Schwarzschild, Droste, and Nordström Solve Einstein’s Field Equations

This chapter examines the contributions of Karl Schwarzschild, Johannes Droste, Hans Reissner, and Gunnar Nordström. I start with Schwarzschild. Schwarzschild communicated with Einstein, providing two exact solutions to Einstein’s field equations: an exterior solution for the vacuum field equations and an interior solution for the field equations with sources. Schwarzschild and Droste independently derived the same metric to describe the gravitational field outside a spherically symmetric, static mass. However, their approaches diverged: Schwarzschild adhered to Einstein’s determinant-one condition, while Droste explicitly renounced it. This difference reflects their affiliations—Schwarzschild was part of the German scientific community closely aligned with Einstein, while Droste belonged to the Dutch scientific community. Einstein maintained strong connections with Dutch scientists, frequently visiting the Netherlands and engaging with their work. In 1916, Hans Reissner and, independently in 1918, Gunnar Nordström extended the spherically symmetric static solution to include charge, deriving what is now known as the Reissner-Nordström metric. Nordström, a member of the Dutch scientific community, adopted Droste’s methodology. Both Reissner and Nordström communicated their results to Einstein before publication, underscoring Einstein’s influential role at the time as an authority whose approval was sought for significant advancements. In the same paper where Nordström introduced his new metric, he demonstrated that the energy density vanished in unimodular coordinates. As I show in Chap. 3, Nordström conveyed this result to Einstein before publication, highlighting a critical implication: Einstein’s 1916 return to unimodular coordinates in the addendum to his gravitational waves paper—intended to salvage his controversial third type of gravitational waves carrying energy—was incorrect. This interaction further illustrates Einstein’s central role in the discourse surrounding general relativity during this formative period.