What Made Silicon Valley Special?

The Einstein field equation is usually written as:

{\displaystyle R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu }+\Lambda g_{\mu \nu }=\kappa T_{\mu \nu }.}

The \mu and \nu subscripts are placeholder tensor indices—you can use any letters you wish; Carl Rovelli used a and b, but the usual convention is \mu and \nu. R_{\mu\nu} is the Ricci curvature tensor which measures how a shape deforms as it moves along geodesics in space, and {\frac {1}{2}}Rg_{\mu \nu } is the scalar curvature, a measure of the curvature at any point in space. The \Lambda g_{\mu \nu } term is the cosmological term with \Lambda the cosmological constant: this term was added by Einstein in an erroneous attempt to make his equation describe a static universe, but now is interpreted as being responsible for the observed acceleration in the expansion of the universe. On the right side of the equation T_{\mu \nu } is the stress-energy tensor, which gives the density and flux of mass-energy and momentum in spacetime. This is multiplied by \kappa,the Einstein gravitational constant, which is:

{\displaystyle \kappa ={\frac {8\pi G}{c^{4}}}\approx 2.076647442844\times 10^{-43}\,{\textrm {N}}^{-1},}

where G is the Newtonian gravitational constant and c is the speed of light in vacuum. Rovellli expressed the equation in units favoured by general relativity theorists who choose units that set \kappa to 1.

The meaning of the equation is simple and profound: “mass-energy tells spacetime how to curve”. The other half of general relativity is the geodesic equation:

{\displaystyle {d^{2}x^{\mu } \over ds^{2}}+\Gamma ^{\mu }{}_{\alpha \beta }{dx^{\alpha } \over ds}{dx^{\beta } \over ds}=0\ }

which has the meaning, “Spacetime curvature tells mass-energy how to move”.