Correspondence Principle
Everything fits with everything
Correspondence is one of the fundamentals in the foundation over which our Tensorland is built.
One story has physics equations. Another story has more physics equations. If the two stories share one or more variables in common it is possible to start with the equations in one story and move over to the equations in the other story. Everything needs to fit together.
An example of this can be found by starting out with the equations of kinematics and moving over to the equations of energy (kinetic energy and free energy).
As one example of what happens, we can calculate the velocity of a ball that has rolled down a frictionless inclined plane using an equation that uses acceleration (by gravity) and length traveled. We can then calculate the change to potential energy and then put all that into kinetic energy and with the knowledge of the mass you can get velocity from kinetic energy--this velocity will be the same that we calculated from the kinematic equation.
We "move" from one story to another using Algebra.
Correpondence shows up when we compare the equation for something given by Newton to an equation that has algebra for relativistic effects.
They have to both be right. And they are, here's how:
Newton's work assumes velocities that are extremely slow. Fast for Newton was probably 30 miles per hour on a horse and at some point we should compare that to the speed of light, 299 792 458 m/s.
When we scrutinize the algebra on the equation with the relativistic effects, we see that the effect disappears as the velocity goes to zero. 30 miles per hour is extremely close to zero from the perspective of 300,000,000 meters per second.
An equation relating three variables--the physicist asked to have it on his tomb.