Orbits and Light Rays in Schwarzschild Spacetime

1. The Geodesic Equation as an Optimization Problem

Recall from our study of differential geometry that a geodesic is a curve whose tangent vector is parallel-transported along itself (${\nabla }_{\dot{\gamma }}\dot{\gamma }=0$). For the Levi-Civita connection, this is also the curve that extremizes the proper time (for massive particles) or an affine parameter (for massless particles) between two points. This allows us to use the calculus of variations, which is the foundation of Lagrangian mechanics.

The Lagrangian for a particle moving in a spacetime with metric $g_{\mu \nu }$ is:

where the dot denotes differentiation with respect to an affine parameter $\lambda$ (which can be proper time $\tau$ for massive particles). The Euler-Lagrange equations, $\frac{d}{d\lambda }\left(\frac{\partial L}{\partial {\dot{x}}^{\alpha }}\right)-\frac{\partial L}{\partial x^{\alpha }}=0$, give us the geodesic equations.

For the Schwarzschild metric, the Lagrangian is:

(We're using units where $G=c=1$, so $r_s=2M$.)

Also, the quantity $2L=g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }$ is itself a constant of the motion. Its value normalizes the 4-velocity:

• For massive particles (timelike geodesics): $g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }=-1$
• For massless particles (null geodesics): $g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }=0$

2. Conserved Quantities and the Radial Equation

Instead of grinding through all the Euler-Lagrange equations, we can use symmetries, just as in classical mechanics. The Schwarzschild metric is independent of the coordinates $t$ and $\varphi$. This corresponds to spacetime symmetries: time translation invariance and axial (rotational) symmetry. By Noether's theorem, these symmetries imply conserved quantities.

The conserved quantity associated with a coordinate $x^{\alpha }$ is the conjugate momentum, $p_{\alpha }=\frac{\partial L}{\partial {\dot{x}}^{\alpha }}$.

1. Energy: The metric is independent of $t$.
   $p_t=\frac{\partial L}{\partial \dot{t}}=g_{tt}\dot{t}=-\left(1-\frac{2M}{r}\right)\dot{t}$
   We define the conserved energy per unit mass, $\tilde{E}\equiv -p_t$.
   $\tilde{E}=\left(1-\frac{2M}{r}\right)\dot{t}$

2. Angular Momentum: The metric is independent of $\varphi$.
   $p_{\varphi }=\frac{\partial L}{\partial \dot{\varphi }}=g_{\varphi \varphi }\dot{\varphi }=r^2{\sin }^2\theta \dot{\varphi }$
   We define the conserved angular momentum per unit mass, $\tilde{L}\equiv p_{\varphi }$.
   $\tilde{L}=r^2{\sin }^2\theta \dot{\varphi }$

Due to the spherical symmetry, any geodesic that starts in the equatorial plane ($\theta =\pi /2,\dot{\theta }=0$) will remain there. We can therefore simplify our analysis, without loss of generality, by setting $\theta =\pi /2$ and $\sin \theta =1$. This gives $\tilde{L}=r^2\dot{\varphi }$.

Now, we substitute these conserved quantities into the normalization condition $g_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }=-ϵ$, where $ϵ=1$ for massive particles and $ϵ=0$ for photons.

Substitute $\dot{t}=\frac{\tilde{E}}{1-2M/r}$ and $\dot{\varphi }=\frac{\tilde{L}}{r^2}$:

Multiplying by $(1-2M/r)$ and rearranging for ${\dot{r}}^2$ gives us the master radial equation of motion:

This single equation, with the particle's energy ($\tilde{E}$) and angular momentum ($\tilde{L}$) as parameters, contains a vast amount of physics. We can analyze it by treating the second term as an effective potential.

The following reading develops this idea by analyzing the right-hand side as a polynomial in $x=1/r$, which is a very powerful and elegant way to classify all possible orbits.

3. Timelike Geodesics: Orbits of Massive Particles

Let's set $ϵ=1$ for massive particles. The radial equation becomes:

We can define an effective potential $V_{eff}(r)$ such that $\frac{1}{2}{\dot{r}}^2+V_{eff}(r)=\frac{1}{2}{\tilde{E}}^2$. Expanding the term above gives:

This looks very familiar!

• $-\frac{M}{r}$: The Newtonian gravitational potential.
• $\frac{{\tilde{L}}^2}{2r^2}$: The classical centrifugal barrier that prevents a particle with angular momentum from falling straight in.
• $-\frac{M{\tilde{L}}^2}{r^3}$: A new, purely relativistic term. It's an attractive force that dominates at small radii and fundamentally changes the nature of orbits close to the central mass.

By plotting $V_{eff}(r)$ for a fixed $\tilde{L}$, we can visualize the possible orbits for a particle with energy $\tilde{E}$.

A sketch of the effective potential for massive particles for a sufficiently large angular momentum. The horizontal lines represent different energy levels $\tilde{E}$.

Circular Orbits and Stability

Circular orbits are possible at radii where the effective force is zero, i.e., at the extrema of the potential where $d{V}_{eff}/dr=0$. The stability of these orbits depends on the second derivative:

• Stable circular orbits: Occur at minima of $V_{eff}$ ($d^2{V}_{eff}/d{r}^2>0$). A small perturbation will cause the particle to oscillate around the stable radius.
• Unstable circular orbits: Occur at maxima of $V_{eff}$ ($d^2{V}_{eff}/d{r}^2<0$). A small perturbation will cause the particle to either spiral inwards or escape outwards.

As the angular momentum $\tilde{L}$ decreases, the maximum and minimum of the potential move closer together. At a critical value of $\tilde{L}$, they merge at an inflection point. For any $\tilde{L}$ below this value, the potential has no local extrema, meaning no stable circular orbits are possible.

This leads to one of the most important predictions of General Relativity:

The Innermost Stable Circular Orbit (ISCO)
There is a minimum radius at which a massive particle can have a stable circular orbit. Inside this radius, all circular orbits are unstable. This radius is the ISCO.

As derived in the notes, the ISCO is located at:

This has profound astrophysical implications. For matter in an accretion disk spiraling into a black hole, the inner edge of the disk is typically located at the ISCO. Inside this radius, particles can no longer maintain a stable orbit and will quickly plunge into the black hole, releasing a final burst of energy.

4. Null Geodesics: The Path of Light

Now let's analyze the motion of massless particles like photons by setting $ϵ=0$. The radial equation simplifies to:

It is convenient to define the impact parameter, $b\equiv \tilde{L}/\tilde{E}$. For a photon coming from infinity, $b$ represents the perpendicular distance between the initial path and the central mass. Dividing by ${\tilde{E}}^2$, we get:

The turning points of a photon's trajectory occur where $\dot{r}=0$, which gives the condition:

The Photon Sphere

Is a circular orbit for a photon possible? This would require the effective potential for photons to have a peak. Looking at the turning point condition $b^2(r)$, we can find the minimum possible value of $b^2$ for a turning point by finding where $d{b}^2/dr=0$. This calculation yields a unique radius:

This is the radius of the photon sphere. At this exact radius, photons can orbit the black hole in a circle. However, this orbit is highly unstable: any small perturbation will send the photon either spiraling into the black hole or escaping to infinity.

Capture vs. Deflection

The photon sphere defines a critical impact parameter, $b_{crit}$. By substituting $r=3M$ into the turning point equation, we find:

The fate of a photon approaching a black hole depends on how its impact parameter $b$ compares to $b_{crit}$:

• If $b>b_{crit}$, the photon is deflected by the black hole's gravity but escapes back to infinity. This is the phenomenon of gravitational lensing.
• If $b<b_{crit}$, the photon is captured and inevitably plunges past the event horizon.
• If $b=b_{crit}$, the photon asymptotically approaches the circular orbit at the photon sphere, orbiting the black hole many times before (in any realistic scenario) either escaping or being captured due to a tiny perturbation.

The critical impact parameter defines the apparent "size" of the black hole to a distant observer. The dark region we see is not the event horizon itself, but a larger area called the black hole shadow, with a radius of $b_{crit}$.

Conclusion

In this lesson, we have moved from the static geometry of the Schwarzschild metric to the dynamics of particles and light within it. By leveraging the concepts of conserved energy and angular momentum, we derived a powerful radial equation of motion that allowed us to classify all possible trajectories.

Key Takeaways:

• Symmetries of the Schwarzschild metric lead to conserved quantities: energy $\tilde{E}$ and angular momentum $\tilde{L}$.
• The trajectories of massive and massless particles can be analyzed using an effective potential. The relativistic correction term $-M{\tilde{L}}^2/r^3$ is responsible for new phenomena not found in Newtonian gravity.
• For massive particles, there exists an Innermost Stable Circular Orbit (ISCO) at $r=6M$. Inside this radius, no stable circular orbits are possible.
• For photons, there is an unstable circular orbit at the photon sphere at $r=3M$.
• A critical impact parameter $b_{crit}=\sqrt{27}M$ determines whether a photon is captured or deflected. This value defines the size of the black hole's shadow.

Preview of Next Lesson:

We have now laid all the groundwork necessary to confront General Relativity with experimental evidence. Using the geodesic equations we've analyzed today, our next lesson will focus on two of the classic tests of GR: the perihelion precession of Mercury and the deflection of light by the Sun. We will perform the calculations to see how Einstein's theory succeeded where Newton's failed.