Schwarzschild Spacetime: Horizons and Singularities

Welcome back to our study of the Schwarzschild solution. In the previous lesson, we accomplished the significant task of deriving the Schwarzschild metric by solving the vacuum Einstein Field Equations for a spherically symmetric, static spacetime. We arrived at the famous line element:

d s^{2} = - (1 - \frac{2 GM}{c ^{2} r}) c^{2} d t^{2} + (1 - \frac{2 GM}{c ^{2} r})^{- 1} d r^{2} + r^{2} (d θ^{2} + sin^{2} θ d ϕ^{2})

We also introduced the Schwarzschild radius, $r_{s} = 2 GM / c^{2}$ , which simplifies the expression.

This metric is our first window into the profound consequences of general relativity, describing the spacetime around everything from planets to black holes. However, a quick inspection reveals two potential trouble spots: the metric components misbehave at $r = r_{s}$ and at $r = 0$ . What is the physical meaning of these locations?

Today's lesson directly addresses this question. Our learning outcome is to analyze the causal structure of Schwarzschild spacetime, identifying the event horizon and the physical singularity. We will dissect the metric to understand what these 'singularities' truly represent, how they shape the paths of light and matter, and what it means to cross the point of no return.

1. Identifying Singularities: Physical vs. Coordinate

Let's rewrite the metric using $r_{s}$ :

d s^{2} = - (1 - \frac{r _{s}}{r}) c^{2} d t^{2} + (1 - \frac{r _{s}}{r})^{- 1} d r^{2} + r^{2} d Ω^{2}

The metric components exhibit pathological behavior at two radii:

At $r = r_{s}$ , the coefficient of $d t^{2}$ vanishes ( $g_{tt} \to 0$ ), and the coefficient of $d r^{2}$ diverges ( $g_{r r} \to \infty$ ).
At $r = 0$ , the coefficient of $d r^{2}$ also diverges, and the logic seems to break down completely.

It is crucial to distinguish between two types of singularities:

Coordinate Singularity: An artifact of a poor choice of coordinates. The underlying spacetime geometry is perfectly smooth, much like the origin in polar coordinates where the metric $d s^{2} = d r^{2} + r^{2} d θ^{2}$ seems ill-defined but the flat plane itself is fine.
Physical Singularity (or Curvature Singularity): A point where the spacetime geometry itself is ill-defined. Curvature becomes infinite, and the laws of physics as we know them break down. This is a true feature of the manifold, not a coordinate artifact.

To tell them apart, we must compute a quantity that is independent of our coordinate system—a scalar invariant built from the curvature tensor. The simplest choice, the Ricci scalar $R$ , is zero in a vacuum solution. The next simplest is the Kretschmann scalar, $K = R_{μν ρ σ} R^{μν ρ σ}$ . If this scalar diverges, we have found a true physical singularity.

6.1.2 A First Look at the Horizon

The following text, 'A First Look at the Horizon' from David Tong's lecture notes on General Relativity, provides a concise calculation of the Kretschmann scalar for the Schwarzschild metric and uses it to distinguish between the singularities at r=0 and r=r_s.

Please read the section '6.1.2 A First Look at the Horizon'. Pay close attention to the formula for the Kretschmann scalar (Equation 6.258 in the text) and the conclusions drawn from it about the nature of the points r=0 and r=r_s.

As the text shows, the Kretschmann scalar for the Schwarzschild solution is:

K = R_{μν ρ σ} R^{μν ρ σ} = \frac{48 G ^{2} M ^{2}}{r ^{6}} = \frac{12 r _{s}^{2}}{r ^{6}}

Let's analyze this result:

As $r \to 0$ , $K \to \infty$ . The curvature scalar diverges, confirming that $r = 0$ is a true physical singularity. This is a point where our theory of classical gravity fails.
At $r = r_{s}$ , $K = 12 r_{s}^{2} / r_{s}^{6} = 12/ r_{s}^{4}$ . This is a finite, non-zero value. The fact that the curvature is finite strongly suggests that $r = r_{s}$ is merely a coordinate singularity.

The surface at $r = r_{s}$ is known as the event horizon. While the geometry there is locally smooth, it possesses profound global properties that define the boundary of a black hole.

2. Causal Structure at the Event Horizon

Let's investigate the causal structure near $r = r_{s}$ using the Schwarzschild coordinates. The paths of light rays (null geodesics) define the boundaries of the light cones and thus dictate what is causally possible. For a purely radial light ray ( $d θ = d ϕ = 0$ ), we set $d s^{2} = 0$ :

0 = - (1 - \frac{r _{s}}{r}) c^{2} d t^{2} + (1 - \frac{r _{s}}{r})^{- 1} d r^{2}

Rearranging for the slope of the worldline on a spacetime diagram gives:

\frac{d t}{d r} = \pm \frac{1}{c ( 1 - \frac{r _{s}}{r} )}

Let's examine the behavior of this slope:

Far from the mass ( $r ≫ r_{s}$ ): $\frac{d t}{d r} \approx \pm \frac{1}{c}$ , which gives the standard $4 5^{\circ}$ light cones of flat Minkowski space (if we plot $c t$ vs $r$ ).
Approaching the horizon ( $r \to r_{s}^{+}$ ): The term $(1 - r_{s} / r) \to 0$ , so $\frac{d t}{d r} \to \pm \infty$ . The light cones "close up" and become vertical.

This means that for an observer far away, a light ray sent towards the black hole appears to take an infinite amount of coordinate time $t$ to reach the horizon. This is a manifestation of extreme gravitational time dilation.

What happens inside the horizon, for $r < r_{s}$ ? The term $(1 - r_{s} / r)$ becomes negative. This has a dramatic consequence for the character of the coordinates:

The $g_{tt}$ component, $- (1 - r_{s} / r) c^{2}$ , becomes positive.
The $g_{r r}$ component, $(1 - r_{s} / r)^{- 1}$ , becomes negative.

This means that inside the event horizon, the time coordinate $t$ becomes spacelike, and the radial coordinate $r$ becomes timelike! Moving along the $r$ direction is no longer a choice; it is as inevitable as moving into the future. Since the $r$ coordinate is now the timelike direction, and the singularity is at $r = 0$ , all future-directed worldlines must point towards decreasing $r$ . Once you cross the event horizon, you are causally bound to travel to the singularity at $r = 0$ .

The following video masterfully illustrates this analysis.

Relativity 108b: Schwarzschild Metric - Interpretation (Gravitational Time Dilation, Event Horizon)

The video 'Relativity 108b' by eigenchris provides an excellent visualization and derivation of the behavior of light rays near the event horizon. It also explains the swapping of the roles of the time and radial coordinates.

Please watch the segment from 20:08 to 31:36. This section covers: Light-like Geodesics: Derivation of the differential equation for radial light rays. Spacetime Diagrams: Visualization of how ingoing and outgoing light rays behave outside, at, and inside the event horizon. Light Cones and Causal Structure: Explanation of how the light cones tilt and flip inside the horizon. Swapping of Coordinates: A clear explanation of how the time coordinate becomes spacelike and the radial coordinate becomes timelike for r < r_s.

Test your understanding!

Inside the event horizon ( $r < r_{s}$ ), the coordinate $r$ is timelike and $t$ is spacelike. What does this imply about an object's ability to remain at a constant radius $r$ ? How does this relate to the phrase "the singularity at $r = 0$ is a future moment in time, not a place in space"?

Show answer

To remain at a constant radius $r$ means $d r = 0$ . Since $r$ is the timelike coordinate inside the horizon, requiring $d r = 0$ is equivalent to requiring time to stand still. This is impossible for any massive or massless particle. All physical objects must advance along their timelike direction.

Because $r$ is the timelike coordinate and decreases towards the singularity, moving towards $r = 0$ is equivalent to moving forward in time. Therefore, once inside the horizon, the singularity at $r = 0$ is not a location you can avoid by accelerating in a different direction; it is an inevitable future event for your worldline, just like "next Tuesday" is for us.

3. Removing the Coordinate Singularity

The strange behavior in Schwarzschild coordinates—infinite time to cross the horizon, the swapping of coordinate roles—cements the idea that these coordinates are ill-suited for describing the physics at and across the horizon. A better coordinate system should show that an infalling observer crosses the horizon in a finite amount of their own proper time without noticing anything locally peculiar.

The first step towards such a system is to define the tortoise coordinate, $r^{*}$ .

d r^{*} = \frac{d r}{1 - r _{s} / r}

Integrating this gives $r^{*} = r + r_{s} ln \frac{r}{r _{s}} - 1$ . Notice that as $r \to r_{s}$ , $r^{*} \to - \infty$ . This coordinate change "pushes" the horizon infinitely far away, which isn't a full solution but is a useful intermediate step.

The key insight, developed by Eddington and Finkelstein, is to define a new time coordinate based on the paths of light rays. We can define an ingoing (or advanced) null coordinate $v$ as:

v = c t + r^{*}

Surfaces of constant $v$ correspond to ingoing radial light rays. By changing from coordinates $(t, r)$ to $(v, r)$ , we can construct a new metric.

6.1.3 Eddington-Finkelstein Coordinates

The section 'Ingoing Eddington-Finkelstein Coordinates' from David Tong's notes formally carries out this coordinate transformation and presents the resulting metric.

Please read the subsection 'Ingoing Eddington-Finkelstein Coordinates' and the following one, 'The Finkelstein Diagram'. Focus on how the new metric (Equation 6.263) is no longer singular at r = r_s. Then, examine the Finkelstein diagram (Figure 44) to see how this new coordinate system depicts the causal structure, showing that the horizon is now traversable.

The resulting metric in ingoing Eddington-Finkelstein coordinates is:

d s^{2} = - (1 - \frac{r _{s}}{r}) d v^{2} + 2 d v d r + r^{2} d Ω^{2}

Look at what has been achieved:

None of the metric components diverge at $r = r_{s}$ . The coordinate singularity has been removed.
The metric is not diagonal. The $d v d r$ cross-term is crucial; it ensures the metric remains non-degenerate at the horizon.
The Finkelstein diagram, which plots worldlines in these new coordinates, clearly shows that infalling particles and light rays pass smoothly through the event horizon in finite coordinate time.
The diagram also visually confirms that once inside, the future light cone is tilted such that all paths, including those of "outgoing" light rays, inevitably move towards smaller $r$ and terminate at the singularity.

This confirms our analysis: the event horizon is not a physical barrier but a causal one—a true point of no return.

Conclusion

In this lesson, we have moved from a purely mathematical formula for the Schwarzschild metric to a deep physical understanding of its structure. We have mapped out the causal landscape of the spacetime around a non-rotating black hole.

Key Takeaways:

The Schwarzschild metric has two points of interest: a true physical singularity at $r = 0$ , where curvature diverges, and a coordinate singularity at the event horizon $r = r_{s}$ , where curvature is finite.
In Schwarzschild coordinates, the event horizon appears as a surface that takes infinite coordinate time to reach, an effect of extreme gravitational time dilation.
Inside the event horizon ( $r < r_{s}$ ), the roles of the time and radial coordinates interchange: $r$ becomes timelike and $t$ becomes spacelike. This makes the journey to the singularity at $r = 0$ inevitable.
The coordinate singularity at the horizon can be removed by changing to Eddington-Finkelstein coordinates, which show that the horizon is a smooth, traversable null surface.
The event horizon is a one-way membrane: once crossed, the causal structure of spacetime forbids any escape and dictates that all worldlines end at the physical singularity.

Preview of Next Lesson:

Now that we have a solid grasp of the causal 'arena' of Schwarzschild spacetime, we are ready to analyze the actual trajectories of particles and light within it. In the next lesson, we will solve the geodesic equations in the Schwarzschild metric to derive the paths of planets, stars, and light rays, setting the stage to understand the classical tests of general relativity.

Recommended resources

1. Identifying Singularities: Physical vs. Coordinate

2. Causal Structure at the Event Horizon

3. Removing the Coordinate Singularity

Conclusion