KLEIN BOTTLE AND BLACK HOLE GEOMETRY IN UNDERSTANDING QUANTUM GRAVITY

Abstract

There are two types of curvature in geometry, intrinsic & and extrinsic curvature of space, extrinsic curvature depends on which space or dimension it is embedded into. To create a geometrical structure that can model or be compatible with the Quantum and macroscopic (GR) nature of gravity, we need to be able to connect the topological & Riermannian curvature or metric space. Here we are trying to integrate the topological i.e. Quantum Property of the geometry & the Macroscopic geometrical nature of a Geometrical Space to model Quantum Gravity theory with it. There can be many mathematical approaches to finding quantum geometry. Here we are concentrating on drawing new mathematical structures or approaches from different modified gravity theory that leads to quantum gravity namely Loop Quantum Gravity. Teleparrall gravity, F(g) Modified gravity etc. We are also investigating many mathematical aspects of creating a Quantum gravity theory like the background independence for gravitation force. In LQG there is no background independence, but for the quantum properties of space to exist, there has to be a background dependency in the theory, which can be derived from taking a Klein bottle as a quanta of space rather than the space itself like as the weave of loops creates the space itself for LQG. Also, the concept of time can be derived from the topological property of Klein bottle namely for its sidedness or no boundary for odd numbers which is not in LQG as it doesn't consider time to be a coordinate as it doesn't count the space-time to be 4 dimensional. It can not have a physical time component as it has for the space. Time is generated by the increased number of nodes in the spin network of the weave, which is a mathematical construct For unifying the Topological space with the metric space we can form a mathematical structure where the topology of a Klein bottle is multiplied by the metric tensor of a 5 particular space to impose a topological variable on the geometry itself so that it has a quantum property. Correction in Riemannian geometry is optional. There is an attempt to model the concept of particle & anti particle in hawking radiation through kelin bottle geometry. Also, the concept of time can be derived from the topological property of Klein bottle namely for its one-sidedness or no boundary for odd numbers. Many aspects of black hole geometry can be explain or model by Klein bottle geometry. Using Nash embedding concept to describe the singularity & wormhole Le. ermepr paradox & a new TQFT model to describe Hawking radiation we can understand more about the possible quantum gravity model. The metric of Klein bottle hole is compared with both Kerr metric & sehrchild metric of blackhole to find out a time dimension. The isometric embedding theory of hawking temperature predicts the hawking temperature. Here we tried to do the same with the Klein bottle to prove the connection between the Klein bottle geometry & black hole geometry by predicting the Hawking temperature. In Nash embedding the the minimum possible dimension is possible to describe the klein bottle is 5 dimensions is the same as the klein bottle hole metric, as it also has the 5 component. Some other geometrical description of wormholes is shown here also, along with a different explanation of blackhole creation because of the high gravity along with the explanation of the creation of singularity in plank length.