Deriving the Schwarzschild Metric

Hello! Welcome to the first lesson of our fourth module. We've spent the last three modules building the complete theoretical framework of General Relativity. We now have the Einstein Field Equations (EFE), a deep understanding of spacetime curvature, and a powerful set of tools for analyzing symmetries through Killing vector fields. It's time to put this machinery to its first great test: finding an exact, non-trivial solution to the EFE.

Our goal today is to derive the spacetime geometry outside a non-rotating, uncharged, spherically symmetric body of mass $M$ . This celebrated result, first found by Karl Schwarzschild in 1916, describes everything from the spacetime around the Earth (to a good approximation) to the exterior of a non-rotating black hole.

The learning outcome for this lesson is to derive the Schwarzschild metric as the unique static, spherically symmetric vacuum solution (Birkhoff's theorem).

To achieve this, we will follow a clear path:

We will impose the assumptions of a static and spherically symmetric spacetime to determine the general form of the metric tensor, leaving only a couple of unknown functions.
We will solve the vacuum Einstein Field Equations, $R_{μν} = 0$ , for this metric form. This will yield a set of differential equations for our unknown functions.
We will solve these equations and use the Newtonian weak-field limit to fix the integration constants, revealing the final form of the metric.
Finally, we will discuss Birkhoff's theorem, a powerful result which proves that our solution is unique and that the assumption of a static spacetime is actually a consequence of spherical symmetry.

Let's begin.

1. The Metric Ansatz: Imposing Symmetry

Our starting point is the most general line element for a four-dimensional spacetime. Our first task is to simplify it by applying our symmetry assumptions.

Assumption 1: The spacetime is static.
As we learned in the previous lesson, a static spacetime is one that is not changing in time. This implies the existence of a timelike Killing vector field, which we can align with the time coordinate, $ξ = \partial / \partial t$ . This means that all metric components $g_{μν}$ must be independent of the time coordinate $t$ . Furthermore, we assume the spacetime is invariant under time reversal ( $t \to - t$ ). This symmetry forces the "cross terms" between time and space, $g_{0 i}$ , to be zero.

Assumption 2: The spacetime is spherically symmetric.
This means the geometry is unchanged by any rotation. In other words, the isometry group of the spacetime contains a subgroup isomorphic to $S O (3)$ , the rotation group in three dimensions. The practical consequence is that at any given time and radial distance, the spatial geometry must be that of a 2-sphere. The metric should therefore contain the standard line element of a 2-sphere, $d Ω^{2} = d θ^{2} + sin^{2} θ d ϕ^{2}$ . Also, any unknown functions in the metric can only depend on the radial coordinate $r$ , not on the angles $θ$ or $ϕ$ .

Putting these assumptions together, we arrive at the most general line element for a static, spherically symmetric spacetime.

Chapter 8 The Schwarzschild Solution

The textbook 'Chapter 8 The Schwarzschild Solution' provides a clear and formal argument for the form of the metric. Section 8.3.1 details how the assumptions of a static and spherically symmetric spacetime lead to the required metric ansatz.

Please read section 8.3.1, 'Form of the metric' (pages 116-118). Focus on how the foliation of spacetime into spatial hypersurfaces (due to staticity) and the foliation of these surfaces into 2-spheres (due to spherical symmetry) leads to the final form of the metric in equation (8.38).

As the text shows, these symmetry arguments constrain the metric to the form:

d s^{2} = - e^{2 a (r)} c^{2} d t^{2} + e^{2 b (r)} d r^{2} + r^{2} (d θ^{2} + sin^{2} θ d ϕ^{2})

Here, $a (r)$ and $b (r)$ are two unknown functions of the radial coordinate $r$ that we must determine. The exponential form is a convention chosen to ensure the coefficients are always positive and often simplifies the resulting differential equations. For our derivation, let's use the slightly simpler (but equivalent) notation:

d s^{2} = - A (r) c^{2} d t^{2} + B (r) d r^{2} + r^{2} d Ω^{2}

where $d Ω^{2} = d θ^{2} + sin^{2} θ d ϕ^{2}$ , and we need to solve for $A (r)$ and $B (r)$ . The coordinate $r$ is defined here such that the surface area of a sphere at that coordinate value is exactly $4 π r^{2}$ .

2. Solving the Vacuum Einstein Equations

Now we apply the physics. We are looking for a solution in the vacuum outside the mass $M$ , so the stress-energy tensor $T_{μν}$ is zero. As we've seen, the vacuum Einstein Field Equations (with cosmological constant $Λ = 0$ ) simplify to a wonderfully compact form:

R_{μν} = 0

Our task is now "simply" to calculate the components of the Ricci tensor $R_{μν}$ for our metric ansatz and set them to zero. This will give us the differential equations that determine $A (r)$ and $B (r)$ .

This is a three-step process:

Calculate the Christoffel symbols $Γ_{μν}^{λ}$ from the metric components $g_{μν}$ .
Use the Christoffel symbols to calculate the Ricci tensor components $R_{μν}$ .
Set the non-zero components of $R_{μν}$ to zero.

This is a famous and notoriously long calculation. Doing it by hand is an essential rite of passage, but watching it unfold line-by-line in text can be dry. The following video provides an excellent, detailed walkthrough of the entire derivation, from the initial assumptions to the final solution.

Relativity 108a: Schwarzschild Metric - Derivation

The video 'Relativity 108a: Schwarzschild Metric - Derivation' by eigenchris is a very clear, step-by-step execution of the full calculation. It perfectly matches the level of rigor you appreciate.

Please watch from 05:00 to 24:59. This covers the entire core of the derivation. Follow these stages closely: Metric Ansatz (05:00 - 11:01): This section re-establishes the form of the metric we discussed above, using functions a(r) and b(r) for the g_tt and g_rr components respectively. (Note: his a(r) and -b(r) correspond to our A(r) and B(r)). Christoffel Symbols (11:01 - 17:13): Watch how the 13 non-zero Christoffel symbols are computed from the metric ansatz. You don't need to memorize them, but understand the process. Ricci Tensor Components (17:13 - 22:36): See how the Ricci components R_00, R_11, and R_22 are constructed from the Christoffel symbols and their derivatives. Pay attention to the complexity and the many terms that arise. Solving the Differential Equations (22:36 - 24:59): This is the crucial step. Observe how combining the equations for R_00 and R_11 leads to the powerful simplification a(r) * b(r) = 1, and how this is then used in the R_22 equation to find the final form of the functions.

3. Assembling the Solution

Welcome back. As you saw in the video, the calculation is involved but the strategy is straightforward. Let's summarize the key results of setting $R_{μν} = 0$ . (Note: we will use $A (r)$ for the $g_{tt}$ component and $B (r)$ for the $g_{r r}$ component, and primes denote derivatives with respect to $r$ ).

The three relevant equations obtained from setting the diagonal Ricci components to zero are (up to some factors):

From $R_{tt} = 0$ : $A^{''} - \frac{A ^{'} B ^{'}}{2 B} - \frac{A ^{'} A ^{'}}{2 A} + \frac{A ^{'}}{r} = 0$
From $R_{r r} = 0$ : $- B^{''} + \frac{B ^{'} B ^{'}}{2 B} + \frac{B ^{'} A ^{'}}{2 A} + \frac{B ^{'}}{r} = 0$
From $R_{θ θ} = 0$ : $1 - \frac{r}{2 B} B^{'} + \frac{r}{2 A} A^{'} - B^{- 1} = 0$

The key insight is to combine the first two equations. If you divide the first by $A$ and the second by $B$ and add them, you get:
$\frac{A ^{''}}{A} - \frac{B ^{''}}{B} - \frac{1}{2} (\frac{A ^{'}}{A})^{2} + \frac{1}{2} (\frac{B ^{'}}{B})^{2} + \frac{1}{r} (\frac{A ^{'}}{A} + \frac{B ^{'}}{B}) = 0$
A more direct method shown in the video is to add $B / A \times (Eq. 1) + (Eq. 2)$ , which immediately simplifies to:

\frac{A ^{'}}{A} + \frac{B ^{'}}{B} = 0 ⟹ \frac{d}{d r} (ln A + ln B) = 0 ⟹ \frac{d}{d r} (ln (A B)) = 0

This means $A B = constant$ .

To fix this constant, we apply a physical boundary condition: as $r \to \infty$ , far from the gravitating mass, spacetime must become flat. The metric must approach the Minkowski metric, where $g_{tt} = - c^{2}$ and $g_{r r} = 1$ . This means $A (r) \to 1$ and $B (r) \to 1$ as $r \to \infty$ . Therefore, the constant must be 1.

A (r) B (r) = 1 ⟹ B (r) = \frac{1}{A ( r )}

Substituting $B = 1/ A$ and $B^{'} = - A^{'} / A^{2}$ into the third equation ( $R_{θ θ} = 0$ ) gives a simple first-order ODE for $A (r)$ :

1 - r A^{'} - A = 0 or r A^{'} + A = 1

This can be recognized as $(r A)^{'} = 1$ . Integrating with respect to $r$ gives:

r A (r) = r + C ⟹ A (r) = 1 + \frac{C}{r}

where $C$ is an integration constant.

4. The Newtonian Limit and the Schwarzschild Radius

We have almost found our solution. The metric is:

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

All that remains is to determine the physical meaning of the constant $C$ . We do this by demanding that in the weak-field limit (large $r$ ), our solution reproduces Newtonian gravity.

In Module 3, we saw that for slow-moving particles in a weak, static gravitational field, the geodesic equation reduces to Newton's law of gravity if we identify:

g_{00} \approx - (1 + \frac{2Φ}{c ^{2}})

where $Φ$ is the Newtonian gravitational potential. For a point mass $M$ , this potential is $Φ = - GM / r$ .

Comparing this with our $g_{00}$ component, $g_{00} = - A (r) c^{2}$ , we have $A (r) = - (g_{00} / c^{2}) \approx 1 + 2Φ/ c^{2}$ .
Let's equate our solution for $A (r)$ with this Newtonian limit:

1 + \frac{C}{r} = 1 + \frac{2}{c ^{2}} (- \frac{GM}{r}) = 1 - \frac{2 GM}{c ^{2} r}

This immediately fixes our constant:

C = - \frac{2 GM}{c ^{2}}

This specific combination of constants has units of length and is so important it gets its own name: the Schwarzschild radius, $r_{s}$ .

r_{s} \equiv \frac{2 GM}{c ^{2}}

Substituting this back into our expressions for $A (r)$ and $B (r)$ , we arrive at the final, famous Schwarzschild metric:

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})

Test your understanding!

Calculate the Schwarzschild radius for the Earth and the Sun. What does this tell you about the deviation of spacetime from flatness in our solar system? (Use $G \approx 6.67 \times 1 0^{- 11} Nm^{2} / kg^{2}$ , $c \approx 3 \times 1 0^{8} m/s$ , $M_{E a r t h} \approx 6 \times 1 0^{24} kg$ , $M_{S u n} \approx 2 \times 1 0^{30} kg$ ).

Show answer

For Earth:
$r_{s, E a r t h} = \frac{2 ( 6.67 \times 1 0 ^{- 11} ) ( 6 \times 1 0 ^{24} )}{( 3 \times 1 0 ^{8} ) ^{2}} \approx \frac{8 \times 1 0 ^{14}}{9 \times 1 0 ^{16}} \approx 0.009 m = 9 mm$
For the Sun:
$r_{s, S u n} = \frac{2 ( 6.67 \times 1 0 ^{- 11} ) ( 2 \times 1 0 ^{30} )}{( 3 \times 1 0 ^{8} ) ^{2}} \approx \frac{2.67 \times 1 0 ^{20}}{9 \times 1 0 ^{16}} \approx 2960 m \approx 3 km$
The term $r_{s} / r$ in the metric determines the deviation from flat spacetime. For the Earth (radius $\approx 6400$ km), the correction at its surface is tiny, on the order of $1 0^{- 9}$ . Even for the Sun (radius $\approx 700, 000$ km), the correction at its surface is of order $1 0^{- 6}$ . This confirms that for most astrophysical objects, GR effects are a small correction to Newtonian gravity. The situation becomes drastically different when an object is compressed to a size comparable to its Schwarzschild radius.

5. The Uniqueness of the Solution: Birkhoff's Theorem

We began by assuming the spacetime was both spherically symmetric and static. A remarkable theorem by George Birkhoff (and independently by J.T. Jebsen) proves that the 'static' assumption is redundant.

Birkhoff's Theorem: Any spherically symmetric solution of the vacuum Einstein equations is necessarily static. This means the Schwarzschild metric is the unique spherically symmetric vacuum solution.

The proof is quite elegant and relies on the mathematical structure you've been learning.

Birkhoff's theorem

The provided thesis on Birkhoff's theorem gives a fully rigorous proof. We won't go through every step, but we will use it to understand the logic and appreciate the theorem's power. A sketch of the proof is also in the textbook 'Chapter 8 The Schwarzschild Solution'.

Please read the introduction (page 5) and the statement of the theorem (Theorem 5.1 on page 18). Then, read the proof outline (steps 1-3 on page 18) and the final part of the proof (Section 5.3, pages 23-25). You don't need to follow the Christoffel symbol calculations in detail, but focus on the logic: starting with a more general time-dependent spherically symmetric metric, the vacuum EFE force the time-dependence to be trivial and reducible to the static case through a coordinate change.

The core idea is that if you start with a more general ansatz where the metric components can depend on time, $d s^{2} = - A (t, r) c^{2} d t^{2} + B (t, r) d r^{2} + ...$ , and then impose $R_{μν} = 0$ , the equations force the time-dependence to be of a very specific, trivial form. This time-dependence can then be completely eliminated by a simple redefinition of the time coordinate, returning you to the static Schwarzschild metric.

Physical Implication: This is not just a mathematical curiosity. It means that a spherically symmetric pulsating star does not radiate gravitational waves. The spacetime outside the star remains static and fixed, regardless of the star's radial pulsations. Only a non-spherical motion can generate gravitational radiation.

Conclusion

In this lesson, we have successfully navigated one of the cornerstone derivations in General Relativity. By combining the principles of symmetry with the vacuum Einstein Field Equations, we have derived the geometry of spacetime around a spherical mass.

Key Takeaways:

Assuming a static, spherically symmetric spacetime allows us to write a metric ansatz with two unknown functions, $A (r)$ and $B (r)$ .
Solving the vacuum Einstein equations ( $R_{μν} = 0$ ) for this ansatz leads to a system of differential equations.
A key step in solving these equations is realizing that $A (r) B (r) = 1$ due to the asymptotic flatness boundary condition.
The final integration constant is fixed by comparing the solution in the weak-field limit to the Newtonian potential, $Φ = - GM / r$ .
The final solution is the Schwarzschild metric, which depends only on the mass $M$ of the object through the Schwarzschild radius $r_{s} = 2 GM / c^{2}$ .
Birkhoff's theorem guarantees that this solution is unique for any spherically symmetric vacuum, and that such a solution is necessarily static.

Preview of Next Lesson:

We have derived the metric, but what does it mean? The mathematical form $d s^{2} = - (1 - r_{s} / r) c^{2} d t^{2} + (1 - r_{s} / r)^{- 1} d r^{2} + ...$ is pregnant with meaning. You can already see potential problems: what happens at $r = r_{s}$ ? What about $r = 0$ ? In the next lesson, we will begin to analyze the causal structure of Schwarzschild spacetime, investigating the nature of the event horizon at $r = r_{s}$ and the physical singularity at $r = 0$ .

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