Philosophy, Physics

A Substantivalist Explanation of Dark Matter

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.

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