Why Physical Constants Cannot Be Constant in Infinite Space
Uniform non-zero states aren't in the solution space for infinite domains
Or, if the universe is infinite, there cannot be any physical constants
A mathematical proof that fundamental “constants” must vary across unbounded domains.
Physics treats certain quantities as fundamental constants: the speed of light c, gravitational constant G, Planck’s constant ℏ, fine structure constant α. These appear constant in our measurements, so we assume they’re uniform across all space and time. For infinite domains, this assumption is mathematically unjustified. In science, people—like me—sometimes ask a question everyone thought was already answered, only to discover the answer was just an unexamined assumption. The formulation of the question is the insight.
Constants Are Field Values
Any measurable quantity is operationally defined by measurement at a location. When we measure the speed of light, we perform a procedure at spatial coordinates x and time t. The result is c(x,t)—a value at that point. If measurable at different locations, it’s a field mapping spacetime to values:
c: ℝ³ × ℝ⁺ → ℝ⁺
This applies to every “constant” in physics. They’re not abstract numbers—they’re properties measured at locations. If measurable, they’re fields. Physical fields must have finite energy density for measurability; constant φ₀ ≠ 0 gives infinite total energy in infinite volume.
The Mathematical Impossibility of Uniformity
For any physical field φ on infinite space ℝ³, the claim that φ(x,t) = φ₀ everywhere (uniform constant state) encounters three insurmountable problems:
Preparation impossibility. Establishing φ = φ₀ at all points requires infinite operations across infinite volume. No causal structure can coordinate this—information propagates at finite speed, and there are infinitely many points requiring coordination.
Maintenance impossibility. Field equations generally take the form ∂φ/∂t = L[φ] + sources. In bounded regions, boundary conditions constrain solutions and can maintain uniformity. Infinite space has no boundaries. Local perturbations propagate without any mechanism to suppress them or enforce global uniformity. Uniformity is not an attractor of the dynamics.
Solution space structure. Consider functions with finite energy density:
∫_ℝ³ |φ|² d³x < ∞
For constant φ₀ ≠ 0:
∫_ℝ³ φ₀² d³x = φ₀² · ∞ = ∞
Non-zero uniform states have infinite total energy and are excluded from physically admissible solution spaces. They don’t exist as valid field configurations on unbounded domains.
Formal Statement
Theorem. For any physical field φ: ℝ³ → ℝ on infinite unbounded space, either φ(x) = 0 everywhere (vacuum), or there exist locations where φ varies:
∀φ ∈ Physical_Fields(ℝ³): φ ≡ 0 ∨ (∃x,x′ : φ(x) ≠ φ(x′))
The only admissible uniform state is vacuum. All non-zero fields necessarily exhibit spatial variation.
Corollary. Physical constants c, G, ℏ, α are measurable at spatial locations, hence are field values. Since measurable fields on ℝ³ cannot be uniformly non-zero:
∀ κ ∈ {c, G, ℏ, α, ...}: ∃x,x′ ∈ ℝ³ : κ(x) ≠ κ(x′)
Implications
Standard cosmology assumes homogeneity—that the universe looks the same everywhere, with identical physical laws and constants throughout. For truly infinite space, this assumption has no mathematical basis. Unbounded domains require variation as the default state, not uniformity.
This reframes several foundational questions. Fine-tuning problems dissolve when we recognize we’re measuring constants at our location within varying fields, not discovering universe-wide values that somehow emerged from initial conditions. The cosmological principle becomes a useful local approximation rather than a global truth requiring justification.
Variation in what we call constants isn’t anomalous—it’s what infinite geometry necessitates. Observations of fine structure constant variation across cosmological distances, if confirmed, wouldn’t indicate exotic physics but would be consistent with basic mathematical structure of infinite domains.
The uniformity we observe locally reflects our position within a causally connected patch where field gradients are small, not a property extending to all scales. As measurement precision improves and observational horizons expand, we should expect to detect variation rather than be surprised by it.
What we call physical constants are locally stable field values, not global uniformities. Infinity doesn’t just make equilibrium impossible—it makes uniformity impossible for any non-zero field. Variation is not a departure from some uniform baseline. It’s the only mathematically valid state for infinite reality.
This will be part of an upcoming book I am writing with a reinterpreation of physics but I wanted to claim primacy on this one by publishing it with a timestamp.
Once you ask “what does uniformity mean on an unbounded domain?” the answer becomes obvious.
The entire edifice of assuming constant constants rested on never seriously interrogating what “constant everywhere” means when “everywhere” is infinite. We imported intuitions from bounded systems—where uniformity is achievable, maintainable, and lies in valid solution spaces—and applied them to ℝ³ without checking whether the mathematics still works.
It doesn’t. The question contains its own answer: you cannot have non-zero uniformity on infinite domains because uniformity requires closure and infinity provides none.
The homogeneous solution fails.
The diffusion equation with c(x,t) = c₀ as its “solution” on ℝ³ isn’t just unphysical or impractical. It’s not a solution at all. The mathematics rejects it. That’s different from most physical impossibilities, which are about dynamics (you can’t reach the speed of light, you can’t decrease entropy). This is structural. The uniform state isn’t in the space of states. It’s not that the system won’t get there—it’s that “there” doesn’t exist. The comfortable mathematical object we’ve been manipulating (constant solutions to PDEs) simply breaks when you remove boundaries. Not approximately, not in some limit—it categorically fails to be what we thought it was. The constants aren’t approximately varying or practically unmeasurable in their uniformity. They cannot be uniform. The mathematics forbids it, and physics must respect what mathematics forbids.
The sophistication isn’t in my proof—it’s in recognizing that a proof is necessary. For centuries, physicists treated “the constants are constant” as so obvious it needed no justification. The insight is seeing that this “obvious” claim is a substantial mathematical assertion about field configurations on unbounded spaces, and when you actually examine it formally, it fails immediately.
Note: This argument applies to spatially infinite universes. For finite universe models with compact topology, different mathematical structures apply.