Mathematics plays a major role for Black Holes because it is the only means of "looking" into them. Visual observations are impossible. The event horizon is surrounded by invisible radiation, so mathematics must make the effort.
The use of mathematical means has some similarity to satellites which scientists may launch, in the hope of discovering meaningful answers about the structure of space. In many cases they succeed, sometimes with answers which are stunning. The premise behind that is to employ a precise method of questioning and a careful setting of the parameters, firmly based on facts. These settings must be derived from measurement, carefully undertaken. Moreover, a precise analysis of the results is crucial. If that is all done with a methodical approach, the mathematics can lead to quite acceptable conclusions, having relevance to the real world.
Mathematics is like a Magician, able to produce a rabbit from a top hat (Wiki)
Observed from the Earth, nothing can be seen or measured beyond the event horizon. Deductions can be made only by means of logic and mathematics. Hawking employed the so called "Euclidean Path Integrals‘‘ to penetrate Black Holes by mathematics. The time t was substituted for space in using complex numbers, with help of the imaginary unit i (sometimes denoted by j). The imaginary unit i is a concept which extends the real number system to the complex number system. The core property is that i² = -1. The term "imaginary" is necessary because no real number can have a negative square.
After a process called Wick-transformation, space-time is converted into space having four dimensions. The imaginary unit i makes it possible to transform time into space and vice versa. Calculations undertaken with complex numbers have been proved to produce precise results. Almost every day, we all utilize the imaginary unit by switching on the MP3-Player, digital radio or watching TV broadcasts. Preceding the transmission of the audio- and video content, after first digitizing, a process called "Fourier transformation" is employed in accessing the use of the imaginary unit. Digital sound or frames are transformed between the so called "time domain" and the "frequency domain". In removing some redundant frequencies, the data can then be compressed heavily, to save bandwidth, transmission time and memory. Because this transformation process is quite reversible, the receiver can restore the original signal. That is an amazing performance, gaining benefit from complex numbers.
A further "juggling trick" of mathematics allows deeper penetration into the black hole: with the help of the so called AdS/CFT correspondence, proposed by Juan Maldacena in late 1997. He combined a gravitational theory named "Anti de Sitter space" (AdS) with the "Conformal Field Theory" (CFT). In a nutshell, the ndimensional AdS defines the world of the macro cosmos and link to the CFT, a quantum field theory, describing the world of the micro cosmos. The results are the
"Maldacena duality" or "Gravity Duality", regarded as a powerful toolkit for the calculation of nuclear and condensed-matter physics. This is precisely what is required for revealing the secrets of the highly condensed cores of black holes.
As previously mentioned, many physicists are working on the superstring theory. Several ideas have been combined, leading to the so called M-Theory. This is an extension of the string theory in which 11 dimensions of space time have been identified. All these ideas converge in a model by the name of Quantum Gravity, describing the force of gravity according to the principles of Quantum Mechanics. Work continues and mathematics is the key.
A comprehensive theory of space and time must be able to explain the four fundamental forces to be found in the Universe, namely: the nuclear forces of strong and weak interaction, electromagnetic force and gravity.
It could be that, with the theory of Quantum Gravity, such a global and unambiguous theory will be found. Black holes should be included in such a self-inclusive
"Theory of Everything" Moreover, black holes could become the motivation for finding the key to a complete & unified theory of physics.
Neither of the great theories – the Theory of Relativity and Quantum Mechanics – can explain behaviour of matter under extreme conditions. Nor can they explain the expansion of the Universe. Everything started with the Big Bang (see p. 39) when the primordial Universe, which was in an extreme, hot and dense state, began to expand rapidly.
Before that, the entire early Universe was apparently concentrated in one single point, having all the same characteristics as a black hole: high density of matter (or energy – since Einstein, we do know that that‘s the same), collected into a single point. We speak of at least a 100 billion galaxies containing 100 billion stars (on average) aggregated into a point mass and could call them a singularity. This raises the question of whether or not the Big Bang was itself the result of a super-massive black hole 14 billion of years ago.
Many questions remain:
• In Particle Physics, antimatter (composed of antiparticles) is missing almost entirely in nature. But why? Antimatter, in the form of individual antiparticles is commonly produced by particle accelerators like the Large Hadron Collider at CERN in Geneva. Their lifetime is very brief and scientists claim that antimatter is the costliest material to make.
• The total mass of the Universe contains 5% ordinary atoms, or baryonic matter, 23% dark matter and 72% dark energy. Astrophysicists hypothesized dark matter and energy due to discrepancies between the orbital velocities of stars in the Milky Way and in star clusters, compared with their calculations. But what is the explanation?
Of what is the Universe really made?
The velocity of stars must be influenced by an invisible force in space. Gravity alone them cannot explain their orbital traces. Early theories of dark matter suggested black holes and Neutron stars as possible candidates for such irregularities. However, evidence indicated that such would constitute only a small portion of the discrepancies.
A key feature of Big Bang cosmology is the expansion of space. In the 1920s, astronomers like Edwin Hubble discovered that most galaxies exhibit a Red-Shift in spectra and that the red-shift was greater for more distant galaxies. From these observations they deduced that the Universe must still be expanding progressively. The Universe never was static. Dark Energy is the name given to whatever is causing the Universe`s expansion to accelerate. Black holes might be a source of energy for dark energy. Supposed to be related to black holes, dark energy remains one of the greatest mysteries of science.
Maximillian Fabricius, working at the Max-PlanckInstitut für extraterrestrische Physik (MPE) concluded in 2013:
