The Equations to Solve to Generalize SR

Eugene Shubert

Is This System Solvable?

What is the general solution to the equation set numbered (64) to (66) of http://www.everythingimportant.org/relativity?

I'm also looking for the general solution to the following question, or to a more manageable set of equations that describes the problem.

Suppose that there is a sufficiently differentiable real-valued function of three real variables T(R,S,w), defined everywhere except the point w=0, that has the following properties:

For all X, Y, a, b, such that a is not equal to zero, b is not equal to zero, and a+b is not equal to zero, there exists a unique Z = Z(X,Y,a,b) such that the following identities are always true:

T(X, Y, a) = T(X, Z, a+b)

T(Y, X, -a) = T(Y, Z, b)

T(Z, Y, -b) = T(Z, X, -a-b)

Note that the uniqueness of Z = Z(X,Y,a,b) is quite remarkable in that Z is defined by three different equations!

I am also requiring the symmetry that there is a unique X = X(Y,Z,a,b) that satisfies all three functional equations for all Y, Z, a, b, such that a is not equal to zero, b is not equal to zero, and a+b is not equal to zero. Similarly for Y.

It's easy to see that the function T(R,S,w) = R/tanh(w) – S/sinh(w) has all these properties. However, I'm looking for the most general solution to the problem.

I vaguely remember something about rank and the Jacobian of a transformation being zero in certain circumstances, and I assume that a system of PDEs may arise from my three functional equations from that angle.

Any insights would be greatly appreciated.