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Schwarzschild Metric: Introduction

As previously mentioned, Einstein's general relativity is our current best theory of gravity. In his theory, gravity is not a force but rather a manifestation of the curvature of spacetime caused by the presence of energy-momentum. The Einstein field equations relate the geometry of spacetime, described by the metric tensor , to the energy-momentum content, described by the stress-energy tensor .

Few exact solutions to the Einstein field equations are known, as they are a set of ten coupled, nonlinear partial differential equations. One such solution is the Schwarzschild metric, which is given in spherical coordinates by

This metric describes the spacetime outside a spherically symmetric, non-rotating, and static mass. As such, it can be applied to various astrophysical objects, such as (slowly rotating) stars, planets, and black holes. When applied to the field equations, it makes many predictions, such as gravitational time dilation, the gravitational Doppler effect, the bending of light by gravity, and the precession of planetary orbits. Moreover, it predicts the existence of black holes whose event horizons are located at the Schwarzschild radius

Table of Contents​

Derivation of the Schwarzschild Metric​

Karl Schwarzschild derived this metric in 1916, shortly after Einstein published his field equations in 1915. His results were published in two papers: "On the Gravitational Field of a Point-Mass according to Einstein's Theory" and "On the Gravitational Field of a Sphere of Incompressible Fluid according to Einstein's Theory." Unfortunately, Schwarzschild passed away in 1916 due to complications from a disease he contracted during his service in World War I.

The derivation of the Schwarzschild metric involves several steps, including making assumptions about the symmetry of the spacetime, choosing an appropriate coordinate system, and solving the Einstein field equations under these assumptions.

The first assumption is that outside the mass, the spacetime is vacuum, meaning that the stress-energy tensor . It leads to the Ricci flat condition, as described in the following infobox.

Ricci Flat Condition

Assuming (in non-cosmological scales) and , the Einstein field equations reduce to

Taking the trace of this equation by contracting both sides with gives

Since in four-dimensional spacetime, we have

Finally, substituting back into the original equation yields

Note that a Ricci flat spacetime does not imply a flat spacetime, as the Riemann curvature tensor may still be non-zero. Recall that the Ricci tensor describes the degree to which the volume of a small geodesic ball in a curved space deviates from that in flat space, while the Riemann curvature tensor provides a more comprehensive description of the curvature of spacetime, including things like holonomy and tidal forces.

The second assumption is that the spacetime is spherically symmetric and static. This means that the and components of the metric must be the same as those in flat spacetime, as there is no preferred direction in a spherically symmetric spacetime. The third is that the metric should reduce to the Minkowski metric at large distances from the mass, where gravitational effects are negligible. In other words,

The Minkowski metric in spherical coordinates is given by

Derivation

Let's derive the Minkowski metric in spherical coordinates. In Cartesian coordinates , the Minkowski metric is given by

To convert to spherical coordinates , we use the following transformations:

The basis vectors (which are just the partial derivative operators) can be expanded using the chain rule:

Calculating the partial derivatives, we have

so the metric is

as you can verify by explicitly calculating the dot products. Notice that the and components have factors of . This is because we are not using a normalized coordinate system.

The last assumption is that the metric should be static, meaning that it does not change with time. The consequence is that

  1. (time independence), and
  2. (time reversal symmetry).

A more rigorous definition of a static spacetime is that it possesses a timelike Killing vector field that is hypersurface orthogonal. With the last assumption, the components of the metric must be zero, as they would change sign under time reversal.

To summarize, our assumptions are

  1. Vacuum: ,
  2. Spherical symmetry,
  3. Asymptotic flatness: ,
  4. Static: and .

With these assumptions, we can write the metric as

The and components are zero as must point radially outward, while and point tangentially. Moreover, the and components depend only on due to spherical symmetry and time independence. As such, we can introduce the functions and such that

Lastly, we will make a coordinate shift of to eliminate the function. This leads us to

with

LCC Coefficients​

Our next step is to solve for the functions and by applying the Ricci flat condition . To do so, we need to choose a connection (a way to differentiate tensors on a manifold). In general relativity, we use the Levi-Civita connection (LCC), which is the unique torsion-free connection that is compatible with the metric. A LCC always exists for any (pseudo-)Riemannian manifold as stated by the fundamental theorem of Riemannian geometry.

The LCC coefficients (also known as Christoffel symbols) are given by

I will not show the full derivation here, as it is quite tedious and can be found in many textbooks and online resources. The non-zero LCC coefficients are

Ricci Tensor​

Next we can compute the Ricci tensor using the LCC coefficients. Recall that Einstein's field equations tell us that in vacuum. As such, we can compute the Ricci tensor and set it to zero to solve for and .

Once again I will skip the full derivation. We only need to look at a few components of the Ricci tensor, as shown below. Differentiation is with respect to unless otherwise specified.

We can add the first two equations to eliminate the second derivative term, yielding

As , we have and , so .

Plugging in these findings into the third equation, and doing some algebraic manipulation, we have

and naturally

The constant turns out to be the Schwarzschild radius , which can be determined by comparing the weak-field limit of the Schwarzschild metric to the Newtonian gravitational potential.

Low-Field Limit​

To determine the constant , we can compare the Schwarzschild metric to the Newtonian gravitational potential in the weak-field limit. In this limit, Poisson's equation holds and can be written as

where is the Newtonian gravitational potential. Notice that this equation is similar to the geodesic equation in general relativity, which is given by

if we set , , and . The geodesic equation reduces to

Comparing the two equations, we have

Next, we also need to take the weak field limit of the metric to calculate this LCC coefficient. In this limit, we can write the metric as a perturbation of the Minkowski metric:

There are two assumptions we need to make here:

  1. The perturbation is small: . This means that the product is negligible.
  2. The perturbation's derivatives are small: . This means that and are negligible.

When we differentiate the metric, the Minkowski metric is constant, so we have

As such, the LCC coefficient becomes

And since this must equal , we have

We set the constant to zero, as we want the perturbation to vanish when the potential is zero. For a spherically symmetric mass distribution, the Newtonian gravitational potential is given by

As such, we have

This is true in both Cartesian and spherical coordinates, as the time component is invariant under spatial coordinate transformations. Anyways, we can then compare this to the component of the Schwarzschild metric in the weak-field limit:

This implies that .

Thus, we have derived the Schwarzschild metric:

Summary and Next Steps​

In this note, we derived the Schwarzschild metric by making several assumptions about the spacetime outside a spherically symmetric, non-rotating, and static mass. We started with the Einstein field equations and applied the Ricci flat condition, spherical symmetry, asymptotic flatness, and staticity to simplify the metric. Lastly, we determined the constants in the metric by comparing it to the Newtonian gravitational potential in the weak-field limit.

Next, we will explore how the Schwarzschild metric predicts various phenomena, such as gravitational time dilation, the gravitational Doppler effect, the bending of light by gravity, and the precession of planetary orbits.