If We Are Anything

General Relativity I am now working exclusively on the gravity problem. One thing is certain, that never in my life have I tormented myself anything like this, compared to this problem the original, special relativity theory, is child's play. Albert Einstein, 1912 The above passage is an extract from Einstein’s letter to physicist Arnold Somerfeld. At the time of writing, Einstein knew full well that he would need to pull together every resource at his disposal if he was to have any chance of constructing a theory of gravitation. So in this section, I try to convey both a snapshot of the history and the way in which Einstein developed the theory. First, both special and general relativity were conceptualised and pieced together by Einstein as a result of many other events, mathematical works, and experimental evidence. A complete history of either theory would be too vast to include in this book so I must be clear and upfront when I say that there were many contributors to both theories, both before and after publication. Some of the earliest were Greek philosophers, and in the case of general relativity, material would need to include a sizable portion of both the evolution of geometry and mathematics too. I also draw on some of the lesser-know contributors to general relativity, as these people are often overlooked when highlighting the historical pathway. So with this caveat, and the understanding that we are only covering a few key contributors, we should proceed to a logical start point … and since general relativity, or gravitation, is essentially the curvature of spacetime, then a basic rundown on the key developments in geometry should help us understand how it all came about. Leonhard Paul Euler was a Swiss physicist and mathematician who spent most of his life in Russia and Germany. He was considered to be one of the greatest mathematicians of all time and featured prominently throughout the second half of the 18th century. Among many other works, Euler mathematically modeled the curvature of spheres. We all know that the shortest distance between two points when drawn on a piece of paper is a flat straight line; however, what is the shortest distance between two points on the surface of a curve? The answer is simple: a straight line which follows the curvature of the sphere. So Euler conducted complex mathematical modeling of curvature as viewed from outside an object to define area and relationships to the curvature. Enter Johann Carl Friedrich Gauss, a German mathematician, scientist and child prodigy, who asked a simple but very complex mathematical question on this topic: we can measure the curvature of a straight line on a sphere from outside the sphere – this is called extrinsic curvature – but can we measure the curvature of a sphere from inside the sphere itself, or intrinsic curvature? And yes, in fact, it can be derived from the way distances are measured on the surface, known today as Gaussian curvature. The last in our understanding of curvature is yet another German mathematician, Georg Friedrich Bernhard Riemann, a student of Gauss’. On the look-out for a worthy successor, Gauss asked Riemann to prepare a presentation and paper on the foundations of geometry. If anyone was up to the task, it was Riemann. The following year, he presented his paper and seminar but not on the foundations of geometry as requested by Gauss, but instead he hypothesised and showed mathematical formulations of an underlying mathematics of geometry itself. Gauss had found his worthy successor. In the ensuing years, Riemann developed a theory of manifolds (‘many folds’) and eventually, the Riemann Curvature Tensor. This new geometry, in which curvature can be expressed by a set of numbers called a tensor, offered perhaps the single greatest insight into Einstein’s theory of gravity. A tensor – sometimes referred to as a metric or metric tensor – defines curvature at any given location within a curved space and when Einstein realised that gravity was simply curvature in space and time, he knew immediately that a suitable metric tensor would be needed to calculate the curvature at all possible points. One of Einstein’s closest colleagues was Georg Pick. In the years when Einstein worked in Prague, Pick and he met almost daily and discussed many problems. In the course of long walks Einstein confided in Pick the mathematical difficulties that confronted him in his attempts to generalise his theory of relativity. What Einstein needed was a way to write partial differential equations in a coordinate-independent form which could separate the changing metric tensor from the forces which were required to remain constant. (This basically means that while the forces remain constant and unchanging, the space and time in which they operate are dynamic, and curve when in the presence of mass/matter.) Pick suggested that the appropriate mathematical instrument was ‘absolute differential calculus’ or ‘covariant calculus’, developed by the Italian mathematicians Gregorio Ricci-Curbastro and his student Tullio Levi-Civita. Let’s take a quick look at what Ricci-Curbastro and Levi-Civita accomplished. This formulation, unknown to Einstein at the time, allowed forces to remain constant and independent of the changing geometry of spacetime, represented by a general [symmetric metric] tensor at this time. Accomplishing this mathematical work-around accounted for the consistency of the speed of light represented in both flat and curved four-dimensional spacetime structures so the forces would remain constant, even though the metric changed as a result of spacetime curvature. The metric tensor would need to enter the general process of differentiation with the requirement that the outcome of the differentiation process of the tensor must produce yet another tensor: the derivative itself. The Christoffel symbols, representing force, would be included at this time, but there was a catch: the differentiation would depend on the order in which it was inserted (meaning that 2x4 = 8, but 4x2 ≠ 8) and that the original symmetric metric was now transformed into a non-commutable, non-linear metric tensor, partly dynamic (the changing metric) and partly static (the constancy of the forces). It is not known, and is often a topic of some debate, whether Einstein was capable of this level of mathematics in the first place. Special relativity was indeed a cinch in comparison to the mathematical complexity of general relativity, but in any case, what Ricci-Curbastro and Levi-Civita had done was undoubtedly a remarkable piece of mathematical work. It seemed that Ricci-Curbastro and Levi-Civita were hot on the trail and had already completed the next step. This breakthrough should have been no surprise, because Ricci-Curbastro was not just a mathematician, he was also a physicist, and his deep insight into this type of calculus came as a result of understanding Maxwell’s equations in both a mathematical and practical sense. Ricci-Curbastro had become well versed with Maxwell’s equations of electromagnetism, as he had previously transcribed them from English into Italian at the request of fellow mathematician Enrico Betti. Maxwell’s equations clearly showed that magnetic fields are independent of space and time. One of Einstein’s teachers famously referred to him as a ‘lazy dog’. This teacher was his professor of mathematics at Eidgenössische Polytechnikum, known today as Zurich’s ETH. The teacher, Hermann Minkowski, was no ordinary mathematician and no ordinary professor. He was another child prodigy and contributed greatly to Einstein’s future theory of gravity. He was also a close colleague of yet another German mathematician, David Hilbert, who also contributed one of the final pieces of the gravitational puzzle (Field Equations) and was rumored to have been only three weeks behind Einstein in delivering his own theory. At the time of his ‘lazy dog’ statement Minkowski was right: Ein