Solution to GR Field Equations

[zeronem]

I understand that the Black Hole and the worm hole are solutions to Einstein's Field Equations. However, what does it mean by solution? I want to understand what it means by solution to the Field Equations. Can the solution be anything that forms a gravitational field? Can a solution in Einstein field equations be any kind of warped Space-Time?


For instance the solution to this quadratic equation x^2-4x+3 = 0 is x=1,3

So what makes the Black Hole the solution to Field Equations? The Mechanical Understanding of the Equations is what I'm looking for.

[DW]

Einstein's field equations are coupled second order nonlinear differential equations for the metric tensor's elements given those of the stress energy tensor as a source. A solution to Einstein's field equations is a metric tensor that according to his field equations is consistent with a given stress energy tensor. The Schwarzschild solution is the metric that is consistent with the exterior of a spherically symmetric nonrotating noncharged matter distribution. Other black holes are consistent with other stress energy tensors containing for example electric field energy or rotating matter.

[pmb_phy]

I understand that the Black Hole and the worm hole are solutions to Einstein's Field Equations. However, what does it mean by solution? I want to understand what it means by solution to the Field Equations. Can the solution be anything that forms a gravitational field? Can a solution in Einstein field equations be any kind of warped Space-Time?


For instance the solution to this quadratic equation x^2-4x+3 = 0 is x=1,3

So what makes the Black Hole the solution to Field Equations? The Mechanical Understanding of the Equations is what I'm looking for.
A solution to an equation such as x^2-4x+3 = 0 is a number. The equations of physics are differential equation. The solutions to differetial equations aree functions. For example: in Newton's theory the solution to the Newtonian field equations, i.e. Poisson's equation, is a vector equation, i.e. a vector function of a vector variable. In GR Einstein's field equations (EFE) are equations between geometric objects called tensors. The EFE have a tensor as a solution, the metric tensor. When one chooses a coordinate system then these tensor equations become differential equations and as such the solution, i.e. the metric tensor, can be thought of as a set of ten gravitational potentials which are functions of events, i.e. of space and time.

Pete