I know it’s a crazy title. It has no place in a physics blog, but then I am sure this article will go elsewhere. […] Well… […] Let me be honest: it’s probably gonna go nowhere. Whatever. I don’t care too much. My life is happier than Wittgenstein’s. 🙂
My original title for this post was: discrete spacetime. That was somewhat less offensive but, while being less offensive, it suffered from the same drawback: the terminology was ambiguous. The commonly accepted term for discrete spacetime is the quantum vacuum. However, because I am just an arrogant bastard trying to establish myself in this field, I am telling you that term is meaningless. Indeed, wouldn’t you agree that, if the quantum vacuum is a vacuum, then it’s empty. So it’s nothing. Hence, it cannot have any properties and, therefore, it cannot be discrete – or continuous, or whatever. We need to put stuff in it to make it real.
Therefore, I’d rather distinguish mathematical versus physical space. Of course, you are smart, and so you now you’ll say that my terminology is as bad as that of the quantum vacuumists. And you are right. However, this is a story that I am writing, and so I will write it the way I want to write it. 🙂 So where were we? Spacetime! Discrete spacetime.
Yes. Thank you! Because relativity tells us we should think in terms of four-vectors, we should not talk about space but about spacetime. Hence, we should distinguish mathematical spacetime from physical spacetime. So what’s the definitional difference?
Mathematical spacetime is just what it is: a coordinate space – Cartesian, polar, or whatever – which we define by choosing a representation, or a base. And all the other elements of the set are just some algebraic combination of the base set. Mathematical space involves numbers. They don’t – let me emphasize that: they do not!– involve the physical dimensions of the variables. Always remember: math shows us the relations, but it doesn’t show us the stuff itself. Think of it: even if we may refer to the coordinate axes as time, or distance, we do not really think of them as something physical. In math, the physical dimension is just a label. Nothing more. Nothing less.
In contrast, physical spacetime is filled with something – with waves, or with particles – so it’s spacetime filled with energy and/or matter. In fact, we should analyze matter and energy as essentially the same thing, and please do carefully re-read what I wrote: I said they are essentially the same. I did not say they are the same. Energy and mass are equivalent, but not quite the same. I’ll tell you what that means in a moment.
These waves, or particles, come with mass, energy and momentum. There is an equivalence between mass and energy, but they are not the same. There is a twist – literally (only after reading the next paragraphs, you’ll realize how literally): even when choosing our time and distance units such that c is numerically equal to 1 – e.g. when measuring distance in light-seconds (or time in light-meters), or when using Planck units – the physical dimension of the c2 factor in Einstein’s E = mc2 equation doesn’t vanish: the physical dimension of energy is kg·m2/s2.
Using Newton’s force law (1 N = 1 kg·m/s2), we can easily see this rather strange unit is effectively equivalent to the energy unit, i.e. the joule (1 J = 1 kg·m2/s2 = 1 (N·s2/m)·m2/s2 = 1 N·m), but that’s not the point. The (m/s)2 factor – i.e. the square of the velocity dimension – reflects the following:
- Energy is nothing but mass in motion. To be precise, it’s oscillating mass. [And, yes, that’s what string theory is all about, but I didn’t want to mention that. It’s just terminology once again: I prefer to say ‘oscillating’ rather than ‘vibrating’. :-)]
- The rapidly oscillating real and imaginary component of the matter-wave (or wavefunction, we should say) each capture half of the total energy of the object E = mc2.
- The oscillation is an oscillation of the mass of the particle (or wave) that we’re looking at.
In the mentioned publication, I explore the structural similarity between:
- The oscillating electric and magnetic field vectors (E and B) that represent the electromagnetic wave, and
- The oscillating real and imaginary part of the matter-wave.
The story is simple or complicated, depending on what you know already, but it can be told in an abnoxiously easy way. Note that the associated force laws do not differ in their structure:
The only difference is the dimension of m versus q: mass – the measure of inertia -versus charge. Mass comes in one color only, so to speak: it’s always positive. In contrast, electric charge comes in two colors: positive and negative. You can guess what comes next, but I won’t talk about that here. Just note the absolute distance between two charges (with the same or the opposite sign) is twice the distance between 0 and 1, which must explains the rather mysterious 2 factor I get for the Schrödinger equation for the electromagnetic wave (but I still need to show how that works out exactly).
The point is: remembering that the physical dimension of the electric field is N/C (newton per coulomb, i.e. force per unit of charge) it should not come as a surprise that we find that the physical dimension of the components of the matter-wave is N/kg: newton per kg, i.e. force per unit of mass. For the detail, I’ll refer you to that article of mine (and, because I know you will not want to work your way through it, let me tell you it’s the last chapter that tells you how to do the trick).
So where were we? Strange. I actually just wanted to talk about discrete spacetime here, but I realize I’ve already dealt with all of the metaphysical questions you could possible have, except the (existential) Who Am I? question, which I cannot answer on your behalf. 🙂
I wanted to talk about physical spacetime, so that’s sanitized mathematical space plus something. A date without logistics. Our mind is a lazy host, indeed.
Reality is the guest that brings all of the wine and the food to the party.
In fact, it’s a guest that brings everything to the party: you – the observer – just need to set the time and the place. In fact, in light of what Kant – and many other eminent philosophers – wrote about space and time being constructs of the mind, that’s another statement which you should interpret literally. So physical spacetime is spacetime filled with something – like a wave, or a field. So how does that look like? Well… Frankly, I don’t know! But let me share my idea of it.
Because of the unity of Planck’s quantum of action (ħ ≈ 1.0545718×10−34 N·m·s), a wave traveling in spacetime might be represented as a set of discrete spacetime points and the associated amplitudes, as illustrated below. [I just made an easy Excel graph. Nothing fancy.]
The space in-between the discrete spacetime points, which are separated by the Planck time and distance units, is not real. It is plain nothingness, or – if you prefer that term – the space in-between in is mathematical space only: a figment of the mind – nothing real, because quantum theory tells us that the real, physical, space is discontinuous.
Why is that so? Well… Smaller time and distance units cannot exist, because we would not be able to pack Planck’s quantum of action in them: a box of the Planck scale, with ħ in it, is just a black hole and, hence, nothing could go from here to there, because all would be trapped. Of course, now you’ll wonder what it means to ‘pack‘ Planck’s quantum of action in a Planck-scale spacetime box. Let me try to explain this. It’s going to be a rather rudimentary explanation and, hence, it may not satisfy you. But then the alternative is to learn more about black holes and the Schwarzschild radius, which I warmly recommend for two equivalent reasons:
- The matter is actually quite deep, and I’d recommend you try to fully understand it by reading some decent physics course.
- You’d stop reading this nonsense.
If, despite my warning, you would continue to read what I write, you may want to note that we could also use the logic below to define Planck’s quantum of action, rather than using it to define the Planck time and distance unit. Everything is related to everything in physics. But let me now give the rather naive explanation itself:
- Planck’s quantum of action (ħ ≈ 1.0545718×10−34 N·m·s) is the smallest thing possible. It may express itself as some momentum (whose physical dimension is N·s) over some distance (Δs), or as some amount of energy (whose dimension is N·m) over some time (Δt).
- Now, energy is an oscillation of mass (I will repeat that a couple of times, and show you the detail of what that means in the last chapter) and, hence, ħ must necessarily express itself both as momentum as well as energy over some time and some distance. Hence, it is what it is: some force over some distance over some time. This reflects the physical dimension of ħ, which is the product of force, distance and time. So let’s assume some force ΔF, some distance Δs, and some time Δt, so we can write ħ as ħ = ΔF·Δs·Δt.
- Now let’s pack that into a traveling particle – like a photon, for example – which, as you know (and as I will show in this publication) is, effectively, just some oscillation of mass, or an energy flow. Now let’s think about one cycle of that oscillation. How small can we make it? In spacetime, I mean.
- If we decrease Δs and/or Δt, then ΔF must increase, so as to ensure the integrity (or unity) of ħ as the fundamental quantum of action. Note that the increase in the momentum (ΔF·Δt) and the energy (ΔF·Δs) is proportional to the decrease in Δt and Δs. Now, in our search for the Planck-size spacetime box, we will obviously want to decrease Δs and Δt simultaneously.
- Because nothing can exceed the speed of light, we may want to use equivalent time and distance units, so the numerical value of the speed of light is equal to 1 and all velocities become relative velocities. If we now assume our particle is traveling at the speed of light – so it must be a photon, or a (theoretical) matter-particle with zero rest mass (which is something different than a photon) – then our Δs and Δt should respect the following condition: Δs/Δt = c = 1.
- Now, when Δs = 1.6162×10−35 m and Δt = 5.391×10−44 s, we find that Δs/Δt = c, but ΔF = ħ/(Δs·Δt) = (1.0545718×10−34 N·m·s)/[(1.6162×10−35 m)·(5.391×10−44 s)] ≈ 1.21×1044 N. That force is monstrously huge. Think of it: because of gravitation, a mass of 1 kg in our hand, here on Earth, will exert a force of 9.8 N. Now note the exponent in that 1.21×1044 number.
- If we multiply that monstrous force with Δs – which is extremely tiny – we get the Planck energy: (1.6162×10−35 m)·(1.21×1044 N) ≈ 1.956×109 joule. Despite the tininess of Δs, we still get a fairly big value for the Planck energy. Just to give you an idea, it’s the energy that you’d get out of burning 60 liters of gasoline—or the mileage you’d get out of 16 gallons of fuel! In fact, the equivalent mass of that energy, packed in such tiny space, makes it a black hole.
- In short, the conclusion is that our particle can’t move (or, thinking of it as a wave, that our wave can’t wave) because it’s caught in the black hole it creates by its own energy: so the energy can’t escape and, hence, it can’t flow. 🙂
Of course, you will now say that we could imagine half a cycle, or a quarter of that cycle. And you are right: we can surely imagine that, but we get the same thing: to respect the unity of ħ, we’ll then have to pack it into half a cycle, or a quarter of a cycle, which just means the energy of the whole cycle is 2·ħ, or 4·ħ. However, our conclusion still stands: we won’t be able to pack that half-cycle, or that quarter-cycle, into something smaller than the Planck-size spacetime box, because it would make it a black hole, and so our wave wouldn’t go anywhere, and the idea of our wave itself – or the particle – just doesn’t make sense anymore.
This brings me to the final point I’d like to make here. When Maxwell or Einstein, or the quantum vacuumists – or I 🙂 – say that the speed of light is just a property of the vacuum, then that’s correct and not correct at the same time. First, we should note that, if we say that, we might also say that ħ is a property of the vacuum. All physical constants are. Hence, it’s a pretty meaningless statement. Still, it’s a statement that helps us to understand the essence of reality. Second, and more importantly, we should dissect that statement. The speed of light combines two very different aspects:
- It’s a physical constant, i.e. some fixed number that we will find to be the same regardless of our reference frame. As such, it’s as essential as those immovable physical laws that we find to be the same in each and every reference frame.
- However, its physical dimension is the ratio of the distance and the time unit: m/s. We may choose other time and distance units, but we will still combine them in that ratio. These two units represent the two dimensions in our mind that – as Kant noted – structure our perception of reality: the temporal and spatial dimension.
Hence, we cannot just say that c is ‘just a property of the vacuum’. In our definition of c as a velocity, we mix reality – the ‘outside world’ – with our perception of it. It’s unavoidable. Frankly, while we should obviously try – and we should try very hard! – to separate what’s ‘out there’ versus ‘how we make sense of it’, it is and remains an impossible job because… Well… When everything is said and done, what we observe ‘out there’ is just that: it’s just what we – humans – observe. 🙂
So, when everything is said and done, the essence of reality consists of four things:
- Mass, i.e. something, or not nothing
- Movement (of something), from nowhere to somewhere.
- Us: our mind. Or God’s Mind. Whatever. Mind.
The first is like yin and yang, or manicheism, or whatever dualistic religious system. As for Movement and Mind… Hmm… In some very weird way, I feel they must be part of one and the same thing as well. 🙂 In fact, we may also think of those four things as:
- 0 (zero)
- 1 (one), or as some sine or a cosine, which is anything in-between 0 and 1.
- Well… I am not sure! I can’t really separate point 3 and point 4, because they combine point 1 and point 2.
So we’ve don’t have a quadrupality, right? We do have a Trinity here, don’t we? […] Maybe. I won’t comment, because I think I just found Unity here. 🙂