c·t = CONST, THAT CLICKS EPISODE 42 ／ Part V — where the tool breaks
The one that does not move cannot be measured
A horizon no conformal
transformation can remove
The event horizon has weight 0 — it is fixed by causal structure alone.
But the apparent horizon moves. And inside, the contents still cannot be read.
Last time the end state turned out to be Step 3, where the tool does not reach. This time we go inside it — the interior of a black hole. The conclusion first: the event horizon has weight 0, so a conformal transformation can neither create nor destroy it. But the apparent horizon does move. And once inside, the \(1.5\times10^{77}\) bits still cannot be read.
01Can a conformal transformation remove a horizon?
02But the apparent horizon is not conformally invariant
\(\theta=0\) is not preserved — the apparent horizon moves
| Horizon | Defined by | Weight | Conformal transformation |
|---|---|---|---|
| Event horizon | causal structure (global) | \(0\) | does not move |
| Apparent horizon | \(\theta=0\) (local) | \(\ne0\) | moves |
Conclusion of §02
The difference between the two horizons is exactly a difference of weight.
But locating the event horizon requires knowing the entire future —
the one that does not move cannot be measured, and the one that can be measured moves.
03Penrose diagrams — where notation does real work
Conclusion of §03
It is the twin of Episode 1's claim that "\(c\cdot t=\)const is a notation, not a model" —
the Penrose diagram is not a model either. And it is one of the most useful tools in relativity.
── Notation is not worthless. It is simply not a claim.
04The core — even inside, the contents cannot be read
| Black hole | tide at the horizon [g] | horizon → singularity [s] |
|---|---|---|
| Solar mass | \(2.10\times10^{9}\) | \(1.55\times10^{-5}\) |
| Sgr A\(^*\) (\(4.3\times10^6M_\odot\)) | \(1.14\times10^{-4}\) | \(66.6\) |
| M87\(^*\) (\(6.5\times10^9M_\odot\)) | \(4.97\times10^{-11}\) | \(1.01\times10^{5}\) |
At solar mass it is \(2\times10^9\) g — you are torn apart atom by atom. But at M87\(^*\) it is \(5\times10^{-11}\) g and you feel nothing at all. The horizon is not a place where something happens — at a large black hole you would not notice crossing it.
| Black hole | readable [bits] | the hole's \(S\) [bits] | shortfall [bits] |
|---|---|---|---|
| Solar mass | \(8.37\times10^{48}\) | \(1.51\times10^{77}\) | \(93.9\) |
| Sgr A\(^*\) | \(3.60\times10^{55}\) | \(2.80\times10^{90}\) | \(115.9\) |
| M87\(^*\) | \(5.44\times10^{58}\) | \(6.40\times10^{96}\) | \(126.5\) |
The main point of this episode
In every case it falls short by orders of magnitude.
── Long before the question of whether information is "lost", there is no time to extract it.
The bigger the hole, the longer you have inside — and the larger the shortfall (93.9 → 126.5 bits).
Figure: as the mass varies, the tide at the horizon (whether you feel it) and the shortfall in information you could read inside. Move the slider — the bigger it is, the less it hurts, and the less you can read.
05The evaporation clock
| Black hole | \(t_{\rm evap}\) [yr] | × the age of the universe |
|---|---|---|
| Solar mass | \(2.10\times10^{67}\) | \(1.5\times10^{57}\) |
| M87\(^*\) | \(5.76\times10^{96}\) | \(4.2\times10^{86}\) |
| Mass finishing evaporation now | \(M=1.73\times10^{11}\) kg (one mountain's worth) | |
Anything of stellar mass or above takes at least \(10^{57}\) times the age of the universe — effectively, it does not evaporate. The mass range finishing now (\(10^{11}\) kg) is exactly where primordial black holes are being searched for.
06The Page time — if information comes out, when?
On the Page curve, the entanglement entropy of the radiation turns over when it equals the black hole's entropy. The frequently quoted "0.54" comes from a different convention; here we took the moment the area halves. Either way it is the middle of the evaporation — if information comes out, it comes out after that. For a solar mass, \(1.4\times10^{67}\) years from now.
07Checking against the weight table
| Quantity | Weight | Why |
|---|---|---|
| Whether an event horizon exists | \(0\) | fixed by causal structure alone |
| \(S=A/4\ell_P^2\) | \(0\) | Episode 40 §01 |
| The tide measured in \(g\) | \(0\) | acceleration ÷ acceleration |
| \(t_{\rm evap}/t_0\) | \(0\) | time ÷ time |
| The Page fraction | \(0\) | time ÷ time |
| The location of the apparent horizon | \(\ne0\) | the one exception |
(1) §01's "conformal transformations preserve causal structure" has a condition. \(\Omega\) must be positive and smooth everywhere; where \(\Omega\) goes to zero or diverges — precisely the boundary a Penrose diagram uses to bring infinity in — the story changes. Handling that boundary is the subtlest part of conformal geometry, and both Episode 31's CCC and Episode 41's hypothesis were demands made at exactly such a place.
(2) §04's "readable information" is an upper bound on an upper bound. Episode 24's channel capacity (the Margolus–Levitin limit) is a speed limit on ideal quantum computation, not a claim that a real observer could take information in at that rate — the conclusion is strengthened by the fact that even this most generous estimate falls short, but the numbers themselves are indicative.
(3) §04's tides and infall times are Schwarzschild values. Real black holes rotate (Episode 39) and have a different interior — read these as order-of-magnitude calculations. They are also the maximum proper time for free fall from rest; other trajectories are shorter.
(4) §06's Page curve is not fully settled. Recent "island" computations reproduce the curve, but the mechanism by which information comes out is unresolved. This document treats only the timing of the turnover and makes no claim about whether information actually emerges.
(5) §05's evaporation times assume nothing falls in. In fact the CMB at 2.7 K is hotter than a stellar-mass hole's Hawking temperature (around \(10^{-8}\) K), so in today's universe these holes do not evaporate at all — they absorb and grow. Evaporation begins only once the universe is far colder.
Exercises
- Can a conformal transformation remove an event horizon? Why?
Show the answer
No. Conformal transformations do not move light cones, and the event horizon is fixed by causal structure alone ("the boundary of the causal past of future null infinity") — so it has weight 0. Per caveat (1), this requires \(\Omega\) to be positive and smooth everywhere. - What about the apparent horizon?
Show the answer
It moves. It is fixed by the local condition \(\theta=0\), and under a conformal transformation \(\theta\to\Omega^{-1}(\theta+2l^\mu\partial_\mu\ln\Omega)\), so \(\theta=0\) is not preserved. ── The one that does not move cannot be located without knowing the whole future; the one that can be located moves. - How much does crossing M87\(^*\)'s horizon hurt?
Show the answer
Not at all — \(5\times10^{-11}\) g. At solar mass it is \(2\times10^9\) g and you are torn apart atom by atom, but the tide falls as \(1/M^2\). The horizon is not a place where something happens. - Could you read a black hole's information by going inside?
Show the answer
No. Even on Episode 24's most generous channel-capacity estimate you fall short by 93.9 bits at solar mass and 126.5 bits at M87\(^*\) — long before the question of whether information is "lost", there is no time to extract it. And the bigger the hole, the longer you have inside and the larger the shortfall. - (Harder) What do Penrose diagrams and \(c\cdot t=\)const have in common?
Show the answer
Both are notation, not models. A Penrose diagram is a conformal transformation: distances on the page change (bookkeeping), the tilt of the light cones does not (physics). And it is one of the most useful tools in relativity — notation is not worthless; it is simply not a claim.
Summary: the one that does not move cannot be measured
\(g\to\Omega^2g\) does not move the light cones, and the event horizon is fixed by causal structure alone — so the event horizon has weight 0 and can be neither created nor destroyed. That is why this series' tool cannot make a black hole go away.
But the apparent horizon moves: it is fixed by the local condition \(\theta=0\), and \(\theta\to\Omega^{-1}(\theta+2l^\mu\partial_\mu\ln\Omega)\). The difference between the two horizons is exactly a difference of weight — and the one that does not move cannot be located without knowing the whole future, while the one that can be located moves.
A Penrose diagram is a conformal transformation. Distances on the page change (bookkeeping); the tilt of the light cones does not (physics) — the twin of Episode 1's "\(c\cdot t=\)const is a notation, not a model", and one of the most useful tools in relativity. Notation is not worthless. It is simply not a claim.
Inside: at solar mass the tide at the horizon is \(2\times10^9\) g and tears you apart atom by atom, but at M87\(^*\) it is \(5\times10^{-11}\) g and you feel nothing — the horizon is not a place where something happens. Can you read the information there? On the most generous estimate you still fall short by 93.9 bits at solar mass and 126.5 at M87\(^*\). Long before information can be "lost", there is no time to extract it — and the bigger the hole, the longer you have and the larger the shortfall.
Waiting outside is no better: a stellar-mass hole takes at least \(10^{57}\) times the age of the universe to evaporate, and if information emerges it does so after the Page time (65 per cent of the way) — \(1.4\times10^{67}\) years from now for a solar mass. Everything about black holes sat in the weight-0 column, except the apparent horizon.
This document is Episode 42 of "c·t = const, That Clicks" (the sixth of Part V), written for physics-minded high-school and university readers. That conformal transformations preserve causal structure, the different characters of event and apparent horizons, Penrose diagrams, Hawking evaporation and the Page curve are all standard, and nothing here is a new claim — the numbers are computed in kenshou/calc46.py. §01's statement requires \(\Omega\) to be positive and smooth everywhere; where \(\Omega\) vanishes or diverges — precisely the boundary a Penrose diagram uses — the story changes, and handling that boundary is the subtlest part of conformal geometry. §04's "readable information" is an upper bound on an upper bound: the Margolus–Levitin limit is a speed limit on ideal quantum computation, not a claim about real observers — the conclusion is strengthened by even this generous estimate falling short, but the numbers are indicative. §04's tides and infall times are Schwarzschild values (real black holes rotate) and are the maximum proper time for free fall from rest. §06's Page curve is not fully settled; recent "island" computations reproduce it but the mechanism by which information emerges is unresolved — this document treats only the timing and makes no claim about whether information actually comes out. §05's evaporation times assume nothing falls in; in fact the 2.7 K CMB is hotter than a stellar-mass hole's Hawking temperature, so in today's universe such holes absorb and grow rather than evaporate. ── To make a PDF, use your browser's Print dialogue (sliders freeze and answers are hidden in the print version).