Strong and Weak Gravitational Lensing

Gravitational lensing changes a waveform because spacetime supplies more than one propagation path, not because the source radiates a different intrinsic signal. In geometric optics, every stationary path produces an image with a magnification, arrival time, parity, and Morse phase. Weak lensing usually appears as a small focusing or shear correction; strong lensing produces resolvable or coherently interfering images near critical structure. This page develops those propagation signatures in the thin-lens limit. It does not construct detector likelihoods, and it assumes general-relativistic, luminal tensor waves rather than theory-specific dispersion.

Required background. Waves, Green Functions, and Geometric Optics supplies stationary phase and ray transport; Curvature supplies null-geodesic bending and focusing.

Helpful background. Einstein Equations supplies the weak-field lens metric; Relativistic Cosmology supplies angular-diameter distances and lens redshift.

Fermat delay and image formation

Adopt the thin-lens and small-angle approximations. Let $\boldsymbol{x}$ be a dimensionless position in the lens plane and $\boldsymbol{y}$ the corresponding source position. Write the observed delay as

$$t_d(\boldsymbol{x},\boldsymbol{y}) =t_0T(\boldsymbol{x},\boldsymbol{y}), \qquad T=\frac12\lvert\boldsymbol{x}-\boldsymbol{y}\rvert^2 -\psi(\boldsymbol{x})-\phi_m(\boldsymbol{y}),$$

where $t_0>0$ contains the lens-redshifted mass and distance scale. Here $\phi_m(\boldsymbol{y})$ is defined as the minimum, over $\boldsymbol{x}$, of the preceding geometric-plus-potential terms, so the global minimum has $T=0$. Some conventions instead give the additive shift itself the name $\phi_m$; the explicit minus sign above fixes the translation. Only delay differences are observable, but declaring the zero prevents a hidden frequency-dependent overall phase when formulas are compared.

Images are stationary points of the Fermat potential:

$$\boldsymbol{0}=\boldsymbol{\nabla}_{\!x}T, \qquad \boldsymbol{y}=\boldsymbol{x}-\boldsymbol{\nabla}\psi(\boldsymbol{x}).$$

The inverse magnification matrix is $\mathsf A=\partial\boldsymbol{y}/\partial\boldsymbol{x}$. Its determinant gives the signed magnification

$$\mu_j=\frac{1}{\det\mathsf A_j}.$$

Positive parity means the mapping preserves orientation; negative parity means it reverses orientation. A vanishing determinant marks a critical curve, whose source-plane image is a caustic. The geometric amplitude $\sqrt{\lvert\mu_j\rvert}$ then diverges, signaling failure of geometric optics rather than infinite physical strain. Takahashi and Nakamura give the diffraction integral, its stationary images, and this normalization in Takahashi and Nakamura 2003, §2, pp. 3–8, PDF.

A solved point-mass lens

For a point lens choose $\psi(\boldsymbol{x})=\ln\lvert\boldsymbol{x}\rvert$. Rotational symmetry places the two images on the source–lens line. For $y>0$, the scalar lens equation is

$$y=x-\frac1x,$$

with solutions

$$x_\pm=\frac12\left(y\pm\sqrt{y^2+4}\right).$$

The outer image $x_+>0$ is a minimum of $T$ and has positive parity. The inner image $x_-<0$ is a saddle and has negative parity. Their signed magnifications are

$$\mu_\pm =\frac12\pm \frac{y^2+2}{2y\sqrt{y^2+4}}.$$

Thus $\mu_+>0$, $\mu_-<0$, and the signed identity $\mu_++\mu_-=1$ holds exactly. It is a useful local flux-mapping check; the observable sum of absolute image fluxes is generally larger than one. With the minimum delay set to $T_+=0$, the other image arrives later by

$$\Delta T=T_--T_+ =\frac{y}{2}\sqrt{y^2+4} +\ln\!\left( \frac{\sqrt{y^2+4}+y}{\sqrt{y^2+4}-y} \right)>0.$$

For a point lens, $t_0=4GM_L(1+z_L)$, so $\Delta t_d=t_0\Delta T$. The source and detector use cosmological rest frames, and the lens factor $(1+z_L)$ converts the lens-frame delay to the detector frame.

Waveform copies and Morse phases

Use the transform pair consistent with $e^{-i\omega t}$,

$$h(t)=\int_{-\infty}^{\infty}\tilde h(f)e^{-2\pi ift}\,df, \qquad \tilde h(f)=\int_{-\infty}^{\infty}h(t)e^{2\pi ift}\,dt.$$

A delayed copy therefore carries $e^{2\pi ift_j}$. In the high-frequency stationary-phase limit, the lensed waveform is

$$\tilde h_L(f) =\sum_j\sqrt{\lvert\mu_j\rvert}\, \exp\!\left(2\pi ift_j-i\pi n_j\operatorname{sgn}f\right)\tilde h(f),$$

where $n_j=0$ for a minimum, $1/2$ for a saddle, and $1$ for a maximum. A saddle supplies $-i$ for positive frequency and $+i$ for negative frequency. This continuation ensures $F(-f)=F(f)^*$, as required to map each real polarization into a real polarization. The stationary-phase approximation is not a prescription at $f=0$. For the point lens and $f\gt0$,

$$\tilde h_L =\left[ \sqrt{\mu_+} -i\sqrt{\lvert\mu_-\rvert} e^{2\pi if\Delta t_d} \right]\tilde h$$

after removing the common minimum arrival time. Each image has the same source-frame dynamics and the same cosmological redshift. Lensing changes propagation amplitude, delay, screen transport, and Morse phase; it does not manufacture an independent chirp mass, spin evolution, or additional physical polarization.

For a broadband real signal, the saddle operation is a frequency-sign-dependent quadrature transform, not multiplication of the entire time series by $-i$. Only a narrowband oscillation can generally absorb it into one constant waveform phase. The positive-frequency image expression is Takahashi and Nakamura 2003, p. 6, eqs. (15) and (18), PDF; the negative-frequency factor follows by complex conjugation.

Weak and strong regimes

The comparison axes are the number of relevant stationary paths, the size of the lens-map distortion, the resulting waveform transformation, and the approximation’s failure signal.

Regime	Lens map	Waveform consequence	Failure signal
Weak lensing	One dominant branch with small convergence and shear	Nearly unlensed waveform with a modest amplitude and direction change	A secondary branch or caustic becomes important
Strong lensing	Several relevant stationary paths, often with different parities	Delayed, magnified copies with fixed Morse factors	Images merge at a caustic
Coherent overlap	Delays are short enough that copies overlap in frequency analysis	Frequency-dependent interference fringes	A discrete image sum fails when diffraction is important

Strong lensing here means multiple imaging; it does not require a strong local gravitational potential. The point-mass thin-lens calculation can describe strong lensing while every relevant ray remains far outside the lens’s gravitational radius.

For the point lens, $y\gg1$ gives $\mu_+\to1$, $\mu_-\to0^-$, so both the signed sum and the total absolute magnification approach the unlensed value. Near $y=0$, geometric magnification diverges and wave optics must replace the image sum. More generally, global flux conservation is an ensemble statement requiring the full sky and all images; it should not be inferred from one magnified line of sight.

This image construction assumes the same short-wavelength hierarchy that underlies null-ray propagation and amplitude focusing Flanagan and Hughes 2005, §5, pp. 23–29, PDF. When the lens varies appreciably across one wavelength, stationary rays cease to be a complete description even if the background remains weak.

Intrinsic waveform structure can mimic some individual effects—for example, a phase change can resemble a Morse factor over a restricted band. The propagation hypothesis is more restrictive: several images must be related by a common emitted waveform, consistent redshifted intrinsic scales, and lens-compatible amplitude and delay relations. Turning those relations into a statistical detection test belongs to gravitational-wave data analysis, not to this propagation derivation.

Exercise

For a point lens with $y=1$, compute $x_\pm$, determine each parity, verify the signed-magnification identity, and identify which image arrives first and which Morse factor multiplies the later image.

Solution

The image positions are $x_+=(1+\sqrt5)/2$ and $x_-=(1-\sqrt5)/2$. The outer solution is a minimum with positive parity; the inner solution is a saddle with negative parity. Their magnifications are

$$\mu_+=\frac12+\frac{3}{2\sqrt5}, \qquad \mu_-=\frac12-\frac{3}{2\sqrt5},$$

so $\mu_++\mu_-=1$. By the declared delay zero, the minimum has $T_+=0$ and arrives first. The later saddle has $n_-=1/2$ and hence the positive-frequency Morse factor $e^{-i\pi/2}=-i$. Its absolute strain amplification is $\sqrt{\lvert\mu_-\rvert}$.

References

• Éanna É. Flanagan and Scott A. Hughes, “The Basics of Gravitational Wave Theory,” New Journal of Physics 7 (2005) 204, arXiv:gr-qc/0501041v3, DOI, Open PDF.
• Ryuichi Takahashi and Takashi Nakamura, “Wave Effects in the Gravitational Lensing of Gravitational Waves from Chirping Binaries,” The Astrophysical Journal 595 (2003) 1039–1051, arXiv:astro-ph/0305055v2, DOI, Open PDF.