RST Rule — Time–Temperature Non-Inversion

Boxed RST Rule for the book

RST Rule — Time–Temperature Non-Inversion

Time is an operational rate. Temperature is a rate of state exploration per unit local operational time. They are correlated because both depend on substrate state, but neither causes the other.

Admissible dependency (locked):

substrate state → local clock rate → transition rate → temperature

Inadmissible inversion:

temperature → time

(or any claim that heat “generates” time, or that cooling “halts” time).

Interpretive constraint: Any argument that requires global reversibility, environment-independent clocks, or unlimited coherence bandwidth in order to make temperature or unitarity “fundamental” is physically inadmissible under RST.

Placement note: This can be used as a boxed rule in Chapter 2 or as a recurring “constraint reminder” at the start of Chapters 5–6.

Concrete case walkthrough (step-by-step): Near-horizon equilibrium

Near-horizon is the sharpest test because it forces local time, global coordination, and temperature to remain distinct (and prevents accidental inversion).

Setup (what we assume, RST-consistent)

• We have a static gravitational field (e.g., outside a non-rotating black hole; same logic for any static potential).
• We consider “static observers” hovering at different radii r.
• Each observer has:
  • a local clock (operational time),
  • a local thermometer (temperature defined by local state exploration per local time),
  • access only to local measurements.
• RST stance: keep GR’s metric structure unchanged, but treat it as descriptive of organized substrate stress (not a thing that “makes time flow”).

Step 1 — GR gives a redshift factor (coordination tool)

In a static spacetime, you can write:

dτ = √(−g_tt(r)) · dt

• t = global coordination parameter (bookkeeping)
• τ = local proper time (what a physical clock measures)
• As r approaches the horizon, √(−g_tt(r)) decreases.

RST interpretation: this is not “time slowing.” It is local operational rate suppression because the substrate is more stressed/saturated closer to the horizon.

Step 2 — Local clock rate changes (physical time is local)

Take two observers:

• A far away: r = r_∞
• B closer in: r = r_B

They compare rates via coordination:

dτ_B/dt = √(−g_tt(r_B))
dτ_∞/dt = √(−g_tt(r_∞)) ≈ 1

So B’s local clock ticks more slowly per unit coordination time.

RST reading: Clock rate is downstream of substrate state:
substrate stress near horizon → reduced local transition rate → reduced clock rate
No “flow of time” entity needed.

Step 3 — Temperature is defined per unit local proper time

Temperature in RST is not “energy alone.” It is the rate of microstate exploration per unit local operational time. So observer B defines T_B using τ_B, and observer ∞ defines T_∞ using τ_∞.

Important: Both are valid, but they are not the same quantity unless you specify the frame/time normalization.

Step 4 — What equilibrium means in RST

Equilibrium is often naively stated as “temperature is uniform.” RST says that’s only true in trivial cases. In a gravitational field, equilibrium means:

State exploration per unit local proper time is consistent across the stationary configuration once you correctly account for local clock rate differences.

This reproduces the Tolman–Ehrenfest relation:

T(r) · √(−g_tt(r)) = constant

Equivalently:

T(r) = T_∞ / √(−g_tt(r))

So local temperature increases as you hover closer to the horizon.

Step 5 — Why this does not mean temperature causes time

The trap is to see T(r) rise while √(−g_tt(r)) falls and conclude: “Higher temperature is associated with slower time, so temperature causes time dilation.”

RST blocks that inversion.

What’s actually happening is:

1. Substrate stress increases closer to the horizon.
2. That stress suppresses coherent operational rates (clock rate).
3. The same stress structure sets local conditions for thermal sampling.
4. To maintain equilibrium across a redshift gradient, local temperature must scale accordingly.

So both effects share a common cause: substrate state (encoded descriptively by g_tt(r)). Temperature is compensating for rate differences to preserve equilibrium; it is not generating those rate differences.

Step 6 — What “Hawking temperature” becomes under RST (interpretive constraint only)

Standard semiclassical physics gives a Hawking temperature for a black hole as seen from infinity, T_H. RST does not change that derivation, but interprets the near-horizon region as boundary-limited:

• substrate response approaches saturation,
• coherence becomes exhaustible,
• effective field descriptions remain valid up to a boundary.

The thermal character near the horizon is read as finite response under extreme stress and an equilibrium relation that must respect local operational time, not global reversibility fantasies.

Step 7 — What RST forbids you from saying here

• “Heat generates time.” ❌
• “Time dilation is caused by temperature.” ❌
• “Temperature is more fundamental than proper time.” ❌
• “Global unitarity must be preserved through horizon saturation.” ❌
• “Equilibrium requires a single global temperature.” ❌

One-sentence closure (reusable)

Near a horizon, clock rate and temperature co-vary because both track substrate stress; equilibrium is the invariance of state-sampling per local proper time under redshift, not a global uniform temperature.

RST is a Slide Ruler.

Not an engine, not a detector — a discipline tool.

“Does anyone actually think temperature generates time?”

The careful answer is: Almost no one says it that bluntly — but many influential ideas quietly rely on exactly that inversion.

Physicists are generally too careful to write a paper titled "Temperature causes time to exist." However, the landscape is littered with subtle interpretive moves where time is treated as emergent from thermal structure, or where the absence of thermal activity is treated as the absence of time itself.

Where the Inversion Hides

1. “Thermal Time” (Connes–Rovelli)

Popular interpretations of thermal time proposals often suggest time emerges from the thermodynamic state. RST does not reject the math; it rejects the inversion. Time is a clock-normalized operational parameter used to define thermal averages—it cannot be produced by them.

2. Entropic-Time Rhetoric

Claims like “Time is just entropy increase” promote entropy into a generator. RST’s response is sharp: Entropy requires time normalization. A quantity defined per unit time cannot be the source of time.

3. Cosmological Popularizations

Phrases like “Cooling sets the pace of cosmic time” hardens shorthand into false ontology. In RST, cooling tracks changing interaction rates measured by local clocks—there is no global clock for temperature to "pace."

4. Unruh/Hawking Misunderstandings

Because acceleration produces a thermal response, the temptation is to assume heat is responsible for time dilation. RST classifies this inference as natural, tempting, and wrong.

Why the "Slide Ruler" Metaphor Matters

Physics doesn’t break from explicit, loud claims—it breaks from unnoticed defaults. When a framework allows the sentence “Temperature is more fundamental than time in this regime” to stand, clocks become mere bookkeeping devices and reversibility sneaks back in where it doesn't belong.

The Book-Safe Synthesis:

While few physicists explicitly claim that temperature generates time, many common interpretive practices implicitly reverse the dependency by treating thermal or entropic structure as more fundamental than operational clocks; Reactive Substrate Theory exists precisely to prevent that inversion from silently hardening into ontology.

RST asks: Does the math still slide smoothly? Do the markings stay aligned?
Acceleration and cosmology are where most frameworks flip the numerator and denominator.
RST doesn't.