Albert Einstein’s scientific career began in 1901 when, at the age of 21, he published a paper on an effect known as capillary rise. Four years later, in 1905, Einstein submitted his doctoral dissertation to the University of Zurich, in which he derived an estimate of the size of molecules. During 1905 Einstein also published four groundbreaking papers, leading many to refer to this as his ‘annus mirabilis’ or miracle year. The first of these papers provided an explanation for the so-called photoelectric effect, whereby light hitting certain metals causes these metals to give off electrons. Building on the work of his fellow German physicist Max Planck, Einstein explained the photoelectric effect by postulating that light is made up of particles. Einstein would be awarded the Nobel prize for this work in 1921.
The second of these papers was on thermodynamics: the study of heat and how it moves and interacts with matter. Einstein showed that if a dust particle is diffusing randomly through the air, then the distance that the particle can travel from its starting point can be determined by a simple mathematical equation now known as the Einstein relation. This relation is expressed as D = mkT, where D is the diffusion coefficient, m is the mobility of the particle, k is the Boltzmann constant, named after German physicist Ludwig Boltzmann, and T is the air temperature. In this paper Einstein was able to explain a phenomenon known as Brownian motion, named after Scottish botanist Robert Brown, which refers to the apparently random motion of particles suspended in a medium.
The third of these papers was on the special theory of relativity. During the 19th century, physicists believed that light waves were ripples in a substance pervading the whole of space, which they referred to as the ether. The existence of this substance implies a preferred frame of reference for the universe. Einstein hated this idea, arguing instead that all frames of reference are equally valid – a notion he referred to as the principle of relativity. There was a problem with this however. In 1861, Scottish physicist James Clerk Maxwell put forward his theory of electromagnetism, which predicted that the speed of light is a fundamental constant of nature. This is at odds with the principle of relativity, according to which the speed of light should vary according to the reference frame of an observer.
Einstein’s solution was to postulate that light travels at the same speed in all frames of reference. It was this insight that led him directly to the special theory of relativity. The crux of this theory was a set of equations linking properties of objects in frames of reference moving at different speeds. The theory involved unifying of space and time into a four-dimensional object known as space-time. One consequence of this theory is that two events measured as occurring simultaneously in one frame of reference are not necessarily simultaneous in another. Another is that if faster-than-light travel is allowed then it is possible for the order of cause and effect to be reversed, leading to paradoxical situations in which outcomes happen before the events that caused them.
The fourth and final paper published by Einstein during his annus mirabilis developed the principle of mass–energy equivalence, expressed by the equation E = mc2. This is probably the most famous equation in the whole of physics and drops directly out of the mathematics of special relativity. It expresses a relationship between energy, E, mass, m, and the speed of light, c, and essentially says that energy and mass are equivalent. So, for example, if you burn a lump of coal, the mass of the ash will be less than the original mass of the coal, with the difference in mass multiplied by c2 equal to the energy released during combustion. This paper and equation led to the discovery and use of nuclear power decades later.
Einstein knew that his special theory was incomplete because it applied only to the special case of objects travelling at uniform speed, and not to objects undergoing acceleration. He was also aware that the special theory was inconsistent with gravity. Soon after the development of special relativity it dawned on Einstein that these concepts are intimately entwined: gravity and acceleration are also equivalent. By 1912 it had occurred to Einstein that one route to incorporating gravity and acceleration into his theory could be to bend the flat space-time of the special theory. With the help of a friend and colleague, Swiss mathematician Marcel Grossman, he turned to a branch of mathematics developed in the latter half of the 19th century known as differential geometry.
Differential geometry is a way of using equations to represent how points an surface are connected to one another. This is done using a mathematical concept called a tensor: a multidimensional array of numbers which generalizes vectors and matrices to arbitrarily many dimensions. Einstein defined a tensor which describes the gravitational curvature of space-time, which is now known as the Einstein tensor. All that remained now was to link the Einstein tensor to the contents of space-time, which could be described using another tensor known as the energy-momentum tensor. The so-called Einstein Field equation is the formula which links these two tensors and became the lynchpin of the general theory of relativity.
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