# Have fractional order differential models been explored as an alternative to standard gravitational field theory?

Since Einstein introduced his field equations and general theory of relativity, experimental evidence, at least on the cosmic scale has repeatedly supported the theory. Nevertheless, many seeking to extend the theory (Grand theories that combine relativity and quantum mechanics) or to explain gaps that have so far been attributed to TBD effects of dark matter, dark energy, etc. have failed to find anything better than GR. Recently a paper presented data from observations in our own galaxy that refutes the modified gravity theory of Moffat.

Correct me if I'm wrong, but the underlying mathematics that model the field equations of general relativity have a fundamental basis in integral calculus. Right?

To the point of my question, have any theorists explored the application of models based in fractional order calculus in consideration of gravitation field theory?

• The a-th derivative of a function f(x) at a point x is a local property only when a is an integer; this is not the case for non-integer power derivatives. I guess that this non-locality could be a problem – Wolphram jonny Oct 20 '18 at 17:03
• Related: physics.stackexchange.com/q/89922/2451 and links therein. – Qmechanic Oct 20 '18 at 17:23
• @Wolphramjonny thanks - but please point me to references that further explain why this is so. – docscience Nov 16 '18 at 14:37
• I can add that in realizing a fractional order system you generally have to come up with an approximation. For example control theory has tried to realize fractional order PID controllers. To realize in terms of rational polynomial functions (for practical implementation) it takes much higher order polynomials to approximate than the simple 1st order polynomial you get with integral PID. – docscience Nov 16 '18 at 14:42
• There are exact representations of fractional order systems, mathematically, but to realize as an engineering system, one needs to approximate them. Probably because they are infinite dimensional. Nature seems to get away with that. – docscience Nov 16 '18 at 14:45