Relativity That ClicksEpisode 6 / Gravity Is the Curvature of Spacetime ── Why c Is the Limit

From flat spacetime (special relativity) to gravity ── and the payoff of the homework held since Episode 1, ⟨why c is the limit⟩

Gravity Is the Curvature of Spacetime Inside an elevator, gravity and acceleration cannot be told apart (the equivalence principle). From here, gravity is not a "force" but
the curvature of spacetime itself, and objects are merely going "straight" through curved spacetime ── and the reason c is the limit is here too.

Tools you'll need: the s² of Episode 3, the light cone of Episode 5, the time dilation of Episode 2 This episode's core: gravity = curvature, c = the skeleton of the light cone

Up through Episode 5, everything was about flat spacetime without gravity (special relativity). This episode steps into gravity. The starting point is the single realization Einstein called "the happiest thought of my life" ── when you are in free fall, gravity vanishes (that floating you see with astronauts). Conversely, inside a rocket accelerating through space, you're pressed to the floor and feel "gravity." In other words, from the inside, gravity and acceleration cannot be distinguished (the equivalence principle). From here, an astonishing conclusion tumbles out ── gravity is not a "force" that pulls objects, but the curvature of spacetime itself. Objects are merely going "straight" through curved spacetime, and that looks like "falling." And the answer to the homework we've been holding since Episode 1 ── ⟨why is $c$ the absolute limit⟩ ── also lies in this very geometry of spacetime.

01The happiest thought ── when you're falling, gravity vanishes

When an elevator cable snaps and it free-falls, the person inside and the floor fall at the same acceleration, so the floor stops pushing on their feet ── weight vanishes (weightlessness). The International Space Station is "weightless" for the same reason: the whole station keeps free-falling toward Earth (that's what orbiting is). Inside a falling box, gravity can be cleanly erased. If it can be "erased," then it is a cousin of the inertial forces that "Force That Clicks" dealt with in Episode 4 (apparent forces that can be erased by a change of coordinates) ── gravity is not an absolute force.

02Inside the box, gravity and acceleration can't be told apart

The equivalence principle ── gravity = acceleration

From experiments confined to a windowless box alone, you cannot distinguish at all between "at rest on the ground under gravity $g$" and "rising at acceleration $g$ in gravity-free space." Let go of a ball and, in either case, it falls to the floor the same way. Gravity and acceleration are, locally, the same thing.

It looks obvious, but it runs deep. A coincidence that was a mystery in Newton ── why an object's "resistance to being moved (inertial mass)" and its "susceptibility to gravity (gravitational mass)" are exactly equal (which is why in a vacuum all objects fall at the same rate, a feather and an iron ball alike) ── becomes natural under the equivalence principle. If gravity and acceleration are the same thing, then it's obvious that how fast things fall doesn't depend on the object.

03So light bends, and time runs at different rates in different places

The equivalence principle immediately spits out testable predictions. Shine light sideways across an accelerating rocket. While the light crosses the box, the rocket moves upward under acceleration, so the light appears to bend toward the floor ── that's obvious. But by the equivalence principle, the same thing must happen with gravity too.

Predictions of the equivalence principle

•Gravity bends light (acceleration bends light = gravity bends light). During the 1919 solar eclipse, the positions of stars near the Sun were displaced, and this was confirmed.
•Time runs more slowly the stronger the gravity (gravitational time dilation). A clock on a lower floor runs slightly slower than one on a higher floor.

The size of the time dilation can be written as a dimensionless ratio ── the gravitational potential $\Phi$ divided by $c^2$, namely $\Phi/c^2$.

Try it ── gravitational time dilation (dimensionless) $$\frac{\Delta(\text{rate of time's passing})}{\text{time}}\approx\frac{\Delta\Phi}{c^2}=\frac{g\,h}{c^2}$$

The difference in clock rates between the ground and a height $h$ is $gh/c^2$. This dimensionless ratio is minuscule on the ground (because $c^2$ is enormous) ── which is why we don't notice it in daily life. But for GPS satellites (altitude 20,000 km), $\Phi/c^2\approx5\times10^{-10}$, and their clocks run ahead by about 45 μs per day; without correction, positions would be off by several kilometers. Combined with the speed-based dilation of Episode 2 (special relativity), GPS simply cannot work without relativity. Here too, what matters is the unitless ratio $\Phi/c^2$.

04Let's play with it ── the equivalence-principle box

Below is the inside of a windowless box. Let go of a ball and it falls to the floor; light crossing the box bends slightly downward. With the button, you can switch whether this box is "at rest on the ground under gravity $g$" or "rising at acceleration $g$ in gravity-free space" ── but what you see inside is exactly the same. Confirm with your own eyes that you cannot tell them apart. Strengthen $g$ and both the fall and the light's bend grow larger.

Figure: a windowless box. The ball falls to the floor, and light crossing it bends downward. The button switches between "at rest under gravity" / "accelerating in weightlessness" ── what's seen inside is identical (the equivalence principle). So gravity bends light too (confirmed at the 1919 eclipse).

strength of g 1.0

ball (falls to the floor) light (bends downward)

05Gravity is the curvature of spacetime ── and "straight" is falling

Light bends, and time runs at different rates in different places ── these are manifestations of spacetime being curved. General relativity in one sentence goes like this ── matter tells spacetime how to curve, and curved spacetime tells matter how to move. An object travels the straightest possible path (a geodesic) through curved spacetime. That, to us, looks like "falling under gravity."

Why "straight" is falling ── what curves is mainly "time" It sounds strange, but the key is Episode 1. An object moves through spacetime almost entirely in the time direction (in daily life $\beta\approx0$ = the world line is nearly straight up). So a slight curvature of the time direction matters more than a slight curvature of space. Even with gravity as weak as Earth's, the rate of a clock differs a tiny bit with height ($\Phi/c^2$) = the time axis is faintly curved. For an object racing along almost purely in the time direction (at $c$!), that minute curvature of time bends the world line a great deal ── that is "falling." An apple falls because of the curvature of time, not of space. The famous picture of a rubber sheet dented by a weight, deforming only space, is an incomplete metaphor that cannot depict this "curvature of time."

And the answer to the homework we've been holding since Episode 1 ── ⟨why is $c$ the absolute limit⟩ ── is here too. $c$ is not "the top speed of a fast vehicle." At each point of spacetime there is a light cone (Episode 5), the framework of causality, and $c$ sets how wide that cone opens ── it is the very structure of spacetime. By the equivalence principle, no matter how curved spacetime is, if you look at the immediate vicinity of any point, you always return to that flat spacetime (the light cone of $s^2$). So $c$ is necessarily the same locally everywhere in the universe ── it is not a speed limit imposed on objects, but the exchange rate that defines the causal framework of spacetime (Episode 1), and so it is absolute and cannot be crossed. Even when gravity curves spacetime, the light cone is preserved at each point and causality is protected. $c$ is the limit because it is an invariant woven into the geometry of spacetime.

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The honest line ── the equivalence principle is "local" only

The equivalence principle (locally, gravity and acceleration cannot be distinguished), the equality of inertial mass and gravitational mass, the gravitational bending of light (the 1919 eclipse and later high-precision tests), gravitational redshift / gravitational time dilation $\approx\Phi/c^2$ (the Pound–Rebka experiment, GPS's ~45 μs/day), gravitational waves (LIGO/Virgo, 2015 onward), black holes, and "gravity = the curvature of spacetime" (general relativity) ── all of these are established physics, precisely verified.

However, the equivalence principle is local ── it's a story only inside a sufficiently small box. In a large box, the "narrowing" of gravity toward Earth's center makes two side-by-side balls draw slightly closer, while up-and-down ones move apart (tidal forces). This tidal effect is precisely the real curvature of spacetime, and it cannot be erased by acceleration ── uniform acceleration and gravity are exactly the same only locally. Also, the bending of light is only half from the equivalence principle alone; adding the curvature of the "space" part of spacetime gives general relativity's correct value (about 1.75 arcseconds at the Sun). "Curvature of time dominates" holds for weak gravity; in strong fields such as near a black hole, the curvature of space matters a great deal too. The full mathematics (Einstein's equations) is serious geometry built atop this intuition.

Practice problems (solvable with this episode's ideas)

What happens to gravity inside a free-falling elevator? Why is the International Space Station weightless?
See the answer
In free fall the floor doesn't push on your feet, and gravity vanishes (weightlessness). The ISS is also continuously free-falling toward Earth (orbiting = free fall), so its interior is weightless. Gravity can be locally erased by free fall.
Why do all objects fall at the same rate in a vacuum? Explain with the equivalence principle.
See the answer
If gravity and acceleration are the same thing, the box's acceleration acts equally on every object inside = how fast things fall doesn't depend on the object. The equality of inertial mass and gravitational mass also becomes natural this way.
For a GPS satellite (Φ/c²≈5×10⁻¹⁰), how much does the gravitational clock offset amount to in one day (1 day ≈ 8.6×10⁴ s)?
See the answer
$5\times10^{-10}\times8.6\times10^4\ \text{s}\approx4.3\times10^{-5}$ s = about 43 μs (a rough estimate; the actual GR contribution is about 45 μs/day). Without correction, positions drift by several kilometers.
Why is "c the limit"? How does it differ from a speed limit, in the language of spacetime geometry?
See the answer
c is the very structure of spacetime that defines the light cone (the causal framework) at each point. Since the equivalence principle guarantees the local region always returns to flat spacetime, c is the same invariant everywhere. It's not a speed limit imposed on objects but an exchange rate woven into the geometry of spacetime, and so it is absolute and cannot be crossed.

Episode 6 SummaryGravity is curvature, straight is falling, c is the skeleton of geometry

Free-fall and gravity vanishes ── so gravity is not an absolute "force." In a windowless box, gravity and acceleration cannot be distinguished (the equivalence principle). From here the mystery of inertial mass = gravitational mass is solved, gravity bends light (the 1919 eclipse), and time runs slower where gravity is stronger ($\approx\Phi/c^2$, about 45 μs/day for GPS). These are all manifestations of spacetime being curved ── matter curves spacetime, and curved spacetime determines how objects move. An object travels the "straightest path (a geodesic)" through curved spacetime, and that looks like falling. And what curves is mainly time (because it races at c almost purely in the time direction, a minute curvature of time matters).

And the payoff of the Episode 1 homework ── $c$ is the limit because it is the structure of spacetime that defines the light cone (the causal framework) at each point. Since the equivalence principle guarantees the local region always returns to flat spacetime ($s^2$), $c$ is the same invariant everywhere. It's not a speed limit on objects but an exchange rate woven into the geometry of spacetime (Episode 1), so it is absolute and cannot be crossed. Even gravity was not a unit-bearing "force" but the geometry of spacetime, the very stage itself ── here we shake hands with "Force That Clicks" Episode 5, "Gravity Is Geometry."

Coming next ── Episode 7 (the finale) "You may as well say the speed of light changes ── a varying c and curved space are equivalent." We've drawn gravity as "spacetime curving." But you can restate exactly the same phenomenon as "the speed of light varies from place to place" = gravity has a refractive index $n(x)$, and the bending of light and gravitational time dilation come out exactly the same. Curvature and variable $c$ are merely two descriptions of the same single invariant (the geometry of spacetime) ── a difference in choice of coordinates. The locally measured $c$ and $\alpha$ are invariant in either language. On to the finale, seamless with your "Cosmology That Clicks," $c\cdot t=$constant.

This document is Episode 6 of the "Relativity That Clicks" series, a reading piece for physics-loving high-schoolers and undergraduates. The equivalence principle, the equivalence of inertial and gravitational mass, the gravitational deflection of light (Eddington's 1919 eclipse observation, about 1.75 arcseconds at the Sun), gravitational redshift and gravitational time dilation ($\Delta\nu/\nu\approx\Delta\Phi/c^2$, the Pound–Rebka experiment, GPS's general-relativistic contribution of about +45 μs/day, giving a net of about +38 μs/day combined with special relativity), the direct detection of gravitational waves, and gravity as the curvature of spacetime (general relativity) ── all of these are established standard physics. That the equivalence principle is local and that tidal forces (true curvature) remain globally, that the deflection of light is only half from the equivalence principle alone and general relativity including spatial curvature gives the correct value, that "gravity is mainly the curvature of time" is a weak-field characterization while spatial curvature matters in strong fields too, and that the rubber-sheet model is an incomplete metaphor of spatial curvature only, are all noted in the "honest line" section. The figure is a schematic of the equivalence principle (the local equivalence of uniform gravity and uniform acceleration) and the deflection of light, with the light path's bend exaggerated for visibility. ── To print, use your browser's "Print" and "Save as PDF" (in the print version the animation and answers are frozen and hidden). Neighboring episodes: Episode 5 Simultaneity and Causality / Table of Contents / sister series Force That Clicks.

Print / Save as PDF: ⌘+P (Ctrl+P on Windows). On screen, even when you switch "at rest under gravity / accelerating in weightlessness," the inside of the box (falling ball, bending light) is the same. Click "See the answer" to open a solution.