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

**. 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 can**

*quantum vacuum**not*have any

*properties*and, therefore, it can

*not*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**: the physical dimension of energy is kg·m

*physical*dimension of the*c*^{2 }factor in Einstein’s E = m*c*^{2 }equation doesn’t vanish^{2}/s

^{2}

**.**

Using Newton’s force law (1 N = 1 kg·m/s^{2}), we can easily see this rather strange unit is effectively equivalent to the energy unit, i.e. the *joule* (1 J = 1 kg·m^{2}/s^{2} = 1 (N·s^{2}/m)·m^{2}/s^{2 }= 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 = m*c*^{2}. - 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 *dis*continuous.

**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
*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)] ≈ 1.21×10^{44}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 1.21×10^{44}number. - If we multiply that monstrous force with Δs – which is extremely tiny – we get the Planck energy: (1.6162×10
^{−35}m)·(1.21×10^{44}N) ≈ 1.956×10^{9}*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:

- Nothing
- Mass, i.e.
*something*, or*not*nothing - Movement (of
*something*), from nowhere to*some*where. - 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
- Well… I am not sure!

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. 🙂