We are at a turning point in the history of physics. Fundamental problems at the foundation of physics are being shaken to their core.

These foundations are quantum mechanics (QM) and Einstein's general theory of relativity (GRT). The former is an extremely accurate description of small-scale physics, such as the behaviour of subatomic particles or electromagnetic radiation.

This theory is being tested in all disciplines of physics. The second describes the large-scale world, for example the movement of celestial bodies due to gravitational forces or the expansion of our universe.

Einstein's theory has also been very well confirmed by experiments. Their mathematical language is very elegant, but very different.

GRT is a deterministic theory, which means that the state of the universe at any given time and the fundamental laws of physics fully determine the past history and future evolution of the universe. QM, on the other hand, is based on probabilities, which means that reality arises when the observer participates in the event.

One can accept this fundamental difference and continue research in both fields. However, physicists have an irresistible urge to fathom this difference.

This is the "holy grail", a "theory of everything" (TOE). To date, there is no TOE.

Unified mathematics must describe a new field of research, the field of quantum gravity. To achieve unification, one can adapt GRT or QM.

Both methods have their advantages and pitfalls. An obvious question is why is it so difficult to achieve these unifications? A huge number of prominent theoretical physicists are working in this area of theoretical physics, a very complicated task.

First of all, you use approximation schemes, which are not always the best method. For example, you come up against the non-renormalizability of the fields.

We know that QM is the best model we have. It is a solid model and it explains the behaviour of elementary particles to a very high degree of accuracy.

So it is a logical step to try to incorporate gravitational effects into an extended quantum model. The discovery of the graviton, the intended "carrier" of the gravitational force and comparable to the photon, made this route plausible.

But the problems along this path are enormous. Adapting QM is like swearing in church.

Are there alternatives?In this book we try to explain that there is an alternative. We will focus on black hole physics, because it is thought that understanding the quantum effects near the horizon of a black hole will lead to a quantum gravity theory.

First, the model must retain unitarity and renormalizability. This can be done by imposing the topological boundary condition of an antipodal mapping.

Secondly, local conformal symmetry is applied, which is spontaneously broken when a mass term is added to the Lagrangian. Finally, we build our model on a warped five-dimensional spacetime.

This opens up new possibilities. It turns out that an exact instanton solution can be found, valid on the warped spacetime and the effective 4D induced Riemannian spacetime.

The antipodal boundary condition can then be enforced on the embedded hypersurface of a Klein bottle. This approach solves many problems that plague the description of Hawking radiation.

We exploit the non-orientability of the Klein surface to keep the incoming Hawking particle pure. We also treat the self-gravitating cosmic string, because this interesting object is closely related to the axially symmetric Kerr black hole.

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