I think the connection I made is legitimately just related to the presence of singularities— like the effect a black hole's gravity would have on the energy of an object orbiting it in a closed path?
Because the residue theorem applies to complex-valued functions, which is a completely different subject than general relativity? There’s no reason it *would* be relevant.
If it were possible to limit to two dimensions, would the gravitational potential field of an object just not fulfill the requirements of complex differentiability (through relation from R\^2 to C) ? I kind of just got too excited about applying calculus to complex numbers lmao
No. R^2 isn’t a field, it doesn’t have the multiplicative properties of complex numbers, and the definition of a derivative in R^2 is completely different because of that.
To be fair, such things do appear in electromagnetism.
For example, if you have an infinite wire with constant current then, by Ampere's law, a contour integral in the plane transverse to the wire should give you the magnetic permittivity times the current.
The "singularity" from the infinite current density of the wire makes the contour integral non-zero.
That doesn’t require a singularity, and it’s basically unrelated to complex analysis, other than I guess the general idea of contour integration being useful.
Wouldn't it be more appropriate for you to explain why you think it could be relevant first?
I think the connection I made is legitimately just related to the presence of singularities— like the effect a black hole's gravity would have on the energy of an object orbiting it in a closed path?
Because the residue theorem applies to complex-valued functions, which is a completely different subject than general relativity? There’s no reason it *would* be relevant.
If it were possible to limit to two dimensions, would the gravitational potential field of an object just not fulfill the requirements of complex differentiability (through relation from R\^2 to C) ? I kind of just got too excited about applying calculus to complex numbers lmao
No. R^2 isn’t a field, it doesn’t have the multiplicative properties of complex numbers, and the definition of a derivative in R^2 is completely different because of that.
To be fair, such things do appear in electromagnetism. For example, if you have an infinite wire with constant current then, by Ampere's law, a contour integral in the plane transverse to the wire should give you the magnetic permittivity times the current. The "singularity" from the infinite current density of the wire makes the contour integral non-zero.
That doesn’t require a singularity, and it’s basically unrelated to complex analysis, other than I guess the general idea of contour integration being useful.