The Schwarzschild Metric
This metric is named after Karl Schwarzschild, a German astrophysicist, who found a solution to Einstein’s field equations for an uncharged, non-rotating, spherically symmetric body of mass [M]. In many respects, it might be argued that this metric, more than any other, highlights the relativistic effects of mass on spacetime. Therefore, from a learning perspective, this metric is worthy of some investigation because it allows the mathematical abstraction of general relativity to be anchored in physical phenomena, which science seeks to understand and verify.
Schwarzschild was born in 1873 and published his first paper on celestial mechanics at the age of just 16, before going on to gain his doctorate in 1896. From 1901-1909, he held the position of professor at the prestigious Göttingen institute, working along side some significant mathematicians of the day, e.g. David Hilbert and Hermann Minkowski. Later, Schwarzschild became the director of the observatory in Göttingen. In 1915, Einstein published his theory of general relativity, containing the famous, or possibly infamous, field equations given their apparent complexity. Therefore, it came, as somewhat of a surprise that somebody was able to derive an exact solution in the same year. While a solution for a charged, spherical, non-rotating body was discovered shortly after Schwarzschild's death in 1916, the exact solution was to remain unsolved until 1963, when resolved by Roy Kerr. Therefore, in retrospect, Schwarzschild’s solution is even more remarkable given the conditions under which he was working. At the time, Schwarzschild was serving on the Russian front, during the First World War, as an artillery officer in the German army. During this time, he wrote two papers, addressing the exterior and interior space of a star. The first, which has become the most famous, addresses the idea of exterior space and outlines the concept of `Schwarzschild Geometry` that is applicable to the Earth, Sun and many other types of stars, as well as black holes, albeit without rotation. He sent both papers to Einstein to which Einstein replied, as follows, just before Schwarzschild’s early death in May 1916:
I have read your paper with the utmost interest. I had not expected that one could formulate the exact solution of the problem in such a simple way. I liked very much your mathematical treatment of the subject. Next Thursday I shall present the work to the Academy with a few words of explanation |
To-date, there have been thousands of dissertations, articles, and books written on the subject of Schwarzschild's solution to the Einstein field equations. In this context, this discussion cannot and does not aspire to be a critique of the `Schwarzschild Metric`, simply an honest attempt to come to some understanding of its meaning and implications. As such, there will be the usual bias towards presenting some of the foundation ideas in terms of Newtonian mechanics, not because they are thought to be better, but simply because these concepts are more widely understood and may help highlight where modern science continues to deviate from classical determinism. Initially, the Schwarzschild solution was originally seen as a curiosity, but was to later underpin the hypothesis of a gravitational black hole. However, the roots of this solution can still be shown in terms of Newton's laws of motion and universal gravitational, although the ideas are really based on Einstein’s field equations within general relativity.
It has been known from the time of Newton and Kepler that a planet
orbiting the sun, under the influence of gravity, would follow an elliptical
orbit. In addition, the presence of the other planets could cause an
elliptical orbit to rotate or precess. Over the centuries, the rate
of precession of the planets has been very accurately measured, although
this assumption needs to be founded on a full understanding of all the
gravitational source acting in, and on, the solar system. Given this
caveat, the measured observation of
Mercury’s precession about the Sun
does not exactly match the Newtonian theory, albeit by the apparently
minuscule deviation of 43 arcseconds of rotation every century. Initially,
in defence of Newton's laws, it was suggested that unobserved gravitational
bodies might exist in the solar system, while others even considered
the possibility that the inverse square law of gravitation may not be
an exact power of 2. However, it would be the publication of the special
theory of relativity, in 1905, which suggested that nothing could
go faster than the speed of light that began to highlight the most serious
conflict with Newtonian mechanics. For if gravity could not propagate
faster than light, then it also suggested that some of the energy associated
with gravity must lie in the field of propagation itself and not just
within the concept of mass.