Black Holes That ClickEpisode 1 ／ The Horizon ── the surface information can't leave

Toward the diagonal of the physics "cube" ── starting with the one-way membrane that not even light can leave

The Horizon ── the surface information can't leave The radius at which the escape velocity reaches the speed of light c is the event horizon. Seen from outside, time freezes there (a frozen star).
But for the one who falls in, they slip right through without noticing a thing ── "stopping" depends on the observer. A one-way membrane for information.

Tools you'll need: escape velocity, Relativity That Clicks #5 & #6 (simultaneity, gravitational time dilation) This episode's ratio: R_s/r ＝ closeness to the horizon

This series heads for the diagonal of the physics "cube" ── the corner where c, ℏ, and G all matter at once, which is quantum gravity ── using black holes as our window. There, physics eventually turns entirely into "dimensionless ratios of information (bits)." But before that, let's begin with a black hole's single most defining feature ── not even light can leave. Gravity is so strong that there is a region where the escape velocity exceeds the speed of light $c$. The boundary of that region is the event horizon. It is not a physical wall but a one-way membrane for information. You can go in, but you cannot come out (cannot send word to the outside). And here is the fascinating part ── seen from outside, time freezes at the horizon, yet the one falling in slips through feeling nothing. Whether they are "stopped" or "not stopped" disagrees between observers. "Simultaneity depends on the observer," from Relativity That Clicks, shows itself here in its most extreme form.

01Not even light can leave ── the surface where the escape velocity equals c

For a rocket to escape Earth, it needs a certain speed (the escape velocity). The heavier and smaller a star is, the larger its escape velocity. So then ── what if we keep compressing a star smaller and smaller? The escape velocity keeps climbing, until finally it reaches the speed of light $c$. Past that, not even light can break free. The radius at which "escape velocity = c" is the event horizon.

LET'S TRY IT ── set the escape velocity equal to c

Put v = c into the escape-velocity formula

$$v_{\text{esc}}=\sqrt{\frac{2GM}{R}}=c\ \Rightarrow\ R_s=\frac{2GM}{c^2}$$

This $R_s$ is the Schwarzschild radius (the radius of the horizon). For the Sun it is about 3 km, for the Earth about 9 mm ── crush it that small and it becomes a black hole. Notice that the formula needs both $G$ (gravity) and $c$ (relativity). The horizon lives on the "c–G face" of the cube.

This episode's ratio is closeness to the horizon $R_s/r$ (or "compactness" $2GM/rc^2$). At the Sun's surface $R_s/R\sim4\times10^{-6}$ (very diffuse), for a neutron star $\sim0.3$, and exactly at the horizon $R_s/r=1$. The closer this ratio gets to 1, the more extreme the relativistic effects become.

AN HONEST ASIDE ── why Newton's calculation happens to be right The $R_s=2GM/c^2$ above is a "cheat": we forced $v=c$ into the Newtonian escape velocity. Newtonian mechanics really doesn't apply to light. And yet, when you solve it properly with Einstein's general relativity, the radius of the horizon comes out to be exactly the same $2GM/c^2$ (matching right down to the factor of 2). It looks like a coincidence, but it's famous, and it's handy. So "escape velocity = c" is a good doorway to the correct answer (though the honest line is that the real reasoning is relativity).

02The frozen star ── seen from outside, time freezes at the surface

Near the horizon, the gravitational time dilation from Relativity That Clicks #6 becomes extreme. The stronger gravity is, the more slowly clocks run ── and at the horizon, seen by an outside observer, time's advance goes to zero. An object falling in slows down more and more as it nears the horizon, redshifts and dims, and finally appears to stop, frozen on the surface. That's why black holes were once called frozen stars.

Time dilation seen from outside (infinite at the horizon)

$$\frac{\text{outside time}}{\text{their own time}}=\frac{1}{\sqrt{1-R_s/r}}\ \xrightarrow[r\to R_s]{}\ \infty$$

At the horizon $r=R_s$, this factor diverges to infinity. One second of their time is stretched out infinitely on the outside ── so "time (and calculation) appears to stop at the surface." Light is redshifted by the same factor, growing infinitely red and dim until it fades away.

03Play with it ── a clock falling to the horizon

The figure below. A clock falls toward the horizon (the black band at the bottom). The left column is the ticking of the falling clock itself (evenly spaced ── for the one falling, time runs normally). The right column is the interval at which the light of those ticks reaches a distant observer ── the closer to the horizon, the more it is stretched out by a factor of $1/\sqrt{1-R_s/r}$ and reddened. Use the slider to bring the clock's position $r/R_s$ closer to the horizon.

The ticks of their own clock don't change, yet the ticks arriving outside grow more and more spread out, and at $r\to R_s$ they are stretched infinitely and freeze. The outside observer can never see the moment the clock reaches the horizon (a frozen star). But ── by their own clock, the one falling reaches the horizon in a finite time. That's the setup for the next section.

Figure: a clock falling to the horizon. Left = ticks of their own clock (evenly spaced), right = ticks arriving at the outside observer (stretched by 1/√(1−R_s/r) plus redshift). As r→R_s they stretch infinitely and freeze at the surface (frozen star)

Clock position r / R_s (horizon ＝ 1) 3.00

their own clock (evenly spaced) ticks arriving outside (stretched, redshifted)

04But the one falling slips through ── "stopping" depends on the observer

Here is this series' first "reveal." "Time stops at the horizon" is only the view from outside. For the one falling in, the horizon is a perfectly ordinary place ── no special wall, no pain, no sensation of a "boundary," and they simply pass through it in a finite time (the equivalence principle from Relativity That Clicks #6: "no drama"). That's because the horizon is not a thing you can stand on and touch, but a causal boundary line meaning "light from beyond here can never get out again."

The two viewpoints are compatible (not a contradiction)

・Outside observer: the clock freezes at the horizon and appears never to reach it (a frozen star).
・The one falling in: passes through the horizon in a finite proper time, feeling nothing.
Both are correct. That "stopped / not stopped" disagrees between observers is the gravitational version of "simultaneity depends on the observer" from Relativity That Clicks #5.

And the most important property of the horizon ── it is a one-way membrane for information. Light inside cannot get out (it is causally cut off). Matter, light, and information can go in but cannot come out to tell of it. So where does that information go? Does it vanish? Does it stick to the surface? ── This question is the backbone that pulls the whole series along to the very end (Episode 6, the information paradox). The horizon is the doorway into quantum gravity.

◇ ◇ ◇

THE HONEST LINE ── what's established, and what to watch, around the horizon

The Schwarzschild radius $R_s=2GM/c^2$ (the same coefficient even in the exact general-relativistic solution); the horizon being a causal boundary that not even light can escape; the gravitational redshift and time dilation that become infinite in external coordinates (the frozen star); and a falling observer passing through the horizon in a finite proper time with nothing locally singular there (the equivalence principle, no drama) ── all of these are established general relativity. The real existence of black holes is likewise established, through gravitational waves (LIGO/Virgo, 2015 onward), the Event Horizon Telescope images (M87* and Sagittarius A*, 2019/2022), and the orbits of stars at the galactic center (S2).

Caveats. ① The Newtonian "escape velocity = c" derivation only happens to give the right answer; the correct reasoning is relativity (Newtonian mechanics can't be applied to light). ② The "frozen star" is how it looks in external coordinates; the horizon is only an apparent singularity of the coordinates, and spacetime itself is smooth at the horizon (the real singularity is at the center). ③ This episode is about the outside of the horizon. The interior, the singularity, and tidal forces (spaghettification) are not covered. ④ For rotating or charged black holes (the Kerr solution, etc.) the horizon is a bit more complicated, but the essence is the same.

EXERCISES (solvable with just this episode's formulas)

If you double the mass, what happens to the Schwarzschild radius $R_s=2GM/c^2$? Given the Sun (about 3 km), what about ten Suns' worth?
See the answer
Since $R_s\propto M$, it doubles. Ten Suns' worth would be about 30 km. The horizon grows in proportion to the mass (in Episode 2, area ∝ M² comes into play).
When a clock is at $r=1.01\,R_s$ (just outside the horizon), roughly what is the time dilation $1/\sqrt{1-R_s/r}$ seen from outside?
See the answer
$1/\sqrt{1-1/1.01}=1/\sqrt{0.0099}\approx10$ times. One second of their time is about 10 seconds outside. It diverges as you approach the horizon, becoming infinite at $r\to R_s$ (frozen).
Why is it no contradiction that "time stops at the horizon" and "the one falling passes through in a finite time"?
See the answer
"Stops" is how it looks in the outside observer's coordinates; "passes through in a finite time" is in terms of the faller's own proper time. The two are descriptions by different observers ── just different viewpoints (the gravitational version of Relativity That Clicks #5). Both are correct.
Why is the horizon called a "one-way membrane for information"?
See the answer
From inside the horizon not even light can get out (it is causally cut off), so information inside cannot be signaled to the outside. You are free to go in, but it's a one-way trip you can't come out of. This question of "where does the information go" leads to the information paradox (Episode 6).

Episode 1 summaryThe horizon ── a one-way causal boundary that neither light nor information can leave

A black hole is everything inside the radius where the escape velocity reaches the speed of light ── the Schwarzschild radius $R_s=2GM/c^2$ (which needs both $G$ and $c$). Its boundary is the event horizon, a one-way membrane for information that not even light can leave. This episode's ratio is closeness to the horizon $R_s/r$ (1 at the horizon). The closer you get, the more extreme relativity becomes, and from outside the gravitational time dilation goes to infinity ── a falling clock freezes on the surface, redshifts, and fades away (a frozen star).

But the one falling in slips through feeling nothing, in a finite proper time (the equivalence principle, no drama). The disagreement over "stops / doesn't stop" is the gravitational version of "simultaneity depends on the observer" from Relativity That Clicks #5 ── not a contradiction. The heart of the horizon is that information goes one way ── the question of where the information that went in goes is the backbone that runs all the way to quantum gravity (Episode 6). The door to the diagonal of the physics cube opens from this surface.

NEXT TIME ── Episode 2 "Entropy = number of bits ── the formula where the four constants gather." If information can't get out, the "disorder" inside the horizon only keeps piling up ── so what is a black hole's entropy (amount of information)? The answer is one of the most beautiful formulas in physics ── $S=k_B c^3 A/(4G\hbar)$. The one formula where c, ℏ, G, and k_B all line up in a single row. And divide by the Planck area and $S/k_B=A/(4\ell_P^2)$ ＝ the number of bits of the horizon. Entropy turned out to be just a dimensionless number: the area divided by the Planck area. And this skeleton, too, we can derive with only our "ratios."

This document is Episode 1 of the "Black Holes That Click" series, a piece of reading for physics-loving high-school and university students. The Schwarzschild radius $R_s=2GM/c^2$ (matching the exact general-relativistic solution); the event horizon being a causal boundary that light cannot escape; the gravitational redshift and time-dilation factor $(1-R_s/r)^{-1/2}$ in external Schwarzschild coordinates diverging at the horizon (the frozen-star picture); a free-falling observer passing through the horizon in a finite proper time, the horizon being only a coordinate singularity (the curvature is finite, the true singularity is at the center); and the observational establishment of black holes (gravitational waves from GW150914 onward, the EHT images of M87* and Sgr A*, the orbit of the star S2) ── all of these are established physics. That the escape-velocity Newtonian derivation only happens to match in its coefficient while the legitimate derivation is general relativity, that the frozen star is an external-coordinate-dependent appearance, and that the interior, the singularity, tidal forces, and rotating/charged (Kerr / Reissner–Nordström) solutions are outside this episode's scope, are noted in the main text's "The honest line." The figure is a schematic of the stretching of signal intervals due to gravitational time dilation. ── To print, use your browser's "Print" and "Save as PDF" (in the print version the slider and answers are static and hidden). Related: Table of contents ／ sister series Relativity That Clicks #6 (gravity = curvature).

Print / PDF: ⌘+P (Ctrl+P on Windows). On screen, bring the clock toward the horizon and the ticks arriving outside stretch out infinitely and freeze. "See the answer" opens each solution.