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As far as I know, just like matter and energy, information density can also contribute to a black hole. A real number has infinite precision, so even a single

As far as I know, just like matter and energy, information density can also contribute to a black hole. 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?

eigenspace 3y ago
No.
- Aardwolf 3y ago
  > No. Please elaborate
  - eigenspace 3y ago
    Because what you're describing is coordinate or unit system dependent. I could invent a unit system where the mass of an electron is pi units, and another unit system where the mass of the electron is 1 unit. Me using a different unit system doesn't affect the physics of the system.
    - Aardwolf 3y ago
      But the particles can move around, they won't always be exactly on values of pi, 1, etc... in any coordinate system. And pi and 1 happen to be examples that can be described with a finite amount of symbols. But a continuous particle has to move through any value in between.
      - aeonik 3y ago
        Your idea interests me. 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 https://en.wikipedia.org/wiki/Planck_units#Planck_length Very cool idea though.
      - PH95VuimJjqBqy 3y ago
        The point is that you're discussing precision as it relates to a coordinate system trying to model position. That really and truly has no bearing on reality except insofar as the coordinate system has attempted to map reality.
ko27 3y ago
Particles don't have infinite precision. Position and momentum precision is limited by Heisenberg uncertainty principle. Weight precision is also limited because energy is quantized. That, however, doesn't mean that spacetime is not a continuum.
- Aardwolf 3y ago
  Then what does spacetime being a continuum mean? 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?
  - ko27 3y ago
    Gravity is continuous, and to disprove that you would need to disprove Lorentz invariance and Einstein's relativity. 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.
    - Aardwolf 3y ago
      But gravity is caused by the discrete particles (those with mass), so is it really continuous then? A discrete particle could only move in discrete steps, causing discrete jumps in gravity, would that be right? And if gravity is continuous, could it store infinite information (for all the same reasons, having arbitrarily precise values) then?
      - ko27 3y ago
        > But gravity is caused by the discrete particles (those with mass), so is it really continuous 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.
ben_w 3y ago
I'm not certain, but I think the only fundamental thing contributing to a black size is energy density[0], mass only contributes because it has an energy density (that's what E=mc^2 is), and information because it has to be carried by stuff which itself has a minimum energy per bit. 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
alecst 3y ago
You got a glib answer from another guy, but I like the question. It's kind of like you're saying: how much information can I store in a particle? And should there be a limit? (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!
- alecst 3y ago
  Alright: the kicker (and the reason why I don't think this will result in a black hole) is that in order to extract that energy from the particle, you have to continuously re-heat it and add more energy back each time you want kT \ln 2 units of work done. Remember that the work is done isothermally, so the particle's energy is actually constant throughout this process. 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.
falcor84 3y ago
Particle positions aren't stored by the universe as REAL but rather as DECIMAL(62, 35)
Kranar 3y ago
Infinite precision does not imply an infinite amount of information.
chongli 3y ago
The amount of information in a real number is not determined by the number of digits. For example, 1/3 = 0.333333… but we wouldn’t say 1/3 has “infinite 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.
- rnhmjoj 3y ago
  Yes, but the real numbers that can be computed to arbitrary precision with an algorithm are countable: this means almost all the reals contain an infinite amount of information. The real numbers are useful to reason and construct proofs, but as far as the physical world go, only very few of them are actually meaningful.
  - chongli 3y ago
    The non-computable reals are also not uniquely representable. It’s this property which I think is the most reasonable explanation for why these real numbers don’t cause an information black hole.
- Aardwolf 3y ago
  But 1/3 counts as discrete in a sense, you can describe it with a finite amount of symbols. To move from one point to another, a particle has to move through all of them though, including the non computable ones
  - chongli 3y ago
    If you're talking about a particle in real life, then you're talking about quantum mechanics, which is discrete, so the real numbers don't even come into play in that realm. Then, the use of real numbers in physics is a conceptual and calculational convenience which is not quite correct.
rnhmjoj 3y ago
It is, this idea is explored a bit in this paper: https://arxiv.org/abs/1803.06824 https://arxiv.org/abs/1803.06824. It also argues that using real numbers for physical quantities makes classical mechanics misleadingly deterministic, that is, they're hiding information that is impossible to measure in the initial conditions.

eigenspace 3y ago

No.

Aardwolf 3y ago
> No. Please elaborate
- eigenspace 3y ago
  Because what you're describing is coordinate or unit system dependent. I could invent a unit system where the mass of an electron is pi units, and another unit system where the mass of the electron is 1 unit. Me using a different unit system doesn't affect the physics of the system.
  - Aardwolf 3y ago
    But the particles can move around, they won't always be exactly on values of pi, 1, etc... in any coordinate system. And pi and 1 happen to be examples that can be described with a finite amount of symbols. But a continuous particle has to move through any value in between.
    - aeonik 3y ago
      Your idea interests me. 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 https://en.wikipedia.org/wiki/Planck_units#Planck_length Very cool idea though.
    - PH95VuimJjqBqy 3y ago
      The point is that you're discussing precision as it relates to a coordinate system trying to model position. That really and truly has no bearing on reality except insofar as the coordinate system has attempted to map reality.

ko27 3y ago

Particles don't have infinite precision. Position and momentum precision is limited by Heisenberg uncertainty principle. Weight precision is also limited because energy is quantized. That, however, doesn't mean that spacetime is not a continuum.

Aardwolf 3y ago
Then what does spacetime being a continuum mean? 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?
- ko27 3y ago
  Gravity is continuous, and to disprove that you would need to disprove Lorentz invariance and Einstein's relativity. 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.
  - Aardwolf 3y ago
    But gravity is caused by the discrete particles (those with mass), so is it really continuous then? A discrete particle could only move in discrete steps, causing discrete jumps in gravity, would that be right? And if gravity is continuous, could it store infinite information (for all the same reasons, having arbitrarily precise values) then?
    - ko27 3y ago
      > But gravity is caused by the discrete particles (those with mass), so is it really continuous 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.

ben_w 3y ago

I'm not certain, but I think the only fundamental thing contributing to a black size is energy density[0], mass only contributes because it has an energy density (that's what E=mc^2 is), and information because it has to be carried by stuff which itself has a minimum energy per bit. 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

alecst 3y ago

You got a glib answer from another guy, but I like the question. It's kind of like you're saying: how much information can I store in a particle? And should there be a limit? (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!

alecst 3y ago
Alright: the kicker (and the reason why I don't think this will result in a black hole) is that in order to extract that energy from the particle, you have to continuously re-heat it and add more energy back each time you want kT \ln 2 units of work done. Remember that the work is done isothermally, so the particle's energy is actually constant throughout this process. 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.

falcor84 3y ago

Particle positions aren't stored by the universe as REAL but rather as DECIMAL(62, 35)

Kranar 3y ago

Infinite precision does not imply an infinite amount of information.

chongli 3y ago

The amount of information in a real number is not determined by the number of digits. For example, 1/3 = 0.333333… but we wouldn’t say 1/3 has “infinite 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.

rnhmjoj 3y ago
Yes, but the real numbers that can be computed to arbitrary precision with an algorithm are countable: this means almost all the reals contain an infinite amount of information. The real numbers are useful to reason and construct proofs, but as far as the physical world go, only very few of them are actually meaningful.
- chongli 3y ago
  The non-computable reals are also not uniquely representable. It’s this property which I think is the most reasonable explanation for why these real numbers don’t cause an information black hole.
Aardwolf 3y ago
But 1/3 counts as discrete in a sense, you can describe it with a finite amount of symbols. To move from one point to another, a particle has to move through all of them though, including the non computable ones
- chongli 3y ago
  If you're talking about a particle in real life, then you're talking about quantum mechanics, which is discrete, so the real numbers don't even come into play in that realm. Then, the use of real numbers in physics is a conceptual and calculational convenience which is not quite correct.

rnhmjoj 3y ago

It is, this idea is explored a bit in this paper: https://arxiv.org/abs/1803.06824 https://arxiv.org/abs/1803.06824. It also argues that using real numbers for physical quantities makes classical mechanics misleadingly deterministic, that is, they're hiding information that is impossible to measure in the initial conditions.