# Airy's Experiment and Relativity **Franco Ligabue** — Slides/Presentation (2025) Source: airy_relativity-1.pdf (14 slides) --- ## Summary Franco presents a relativistic treatment of stellar aberration and the Airy water telescope experiment using Lorentz transformation of the electromagnetic wave 4-vector. His central argument is that the correct way to analyze light in a moving medium is through wavevector transformations and boundary conditions at interfaces, not through geometric "timing" arguments. He derives the aberration angle in both vacuum and water, shows that Snell's law holds in both frames (to first order), and demonstrates that the Fresnel drag result emerges naturally from the relativistic velocity addition of wave packets. --- ## Slide by Slide Breakdown ### Slide 1 — Light Propagation (Plane Waves) ![[slide-01.png]] - Monochromatic plane wave: $\mathbf{E}(\mathbf{r},t) = \mathbf{E}_0 \exp\!\bigl(i(\mathbf{k}\cdot\mathbf{r} - \omega t)\bigr)$ - In a stationary, non-dispersive dielectric of constant $n$: $k = 2\pi/\lambda = n\omega/c$ - Phase velocity equals group velocity for a non-dispersive medium: $v = \omega/k = c/n$ - The wave vector $\mathbf{k}$ gives the propagation direction (perpendicular to wavefronts). ### Slide 2 — Special Relativity and 4-Vectors ![[slide-02.png]] - SR requires $x^\mu \equiv (ct,\,\mathbf{r})$ and $k^\mu \equiv (\omega/c,\,\mathbf{k})$ to transform as 4-vectors. - The phase $\mathbf{k}\cdot\mathbf{r} - \omega t = -k^\mu x_\mu$ is a Lorentz scalar. - Boost along the $x$-axis with velocity $V = \beta c$: $k'_x = \gamma(k_x + \beta\,\omega/c) \;\Longrightarrow\; k'\sin\theta' = \gamma k(\sin\theta + \beta/n) \qquad(3)$ $k'_y = k_y \;\Longrightarrow\; k'\cos\theta' = k\cos\theta \qquad(4)$ $\omega' = \gamma(\omega + \beta c\, k_x) \;\Longrightarrow\; \omega' = \gamma(\omega + \beta c\, k\sin\theta) \qquad(5)$ - General angle transformation: $\tan\theta' = \frac{\gamma(\sin\theta + \beta/n)}{\cos\theta} \qquad(6)$ ### Slide 3 — Special Case A: Stellar Aberration ![[slide-03.png]] - In the frame where the medium moves, light comes along the $y'$-axis ($\theta' = 0$). - From eq (6): $\sin\theta = -\beta/n \qquad(7)$ - For $n = 1$ (vacuum/air): $\sin\theta = -\beta$ — the standard aberration formula. - For $n > 1$ (water): $\sin\theta = -\beta/n$ — aberration angle is **reduced** by a factor of $n$. - Diagram shows the wavevector tilting in the medium's frame vs straight down in the Sun's frame. ### Slide 4 — Special Case B: Fizeau's Experiment ![[slide-04.png]] - Medium's velocity parallel to light propagation ($k_y = 0$ in both frames). - From eqs (3) and (5): $v' = \frac{\omega'}{k'} = \frac{c}{n}\,\frac{1 + n\beta}{1 + \beta/n} = \frac{v + V}{1 + vV/c^2}$ - This is the relativistic velocity addition formula — identical to the Fresnel drag result to first order. - The wavevector transformation naturally produces the Fizeau result. ### Slide 5 — Special Case C: Airy Situation (Interface Parallel to Velocity) ![[slide-05.png]] - Light crosses a vacuum/water interface where the water velocity is parallel to the interface surface. - Boundary conditions: $\omega_1 = \omega_2$ (frequency conserved) and $k_1 \sin\theta_1 = k_2 \sin\theta_2$ (tangential wavevector conserved) — Snell's law. - Applying the Lorentz transform to both media: $k'_2 \sin\theta'_2 = \gamma(k_2 \sin\theta_2 + \beta\,\omega_2/c) = \gamma(k_1 \sin\theta_1 + \beta\,\omega_1/c) = k'_1 \sin\theta'_1$ $\omega'_2 = \omega'_1$ - **The frequency and tangential wavevector component are conserved in both frames.** ### Slide 6 — Airy Case: Interface Parallel to Velocity (Results) ![[slide-06.png]] - In the medium's frame (water stationary): - $\sin\theta_1 = -\beta$ (vacuum aberration) - $\sin\theta_2 = -\beta/n$ (water aberration, consistent with Snell's law) - In the Sun's frame (water moving), for plane waves with $\theta' = 0$ in vacuum, both $\theta'_1$ and $\theta'_2$ are determined by the wavevector transformation. ### Slide 7 — Tilted Interface (General Case) ![[slide-07.png]] - When the interface is not parallel to the boost velocity, rotations complicate the analysis. - Solved to first order in $\theta$ (small angle approximation). - Modified boundary condition: $k_1(\theta_1 - \alpha) \approx k_2(\theta_2 - \alpha)$ where $\alpha$ is the tilt angle. - If the tilt compensates the vacuum aberration ($\alpha = \theta_1 = -\beta$): - $\theta'_1 = 0$ (vacuum light enters normally in the Sun's frame) - $\theta'_2 = \frac{1 - n}{n}\,\beta$ (light bends according to Snell's law in the Sun's frame) - **Key claim**: in the Sun's frame, light doesn't just slow down — it bends. The interface imposes Snell's law even in the moving frame (to first order). ### Slide 8 — Airy Conclusion (Plane Waves) ![[slide-08.png]] - For small angles and velocities, Snell's law holds at the water surface in both frames. - If the telescope is tilted for normal incidence in the telescope's frame, light proceeds undeviated inside. - In the star's frame, it will be deviated according to Snell's law. - **However**: Snell's law does NOT hold in the general case in the frame where water is moving, because the continuity of the tangential wavevector component is not preserved under the Lorentz boost (eqs 3--5) for arbitrary angles. ### Slide 9 — Airy Telescope Calibration ![[slide-09.png]] - The conversion factor $\Delta\theta_1/\Delta x$ is obtained by measuring transit time $\Delta t$. - $\Delta\theta_1 = \omega \times \Delta t$, where $\omega$ is the angular rate of Earth's orbit. - The angle measured by the crosshairs is that of the **incident light** relative to the telescope axis direction. - With water: the effective focal length becomes $L' = L/n$, and $\Delta\theta_1 \approx n\,\Delta\theta_2 \approx n\,\Delta x / L = \Delta x / L'$. - The calibration accounts for the refraction at the water surface. ### Slide 10 — Wave Packet Propagation ![[slide-10.png]] - For localized signals (wave packets), group velocity is the correct description. - Group velocity: $v_i = \partial\omega / \partial k_i$ - In a non-dispersive, isotropic medium: $n^2 \omega^2 = c^2(k_x^2 + k_y^2 + k_z^2)$ - Differentiating: $v_i = c^2 k_i / (n^2 \omega) = (c/n)(k_i / k)$ - So the group velocity vector is $\mathbf{v} = (c/n)\,\hat{\mathbf{k}}$ — parallel to $\mathbf{k}$ and magnitude $c/n$. - Phase velocity equals group velocity in non-dispersive media. ### Slide 11 — Group Velocity in a Moving Medium ![[slide-11.png]] - If the medium is moving, isotropy is lost. Frequency depends on propagation direction. - Rewriting eqs 3--5 for $k^\mu = (\omega/c,\,\mathbf{k}) = (k/n,\,\mathbf{k})$ in the medium's rest frame, boosted to the frame where medium moves at $V = \beta c\,\hat{\mathbf{x}}$: $c\,k'_x = \gamma\omega(n\sin\theta + \beta) \qquad(13)$ $c\,k'_y = \omega\, n\cos\theta \qquad(14)$ $\omega' = \gamma\omega(1 + \beta\, n\sin\theta) \qquad(15)$ - Differentiating to find group velocity components $\partial\omega'/\partial k'_x$ and $\partial\omega'/\partial k'_y$. ### Slide 12 — Group Velocity Components (Derivation) ![[slide-12.png]] - Setting $dk_y = 0$: $\frac{\partial\omega'}{\partial k'_x} = \frac{c(\sin\theta + \beta n)}{n + \beta\sin\theta} = \frac{(c/n)\sin\theta + V}{1 + (V/cn)\sin\theta}$ - Setting $dk_x = 0$: $\frac{\partial\omega'}{\partial k'_y} = \frac{c\cos\theta}{\gamma(n + \beta\sin\theta)} = \frac{(c/n)\cos\theta}{\gamma\bigl(1 + (V/cn)\sin\theta\bigr)}$ - These are the **relativistic velocity addition** formulas. - For wave packets: $\tan\theta' = v_x / v_y = \gamma(\sin\theta + \beta n)/\cos\theta$ - Note: this differs from the plane wave angle transformation by a factor of $n$ in the $\beta$ term ($\beta n$ vs $\beta/n$). ### Slide 13 — Refraction and Airy's Case (Summary + Wave Packet Example) ![[slide-13.png]] - **Stellar aberration**: the yearly variation of relative velocity between Earth and star causes starlight to arrive at a periodically changing angle. For $n = 1$, plane wave and wave packet descriptions agree. - **Refraction through a moving medium**: for plane waves (distant starlight), the wavevector transformation and boundary conditions are the correct approach. For localized pulses, relativistic velocity addition applies. - The velocity of the wave packet in the moving medium is angle dependent and the direction after crossing an interface is non-trivial — boundary conditions at the interface (Snell's law in the medium's rest frame) are the correct method. - **"Timing" reasonings are incorrect**, since they unjustifiedly assume the direction and value of speed of light in water in the Sun's frame. - **Any argument regarding the angle relative to the telescope which doesn't take boundary conditions into account is flawed.** - Light pulse from distant star along the $y$-axis in the Sun's frame. - In Earth's frame: incident angle $\theta = -\beta \approx 20''$. - Entering water filled vertical telescope: $\theta_2 = -\beta/n \approx -15''$. - Transforming back to the Sun's frame using relativistic velocity addition: $\theta'_2 \approx \theta_2 + \beta n = \beta(n - 1/n) \approx 12''$ - For plane waves, this was $0$ (no deflection in the Sun's frame). - For wave packets, the pulse is dragged by the moving water by $\beta(n - 1/n)$. - **"The value of the speed is left as an exercise to the reader, but it will not be $c/1.33$."** ### Slide 14 — Fresnel Drag Timing Argument ![[slide-14.png]] - Presents the geometric/timing derivation for comparison: - **Vacuum**: light travels vertically at $c$ through telescope of length $L$. Transit time $t = L/c$. Telescope moves $x_T = vL/c$. Angle: $\theta_0 = v/c = \beta$. - **Water + Fresnel drag**: vertical speed $c/n$, transit time $nL/c$. Telescope moves $x_T = vnL/c$. Light dragged horizontally by $x_L = v(1 - 1/n^2) \times nL/c = (n - 1/n)\,vL/c$. Net displacement: $vL/(nc)$. Angle: $\theta_w = v/(cn) = \theta_0/n = \beta/n$. - "That is, Snell's law" — the timing argument with Fresnel drag reproduces the Snell's law result. --- ## See Also - [[1895_Lorentz_Versuch_Airy_Sections|Lorentz 1895]] — Franco's wavevector argument is the relativistic version of Lorentz's "corresponding states" - [[1921_Pauli_Theory_of_Relativity|Pauli 1921]] — Pauli's §36(γ) makes the same "normal incidence" argument Franco makes - [[1871_Airy_Supposed_Alteration_Aberration|Airy 1871]] — The experiment Franco's slides analyze - [[1867_Klinkerfues_Aberration_Wave_Theory|Klinkerfues 1867]] — The wave theory prediction Franco's framework cancels - [[1976a_Jones_Aether_Drag_Transverse_Improved|Jones 1976]] — Player-Rogers correction breaks Franco's "exact cancellation" - [[Franco|Response to Franco]] — Our 7-part rebuttal ## Franco's Core Position (Steel Man) 1. **The correct framework** for analyzing light in a moving medium is the Lorentz transformation of the electromagnetic wave 4-vector, not geometric timing arguments. 2. **Snell's law at the interface** determines what happens when light enters the water filled telescope. In the telescope's rest frame (Earth), the medium is stationary and isotropic, so standard Snell's law applies directly. The angle in water is $\theta_2 = \beta/n$. 3. **What Airy measured** was the apparent position of stars — determined by the direction the telescope must point. This is set by the vacuum aberration angle $\beta$, not the internal water angle $\beta/n$. The water doesn't change the observation because the telescope calibration inherently accounts for the refraction. 4. **Timing arguments are wrong** because they assume light maintains its direction and travels at $c/n$ straight down through the moving water (Sun's frame). In reality, the direction and speed in the moving medium are determined by the relativistic wavevector transformation, and they are not simply $c/n$ in the vertical direction. 5. **The Fresnel drag timing argument** (slide 14) happens to give the right answer ($\beta/n$) for the water angle, but this is because the Fresnel drag coefficient $(1 - 1/n^2)$ was historically calibrated to match the relativistic result. The timing argument is a convenient shortcut, not the fundamental physics. 6. **Plane waves vs wave packets** give different predictions for the Sun's frame angle inside water ($0$ vs $\beta(n - 1/n)$), but both give the same observable: the telescope measures the vacuum aberration angle $\beta$ regardless.