Thinking in Waves Episode 46 / Part XVI — Horizons, as Waves
A machine for lowering frequency exponentially
Black Hole Evaporation —
The information did not vanish; in 1.5 milliseconds it went out of range
The Hawking temperature is Episode 45's Unruh temperature with \(a\) replaced by the surface gravity \(\kappa\). Nothing more.
★ The entropy is area ÷ Planck area (\(1.05\times10^{77}\) for a solar mass) — 19 orders of magnitude above a star of the same mass.
★★ And near the horizon \(\omega(t)=\omega_0e^{-\kappa t/c}\) — for a solar mass, 1.5 milliseconds takes visible light down to a frequency whose single period outlasts the age of the universe.
Episode 45 reached "discard what is beyond a horizon and heat appears."
★ Now that horizon evaporates by itself — and the question arises: where did what fell in go?
01The Hawking temperature is the Unruh temperature
| Mass | \(r_s\) [m] | \(\kappa\) [m/s²] | ★ \(T\) [K] |
|---|---|---|---|
| solar mass | 2.9540e+3 | 1.5212e+13 | 6.1687e−8 |
| \(10^6\) solar masses | 2.9540e+9 | 1.5212e+7 | 6.1687e−14 |
| Earth mass | 8.8701e−3 | 5.0662e+18 | 2.0544e−2 |
| \(10^{12}\) kg | 1.4852e−15 | 3.0256e+31 | ★ 1.2269e+11 |
The point of §01
★ Only \(a\to\kappa\) — Episode 45 already established that heat is logarithmic time, so there is nothing new to say here. Where there is a horizon, time near it goes logarithmic.
★★ The mass in equilibrium with \(T=2.725\) K (the CMB) is \(4.50\times10^{22}\) kg (0.61 lunar masses) — anything heavier does not evaporate at all; it grows by absorbing the background.
★ So real stellar black holes will not begin evaporating until the universe is much colder.
02The entropy is area — Episode 39's holography returns
| Mass | area \(A\) [m²] | \(A/(4\ell_p^2)\) | \(4\pi GM^2/(\hbar c)\) |
|---|---|---|---|
| solar mass | 1.0966e+8 | 1.0494e+77 | 1.0494e+77 |
| Earth mass | 9.8871e−4 | 9.4621e+65 | 9.4621e+65 |
| \(10^{12}\) kg | 2.7720e−29 | 2.6529e+40 | 2.6529e+40 |
The point of §02
★ "Area ÷ Planck area" is the number of degrees of freedom. The same counting as Episode 39's \((L/\varepsilon)^{d-1}\) — and here the horizon fixes that \(\varepsilon\).
★★ And for the same mass, becoming a black hole raises the entropy by 19 orders (\(\sim10^{58}\) for the Sun's thermal entropy against \(1.05\times10^{77}\)).
★ "Gravity is the most disordered configuration" — this feeds straight into Episode 21's account of information: the cheapest place to throw things away is gravity.
03Evaporation — negative heat capacity makes the ending accelerate
| Mass | \(t_{\rm evap}\) [yr] | relative to the age of the universe |
|---|---|---|
| solar mass | 2.0972e+67 | 1.52e+57 |
| Earth mass | 5.6778e+50 | 4.11e+40 |
| \(10^{11}\) kg | 2.6655e+9 | 0.193 |
| \(10^{8}\) kg | 2.6655 | 1.93e−10 |
★ The mass that evaporates in exactly the age of the universe is \(1.73\times10^{11}\) kg (with \(r_s=2.6\times10^{-16}\) m and \(T=7.1\times10^{11}\) K at that point) — if such objects formed in the early universe, their final flash should be visible now.
| Time remaining | Mass left [kg] | \(T\) [K] | \(r_s\) [m] |
|---|---|---|---|
| half | 7.9370e+11 | 1.5458e+11 | 1.18e−15 |
| \(10^7\) s | 4.9171e+7 | 2.4952e+15 | 7.30e−20 |
| 1 s | 2.2823e+5 | 5.3757e+17 | 3.39e−22 |
The point of §03
★ The last second releases \(2.05\times10^{22}\) J — about \(3.3\times10^8\) Hiroshima bombs.
★★ Because the heat capacity is negative (lighter means hotter), it must run away and end.
★ Ordinary objects stop when they cool; this is the opposite — close to what this series called "negative \(\mathrm{Re}\,Z\)" (Episodes 17–18), except that nothing is driving it from outside; it is a property of gravity itself.
04The Page curve — up, then down
| \(t/T\) | \(M/M_0\) | emitted | remaining | ★ \(S_{\rm rad}\) |
|---|---|---|---|---|
| 0.1 | 0.965489 | 0.067830 | 0.932170 | 0.067830 |
| 0.5 | 0.793701 | 0.370039 | 0.629961 | 0.370039 |
| 0.6464 | 0.707138 | 0.499956 | 0.500044 | ★ 0.499956 (peak) |
| 0.8 | 0.584804 | 0.658005 | 0.341995 | 0.341995 |
| 1.0 | 0 | 1.000000 | 0 | ★ 0 |
The point of §04
★ The Page time is where \(S_{\rm BH}(M)=S_{\rm BH}(M_0)/2\), i.e. \(M=M_0/\sqrt2\). In time, \(t/T=1-2^{-3/2}=0.646447\) — 64.6 % of the lifetime.
★★ It rises to there and falls after. Returning to 0 at the end would mean the information came out.
★ But this is a model. It is the shape one gets by assuming that the whole evaporation is unitary, and this article has not derived it.
05★★ Where did the information go — read it as redshift
| Mass | \(\kappa/c\) [1/s] | one e-fold [s] | ★ 74.5 e-folds [s] |
|---|---|---|---|
| solar mass | 5.0743e+4 | 1.9707e−5 | ★ 1.4662e−3 |
| \(10^6\) solar masses | 5.0743e−2 | 1.9707e+1 | 1.4662e+3 |
| Earth mass | 1.6899e+10 | 5.9175e−11 | 4.4026e−9 |
The point of §05 — ★★★ the core of this episode
Going from visible light (\(5\times10^{14}\) Hz) to the inverse age of the universe (\(2.3\times10^{-18}\) Hz) takes 74.5 e-folds.
★★★ For a solar mass, that is \(1.47\times10^{-3}\) seconds.
In 1.5 milliseconds, the information in visible light falls to a frequency
whose single period outlasts the age of the universe.
★ Episode 5 wrote that "a period longer than the age of the universe is indistinguishable from a constant." What we are watching here is information being swept into exactly that region.
★★ Not "the information vanished" but "the information was moved out of the measurable band" — that is how wave language reads it.
06"No hair," read as a filter
| Before falling in | After | In wave language |
|---|---|---|
| a book (\(10^7\) bits) | the mass went up | ★ an extreme low-pass |
| a rock of the same mass | indistinguishable | ★ the same output |
| something charged | the charge goes up too | three components pass |
The point of §06
★ "No hair" does not mean the information was destroyed; it means only three components survive in the outside observer's band.
★★ As §05 showed, the rest was merely moved to exponentially lower frequency. Whether it is in principle recoverable is not settled here.
★ "Cannot be measured" and "does not exist" are different — the same caution as Episode 38's exercises. Here too, what can be said reaches only as far as the limit of measurement.
07What stays unresolved
| Question | Verdict | Basis |
|---|---|---|
| The Hawking temperature itself | ◎ one line | same as Episode 45's Unruh |
| Why entropy is area | △ only counted | the same "translation" as Ep. 39 |
| ★ Whether the Page curve really comes down | △ model assumption | ★ follows from assuming unitarity |
| ★ Whether the information is recoverable | × unresolved | ★ redshifted ≠ unrecoverable |
| ★ What is left when evaporation ends | × no | ★ a quantum-gravity question (Ep. 38) |
The point of §07 — ★ honestly
This episode is translation too. The information paradox is not solved.
★ What could be said is — not "it vanished" but "it went somewhere unmeasurable." The one contribution here is turning "how fast" into the number 1.5 milliseconds.
★★ As decided in Episode 43, organizing is not discovering.
Exercises
- Is §05's "1.5 milliseconds" evidence that information was lost?
Show answer
No — it is an estimate of how hard it is to measure. Exponential reddening never makes the amplitude zero; mathematically the information is still there.
★ The issue is that observing even one period would take longer than the age of the universe, which is impracticable, not impossible.
★★ In practice, though, an exponentially shrinking signal eventually falls below one photon — past that point "cannot be measured" approaches "does not exist" arbitrarily closely. ★ Episode 9's bandwidth theorem and Episode 21's limits on information take over there. - §02 found a black hole's entropy 19 orders above a star's. What does that mean?
Show answer
That gravity possesses the degrees of freedom to be maximally disordered. At the same mass and volume, the gravitationally collapsed state has overwhelmingly more microstates.
★ By the second law, matter left alone wants to become a black hole — and it does (gravity is the one force that grows more disordered as it clumps).
★★ And since \(S\propto M^2\), merging two of them quadruples rather than doubles \(S\) — mergers always increase entropy. ★ Episode 21 said information has a price; the cheapest place to dump it is a horizon. - §01 said black holes heavier than \(4.5\times10^{22}\) kg grow. What does that mean for the universe today?
Show answer
That stellar black holes are not evaporating now. Against their \(6\times10^{-8}\) K, the surroundings are at 2.725 K — absorption wins by an enormous margin.
★ Evaporation can only begin once the universe has cooled to \(6\times10^{-8}\) K, which requires a further redshift of \(\sim10^{8}\) — some \(10^{17}\) years away at least.
★★ That is negligible next to §03's \(10^{67}\)-year lifetime, but "not evaporating now" matters — the only thing worth searching for observationally is the small \(\sim10^{11}\) kg objects from the early universe.
Summary — a horizon is a machine for lowering frequency exponentially
The Hawking temperature is Episode 45's Unruh temperature with \(a\) replaced by the surface gravity \(\kappa=c^4/4GM\) — where there is a horizon, time near it goes logarithmic. That is all. ★ The mass in equilibrium with 2.725 K is \(4.50\times10^{22}\) kg (0.61 lunar masses), so everything heavier is still growing.
The entropy is area ÷ Planck area (\(1.05\times10^{77}\) for a solar mass, by two agreeing calculations) — Episode 39's holography returns, now with a concrete \(\varepsilon\): the horizon. ★ 19 orders above a star of the same mass — gravity is the most disordered configuration.
The evaporation time goes as \(M^3\), and exactly the age of the universe corresponds to \(1.73\times10^{11}\) kg. Negative heat capacity accelerates the ending: the last second releases \(2.05\times10^{22}\) J (some \(3.3\times10^8\) Hiroshima bombs).
The Page curve peaks at \(M=M_0/\sqrt2\), i.e. at 64.6 % of the lifetime (\(1-2^{-3/2}\)) — ★ but that is a model assuming unitarity, not something derived here.
★★★ The core — near the horizon, \(\omega(t)=\omega_0e^{-\kappa t/c}\).
Visible light needs 74.5 e-folds to fall below the inverse age of the universe,
which for a solar mass is a mere 1.5 milliseconds.
★ Episode 5 wrote that "a period longer than the age of the universe is indistinguishable from a constant." A horizon is the machine that sweeps information into that region in 1.5 milliseconds. "No hair" says the same thing differently — nothing vanished; only three components survive in the outside band.
★ Honestly: this episode is translation. The information paradox is not solved, and the gap between "unmeasurable" and "non-existent" is not closed. ★ The one contribution is putting a number on how fast it becomes unmeasurable: 1.5 milliseconds.
★ Next time the wave becomes its own medium — the universe itself, as a wave.
★★ Inflation is exactly Episode 44's time modulation (the scale factor plays the role of \(\varepsilon(t)\)), and we read why the fluctuation spectrum comes out at \(n_s\approx0.965\) — "almost 1 but not 1" — in the language of order.
This document is Episode 46 of the "Thinking in Waves" series. Computations are in kenshou/bh.js. On the numbers here: the Hawking temperatures, Schwarzschild radii, surface gravities, the two entropy calculations, evaporation times, the end-stage masses and temperatures, the Page time, and the redshift e-folding times were all computed here. Hawking radiation and \(T=\hbar\kappa/2\pi ck_B\), the Bekenstein–Hawking entropy \(S=A/4\ell_p^2\), the \(M^3\) evaporation time, negative heat capacity, the no-hair theorem, the Page curve and the Page time (\(1-2^{-3/2}\) of the lifetime), and the exponential redshift near a horizon are all well-known results and none are discoveries of this article — what this article did was re-shelve them under Episode 45's "logarithmic time" and Episode 5's "anything longer than the age of the universe is indistinguishable from a constant," and put the number 1.5 milliseconds on the time it takes information to leave the measurable band. The information paradox is not solved. The Page curve of §04 is a coarse-grained model conditional on assuming that the whole evaporation is unitary; it is not derived here, and the actual shape is model-dependent. The "Sun's thermal entropy \(\sim10^{58}k_B\)" in §02 is an approximate literature figure. The evaporation time in §03 is the standard expression for a single massless species and in reality depends on how many particle species are emitted. The \(\omega(t)=\omega_0e^{-\kappa t/c}\) of §05 is the standard near-horizon approximation for outgoing modes; what a distant observer actually receives depends on the details of the radiation. — To print, use your browser's Print and "Save as PDF."