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Chapter 3.6 Einstein Field Equation

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When expressed mathematically, Einstein's gravitation theory yields a set of formulas known collectively as the Einstein Field Equations. In a nutshell, this equation describes the properties of a gravitational field surrounding a given mass. Stated simply, the equation describes how an object curves space and how the curvature, in turn, stretches or squeezes matter in three spatial directions: north-south, east-west and up-down.
However, applying the equations is no easy task. Mathematically, General Relativity is built upon ten so-called "coupled hyperbolic-elliptic nonlinear partial differential equations" which take many pages to write down (and a deep breath just to say!). For a mathematician, the difficulties lie in the fact that the equations are nonlinear and coupled also and in their sheer number of terms.

After he published his famous paper in 1916, Einstein later conceded that the mathematical difficulties of his General Theory of Relativity were a "very serious" impediment to its further development. So serious, in fact, that it took nearly 75 years before the best minds in the field could come close to solving the equations stated by the theory. Now, high performance computers permit more accurate modeling of the distortions of spacetime by massive objects, including black holes.

Physicists are motivated to grapple with the complexity of Einstein's field equations because they can be used, theoretically at least, to describe all possible spacetime scenarios, from the collisions of black holes to the gravitational interactions of irregular chunks of matter. In practice, researchers had to first restrict their calculations to very simple systems in order to manage the equations.

This elegant symbolic formulation of Einstein's general theory of relativity cannot be used for actual calculations, but it clearly shows the principle that "matter tells spacetime how to curve, and curved space tells matter how to move" (John Wheeler, Princeton University and the University of Texas at Austin) . The left side of the equation contains all the information about how space is curved, and the right side contains all the information about the location and motion of the matter. General relativity is beautiful and simple (to a physicist), but mathematically it's very complicated and subtle.

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