# The General Theory of Relativity

Einstein’s general theory of relativity is one of the major achievements of 20th-century physics. Published in 1916, it postulates that gravitational fields can be interpreted as manifestations of the space-time curvature. In this picture, space and time, as physical constructs, had to be combined into a new mathematical/physical entity called ‘space-time’. This new concept about space and time was first introduced by Hermann Minkowski.

 The equations of relativity show that both space and time coordinates of any event must be combined mathematically to accurately describe an event. Space-time is a 4-dimensional entity since space consists of 3 dimensions and time is 1-dimensional. It is believed to be a ‘continuum’ because there are no missing points in space or instants in time, and both can be subdivided without any apparent limit in size or duration. Physicists consider our world to be embedded in a 4-dimensional space-time continuum, and all events, places, moments in history, actions and so on are described in terms of their location in space-time.

Astronomical objects change the geometry of space-time by stretching and warping it, forming ridges, peaks and valleys that cause bodies moving through it to weave and curve. In Einstein’s picture, although Earth appears to be pulled towards the sun by gravity, there is no such force. It is simply the geometry of space-time around the sun that tells the Earth how to move.

It took Einstein almost ten years to formulate his famous field equations (Einstein’s Field Equations, or EFE) in general relativity that described the fundamental interaction of gravitation as a result of curvature of space-time by matter and energy. In 1917, Einstein used these field equations to model a static universe — the prevailing belief at the time was the universe was static and unchanging (the Steady State picture). But to do so, he had to add a cosmological constant in his field equations by hand, effectively creating a repulsive force to balance the inward pull of gravity.

 The history of the cosmological constant is another testament to Einstein’s towering genius. Einstein invented the constant to expand the fabric of space-time after his field equations would not allow for the cosmos to remain static. When it was later discovered that the universe was indeed not static, but expanding, Einstein abandoned the cosmological constant, deeming it unnecessary. Interestingly, it was resurrected recently when astronomers discovered that not only was the universe expanding, but that the rate of expansion was accelerating due to an unknown force called dark energy. The constant had to be added back to the general relativity field equations to predict a universe that is flying apart at an increasing speed. As with inflation, the repulsion is part of space itself: the bigger the universe gets, the more powerful is its push, resulting in an exponential expansion. The cosmological constant is now one of the leading ideas to account for dark energy, and if proven correct, Einstein will be right once again even when he thought he was wrong.

In 1922, Alexander Friedmann published a set of remarkable solutions to Einstein’s equations (similar solutions were worked on independently by Willem de Sitter) by incorporating simplifying assumptions of homogeneity and isotropy of space known as the Cosmological Principle. Friedmann’s solution raised the possibility of a dynamic universe that changed in size over time, and it was Friedmann who first introduced the concept of expanding universe. One of his solutions modeled a cosmos which began in a singularity — an infinitesimally small point with an expansion rate that increased over time.

 Friedman needed an assumption about how the matter in the universe was distributed. The simplest assumption he could make was, viewed on sufficiently large distance scales, there are no preferred directions or preferred locations in the Universe. It would appear roughly the same everywhere and in every direction. In other words, when averaged over very large scales, the matter in the universe is homogeneous and isotropic. This is called the Cosmological Principle.

According to Friedmann’s model, the ability of the matter (or energy) in the universe to halt the expansion depends on the mean energy density. If the mean energy density is too low, the universe will have enough momentum from the Big Bang explosion to escape the pull of gravity.

 Figure 1. The curvature of the universe determined by the cosmological parameter, Ω0. Image credit: NASA/GSFC

The geometry (shape) of the universe is determined by its mean energy density and allowed three cases of positive, negative, and zero curvature. Figure 1 shows the geometry using the cosmological parameter $\sigma_0$, which is the ratio of the mean energy density of the universe $(\rho_0)$ to the critical density $(\rho_c): \Omega_0=\frac{\rho_0}{\rho_c}$ where $\rho_c$ is equivalent to roughly 10 hydrogen atoms per cubic meter. Although this is an incredibly small number, it must be remembered that most of space is empty. The cosmological parameter $\Omega_0$ is crucial in the development of modern cosmology, as its value, which can be determined by an experiment, affects the overall geometry of the universe. Table 1 shows how the cosmological constant affects the geometry of the universe.

Friedmann’s solution gave two verifiable predictions: (a) the notion of expanding universe (b) the geometry of the universe, which involves measurement of the cosmological parameter $\Omega_0$. Furthermore, one of his solutions modeled a cosmos that appeared to have a beginning in a Big Bang.

 Table 1: Cosmological Models Geometry $\Omega_0$ Fate of Universe Flat =1 Open Universe: The universe has zero curvature if $\rho_0=\rho_c$. This model corresponds to a universe in which the universe will continue to expand forever. Hyperbolic <1 Open Universe: The universe has negative curvature if $\rho_0<\rho_c$. The density of the universe is not big enough to halt the expansion and the universe will continue to expand for ever. Spherical >1 Closed Universe: The universe has positive curvature if $\rho_0>\rho_c$. In the Closed Universe Model, the density of the universe is great enough to halt the expansion and start a contraction. The universe will collapse in a Big Crunch’, which will resemble the reverse process of the Big Bang’.

Initially, Einstein did not endorse Friedmann’s expanding universe solution but later agreed that in fact Friedmann’s results were correct. Unfortunately Friedmann did not live long enough to enjoy the great man’s endorsement. He died prematurely at the age of 37. At the time of his death his wife was pregnant with his only son. In spite of his groundbreaking work, Friedmann was virtually unknown in the west largely because of his untimely death. Alexander Friedmann was the unsung hero of cosmology.

The expansion of the universe was finally corroborated several years later by Edwin Hubble’s observations in 1929. The cosmological parameter, $\Omega_0$, was also measured by sensitive detectors mounted on satellites. Cosmologists finally started piecing together a consistent picture of the cosmos from the wealth of data available, and Alexander Friedmann’s contributions to cosmology have become the cornerstone for our understanding of the universe.

In 1927, unaware of Friedmann’s earlier work, the Belgian astrophysicist and priest Georges Lemaître derived the same solutions. Lemaître proposed the concept that the universe began as a primeval atom. His theory suggested that all of the mass-energy (1051 kg) of the universe was concentrated in a single super-atom about one astronomical unit 5 across. The primeval atom would then fragment, and the universe expand. Lemaître’s concept was a predecessor to the Big Bang model.

5 1 astronomical unit = 149597870700 meters