Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

=== There are two complementary (and standard) GR routes that both tie into your “extra optical path = slower local time” picture. ===

==== Schwarzschild (static, spherically symmetric): ====

ds2=−(1−2GMrc2)c2dt2+(1−2GMrc2)−1dr2+r2dΩ2.ds^2 = -\left(1-\frac{2GM}{rc^2}\right)c^2dt^2 + \left(1-\frac{2GM}{rc^2}\right)^{-1}dr^2 + r^2 d\Omega^2.ds2=−(1−rc22GM)c2dt2+(1−rc22GM)−1dr2+r2dΩ2.
A local proper time tick dτd\taudτ relates to coordinate time dtdtdt via

dτdt=1−2GMrc2    ⇒    Γ(r)Γ(∞)=1−2GMrc2,\frac{d\tau}{dt}=\sqrt{1-\frac{2GM}{rc^2}}
\;\;\Rightarrow\;\;
\frac{\Gamma(r)}{\Gamma(\infty)}=\sqrt{1-\frac{2GM}{rc^2}},dtdτ=1−rc22GM⇒Γ(∞)Γ(r)=1−rc22GM,
if you identify exchange/clock rate Γ\GammaΓ with the local proper rate. For weak fields,

δΓΓ≈−GMrc2.\frac{\delta \Gamma}{\Gamma} \approx -\frac{GM}{rc^2}.ΓδΓ≈−rc2GM.
This is a metric derivation, not a heuristic.

==== Light in a static metric propagates as if in a refractive index ====

n(r)=11−2GMrc2  ≈  1+GMrc2.n(r)=\frac{1}{\sqrt{1-\frac{2GM}{rc^2}}}\;\approx\;1+\frac{GM}{rc^2}.n(r)=1−rc22GM1≈1+rc2GM.
Coordinate propagation time from rar_ara to rbr_brb along a nearly radial path:

t  =  1c∫rarbn(r) dr  ≈  (rb−ra)c  +  GMc3ln⁡ ⁣rbra.t \;=\; \frac{1}{c}\int_{r_a}^{r_b} n(r)\,dr
\;\approx\; \frac{(r_b-r_a)}{c} \;+\; \frac{GM}{c^3}\ln\!\frac{r_b}{r_a}.t=c1∫rarbn(r)dr≈c(rb−ra)+c3GMlnrarb.
Hence a small change of the outer radius RRR at fixed inner radius rar_ara yields

δtt  ≈  δRR⏟pure geometry  +  GMRc2 1ln⁡(R/ra)⏟GR optical correction,\frac{\delta t}{t}
\;\approx\;
\underbrace{\frac{\delta R}{R}}_{\text{pure geometry}} \;+\;
\underbrace{\frac{GM}{Rc^2}\,\frac{1}{\ln(R/r_a)}}_{\text{GR optical correction}},tδt≈pure geometryRδR+GR optical correctionRc2GMln(R/ra)1,
to leading order when R≫raR\gg r_aR≫ra. This shows precisely when your
δt/t≈δR/R\delta t/t \approx \delta R/Rδt/t≈δR/R holds (flat-space limit or when the GR term is small) and how to add the GR correction systematically.

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