A real number has infinite precision, so even a single particle with a real number position or weight, would have infinite information and create a black hole.
Would that be a valid reasoning to say why it can't be real numbers?
(I'm assuming you're talking about in the classical case, since the article also also considers classical dynamics and the problems associated with it.)
I'm actually gonna make a note to think about this later because this reminds me a lot of Charles Bennett's work on information and entropy, and I don't think it's a bad question at all. I'm considering the idea of representing the particle's position as x = .01010111001 and then idealizing a "machine" that lets it do work by erasing one bit of information. I have to work so I can't think about this right now but it's a cool thought.
Edit: I did think about and you're right! You can extract infinite energy if you know a particle's position to perfect precision. Very interesting!
Make the particle an ideal gas in a box at temperature T. At some instant in time, measure the position of the particle to N bits precision. Now place membranes around the particle to keep it to within the precision we measured it. We can kill one bit of precision by allowing the particle to isothermally expand the membranes into a volume with half the precision that we measure (so if we measure down to 1mm precision, we let it expand into 2mm.) The amount of work we can extract is:
W = \int p dV = \int kT dV V = kT ln 2
So for N bits precision you can extract NkT ln2 units of energy. I'll leave it to someone else to comment on whether this makes a black hole or not but this is exactly in the spirit of the article!
What's really happening is that if we consider the positional information to be continuous, then the particle's positional entropy is basically infinite. We can reduce this entropy (locally, to the particle) with a measurement, and use that lowered entropy to extract as much isothermal work as we want. But its energy density is unaffected by the positional information.
Perhaps a better way to characterize the amount of information in a number is to look at the length of the program necessary to compute it. Of course, this depends on the programming language, but I think it’s fair to say that any particular program to compute a number is in some loose sense an upper bound on the information in the number.
So, most of the irrational numbers we’re familiar with (pi, e, phi) are rather low on information since they may be computed by very short programs, despite having infinitely many non-repeating digits.
To move from one point to another, a particle has to move through all of them though, including the non computable ones
That, however, doesn't mean that spacetime is not a continuum.
What's the point of being a continuum if nothing in it can have arbitrary values. If everything in the spacetime has discrete values, what's there still to be continuous?
It's actually amusing that, just few weeks ago, people here on HN were arguing on why are we trying to quantize gravity and that it makes more sense for it to be non-quantum.
And if gravity is continuous, could it store infinite information (for all the same reasons, having arbitrarily precise values) then?
Of course the field can be continuous, even if the weight is exactly some rational number. "Discrete steps" just means that the value differences in gravitational waves is also discrete. A bigger problem, where relativity clashes with QM, is stuff like singularity and superposition. We don't have an answer for that yet.
If so, no: if any particle were to need infinite precision to be described, that would just mean we were wrong about maximum information density within a minimum boundary surface.
[0] while both charge and angular momentum reduce it
Please elaborate
That really and truly has no bearing on reality except insofar as the coordinate system has attempted to map reality.
I think it's related to the Plank Scale, but what you're talking about would possibly be a "Plank Precision"?
I'm having a hard time seeing if this is fundamentally the same thing as the Plank Length, or if it's something different entirely.
https://en.wikipedia.org/wiki/Planck_units#Planck_length
Very cool idea though.