Chapter 4 — Deflection from Transverse Time-Gradients
Rays bend where the rate of time changes across the path. The bending tracks transverse gradients of \(\Phi\).
Goal of this chapter
- Write the bending angle in terms of transverse gradients of \(\Phi\).
- Show agreement with the classic point-mass result.
- Connect to thin-lens notation used in weak lensing.
Set-up
Work in the Born approximation. The ray is nearly straight. Use an unperturbed path coordinate \(z\) and a transverse impact parameter \(b\). Define the transverse gradient as \(\nabla_{\!\\perp}\), orthogonal to the line of sight \(\hat{\mathbf n}\).
\[
\nabla_{\!\\perp} \;\equiv\; \bigl(\mathbf I - \hat{\mathbf n}\hat{\mathbf n}^{\\top}\bigr)\nabla.
\]
Bending from the temporal–optics map
Fermat’s principle for a slowly varying index gives a small deflection proportional to the cross-path index gradient integrated along the line of sight.
Conventions. The paper shows both \(n=e^{\Phi}\) and \(n_{\gamma}=e^{-2\Phi}\). Use the form that keeps factors correct for your section. They are related by \(n_{\gamma}=n^{-2}\).
Form 1 — using \(n_{\gamma}=e^{-2\Phi}\) (matches the paper’s lensing figures)
\[
\boldsymbol{\alpha}(\hat{\mathbf n})
\;\simeq\; \int \nabla_{\!\\perp}\ln n_{\gamma}\,\mathrm{d}z
\;=\; -\,2\int \nabla_{\!\\perp}\Phi\,\mathrm{d}z.
\]
Form 2 — using \(n=e^{\Phi}\)
In this convention the same leading-order physics appears as
\[
\boldsymbol{\alpha}(\hat{\mathbf n})
\;\simeq\; 2\int \nabla_{\!\\perp}\Phi\,\mathrm{d}\ell,
\]
where the factor of 2 matches the \(\ln n_{\gamma}\) expression above. Use one convention per calculation and keep track of factors.
Worked example: point mass
Take \(\Phi(r)\approx -\,GM/(c^2 r)\) with \(r=\sqrt{b^2+z^2}\). The transverse gradient is \(\nabla_{\!\\perp}\Phi = \partial\Phi/\partial b = (GM/c^2)\, b/r^3\).
\[
\int_{-\infty}^{+\infty} \frac{b\,\mathrm{d}z}{(b^2+z^2)^{3/2}}
\;=\; \frac{2}{b}.
\]
Insert this into the \(n_{\gamma}\) form.
\[
\alpha(b) \;\simeq\; -\,2\int \nabla_{\!\\perp}\Phi\,\mathrm{d}z
\;=\; -\,2\left(\frac{GM}{c^2}\right)\frac{2}{b}.
\]
The direction is toward the mass. The magnitude matches the classic result:
\[
|\alpha(b)| \;=\; \frac{4GM}{c^2 b}.
\]
This agrees with the standard weak-field GR prediction.
Thin-lens language
Project the potential along the line of sight. Define a 2D lensing potential \(\psi\) on the sky. Use angular coordinates \(\boldsymbol{\theta}\) with distance factors absorbed in \(\psi\).
\[
\psi(\boldsymbol{\theta}) \;\propto\; \int \Phi\,\mathrm{d}z,
\qquad
\boldsymbol{\alpha}(\boldsymbol{\theta}) \;=\; \nabla_{\!\\boldsymbol{\theta}}\,\psi(\boldsymbol{\theta}).
\]
Convergence and shear follow as derivatives of \(\psi\).
\[
\kappa \;=\; \tfrac{1}{2}\nabla_{\!\\boldsymbol{\theta}}^2 \psi,
\qquad
\gamma_1,\gamma_2 \;\text{ from second derivatives of }\psi.
\]
This matches weak-lensing notation. It links the temporal potential to standard 2D lens maps.
Validity and small parameters
- Weak field: \(|\Phi|\ll 1\).
- Small angles: the Born approximation holds.
- Quasi-static lens: \(\bigl|\partial_t\Phi/\Phi\bigr|\,T_{\mathrm{path}} \ll 1\).
- Thin lens (optional): \(L_{\mathrm{lens}}/(c\,\tau_{\mathrm{var}})\ll 1\).
Sanity checks
- Point mass gives \(|\alpha|=4GM/(c^2 b)\).
- Double the impact parameter. The deflection halves.
- Set \(\Phi\to 0\). Rays remain straight.
Common pitfalls
- Mixing \(n=e^{\Phi}\) and \(n_{\gamma}=e^{-2\Phi}\) mid-derivation. Convert with \(n_{\gamma}=n^{-2}\).
- Dropping the factor of 2 when switching between \(\nabla_{\!\\perp}\ln n_{\gamma}\) and \(\nabla_{\!\\perp}\Phi\).
- Using a curved path in the integral while assuming the Born approximation.
What you have now
A bending formula tied to cross-path time-gradients. A point-mass check that recovers \(\,4GM/(c^2 b)\,\). A bridge to thin-lens notation.