Abstract. I propose to apply my generalized limit theory and generalized partial differential equations to (unchanged) general relativity equations, changing the set of solutions to a different vector field than real numbers. So, I’ve formulated another variant of general relativity. Whether the resulting theory is non-contradictory in the case of singularities, is yet unknown. This theory possibly exhibits all properties expected from quantum gravity.
About Discontinuous Analysis
I remind that I defined generalized limit of arbitrary function. The limit may be an infinitely big value.
It allows to define derivative and integral of an arbitrary function.
I also defined what are solutions of partial differential equations where such infinities (instead of e.g. real numbers or complex numbers) are defined.
You may see in there that for the simple differential equation y'(x) = -1/x2 we can consider either in a sense arbitrary generalized solutions or generalized solutions with a “pseudodifferentiable” derivative. The first one gives an arbitrary value in the zero point and the second a fixed real value in zero point. See the book for more details.
The Modified General Relativity
If we would instead consider the general relativity Einstein equations, we can get the following description of a set of generalized solutions (in supersingularities):
- We require the solutions to be pseudodifferentiable in time (or rather, timelike intervals).
- We do not require the solutions to be pseudodifferentiable in space (or rather, spacelike intervals).
Conjecture. In the singularity point there would form during time of black hole forming a certain value with an infinite structure in the center which is determined by the values of the variables while the hole was forming but is not a function of the characteristics of the already form black hole.
If that hypothesis is true, we have a solution to the black hole information paradox: the center of a black hole holds not a hole but an infinite value containing the information of how the hole was formed. This value is a constant, but does not depend on the “dead” form of an already formed black hole, rather it contains the information of how the hole was formed.
The proposed theory is a candidate for quantum gravity:
- It is precisely formulated.
- It exhibits exactly the same properties as GR in absence of singularities (possibly, except it may have extra solutions?)
- It is compatible (but probably has extra solutions?) with the usual quantum field theory in curved space. Moreover, in absence of singularities, it is exactly the same as QFT.
- It possibly preserves information even when there are singularities.
If we replace “quantum field theory” with some string theory (such as M-theory), we get yet another quantum gravity, apparently.
I am not sure if my theory may be modified in such a way to exclude Hawking radiation (possibly yet without information loss).
It’s a very interesting hypothesis. Just write a publication on this topic, unless I do it before you. Win a Nobel prize.
2 thoughts on “General Relativity in a generalized differential calculus – a candidate for quantum gravity”
My theory is wrong, because it contradicts to the observations:
In my QG the space is the same for all quantum worlds. (I consider many-world interpretation a proved theory.) Therefore gravity is the same in worlds with different positions of the Sun, what is obviously wrong.
Oh, it seems my theory can be improved to be compatible with the many world-interpretation:
1. Replace GR with my “supersingular GR”.
2. Quantize the resulting theory (introduce gravitons).
3. “Merge” it with QFT in curved space.
4. Replace the resulting theory with the corresponding many-worlds theory.