What is the name of the equations which are a set of partial differential equations produced by Einstein relating to the description of gravity as a geometric property of space and time which went on to replace Isaac Newton's rotating spheres argument which attempts to define that true rotational motion can be defined by observing the tension in the string joining two identical spheres?
field equations
Title: Bäcklund transform
Passage: In mathematics, Bäcklund transforms or Bäcklund transformations (named after the Swedish mathematician Albert Victor Bäcklund) relate partial differential equations and their solutions.  They are an important tool in soliton theory and integrable systems.  A Bäcklund transform is typically a system of first order partial differential equations relating two functions, and often depending on an additional parameter.  It implies that the two functions separately satisfy partial differential equations, and each of the two functions is then said to be a Bäcklund transformation of the other.
Title: General relativity
Passage: General relativity (GR, also known as the general theory of relativity or GTR) is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics.  General relativity is considered probably the most beautiful of all existing physical theories.  General relativity generalizes special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or spacetime.  In particular, the "" is directly related to the energy and momentum of whatever matter and radiation are present.  The relation is specified by the Einstein field equations, a system of partial differential equations.
Title: Rotating spheres
Passage: Isaac Newton's rotating spheres argument attempts to demonstrate that true rotational motion can be defined by observing the tension in the string joining two identical spheres.  The basis of the argument is that all observers make two observations: the tension in the string joining the bodies (which is the same for all observers) and the rate of rotation of the spheres (which is different for observers with differing rates of rotation).  Only for the truly non-rotating observer will the tension in the string be explained using only the observed rate of rotation.  For all other observers a "correction" is required (a centrifugal force) that accounts for the tension calculated being different from the one expected using the observed rate of rotation.  It is one of five arguments from the "properties, causes, and effects" of true motion and rest that support his contention that, in general, true motion and rest cannot be defined as special instances of motion or rest relative to other bodies, but instead can be defined only by reference to absolute space.  Alternatively, these experiments provide an operational definition of what is meant by "absolute rotation", and do not pretend to address the question of "rotation relative to "what"?"  General relativity dispenses with absolute space and with physics whose cause is external to the system, with the concept of geodesics of spacetime.
Rotating spheres General relativity