Relativity That ClicksEpisode 7 (Finale) / A changing c and a curving space are equivalent

We drew gravity as "the curving of space." The very same phenomena can be drawn as "the speed of light changes from place to place" ── two descriptions of one and the same invariant

You may as well say the speed of light changes Rephrase gravity not as "spacetime curves" but as "the speed of light changes place by place (a gravitational index of refraction n(x))," and
the bending of light and the slowing of time come out exactly the same. Curvature and variable c are two descriptions of the same geometry.

Tools you'll need: Episode 6's gravity = curvature, refraction (light bends toward the slower side), and Episode 4's "units are stage machinery" The heart of this episode: c(x)=c/n(x) ≡ curvature

In Episode 6 we drew gravity as "the curving of spacetime itself." In this finale we're going to shake up that picture ── take the very same gravitational phenomena and rephrase them as "space stays flat, but the speed of light changes from place to place," and the bending of light, the gravitational slowing of time, and the light delay (the Shapiro delay) all come out identically, down to the last decimal. All you do is treat "light runs slower where gravity is stronger" as space having an index of refraction $n(x)$. Then the question arises: "Is gravity spacetime curving, or is it the speed of light changing from place to place?" ── and the answer is exactly what this series has said all along. Both are just different descriptions of one single invariant (the geometry of spacetime). Whether you say curvature or variable $c$ is a matter of how you pick coordinates ── of stage machinery ── not of the thing itself. This episode connects straight to the $c\cdot t=$constant of your "Cosmology That Clicks."

01The same phenomenon can be drawn two ways

Let's restate Episode 6's predictions ── light bends in gravity, and time runs slower where gravity is stronger ── in different words. Look at "time runs slower" from light's point of view: at that place, light oscillates slowly, meaning light travels slower. In other words, the stronger gravity is at a place, the slower the speed of light there (in coordinate terms). This is the same situation as matter that slows light down ── a medium with a high index of refraction. Could gravity be drawn as a transparent "medium"?

02Gravity can also be written as an "index of refraction"

The gravitational index of refraction ── c changes from place to place

$$n(x)\approx 1-\frac{2\Phi}{c^2}\ (>1)\qquad c(x)=\frac{c}{n(x)}$$

The deeper the gravitational potential $\Phi$ (negative on the attracting side), the larger $n$, and the slower the local speed of light $c(x)=c/n$ there. Just like glass or water ── a region of strong gravity is a "dense" medium for light. Episode 6's dimensionless ratio $\Phi/c^2$ becomes, directly, the departure of the index of refraction from 1.

Light entering glass at an angle bends toward the slower side (refraction). A desert mirage, too, is light bending because of a density difference ── an index-of-refraction difference ── in the air near the ground. See gravity as a medium of $n(x)$, and by the exact same logic ── light bends toward the stronger-gravity (larger-$n$) side. When light passing near a mass gets pulled in, you can draw it not as "being tugged by gravity" but as "bending because of the gradient in the index of refraction."

03The bending of light and the slowing of time come out the same

It's vivid seen through Huygens' principle (a wavefront is the superposition of little wavelets spreading from each point). For light traveling past a mass, the part of the wavefront nearer the mass has larger $n$ and is slower, so it lags. The wavefront "wears down on one side" and tilts ── and as a result, the direction of travel bends toward the mass. Whether you draw it with curvature or with an index of refraction, the bending angle that comes out is the same.

Three phenomena, said in the language of the index of refraction

• The bending of light: the gradient in $n$ refracts light toward the mass (gravitational lensing).
• The slowing of time: where $n$ is large, light oscillates slowly, so clocks run slow (gravitational redshift).
• The Shapiro delay: radio waves passing near the Sun take a "detour" through the region where $n>1$ and arrive late ── confirmed by measurement.

From one single $n(x)$, all of Episode 6's predictions come out. Whether you say "curvature" or "variable c," the quantities you can observe agree completely.

04Try moving it ── light passing near a mass

The figure below. Around the mass at the center, the index of refraction $n(x)$ spreads out (darker = larger = light is slower). Light coming from the left passes near the mass and ── bends toward the mass. The dotted line is the straight path it would have taken. The gap (the bending angle) is gravitational lensing. Turn up the mass slider to steepen the gradient in $n$, and the bending grows too.

The button switches the words of the explanation ── "explain with the curving of space" / "explain with the index of refraction n(x)." But the light path drawn is exactly the same. That's the core of this episode. One and the same bend can be read as "space is curving" or as "the speed of light changes from place to place" ── not which one is right, but two descriptions of the same invariant.

Figure: the index of refraction n(x) around a mass (dark = light is slower). Light from the left bends toward the mass (dotted = straight-line path). The bending grows with the mass. The button toggles "explain with curvature / explain with the index of refraction" ── the path is identical (two descriptions of the same phenomenon).

Mass (strength of the gradient in n) 1.0

light path (bends toward the mass) straight line (if it hadn't bent)

05Curvature or variable c ── two descriptions of the same invariant

So what is gravity's "true form" ── curved spacetime, or a variable-speed-of-light medium? This series' answer is clear ── that is the wrong question. The two are just the same geometry written in different coordinates, in different words. The things you can observe (bending angle, slowing of time, delay) have the same values in either description. What differs is the description; what stays the same is the invariant. In Episode 4 we said "the number for $c$ with units attached is stage machinery," and in Episode 5 we said "how you slice the surfaces of simultaneity is a gauge choice" ── this is the final form of that. Even things that look so very physical, like "the curvature of space" or "the value of the speed of light," depend on the choice of description (coordinates, gauge). What is truly invariant is only the unitless observables.

The heart of this episode ── the description is interchangeable, the invariant is one

$$\underbrace{\text{curved spacetime}}_{\text{description A}}\ \equiv\ \underbrace{\text{flat space}+c(x)=c/n(x)}_{\text{description B}}\ \longrightarrow\ \text{same observables}$$

In Episode 6 we answered "why c is the upper limit" with "it's the skeleton of spacetime," yet even the value of that $c$ depends on the coordinates. What is invariant is the causal structure of the light cone and dimensionless quantities like $\alpha$.

A perfect handshake with "Cosmology That Clicks" c·t=constant The backbone of your "Cosmology That Clicks," $c\cdot t=\text{constant}$, was exactly the thinking of this episode ── you may say "the speed of light changed over the ages," but that is a choice of coordinates (how you measure time) ── a gauge, and as long as the dimensionless $\alpha$ stays invariant, it leaves no trace in observation. Variable c, on its own, is not physics but a change of description. If you really want to change the physics, you have to move a dimensionless quantity like $\alpha$ ── and atomic clocks and BBN tightly constrain any change in $\alpha$ (Cosmology, Bonus ③). The finale of relativity and the backbone of cosmology connect into one line with "the only thing invariant is the dimensionless quantity."

◇ ◇ ◇

The honest line ── the precise reach of "it can be written with an index of refraction"

That in a static gravitational field the propagation of light (null geodesics) can be described as a medium with index of refraction $n(x)\approx1-2\Phi/c^2$, correctly reproducing the bending of light, gravitational redshift, and the Shapiro delay (the optical analogy of gravity, going back to Eddington); that these give observational results identical to those of "spacetime curvature"; and that the locally measured speed of light is always $c$ (what changes is the "coordinate speed of light" in far-away coordinates), with the dimensionless $\alpha$ staying invariant ── all of these are established physics.

That said, not everything can be written as "gravity = a scalar index of refraction $n$." ① To fully include not just light but the motion of matter, strong gravitational fields, and the slowing of time, you need not one $n$ but the whole metric tensor (its time and space components) ── the index-of-refraction picture is chiefly an approximation, a rephrasing that works cleanly for light in static, weak fields. ② There are also phenomena, such as frame-dragging (a rotating mass dragging spacetime along), that cannot be written as a simple medium. So the claim here is not "gravity = mere variable speed of light," but rather curvature and variable c are (when properly formulated) two coordinate descriptions of the same geometry; neither is the thing itself, and the only invariant is the dimensionless observable. Variable-speed-of-light cosmology (VSL), too, remains a change of description rather than physics unless it moves a dimensionless quantity.

Practice problems (solvable with this episode's way of thinking)

If you draw gravity with an index of refraction $n(x)$, toward which side of the mass does light bend? Why?
See the answer
Toward the mass (the larger-$n$ side). Just like glass, light bends toward the slower side (large $n$). The part of the wavefront nearer the mass lags and tilts (Huygens). Gravitational lensing.
Why is "is gravity the curving of space, or a variable speed of light?" a "wrong question"?
See the answer
Because the two are just the same geometry written in different coordinates and words, and the observables (bending angle, slowing of time, delay) agree completely. What differs is the description; what stays the same is the invariant. There's no meaning in asking which one is right.
When we say "the speed of light changes from place to place," which speed of light is changing? What if you measure it locally?
See the answer
What changes is the "coordinate speed of light" measured in a far-away observer's coordinates. Measured locally (with the local ruler and clock), the speed of light is always c. So variable c is a coordinate word, and the local physics is invariant.
How does "Cosmology That Clicks" c·t=constant connect with this episode? What is the condition for variable c to become "physics"?
See the answer
c·t=constant is the same thinking too: read variable c as a choice of coordinates (how you measure time = a gauge), and if the dimensionless α stays invariant, it leaves no trace in observation. Variable c only truly becomes physics when it changes a dimensionless quantity like α.

Episode 7 summary / Relativity That Clicks, concludedThe description is interchangeable; the only invariant is the dimensionless quantity

Draw gravity not as "the curving of spacetime" (Episode 6) but as "the speed of light changes from place to place = an index of refraction $n(x)\approx1-2\Phi/c^2$, $c(x)=c/n$," and the bending of light, the gravitational slowing of time, and the Shapiro delay come out exactly the same. Light refracts toward the larger-$n$ (stronger-gravity) side ── with curvature or with an index of refraction, the path and the observables agree. So "is gravity curvature or variable c?" is a wrong question, and the two are two descriptions of the same invariant (the geometry of spacetime).

This is the destination of "Relativity That Clicks" as a whole. Speed is β (Episode 1), the slowdown is γ (Episode 2), the invariant is s² (Episode 3), mass is rest energy (Episode 4), simultaneity is gauge and causality is invariant (Episode 5), gravity is curvature (Episode 6) ── and even that curvature and the value of c depend on the choice of description (coordinates, gauge) (Episode 7). Time, distance, speed, and "curvature" with units attached are all stage machinery. What is truly invariant is only the causal structure of the light cone and dimensionless observables like $\alpha$. Think from there, and relativity never gets complicated ── that is everything I wanted to say with six ratios and one caveat. Here it shakes hands completely with "Cosmology That Clicks" $c\cdot t=$constant.

Relativity That Clicks, concluded ── and next comes "Quantum That Clicks" "Relativity That Clicks" ends here. Up to now this was a world built around the speed of light $c$ (the exchange rate between time and space). In the next series, "Quantum That Clicks," another fundamental constant, $\hbar$, takes the lead role ── with the dimensionless ratio of how many $\hbar$'s worth the action is (S/ℏ), we'll unravel the wave function, the Schrödinger equation, and quantum theory in a form where "thinking from there never gets complicated." If $S\gg\hbar$ it's classical; if $S\sim\hbar$ it's quantum. After the story of $c$ comes the story of $\hbar$.

This document is Episode 7 (the finale) of the "Relativity That Clicks" series, a reading piece for physics-loving high-schoolers and undergraduates. That in a static gravitational field the optical analogy of light propagation (effective index of refraction $n(x)\approx1-2\Phi/c^2$, coordinate speed of light $c(x)=c/n$) reproduces the deflection of light, gravitational redshift, and the Shapiro delay, and gives observables identical to the description by spacetime curvature; that the local proper speed of light is always $c$ and the dimensionless constant $\alpha$ is invariant; and that variable speed of light (VSL) remains a change of coordinates (a gauge) unless it changes a dimensionless quantity ── all of these are established physics and standard understanding. That the index-of-refraction picture is a scalar approximation, that fully including the motion of matter, strong gravitational fields, the time component, frame-dragging, and so on requires the whole metric tensor, and that it is chiefly a rephrasing for light in static weak fields ── so that the claim is not "gravity = variable speed of light" but "curvature and variable c are two coordinate descriptions of the same geometry, and the only invariant is the dimensionless observable" ── is spelled out in the body's "The honest line." The figure is a schematic of ray tracing (gravitational lensing) in a gradient-index $n(x)$, with the bending angle exaggerated for visibility. ── To print, use your browser's "Print" and "Save as PDF" (in the print version the slider and answers are frozen and hidden). Adjacent episodes: Episode 6 Gravity / Contents / sister series Cosmology That Clicks (c·t=constant).

Print / make a PDF: ⌘+P (Ctrl+P on Windows). On screen, the mass slider changes the bending of light, and switching between "explain with curvature / explain with index of refraction" leaves the path the same. "See the answer" opens each solution.