Introduction to the Schwarzschild Metric

Previously, we derived Einstein's field equations, which describe how matter and energy influence the curvature of spacetime. These equations are fundamental to our understanding of general relativity and the behavior of gravitational fields. However, they are immensely difficult to solve, with most applications requiring numerical methods or approximations. The first exact solution to these equations was found by Karl Schwarzschild in 1916, and it describes the spacetime geometry around a spherically symmetric, non-rotating mass.

Typically, the method in which a new solution is derived is to assume a specific form of the energy-momentum tensor . This can then be plugged into the Einstein field equations to find the metric tensor . From this, we can then derive various geometric properties, such as the connection coefficients, curvature tensors, and geodesics. We will derive the Schwarzschild metric from Einstein's field equations, and then explore its implications for the behavior of objects in a gravitational field.

Table of Contents

• Table of Contents
• Derivation
  • Connection Coefficients

Derivation

We can assume that the universe is comprised of a single, spherically symmetric mass, with no other matter present. This is a simplification, but it allows us to focus on the essential features of the Schwarzschild metric without getting bogged down in unnecessary complications. We will also assume that the mass is non-rotating, which means that we can ignore the effects of angular momentum.

Recall that the energy-momentum tensor represents a flux of energy and momentum through spacetime. Specifically, represents the -momentum flowing through a unit volume of constant . In the exterior of the mass, there is no matter, so the energy-momentum tensor is zero;

We also assume that the spacetime is asymptotically flat, meaning that at large distances from the mass, the spacetime approaches Minkowski spacetime. This means that the metric tensor approaches the Minkowski metric at large distances.

Finally, we assume that spacetime is static, meaning that the metric tensor does not change with time. Formally, this means that spacetime has a timelike Killing vector field and is irrotational. The former means that, by the definition of a Killing vector field, the Lie derivative of the metric tensor with respect to the Killing vector field is zero, so . The latter means that the spacetime has no rotation, so the vorticity tensor . Without going into the details, the result of these assumptions is that first, the metric tensor obeys , and second, the metric tensor obeys a time-reversal symmetry; .

With the assumptions established, now we can plug the energy-momentum tensor into the Einstein field equations:

At non-cosmological scales, we can ignore the cosmological constant , so we have

Next, we can contract this equation with ,

As , this simply yields

Now we can plug this back into the original equation, yielding

A space in which the Ricci curvature tensor is zero is known as a Ricci-flat space. Note that this does not mean that the Riemann curvature tensor is zero, as it is possible to have a non-zero Riemann curvature tensor while having a zero Ricci curvature tensor. The Ricci tensor only measures the change in volume due to spacetime curvature, which can be nonexistent even in a curved spacetime.

Now we can enforce the asymptotic flatness condition. In spherical coordinates, the Minkowski tensor is

Considering that the system has spherical symmetry, the metric tensor should resemble the Minkowski metric, with a scaling factor that depends on the radial coordinate .

As we know that should not change the metric (this was one of our assumptions), we can consider how the basis vectors change under this time reversal. The basis vector transforms as

The component is by definition the inner product of the basis vector with itself, which becomes

Hence, the -component of the metric is indeed invariant under time reversal. The -components, however, transform with one negative sign, since the spatial basis vectors do not flip:

This means that the -components of the metric tensor must be zero, as they cannot change sign under time reversal. Now we know that the metric tensor roughly takes the form

and to maintain spherical symmetry, the and components must be functions of only. We can denote these as and , so we have

Connection Coefficients

To proceed, we need to find the functions , , and . This can be done by using the Ricci-flat condition, . Therefore, we need to find the connection coefficients, then the Riemann curvature tensor, and finally the Ricci curvature tensor. Recall that in the Levi-Civita connection, the connection coefficients are given by

Notice the leading factor , which is the inverse metric tensor. As the metric tensor is diagonal, the term is zero unless , so we can substitute for in the equation;