Einstein field equations

• 1. Introduction

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  The Einstein field equations sit at the heart of general relativity, a set of ten coupled, nonlinear equations that tell the universe how to curve. Published by Albert Einstein in 1915, they arrive in the form of a single compact tensor equation, and yet when fully written out they constitute a system of hyperbolic-elliptic partial differential equations of breathtaking complexity. What question do they answer? Given a distribution of matter and energy in spacetime, they determine the geometry of that spacetime. And from that geometry, the paths of every freely falling particle and every ray of light can be calculated. That is the bargain at the center of modern physics: matter tells space how to curve; curved space tells matter how to move.

• 2. The Geometry of Gravity

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  The Einstein tensor, one of the two central objects in the equations, is a symmetric divergenceless second-degree tensor. It depends on nothing more than the metric tensor and that metric's first and second derivatives. On the other side of the equation sits the stress-energy tensor, which encodes the local energy, momentum, and stress within spacetime. The metric tensor is itself the unknown the equations are solving for: it describes the shape of spacetime the way a map describes the shape of a terrain. Each of the ten tensors in the equation carries ten independent components, but the four Bianchi identities reduce the number of truly independent equations from ten to six. That leaves four degrees of freedom corresponding to the freedom to choose a coordinate system, which means the equations describe physics rather than any particular map drawn onto it.

• 3. Newton at the Limit

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  One of the most powerful checks on any new theory of gravity is whether it recovers the old one when conditions are ordinary. The Einstein field equations pass this test. Under two simultaneous approximations, a weak gravitational field and velocities much slower than the speed of light, the EFE reduce exactly to Newton's law of gravitation. The gravitational constant that appears in the equations, defined in terms of the Newtonian constant of gravitation and the speed of light, is fixed by demanding that this correspondence hold. This principle, that a more general theory must reproduce a well-tested older theory in appropriate limits, is called the correspondence principle, and the EFE satisfy it cleanly. The equations also imply, as a direct consequence, the geodesic equation, which governs how freely falling test objects travel through spacetime.

• 4. The Cosmological Blunder

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  Einstein's original 1915 publication did not contain the cosmological constant term. He added it later with a specific goal: to allow for a universe that neither expands nor contracts, a steady-state solution. The effort failed on two counts. Any such steady-state solution turned out to be unstable. Then Edwin Hubble's observations showed that the universe is in fact expanding. Einstein abandoned the constant, telling the physicist George Gamow that introducing the cosmological constant had been the biggest blunder of his life. The story did not end there. More recent astronomical observations revealed that the universe's expansion is accelerating, and explaining that acceleration requires a positive value of the cosmological constant. The constant is algebraically equivalent to assigning a vacuum state a fixed energy density and a pressure of opposite sign, which is why cosmologists now use the terms cosmological constant and vacuum energy interchangeably.

• 5. Exact Solutions and Their Worlds

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  Solving the field equations exactly requires simplifying assumptions, usually some form of symmetry, because the equations are nonlinear and generally resist closed-form answers. No complete solution is known for a spacetime containing two massive bodies, which would model a binary star system. The solutions that do exist in exact form are among the most celebrated objects in theoretical physics. Flat Minkowski space is the simplest vacuum solution. The Schwarzschild solution and the Kerr solution, which describes a rotating black hole, are nontrivial examples. A method pioneered by Ellis and MacCallum using orthonormal frames reduces the EFE to a set of coupled, nonlinear, ordinary differential equations; self-similar solutions in this framework, studied by Hsu and Wainwright, are fixed points of the resulting dynamical system. New exact solutions have emerged from this approach through work by LeBlanc and by Kohli and Haslam.

• 6. Linearization and Gravitational Waves

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  Far from any gravitating source, spacetime is nearly flat, and the equations can be approximated by treating the true metric as the Minkowski metric plus a small correction term. Dropping higher-power terms in that correction linearizes the equations, making them tractable. It is this linearized form that opened the door to studying gravitational radiation. The nonlinearity that makes exact solutions so hard to find also distinguishes general relativity from the other foundational theories of physics. Maxwell's equations of electromagnetism are linear in the electric and magnetic fields: the sum of two valid solutions is itself a valid solution. The same is true of the Schrodinger equation of quantum mechanics, which is linear in the wavefunction. General relativity shares no such convenience, and that mathematical fact is inseparable from why gravitational physics is so much harder to compute than electromagnetic physics. Despite containing the inverse of the metric tensor, the equations can also be recast into a polynomial form in the metric and its derivatives, a reorganization that applies equally to the Einstein-Hilbert action from which the equations are derived.