Difficult to evaluate, with potential yellow flags.
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Recently I have been studying a core contradiction about the discreteness of spacetime, that is, the checkerboard problem (checkerboard anisotropy). As long as the coordinate system of space has discrete grid points, one has to answer: "why do the microscopic grid points, after coarse-graining, become isotropic and continuous observational results?"
If we think about it carefully, perhaps the first intuitive assumption hides a trap.
Discrete space = cutting space into grid-like coordinates (square, triangular, or hexagonal grids, and so on).
Causal set theory gives a way to dissolve the contradiction. It uses random sprinkling to statistically preserve Lorentz invariance, but this explanation is not clean — it adds one more ansatz to evade the contradiction, instead of pulling out the root of the contradiction: discreteness should not have been put into the coordinates in the first place.
I think the core contradiction between discrete space and relativity should be solved by shifting the perspective.
Is the container discrete, or the content?
If we distinguish these two kinds of discreteness.
The discreteness of the container is: space has a smallest length, has fixed coordinates one grid at a time, and the content is filled into these containers, and therefore becomes discrete along with them.
And the discreteness of the content is: when we keep cutting the continuous content finer and keep distinguishing it, does there exist a kind of limit that is smallest but not infinitely small. That is to say, we finally reach a state that cannot be distinguished any further.
In quantum mechanics, energy levels and spin are discrete. And the holographic principle says that the amount of information in a region has an upper bound. Both of these refer to "the discreteness of the content", not "the discreteness of the container".
The state space is continuous while the measurement results are discrete, which shows that a discrete state does not necessarily require assuming a discrete container. One only needs to assume "there exists a state that cannot be distinguished any further", whether we call it a bit in the information way, or a state in the physics way.
To say it more precisely, the direction and position of space are continuous, but there is an information-resolution limit, they cannot be divided infinitely, so an observer can only read out discrete results — this is holographic discreteness, not lattice discreteness.
Since space is continuous, then what about time?
A state that cannot be divided any further is exactly the finite information that a finite region can hold. And time is exactly the ordered updating of these finite states. In other words, the passage of time is a macroscopic observation accumulated from "irreversible state transitions".
This actually also fits the claim of some physicists, that time is a thermodynamic arrow, that the increase of entropy provides the irreversible causal partial order, and that the passage of time is a macroscopic readout.
The counter-argument here is also very obvious: the Lorentz metric is built on a four-dimensional manifold in which space and time are coupled. If we decouple space and time, and attribute the discreteness to state updates, then how does the Lorentz metric emerge on the macroscopic level?
The four-dimensional manifold is stitched by the speed of light
Under the perspective above, time is the causal dependency relation between local state updates (event A affects event B). And this dependency relation, plus the fact that the spatial distance advanced by each update is bounded, gives the maximum speed of information propagation c. The above together form a conformal geometry, that is, the light cone in four-dimensional spacetime. That is to say, the four-dimensional manifold can be a macroscopic readout of continuous space + the update order, stitched together by the light speed c.
The reading of causal set theory is partially correct; the point is that we do not need random sprinkling.
The real problem converges to boost invariance — one needs to rigorously and formally prove the derivation framework above: whether a local, confluent update rule, plus continuous space, can make the Lorentz invariance of the macroscopic readout emerge.
This article does not try to propose a formal proof, but to change the perspective on an ontological question, and to contract it into one core narrative.
Recently I have been studying a core contradiction about the discreteness of spacetime, that is, the checkerboard problem (checkerboard anisotropy). As long as the coordinate system of space has discrete grid points, one has to answer: "why do the microscopic grid points, after coarse-graining, become isotropic and continuous observational results?"
If we think about it carefully, perhaps the first intuitive assumption hides a trap.
Discrete space = cutting space into grid-like coordinates (square, triangular, or hexagonal grids, and so on).
Causal set theory gives a way to dissolve the contradiction. It uses random sprinkling to statistically preserve Lorentz invariance, but this explanation is not clean — it adds one more ansatz to evade the contradiction, instead of pulling out the root of the contradiction: discreteness should not have been put into the coordinates in the first place.
I think the core contradiction between discrete space and relativity should be solved by shifting the perspective.
Is the container discrete, or the content?
If we distinguish these two kinds of discreteness.
The discreteness of the container is: space has a smallest length, has fixed coordinates one grid at a time, and the content is filled into these containers, and therefore becomes discrete along with them.
And the discreteness of the content is: when we keep cutting the continuous content finer and keep distinguishing it, does there exist a kind of limit that is smallest but not infinitely small. That is to say, we finally reach a state that cannot be distinguished any further.
In quantum mechanics, energy levels and spin are discrete. And the holographic principle says that the amount of information in a region has an upper bound. Both of these refer to "the discreteness of the content", not "the discreteness of the container".
The state space is continuous while the measurement results are discrete, which shows that a discrete state does not necessarily require assuming a discrete container. One only needs to assume "there exists a state that cannot be distinguished any further", whether we call it a bit in the information way, or a state in the physics way.
To say it more precisely, the direction and position of space are continuous, but there is an information-resolution limit, they cannot be divided infinitely, so an observer can only read out discrete results — this is holographic discreteness, not lattice discreteness.
Since space is continuous, then what about time?
A state that cannot be divided any further is exactly the finite information that a finite region can hold. And time is exactly the ordered updating of these finite states. In other words, the passage of time is a macroscopic observation accumulated from "irreversible state transitions".
This actually also fits the claim of some physicists, that time is a thermodynamic arrow, that the increase of entropy provides the irreversible causal partial order, and that the passage of time is a macroscopic readout.
The counter-argument here is also very obvious: the Lorentz metric is built on a four-dimensional manifold in which space and time are coupled. If we decouple space and time, and attribute the discreteness to state updates, then how does the Lorentz metric emerge on the macroscopic level?
The four-dimensional manifold is stitched by the speed of light
Under the perspective above, time is the causal dependency relation between local state updates (event A affects event B). And this dependency relation, plus the fact that the spatial distance advanced by each update is bounded, gives the maximum speed of information propagation c. The above together form a conformal geometry, that is, the light cone in four-dimensional spacetime. That is to say, the four-dimensional manifold can be a macroscopic readout of continuous space + the update order, stitched together by the light speed c.
The reading of causal set theory is partially correct; the point is that we do not need random sprinkling.
The real problem converges to boost invariance — one needs to rigorously and formally prove the derivation framework above: whether a local, confluent update rule, plus continuous space, can make the Lorentz invariance of the macroscopic readout emerge.
This article does not try to propose a formal proof, but to change the perspective on an ontological question, and to contract it into one core narrative.