# Dark Matter

It is well known that there is a conflict between Newton’s and Einstein’s theories of gravity and the behavior of distant galaxies. Either there is more mass in these galaxies than we see, or the pull of gravity is in these conditions stronger than expected. Rather than second-guess Newton, physicists have invented a thing they called ‘dark matter’, non-luminous matter. Not everyone believes in dark matter, and according to MOND (Modified Newtonian Dynamics), the gravitational attraction between the center and the outside of certain galaxies is stronger than that between the center and outside of the solar system, and closer to 1/r rather than 1/r^2. But MOND is unable to explain the central piece of evidence for the existence of dark matter, a colliding cluster of galaxies known as a “bullet cluster”.

A bullet cluster poses a problem for MOND because, while the majority of the mass of the colliding galaxies lies between them in the form of gas, gravitational lensing shows that gravity is in the galaxies themselves. No mere modification of the strength of gravity will be sufficient to account for this situation, because no such modification can explain why the gravity is in the galaxies rather than the gas. In order to solve this problem without recourse to dark matter, a deeper modification than that offered by MOND required…

According to Newton’s universal law of gravitation, the strength of gravity varies inversely as the square of the distance between two masses… His formula 1/r^2 works very well as far as prediction of the orbits of the planets is concerned, and it was Newton’s formula that underlay the calculations that put the first men on the moon. But one thing that always troubled Newton was that, while he had the maths to account for the phenomenon of gravity, he had no explanation for the action of gravity. He wrote in 1662

That one body may act upon another at a distance through a vacuum without the mediation of anything else, by and through which their action and force may be conveyed from one another, is to me so great an absurdity that, I believe, no man who has in philosophic matters a competent faculty of thinking could ever fall into it.

He declined to “feign” a hypothesis. “It is enough”, he said,

that gravity does really exist and acts according to the laws I have explained, and that it abundantly serves to account for all the motions of celestial bodies.

Einstein made progress on Newton’s problem -and added precision to the prediction of the orbit of mercury- by allowing that gravity is a ‘fictitious’ force, an illusion created by the principle of inertia and the geometry of space. According to the principle of inertia, an object will remain at rest or continue moving uniformly in a straight line unless acted on by a force. If space is warped by the presence in it of matter, then gravitational attraction can be explained by the deflection of an object from a straight path in the vicinity of another massive object. For example, the earth can be understood as following as straight a path as possible given the curvature of space in the vicinity of the sun.

However there is something wrong with Einstein’s modified theory of gravity too, and it concerns the 1/r^2 formula. This is the wrong formula to describe the behavior of galaxies. The planets closer to the sun orbit the sun at a faster rate than those further away. These differences are in accordance with 1/r^2. However when it comes to certain galaxies, there are no such difference – the stars on the outside of the galaxy are rotating at comparable speeds to those closer to the center. Assuming that there is not some additional hidden amount of mass located somewhere within these galaxies, this implies that 1/r^2 is the wrong formula, that the attraction between objects and the centers of galaxies is closer to 1/r than to 1/r^2. In Einstein’s terms, the curvature of space about the center of a galaxy is greater than expected. And 1/r^2 is entirely the wrong formula to describe the orbits of electrons about the nucleus of an atom. According to the early planetary model of the atom, the centrifugal force of the electron balances the centripetal force of the nucleus in the same way that the centrifugal force of the planets balances the centripetal force of the sun. But unlike a planet, an electron is electrically charged, and according to Maxwell’s equations, an electric charge emits energy when it changes velocity or direction. This would cause the electron to spiral into the nucleus. Newton misconceived nature of the relationship between energy and space. He took the view that space was a static background or a stage against which energized objects of one sort or another moved or played. In other words, the universe is an inert and empty space within which dynamic objects have been mysteriously placed. This view goes back at least as far as Democritus and Leucippus, who saw the world as comprised of atoms separated by empty space. Einstein challenged this picture by allowing that space was active rather than passive, by allowing that space could influence the objects within it. His proposal that mass curves space has come to be associated with the proposal that the fundamental condition of the universe is infinitely massive -all of the mass in the universe is supposedly compressed to a point. But since this condition involves no space and time, it is difficult to see how it can be that curvature is a function of mass, which by definition is a spatio-temporal phenomenon. It seems to me that this this is an attempt to hang on to the basic element of Newton’s atomistic picture, which gives space priority over energy. One of various negative consequences stemming from the atomistic picture is that the 1/r^2 formula doesn’t have universal application. It doesn’t work when it comes to galaxies, nor when it comes to the atoms or black holes. Mass is a combination of energy and space, and if we reject atomism, and give priority to energy rather than space, then the fundamental condition of the universe involves no mass, and curvature must in part be deemed a function of energy. If energy is given priority, then it is the energetic rather than the spatial aspects of this combination that produce the initial curvature of the universe. In a region in which there is a balance of energy and space, the law of gravitation that comes out of this re-conception will agree with the old law, but in regions in which this balance is tipped toward energy, the new law will make different predictions. It seems therefore that an energy-centric law of gravitation will agree with Newton and Einstein when it comes to the solar system, but disagree when it comes to distant galaxies and atoms. This is because the solar system is a region in which there is a balance of energy and space, while distant galaxies and atoms are regions in which this balance is tipped toward energy. The tip is evidently extreme in the case of atoms, and so the disagreement will be extreme in atomic regions. But from the example of black holes, we see that it is not energy per se that produces curvature, but imbalances of energy and space, and so we have two forms of curvature, the one arising from an imbalance in favour of energy, and the other from an imbalance in favour of space.

Here then we have the philosophical underpinnings of a solution to the bullet cluster problem that has no need of dark matter. Einstein’s theory of gravity differed from Newton’s in that for Einstein gravity is a fictitious force. According to Einstein, the earth’s orbit around the sun was not to be explained by the existence of a force propagating through space and acting upon these bodies, but by the principle of inertia together with the curvature of the space in which they move. This curvature supposedly arises because of the presence in that space of matter. But the idea that space is curved by matter is a half-truth, and this half-truth is exposed by the behavior of distant galaxies, and in particular by the bullet cluster. It isn’t quite true that matter curves space; rather, imbalances of energy and space curves it, and matter involves energy, but it also involves space. In a region of space where there is a balance of energy and space, Newton’s laws of gravity and motion will apply more or less as written, but when this balance tips too strongly towards energy or space these laws will cease to apply unless they are recast in a more sophisticated mathematical framework than Newton, or Einstein, had at their disposal. The most extreme imaginable imbalance is the singularity at the ‘beginning’ of the universe, when the balance is tipped so strongly toward energy that there is nothing but energy, energy undiluted by space and time, infinite energy. Galaxies are closer to this ideal, unattainable, singular state than is our solar system, and hence they involve more energy and more curvature (the sun is closer to that state than the planets and thus it involves more energy and more curvature). Black Holes represent the opposite extreme… With the use of the right mathematics, the departure of galaxies from the strictures of Newton’s and Einstein’s theories of gravity towards a greater degree of contraction than we are accustomed from our terrestrial perspective can be explained without the benefit of invisible, undetectable, matter. Recall that the bullet cluster poses a problem for MOND because, while the majority of the mass of the colliding galaxies lies between them in the form of gas, the curvature is in the galaxies themselves. As stated above, no mere modification of the strength of gravity will be sufficient to account for this situation, because no such modification can explain why the curvature is in the galaxies rather than the gas. A solution is the following: yes, the mass is in the gas between the galaxies of the bullet cluster rather than in the galaxies themselves, but the energy is in the galaxies, and it is imbalances of energy and mass, not mass, that produce curvature. Arguably, what we observe when it comes to the rotation curves of distant galaxies is exactly what we ought to observe, and there is no need for dark matter. Nor- it is arguable- do Newtonian dynamics need to be modified in the ad hoc way proposed by MOND. Arguably, we simply need an extension of the mathematical principles underlying Newtonian dynamics that takes account of the greater than normal curvature to be encountered in energy-dense parts of a universe that is from a human point of view essentially classical.