclusion can be confirmed using a simple experiment. Hold a bicycle wheel by its axle and start it spinning. Any effort to twist the axle and move the wheel out of its rotational plane is resisted by the angular momentum of the wheel. And, more importantly for our purposes here, when the axle is twisted the applied force acts in opposition to the force of the wheel’s angular momentum, slowing or stopping the wheel. In an exactly analogous manner, the internal convection of the spheroid prevents the object as a whole from attaining any angular momentum, even while spinning. In effect, this object is incapable of possessing angular momentum.

Termination Shock

With no angular momentum, the spheroid’s rotation is governed entirely by the action of the south polar jet. The jet, in turn, is governed both by the spheroid’s convective velocity and by the pressure of the ambient gas in the chamber. As mentioned above, we are assuming that the convective circulation of the spheroid is extremely energetic, sufficient to cause a spin velocity that is very high (a non-trivial fraction of the speed of light). If, then, we examine the system after one full rotation, we will find, surrounding the spheroid, a very dense ring of gas at a distance that marks the termination shock of the jet.

Termination shock occurs, in general, when the forced or fast flow of a substance succumbs to the steady or slow flow of that same substance. The most celebrated example of this phenomenon is the termination shock of the solar wind, way out at the inner edge of the solar system, located approximately 70-90 astronomical units from earth. In fact, due to sudden and pronounced changes in the prevalence of cosmic radiation at its current location, it is believed that the Voyager I spacecraft has recently passed through the termination shock. A far less grandiose but still instructive example of termination shock can be created in a kitchen sink (Figure 8).

Termination Shock

Figure 8. The phenomenon of termination shock prevents a substance from dissipating smoothly all the way out from a high-pressure point of origin. Instead, it stops abruptly at a specific distance from its source.

Termination shock is interesting for many reasons, not least of which is its applicability to the solar wind, but our focus for now is on nothing more than the simple fact that it exists at all. In practice, there will no doubt be good cause to examine it in greater detail. What it means for the system we have been looking at here is that the energetic south polar jet will, at a well-defined distance from the spheroid, suddenly succumb to the slow flow of the gas, driving up the pressure of the gas at that radius. The gas pressure does not dissipate gradually, either smoothly or turbulently, off into the distance with no specific stopping point.

Derivative Axes

To this point, we have an extremely energetic convective spheroid that possesses zero net angular momentum, even once it starts rotating, with a powerful jet that propels high-pressure gas from its south pole, and a termination shock at a stable radius from the spheroid. After one complete rotation, the system resembles Figure 9(a). Once the polar jet comes around for the second rotation, the termination shock from the first rotation will still be present (if slightly dissipated) and will push back against the jet. This high-pressure ring will push the jet off to one side, and because the spheroid has no angular momentum, it will not resist this slight reorientation. As a result, the second rotation will trace a new circumference and create a new termination shock (Figure 9(b)).

Derivative Axis Rotation
Figure 9. After one rotation (a), the termination shock creates a high pressure ring around the spheroid. After multiple rotations (b), the spheroid turns in order to minimize the pressure through which its jet passes, but the jet has no alternative but to pass through the poles of the first derivative axis with each rotation, resulting in two high pressure points.

Extending this reasoning over several more rotations, and we begin to see a pattern emerge. The far end of the jet, the termination shock point, migrates into lower pressure regions, turning the spheroid accordingly. This new rotational axis is derived from the migration of the spheroid’s primary convective axis (its polar jet) over several rotations. Indeed, this derivative axis is easy to locate since it is defined by the two points through which the primary axis passes on each primary rotation (Figure 9(b)).

The most interesting property of the first derivative axis is that its poles possess a much higher pressure than any of the other points along the termination shock. Notice that the jet attempts to minimize the frequency with which it passes through any point on the termination shock—but it cannot avoid the poles of the first derivative axis, which it hits on every single rotation, raising the pressure at those points.

Recall, the spheroid has no angular momentum. Consequently, the increased pressure at the poles of the first derivative axis has the same effect as the south polar jet itself; it introduces yet another, independent rotational motion. And, like the south polar jet, the poles of this new axis migrate into lower pressure regions as they are resisted by the relatively high pressures at various points on the termination shock, ultimately giving rise to a second derivative axis.

Derivative Axis
Figure 10. Each derivative axis sends the south polar jet through its poles more frequently than its equator. Hence the poles of each derivative axis must, like the major axis, rotate such as to reduce the pressure variations. In this diagram, the axes are shown with different radii for clarity. In a real atom, they are all the same size, ending at the termination shock.

We can extend this reasoning to create a series of derivative axes, each produced by rotating the poles of the previous one (Figure 10). With the addition of this series of axes, the system now has all of the properties needed to generate the basic behaviors of hydrogen’s electromagnetic spectrum, specifically, the wavelengths of its spectral lines.

Wavelengths

With each rotation of the spheroid, the north polar jet creates a high-pressure ring on the termination shock. As the high pressure at this ring decompresses outside of the termination shock, it generates a wave in the ambient atmosphere (Figure 11). For the major axis only, the frequency of this wave will be directly related to the rotational velocity of the spheroid. Similarly, the rotation of the first derivative axis also creates high-pressure rings on the termination shock, and the decompression of those rings also creates waves in the ambient atmosphere.

Wave Pattern
Figure 11. The high pressure generated by the spinning gas jet compresses the ambient gas, creating a standard wave pattern.

Now, it is easy to see that there is an exponential relationship between the axes. If, for the sake of simplicity, we assume that each axis must rotate only twice in order to generate the next one, then the first derivative axis implies two rotations of the spheroid, the second derivative axis implies 2 x 2 or 4 rotations, the third, 2 x 2 x 2 or 8, etc., which generates a graph like Figure 12 over the first ten derivative axes.

Since this is simply a notional model, Figure 12 does not exactly replicate any of the series of lines in hydrogen’s EM spectrum. However, this is very obviously the type of phenomenon that could generate them. For the purposes of creating a computer simulation of this model, it would be useful to choose values for the various elements that closely mirror what we know of the hydrogen atom from empirical research.

Hydrogen Electromagnetic Spectrum
Figure 12. The exponential relationship between the derivative axes results in a distribution of wavelengths that closely mirrors any one of the observed electromagnetic spectral series of hydrogen.

Density and Size of the Spheroid: As nearly all the mass of an atom is found in the nucleus, the spheroid will be very small by comparison to the diameter of the termination shock, and its density should be extremely high.

Convective Velocity: This is a non-trivial fraction of the speed of light. This extreme value is necessary both because the density of the atmosphere (see below) is so low, and because the distance between the nucleus and the termination shock is so large.

Ambient Gas Density: The density and pressure of the gel gas in the sealed chamber should be very low, reflecting—to the extent possible—the extreme difference in density between solid matter and “empty” space (the vacuum pressure). This low density will ensure that the termination shock is far from the nucleus.

These values will create an “atom” that roughly approximates the relative sizes of the nucleus (spheroid) and electronic shell (termination shock) that experiment has observed.

Conclusion

Balmer Series Wavelength
Figure 1. The Balmer Series

As compelling as this simple model is, accepting it as anything more than an astonishing coincidence would involve a major reconsideration of existing physics. So, the bar is very high. With that in mind, let’s reconsider the evidence:

  • The physics is very simple and well established—entirely Newtonian.
  • The model works best when we choose values for the variables that reflect what we already know from experiment.
  • In a real hydrogen atom, the south polar jet would manipulate space at the vacuum pressure to generate its electronic shell, a phenomenon that has a compelling analog at the termination shock at the edge of the solar system.
  • The EM waves are generated by a spinning object, which is a very intuitive, trigonometric way to generate waves.