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  • The Most Beautiful Equation

    \[\label{einstein} G_{\mu\nu}+\Lambda T_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}\]

    In November 1915, just hundred years ago, the whole world was shaken by a new visionary work. One man alone brought to humankind a profoundly different perspective on the very nature of, arguably, the most relevant aspects of our perceived reality: space and time. In his “On the General Theory of Relativity” (Einstein 1915), Albert Einsten discussed the intimate connection between space, time, energy and mass. The picture that emerges is extremely elegant. The pinnacle of the theory is equation \ref{einstein}, actually a set of 10 equations called Einstein’s Field Equations. These equations describe gravitation as a result of space-time being curved by matter and energy.

    Space and time can be measured using the so-called metric tensor (a tensor is a mathematical object analogous to but more general than a vector), which enters through the quantity \(G_{\mu\nu}\). The metric tensor is, so-to-speak, a complex ruler with a clock attached. The equation states that the result of making a measurement using the metric tensor doesn’t only depend on space-time itself, as expected in Newtonian mechanics. Instead, mass and energy enter the picture via the quantity \(T_{\mu\nu}\), called stress-energy tensor. The stress-energy tensor measures the density and flux of energy and momentum (the product of mass and velocity of an object).

    So the curvature of space-time is zero only if there is no mass and energy. When mass and energy are present, space and time are affected. The orbit of the Earth around the Sun is then explained by the curved space-time produced by the large mass of our star. The resulting orbital motions of the planets simply follow from the change in the metric according to Equations \ref{einstein}. Gravity is no more a force that is added on top of space-time, as in Newton’s description. Gravity is now just a property of space-time itself. Simple, elegant, beautiful.

    Left: Space-time is bended by the presence of mass. Like marbles in a sink, planetary orbits arise naturally as a consequence of curved space-time. Right: Albert Einstein (1879-1955)

    References

    1. A. Einstein. Zur allgemeinen Relativitätstheorie. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften (Berlin), Seite 778-786. 778-786 (1915).