On Heaven and Dark Matter
Reflections on Florensky's complex Cosmos
Introduction
This is going to be a bit of an unusual and rather scientific article. Normally I am not inclined to mathematically supported metaphysics, but some ideas dawned on my which I wish to share, I do not fully know if I agree with all of these ideas, and I am by no means a physicist, but perhaps someone more mathematically and physically educated than me can expand on them. If you aren’t a physics geek, worry not, I will try to make this article as comprehensible as possible, but keep in mind that this is just as much an article of speculative physics as it is of Christian metaphysics
The Relativistic Limits of Matter
Let us begin, as modern physics often does, with Einstein’s famous equation, the mass-energy equivalence:
This formula is iconic, and surprisingly simple to understand. It simply means that the energy (E) of a given body (that is, its capacity to move, do work) is proportional to its mass (m) multiplied by the constant speed of light (c) squared, thus mass and energy are two different manifestations of the same physical phenomenon. This means that the more mass a body has, the more energy it has, but also rather surprisingly, the more energy it has, the more mass it has. Therefore, a body moving at immensely high speeds and thus having a large amount of kinetic energy, would in fact have its mass increased at these speeds, and have it decreased as it slows down.
This increased mass is described by formula for relativistic mass:
Where m₀ is the mass of the body at its resting state, and m is the mass at the given velocity. It is clear that as the velocity of the object nears the speed of light, the second term in the numerator nears 1, and thus the result value of the whole numerator nears 0. As a consequence, the larger the velocity, the larger the mass as we established. Now if hypothetically an object were to reach the speed of light, it would have infinite mass and thus require infinite energy, which is impossible.
Another interesting consequence of Einstein’s relativity is time dilation, that is, the faster an object is moving, the slower time passes for it, this is defined by the following formula:
Hypothetically, if an object were to reach the speed of light, time would stop entirely for that object, for the same mathemtical reason that the mass of an object moving at the speed of light would be infinite
Interesting, the dimensions of the object are also affected by its speed, the faster an object is moving, the more its length contracts, which is described by this formula:
Hypothetically, if an object were to reach the speed of light, it would contract into nothingness, since its length would be 0, because second of the two terms under the square root would be 1, and thus L₀ would be multiplied by 0.
Einstein took all of this to mean that no object could possibly ever accelerate to or past the speed of light.
The Physics of Heaven
The Russian polymath Pavel Florensky however, in his book Imaginaries in Geometry, begged to differ. He had used special relativity to propose that, because there is no absolute point of reference for motion in the universe, the Earth could be taken as an immobile center in the universe in relation to which all other bodies move. However, to posit this, there would have to be a point in space after which celestial bodies have to move faster than the speed of light to take the Earth as an immobile point of reference because of the fact that they are too far from the Earth, Florensky embraces this and claims that everything beyond the aforementioned point is a heavenly realm in which there are qualitative physical differences, as has been implied for the realm of Divine Ideas and immaterial intellects in the cosmologies of ancient philosophers. Everything before this point, according to Florensky would be a sort of terrestrial realm where the known natural laws apply, and everything past this point would be Heaven. Florensky managed to calculate the radius of the breadth of this terrestrial realm, and placed the border of the Empyrean Heaven and the terrestrial realm between the orbits of Uranus and Neptune. Of course, in modern times, we have space probes like the Voyager 1 which have gone beyond this point, disproving Florensky’s neo-geocentric model, however, Florensky’s descriptions of this Heavenly Realm are intriguing and ought to be studied. I will quickly describe them now.
Suppose that we define the second term in the Lorentz factor (that is, the formula with the square root) as so:
If we hypothetically assume that there are bodies which move faster than the speed of light, the value of β would be larger than 1, this would yield some interesting physical results with the relativistic equations as well as some other equations. First and foremost, mass would be an imaginary number, because the result of subtracting the two terms under the square root in the Lorentz factor would yield a negative number under a square root, that is, an imaginary number.
The length of the object would also be “contracted” to an imaginary number
And time between two given points, having appeared to have stopped for an object when viewed from the outside, would appear to be moving in reverse, where the effect or end (telos) would precede the cause, and the cause would become an ideal goal.
An object moving faster than the speed of light would also have the laws of thermodynamics inversed for it and it would take an infinite amount of energy to slow down the object to the speed of light.
Hypothetically such objects could exist without the need for adopting Florensky’s rather (scientifically) archaic geocentric worldview, although most scientists today would say that since we haven’t observed such objects, there is no need to posit them with our current understanding of the universe. In the following parts of this article, I will argue that we have observed some things which point to them and that we should posit their existence.
The Gravitational Force of Heaven
Would an object with such qualitative physical differences from our common material objects as imaginary mass, imaginary dimensions and reverse movement in time be able to interact directly with our common material objects, and the electromagnetic field? Well, one here mostly has to resort to mathematical and metaphysical speculation, imaginary numbers are typically seen as lateral to real numbers and would thus be “on a different plane”, so one would initially be inclined to say no and accept that only physical quantities with a real quality can affect bodies with real qualities, and vice versa More convincingly, if such heavenly objects really exist, then we have not observed them directly coming into contact with our common material objects. However that does not mean we haven’t observed their affects. There are certain gravitional anomalies in the known universe which cannot be explained unless more matter is present. These gravitational anomalies have been traditionally explained through a hypothetical form of matter called dark matter, which doesn’t interact directly with material bodies, nor with light or the electromagnetic field, but exerts a gravitational force on regular material bodies. Let us see whether mathematically Florensky’s heavenly objects would be able to exert a force on regular matter.
Newton’s law of universal gravitation has been superseeded by Einstein’s general relativity, but Newton’s gravitational constant and formula is entirely valid to describe the gravitational behavior between two objects except when the gravitational force is extremely large (such as the case with black hole) or when the gravitational force is extremely small, but in our case, the quantity of gravitational force is irrelevant, only its quality interests us, so for simplicity’s sake, we will use Newton’s formula.
Let’s suppose that one object is a Florenskyan heavenly body, with a mass defined as some m = ai, and the other is a regular body, with a mass defined as some m = b, with G as the gravitational constant and r as the distance between the two objects.
First we need to define the distance between the Florenskyan body and the regular body. Since one object has dimensions on the imaginary plane, and one object has dimensions on the real plane, we cannot infer immediately whether the distance is real or imaginary.
The distance between two objects is qualitatively the same as the vector of one object to the other, it just doesn’t contain the information about the direction. A vector in Cartesian co-ordinates can easily be defined as:
If we extend the Cartesian co-ordinates on the imaginary plane, and make one of the values imaginary, meaning the direction would be moving both in “real” space and “imaginary” space, then we would get a complex number as the distance between the starting and ending points of the vector, even if we introduce another, two more, three more or however many more dimensions, because the sum of the values for the dimensions on the real plane would be a real number and the sum of the values for the dimensions on the imaginary plane would be an imaginary number:
Now if we apply this to Newton’s formula, we get an interesting result:
The multiplication of the real and imaginary mass as well as the gravitational constant would lead to an imaginary number, so in the end we would have an imaginary number divided by a complex number, in the end the gravitational force would be a complex number. We have to engage in physical speculation here and continue with our assumption that physical quantities with a real quality can affect bodies with real qualities, and vice versa. Accepting this, this would mean that the gravitational force between a Florenskyan body and a regular body, being a complex number, could affect both bodies.
This perfectly fits the description of dark matter as a hypothesis. Therefore the physics of the Florenskyan Heaven describe dark matter surprisingly well. In this sense, dark matter could be interpreted as a substance of which a whole “secondary” universe of objects is made, moving faster than the speed of light, whose physical qualities seem to near the traditional philosophical descriptions of the Empyrean Heaven/realm of Platonic Forms. In this case this Empyrean Heaven described by Florensky would thus not be a realm spatially distanced by a real quantity from the terrestrial realm as Florensky imagined, but rather a different invisible, untouchable stratum of the universe superimposed onto the terrestrial stratum, not coming into contact with it directly but gravitationally affecting it.
Philosophical Note: Can We Even Describe Heaven Mathematically?
One of my primary concerns when reading the works of Florensky, and I imagine the concerns of many others is that these sorts of immaterial Platonic realms are metaphysical notions which would seemingly lose a lot of their distinctiveness from material reality if we were to try to describe them with mathematics. And I still believe this is a fairly sound argument, however, it is worth considering a few counterarguments.
First and foremost, the early Christian philosophers and even later scholastics did perceive some sort of spatial connection between the Heaven of heavens/the Empyrean and the terrestrial sphere, and they acknowledged that mathematics can describe many things in the terrestrial sphere. Many of them also took the laws of logic to be an all-encompassing given and not created unlike the natural laws. As I have previously argued, the laws of logic can be theologically understood as manifestations of God’s uncreated Energy (of course, in a metaphysical and not a physical sense) of Order, which He can instantiate according to His Will. So we have no reason to suppose that the laws of logic do not apply to Heaven. And if we take the laws of logic to be able to describe anything which is intelligible, then we can consider them to apply universally, and thus can use the Fregean argument for mathematical platonism to demonstrate the independent existence of numbers. And going forward from this point, we can note that we often use numbers to describe the count of immaterial notions, and if numbers are truly real independent of us, then they must describe a real plurality in immaterial notions if we take these descriptions to be true, so it would be arbitrary to draw a line to which point mathematics can describe Heaven. We do not need to posit that describing the physics and geometry of Heaven materializes the immaterial angelic intellects, because Church Fathers have spoken about subtle bodies, bodies which lie on the threshold between God’s omnipresence and the real spatio-temporal determination of the material realm, these can be equated with bodies which have imaginary mass and dimensions, as we clearly cannot comprehend these qualities of physical quantities as determinate in the same way we comprehend real qualities of physical quantities.
Of course, God, although typically associated with Heaven, transcends the duality of Heaven and Earth and He would not be described physically, because He is beyond Being, beyond Beauty, beyond Order, beyond all transcendentals.
But even if dark matter cannot be philosophically equated with Heaven, I still leave this article as a modest, and rather surprisingly simple but possible explanation for the physics of dark matter.
But what do you think? Has physics finally encountered Heaven? And do we truly stand at the crossroads of science and mysticism.
What has interested me lately is mathematical aporia like the three body problem. My question would be, as one who has really only theoretical understanding of mathematics--e.g. it functions as a language--is when this language breaks down and ceases to be able to circumscribe the very phenomena under investigation. Any thoughts on this?