General relativity — equivalence principle, curved spacetime (overview): Visual walkthrough

Every symbol below is defined the moment it appears. If you have never seen it, you are in the right place.

Step 1 — Meet the box, and the one rule we are allowed to use

WHAT. Picture a sealed box, no windows, floating in empty deep space — no planet nearby, no gravity. Now a rocket underneath fires and the box accelerates upward at a steady rate we call $g$ .

$g$ = "acceleration" = how fast the box's speed is increasing, measured in metres-per-second added every second ( $m/s^{2}$ ).
We pick the same number $g$ that gravity gives at Earth's surface ( $g \approx 9.8 m/s^{2}$ ) — you'll see why in a moment.

WHY. The Equivalence Principle (parent §1) says: a person sealed in this accelerating box cannot tell, by any experiment, whether they are being pushed by a rocket or sitting still on a planet with gravity $g$ . So — clever move — anything we prove about the rocket box must also be true in real gravity. We are allowed to swap "gravity" for "acceleration" whenever it's convenient. That swap is the engine of this whole page.

PICTURE. The red box is our lab; the red arrow is its acceleration $g$ pointing up.

Figure — General relativity — equivalence principle, curved spacetime (overview)

Step 2 — Fire a light pulse from floor to ceiling

WHAT. The floor sends a flash of light straight up toward the ceiling. Light always travels at speed $c \approx 3 \times 1 0^{8} m/s$ — this never changes, in any box, for anybody. That constant $c$ is the one rock-solid fact we lean on.

WHY light and not a ball? Because light carries a built-in clock: its frequency.

Frequency $f$ = how many wave-crests pass a point each second (unit: hertz, "per second"). A high- $f$ light is bluer; a low- $f$ light is redder.
If the frequency the ceiling measures differs from what the floor sent, that difference is a message about time itself — because "crests per second" is literally a ticking clock.

PICTURE. The red wavy pulse leaves the floor and heads up the height $h$ .

The travel time to cross the gap is $t = \frac{h}{c} .$

$h$ = distance to climb (metres) — the numerator, "how far."
$c$ = light's speed (m/s) — the denominator, "how fast."
$t$ = time taken (seconds). Distance over speed = time. Nothing fancy: same as "300 km at 100 km/h takes 3 hours."

Step 3 — During that flight, the ceiling speeds up

WHAT. The box is accelerating. So while the light is in flight for time $t$ , the ceiling doesn't just move — it keeps getting faster. By the moment the light arrives, the ceiling has gained an extra upward speed: $v = g t .$

WHY this is the crux. The receiver (ceiling) is running away from the light by the time the light reaches it. It's like a friend walking backward as you throw them a ball — they catch it moving away from you, and it "feels" slower on arrival. For light, "slower" doesn't mean lower speed (that's fixed at $c$ ); it means stretched-out crests → lower frequency. That is the Doppler effect, and we compute it next.

PICTURE. Same climb, but now the arrow shows the ceiling's gained speed $v = g t$ at the instant of catch.

Let's substitute the flight time from Step 2 into $v = g t$ : $v = g \cdot \frac{h}{c} = \frac{g h}{c} .$

$g$ = the box's acceleration ( $m/s^{2}$ ).
$h$ = the height climbed (m).
$c$ = light's speed (m/s).
$v$ = the receding speed of the ceiling relative to the floor when the light lands (m/s). Read it as "how fast the catcher is fleeing."

Step 4 — Turn "receding receiver" into a frequency drop (the Doppler step)

WHAT. When a light source and receiver move apart at a small speed $v$ (small meaning $v ≪ c$ ), the received frequency is shifted down by the simple fraction: $\frac{Δ f}{f} = - \frac{v}{c} .$

WHY this exact formula, and why the minus sign?

Why Doppler at all? Because Step 3 showed the catcher moving away. Moving-away always red-shifts (lowers frequency) — think of a siren dropping in pitch as an ambulance passes and recedes.
Why $- v / c$ and not something fancier? For everyday speeds ( $v$ far below $c$ ) the full relativistic Doppler formula flattens to exactly this ratio. It's the first, dominant term; higher terms are of size $(v / c)^{2}$ , utterly negligible here. We use the simplest tool that captures the effect.
Reading each symbol: $Δ f = f_{received} - f_{sent}$ is the change in frequency (the Greek $Δ$ , "delta," just means "change in"). Dividing by $f$ makes it a fraction — a percentage-style stretch, independent of what colour we started with. The minus encodes "received is lower than sent."

PICTURE. Top: crests packed tight (sent). Bottom: crests spread out (received) — the same wave, stretched, because the catcher fled.

Step 5 — Substitute, and read the result

WHAT. Plug the receding speed from Step 3, $v = g h / c$ , into the Doppler fraction from Step 4: $\frac{Δ f}{f} = - \frac{v}{c} = - \frac{1}{c} \cdot \frac{g h}{c} = - \frac{g h}{c ^{2}} .$

WHY this is the answer we wanted. Every quantity is now in things we control: the acceleration $g$ , the height $h$ , and the universal $c$ . And — by the Equivalence Principle promise of Step 1 — this rocket-box result is identical to what happens in real gravity of strength $g$ . So a photon climbing height $h$ in gravity arrives red-shifted by exactly $g h / c^{2}$ .

PICTURE. The two boxes side by side — rocket vs planet — giving the identical red-shifted arrival.

Step 6 — From frequency to time: why the floor clock runs slow

WHAT. Frequency and time are two sides of one coin. If frequency is "crests per second," then the time between crests (call it $τ$ , the Greek "tau," a tick-length) is its reciprocal: $τ = 1/ f$ . A drop in $f$ therefore means a rise in $τ$ — longer ticks. Turning the boxed result inside out: $\frac{Δ τ}{τ} = + \frac{g h}{c ^{2}} .$

WHY the sign flips to plus. Fewer crests per second (frequency down, minus) is the same event as more seconds per crest (period up, plus). The ceiling receives crests spaced further apart in its time — but those crests are the floor clock's ticks. So the floor's ticks arrive stretched: the lower clock is running slower than the upper one.

$Δ τ / τ$ = fractional increase in tick-length seen higher up.
Positive sign = "upper clock sees lower clock slowed."

PICTURE. Two clocks, ceiling and floor. The floor clock (red) is drawn ticking behind — it has fallen out of step.

Step 7 — Edge and degenerate cases (never leave a gap)

We must check the formula behaves sanely at its extremes.

Case $h = 0$ (same height). $Δ f / f = - g \cdot 0/ c^{2} = 0$ . No climb, no shift, no clock difference. ✔ Two clocks at the same level tick together — as they must.

Case $g = 0$ (no gravity / no acceleration — deep-space float, engines off). $Δ f / f = 0$ . With no field and no acceleration there's nothing to red-shift the photon. ✔ This is the "floating lab" half of the Equivalence Principle — gravity truly turned off.

Case light going down (ceiling → floor, $h$ becomes $- h$ ). The sign flips: $Δ f / f = + g h / c^{2}$ , a blueshift. Falling light gains frequency — the catcher (floor) is now rushing toward the source. So the lower clock, watching the upper one, sees it sped up. Consistent: whoever is lower always finds themselves the slow one.

Case very large $h$ (approaching a black hole's edge). Our tiny-shift formula assumed $g h ≪ c^{2}$ so that $v ≪ c$ . When $g h$ grows toward $c^{2}$ , the little-Doppler approximation of Step 4 breaks and we'd need the full curved-spacetime metric (see Black Holes, Spacetime Metric & Minkowski Diagram). At the true horizon the shift becomes infinite — light climbing out is red-shifted to nothing. Our derivation is the correct weak-field first step toward that.

The one-picture summary

Read it left to right: an accelerating box (Equivalence Principle) → light takes time $t = h / c$ to cross → in that time the receiver flees at $v = g h / c$ → Doppler drops the frequency by $v / c$ → giving $Δ f / f = - g h / c^{2}$ → which, flipped, means the lower clock ticks slow. One straight chain from an empty box to bent time.

Recall Feynman retelling — explain the whole walk to a friend

Imagine a lift zooming upward in empty space so you feel "gravity." Someone on the floor shines a torch at the ceiling. While the light is on its way up, the ceiling keeps speeding up — so by the time the light gets there, the ceiling is racing away from where it started. Anything racing away from a light beam catches it "stretched out" — redder, fewer flashes per second. Now here's Einstein's trick: a zooming lift and real gravity feel exactly the same, so this must happen with real gravity too — light crawling up out of gravity gets redshifted. And "fewer flashes per second" is the same as "the sending clock down below is ticking slowly." So the deeper you are in gravity, the slower your clock runs — your basement is younger than your attic. That's the whole idea, and it's the reason GPS satellites have to correct their clocks every single day.

Recall Rebuild the chain from memory

Flight time of the pulse? ::: $t = h / c$ Speed the ceiling gains in that time? ::: $v = g t = g h / c$ Doppler frequency shift for a receding receiver? ::: $Δ f / f = - v / c$ Final result after substituting? ::: $Δ f / f = - g h / c^{2}$ Same result written for time (period)? ::: $Δ τ / τ = + g h / c^{2}$ — lower clock runs slow

Connections

Parent overview — this page derives its boxed redshift result.
Special Relativity — where the $- v / c$ Doppler shift of Step 4 comes from.
Gravitational Time Dilation — the direct application of Step 6.
GPS Corrections — the numbers made real.
Gravitational Lensing, Black Holes — the strong-field limit of Step 7.
Geodesics & Curvature, Spacetime Metric & Minkowski Diagram — the full machinery beyond weak fields.
Newtonian Gravity — the $c \to \infty$ limit where every shift vanishes.