Gravitational Lensing: Deflection of Light

1. The Geodesic Equation for Light

Just as we did for massive particles, we start with the geodesic equations derived from the Schwarzschild metric. The crucial difference lies in the constraint we apply. For a massive particle, the worldline is timelike, and we parameterize it by proper time $\tau$, which enforces the condition $g_{\mu \nu }\frac{d{x}^{\mu }}{d\tau }\frac{d{x}^{\nu }}{d\tau }=-c^2$. For a photon, the worldline is null, meaning the spacetime interval along its path is always zero: $d{s}^2=0$.

We can follow a very similar procedure to the one we used for massive orbits, leveraging the conserved quantities associated with the time-translation and rotational symmetries of the Schwarzschild metric.

Let's summarize the key steps and results from the reading.

1. The Null Constraint: We start with the Schwarzschild metric in the equatorial plane ($d\theta =0,\theta =\pi /2$) and set $d{s}^2=0$:

   $$0=-\left(1-\frac{2M}{r}\right)c^{2}d{t}^2+{\left(1-\frac{2M}{r}\right)}^{-1}d{r}^2+r^{2}d{\varphi }^2$$

   Here, we've restored $c$ and are using $M$ as a shorthand for the mass of the central object, not $GM/c^2$ just yet.

2. Conserved Quantities: The conserved energy $E$ and angular momentum $L$ per unit 'whatever' (since photons have no mass, these are just parameters of the geodesic) are still given by:

   $$E=m{c}^2\left(1-\frac{2M}{r}\right)\frac{dt}{d\lambda },L=m{r}^2\frac{d\varphi }{d\lambda }$$

   where $\lambda$ is an affine parameter along the null geodesic. The ratio of these two conserved quantities, which we can call the impact parameter $b=Lc/E$, is what determines the trajectory.

3. The Orbit Equation: After substituting the conserved quantities and changing variables to $u=1/r$, just as we did for planetary orbits, we arrive at the differential equation for the photon's path:

   $$\frac{d^{2}u}{d{\varphi }^2}+u=\frac{3GM}{c^2}{u}^2$$

   Let's pause to appreciate this equation. Compare it to the one for massive particles: $\frac{d^{2}u}{d{\varphi }^2}+u=\frac{GM}{{\tilde{L}}^2}+\frac{3GM}{c^2}{u}^2$. The term corresponding to the Newtonian inverse-square force ($GM/{\tilde{L}}^2$) is gone. The entire dynamics of light bending is governed by the purely relativistic $u^2$ term!

2. Calculating the Deflection Angle

This equation is non-linear, but for a light ray that only grazes the Sun, the relativistic term is small, making it perfect for a perturbative solution. We assume the path is a small deviation from a straight line.

The zeroth-order solution (when the right-hand side is zero) is a straight line. In polar coordinates, a straight line that passes a distance $b$ from the origin is given by $r\cos \varphi =b$, or:

Here, $b$ is the impact parameter, the closest distance the light ray would have approached the central mass if there were no gravity.

Now, we feed this approximate solution back into the right-hand side of the full equation and solve again. The same resource you just read carries out this calculation beautifully. Following the steps in Tong's notes (Section 1.3.5, from equation 1.54 onwards) leads to the final deflection angle $\Delta \varphi$. The total deflection is the difference between the incoming and outgoing asymptotes. For a straight line, this would be $\pi$ radians. The GR path gives a total angle of $\pi +\Delta {\varphi }_{bend}$. The perturbative calculation yields:

This is the famous result for the gravitational deflection of light. It predicts that the path of a light ray grazing a massive object is bent by this small angle. For a ray just grazing the limb of the Sun, $b\approx R_{\odot }$, the angle is about 1.75 arcseconds.

3. Comparison with the Newtonian Prediction

One of the most striking aspects of this result is that it's exactly twice the "Newtonian" prediction. But what does a Newtonian prediction for light bending even mean? We can arrive at one by treating a photon as a classical particle with some effective mass and calculating its trajectory in a Newtonian $1/r^2$ force field.

As the notes show, the Newtonian calculation gives a deflection angle of:

where $r_{min}$ is the distance of closest approach, equivalent to our impact parameter $b$.

So, $\Delta {\varphi }_{GR}=2\times \Delta {\varphi }_{Newton}$. Why the factor of two?

• One half of the deflection ($\frac{2GM}{b{c}^2}$) can be attributed to the way gravity affects the flow of time (the $g_{tt}$ component of the metric). This is the part that can be loosely reasoned from the Equivalence Principle.
• The other half comes from the curvature of space itself (the $g_{rr}$ component). This is a purely General Relativistic effect, with no Newtonian analogue. In essence, light is deflected because it follows a straight line (a geodesic) through a spacetime that is itself curved in both its temporal and spatial dimensions.

Arthur Eddington's 1919 eclipse expedition measured the deflection of starlight grazing the Sun and found a result consistent with the GR prediction, not the Newtonian one. This was hailed as a major triumph for Einstein's theory.

4. Gravitational Lensing

The deflection of light is not just a theoretical curiosity; it's a fundamental tool in modern astronomy. When a massive object (like a star, a galaxy, or a cluster of galaxies) lies between us and a distant light source, it acts as a gravitational lens, bending the light and distorting the image of the source.

Depending on the alignment and mass distribution, this can lead to spectacular effects.

5. Extreme Bending: The Photon Sphere

The perturbative calculation works for light passing far from the central mass. What happens if the light gets very close? As David Tong's notes briefly mention, the effective potential for photons has a maximum. This corresponds to an unstable circular orbit for light.

The video explains that at a radius of $r=\frac{3}{2}{R}_s$, known as the photon sphere, light can orbit the black hole. However, this orbit is unstable. Any light ray passing just inside this sphere will spiral into the black hole, while any ray passing just outside will be sharply deflected and escape. This extreme light bending near the photon sphere is responsible for the bright ring seen in the Event Horizon Telescope's image of the M87* black hole.

Conclusion

Today we've made the leap from timelike to null geodesics, uncovering another key prediction of General Relativity.

Key Takeaways:

• The orbit of a light ray in Schwarzschild spacetime is governed by the equation $\frac{d^{2}u}{d{\varphi }^2}+u=\frac{3GM}{c^2}{u}^2$.
• A perturbative solution yields a deflection angle of $\Delta \varphi =\frac{4GM}{b{c}^2}$, where $b$ is the impact parameter.
• This result is exactly twice the prediction from a Newtonian model, a difference that was experimentally confirmed by Eddington in 1919. The factor of two arises from the combined curvature of both time and space.
• This deflection is the basis for gravitational lensing, a crucial tool in modern astrophysics that produces phenomena like Einstein rings, multiple images of quasars, and microlensing events.
• In the strong-field limit, there exists an unstable circular orbit for photons at $r=1.5R_s$, known as the photon sphere.

Preview of Next Lesson:

We've now analyzed orbits of both massive and massless particles and seen the success of the Schwarzschild metric. However, we've also encountered strange behavior at $r=R_s$, the Schwarzschild radius. In our next lesson, we will analyze the coordinate singularity at the event horizon and remove it using Kruskal-Szekeres coordinates, paving the way to understand the full, bizarre geometry of a black hole.