# The space expands in a black hole

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#### The darkest secret of gravity

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Relativity and space-time

A first adequate mathematical description of black holes is possible with the general theory of relativity (ART). *General Relativity*, GR). This theory goes back to what is probably the most famous physicist of all: *Albert Einstein* (1879-1955). He developed the famous theory of relativity in two parts: special and general relativity. After studying physics, Einstein was a patent officer in Bern from 1901 and, in addition to this work, found plenty of time to deal with theoretical physics. A key area of interest was light and so, for example, the question of what one would perceive during a mental 'ride on a light beam'. No physicist at the time was able to adequately answer this seemingly harmless question. Finding an answer to this **Thought experiment** led to the special theory of relativity (SRT) *Special relativity*, SR). Einstein published the cornerstones of this theory in the scientific journal *Annals of Physics* with the title *On the electrodynamics of moving bodies* in 1905.

Reference systems - it's all a question of location

The SRT compares the measurement of events, which are determined by location and time coordinates, in different reference systems. A frame of reference is a place from which observations are described. However, it is not only a point of reference or reference in space, but also in time.

In the theory of relativity, both spatial and temporal information are important. In order to clearly define an event, four values are given: three space coordinates and one time coordinate. An illustrative example of these four numbers is an appointment: You go to a specific street (first spatial coordinate), to a specific house number (second spatial coordinate), to a specific floor of the building (third spatial coordinate) at a specific time (the time coordinate). These 'dates' are called in the theory of relativity *Events* or*World points*. They are four-dimensional because they can be clearly defined by four independent numbers.

If an event now takes place, it can be described by different observers. What is important in the theory of relativity is how the observer moves relative to the event. There is an excellent observer who is relative to the event *Not* emotional. This observer is in the so-called **Rest system**. An observer who moves relative to the observed event (which is generally the case) has a (possibly constant) relative speed or relative acceleration to the event. These *relatively moving observer* are in a so-called **Laboratory system**.

An example to make the different frames of reference clear is an airplane. From the point of view of an observer on the ground, the aircraft is moving. So this ground-based observer is in a laboratory system. The crew and passengers on board, on the other hand, move relative to the aircraft*Not*because they are flying with you. You are in the rest system of the aircraft.

Einstein asked himself how one can combine physical observations from the point of view of the various observers. He realized that there are mathematical relationships between the different observers (the Lorentz transformations) that make it possible to compare the observations. Let's stay with the example of the airplane, which flies much slower than the light moves through the vacuum. The observers can compare their observations very well within the framework of classical mechanics. At speeds that are, however, similar to the speed of light in a vacuum, this no longer works. Then you need Einstein's theory and discover with astonishment:

Space and time are relative.

Different frames of reference move relative to one another in the SRT at a uniformly straight speed. It shows that in all these reference systems (inertial systems) the **The speed of light in vacuum is a constant** is. The speed of light in vacuum or the speed of light in empty space is symbolized in physics with the letter c, which simply goes back to the fact that it is constant.* cconstant*) is. The numerical value of c in the Système Internationale is loud

*Committee on Data for Science and Technology*CODATA (2002)

**c = 299 792 458 m / s**

The speed of light in a vacuum is (according to all that physicists know today) a universal natural constant. Initially, this constancy was a postulate of Einstein's, i.e. an assumption or working hypothesis with which he tried to construct a consistent physical theory. In the meantime, this assumption has been confirmed many times in experiments. It is therefore clear: not everything is relative in the theory of relativity - the speed of light is absolute!

The essence of relativity

The constancy of the speed of light has far-reaching consequences: Let us assume that two observers are looking at a moving object from different reference systems. If the speed of light is a constant, then other quantities must vary so that both observers physically describe the observation correctly. It turns out that the postulate of the constancy of the speed of light in a vacuum turns into a **Relativity of time** (also of simultaneity) and one **Relativity of length** flows out. The classically incomprehensible effects of time dilation and length contraction are evidence of the relativity of time and length. This essence of relativity gave the theory of relativity its name. Time lost its absolute character, that already *Aristotle* and later *Sir Isaac Newton* postulated: time is a relative quantity. In addition, time lost its independence: time and space are closely related and form a continuum in the theory of relativity, the so-called **Space-time continuum**. This structure is simply called spacetime for short. This continuum is in the SRT *flat*, i.e. not curved, and is described by the Minkowski metric. The flatness is precisely a consequence of this, because the SRT is a theory in a relativistic vacuum (i.e. the energy-momentum tensor vanishes).

'E equals m c square'

The central equation of the SRT and arguably the most famous equation in physics is that**Mass-energy equivalent**. As a mathematical equation written succinctly as

This equation can be found in the paper *Does the inertia of a body depend on its energy content?*, also published in the *Annals of Physics* in 1905 - in Einstein's year of miracles (*annus mirabilis*). The mass is therefore a form of energy, just like radiation energy, thermal energy or kinetic energy. But besides this statement there is another important conclusion: due to the validity of this equation, a mass that is relatively at rest (relative speed zero) also has a non-vanishing one **Resting energy**. One reads off that even very small masses are caused by the enormous number 'speed of light squared' (about 10^{17} m^{2}/ s^{2}) have an extraordinarily high resting energy.

The concept of spacetime was expanded in the general theory of relativity. Here the four-dimensional manifold, consisting of three spatial dimensions (length, width, height) and a time dimension, *curved* be. This happens precisely when a form of energy (mass, electromagnetic radiation, dust, etc.) is present. It creates a curved spacetime. Physicists say: Energy (mass) is the source of gravity. The curvature becomes particularly great where the energy is located. Einstein's field equations now provide information about where space-time has curvatures and how strong these are. On the other hand, the equations also say how the curvatures affect the energy. This complicated, mutual coupling of space-time to energy and energy to space-time is currently in the **complicated, non-linear character of the field equations**.

Tensors

The development of the general theory of relativity cost Einstein enormous efforts, as one can guess from the much later publication year 1916. This is due to the mathematical formalism of the theory of relativity, the **Tensor calculus**. Although this is already used in the SRT, it becomes more complicated and extensive in the ART. The mathematician *Marcel Grossmann* (1878 - 1936), Einstein's friend and fellow student when he was studying physics at ETH Zurich, taught him how to use tensors. Grossmann must be seen as an important co-founder of the ART, because he was familiar with the work of the following mathematicians:

- The German mathematicians
*Johann Carl Friedrich Gauss*(1777 - 1855) and*Georg Friedrich Bernhard Riemann*(1826 - 1866) developed the fundamentals of differential geometry. - Grossmann also knew the literature of the German physicist and mathematician
*Elwin Bruno Christoffel*(1829 - 1900), who founded the tensor analysis and introduced the Christoffel symbols named after him. Developed on the basis of his work*Ricci-Curbastro*and*Levi-Civita*coordinate-free access to differential calculus. - The Italian mathematician
*Gregorio Ricci-Curbastro*(1853 - 1925) worked, among other things, in the field of differential geometry, which he mainly developed between 1884 and 1894. - His student, the Italian mathematician
*Tullio Levi-Civita*(1873 - 1941) expanded the tensor calculus, dealt with the covariant derivative in 1887 (*Christoffel*following) and with*Ricci*around 1900 the differential calculus.

Tensors are quantities of differential geometry that are defined on a four-dimensional manifold and that satisfy certain transformation laws. The scalars, vectors and matrices of linear algebra are also tensors, but of a lower level. Differential geometry knows far more complicated tensors, which can always be written as a well-defined arrangement of numbers and functions. The tensors of the ART are very clear objects, which with a physical quantity, such as the energy (energy-momentum tensor), the curvature of space (Riemann curvature tensor) or the electromagnetic field (Maxwell tensor resp. *Faraday tensor*) stay in contact. The spacetime itself, the metric, is determined by the *metric tensor* described.

The key property of tensors is theirs **Coordinate independence**, i.e. regardless of the coordinate system in which they are formulated: the physical statement they make is always the same.

Often the physical quantities of the GTR are tensors of the 2nd level, which can be written as a 4 × 4 matrix (an arrangement of 16 numbers or functions in four columns and four rows) and thus get a familiar shape. Physical tensors are usually symmetrical. For a 4 × 4 matrix this means that only 10 components (upper or lower triangular shape) are independent, because the others are determined by the symmetry properties. This also applies to the **metric tensor**, which can alternatively also be described by the line element.

General relativity is in the sense of being *generally* to understand because the relative movement of the reference systems to each other (compared to the SRT) is generalized: the inertial systems can be against each other **accelerated** become. A fundamental postulate of the GTR is the equivalence principle. It says that inert and heavy mass are equivalent, i.e. that it makes no difference in motion whether a mass is accelerated (inertia) or falls in the gravitational field of a body (gravity). The *Eötvös experiment*, a structure of a rotary balance with moving masses on a torsion pendulum, confirmed this principle within the framework of the experimental accuracy. Another brilliant confirmation of the equivalence principle, but also of the GTR itself, is the observed deflection of radiation in the gravitational field from the sun and planets.

Revolutionary views: geometrical view of gravitation

The concept of general relativity replaced the old Newtonian view of things that gravity is the instantaneous (i.e. without loss of time) mediation of forces between masses. According to the GTR, gravity is one **geometric property of spacetime**whose mediation propagates at the speed of light in a vacuum. The fundamental finding of ART is:

**Energy bends space-time**

or written as a tensor equation

That's the so-called **Einstein's field equation** or short **Einstein equation** in the language of physicists (not the undoubtedly more famous formula E = mc^{2} is meant by 'Einstein equation' among physicists!) The Einstein equation is also the more important equation.

The Einstein equation is very compact here as a single equation, but actually that's ten! The Einstein field equations are due to the symmetry of the metric a **System of ten non-linear, coupled, partial differential equations**. On the left is the Einstein tensor**G**which just contains second derivatives of the metric; on the right is the **Energy-momentum tensor****T**that describes the matter (dust, ideal fluid, electromagnetic field, etc.). In the case of a vacuum, i.e. in absence of matter, the energy-momentum tensor vanishes. This is realized especially for electrically uncharged black holes. Hence they are called **Vacuum solutions of Einstein's field equations**.

So the problem remains of making the Einstein tensor disappear. There is no direct method of solving this problem from coupled non-linear, partial differential equations that is immediate**all** Could provide solutions to the problem. This distinguishes partial differential equations from ordinary differential equations. For this reason, the vacuum solutions have historically been found gradually and rather randomly and sometimes even several times. In the **Numerical relativity theory** there are now procedures to prevent the rediscovery of an already known solution in possibly different coordinates. So efforts are made to systematize the solutions of the field equations of gravity. in the **Equivalence problem** it is about deciding whether two metrics g and g 'match. *A. Karlhede* has decisively advanced this problem by using the geometries with basic systems, the so-called. **Four-legged** (Tetrad), systematized. Nowadays, this theoretical research continues on the Einstein equations, e.g. at the Albert Einstein Institute (AEI) in Golm. Such efforts avoid that already known solutions are 'rediscovered', as has been proven to happen at least twenty times with the Schwarzschild solution!

The **Coupling constant** corresponds in the Système Internationale to the product of eight times the circle number π ('Pi') with the gravitational constant G, divided by the speed of light c to the fourth power. In the geometrized units used here (G = c = 1) this is simplified to 8π. The coupling constant necessarily follows from a correspondence principle: In the limiting case of weak gravitational fields and low speeds compared to the speed of light, Einstein's theory must pass over to Newton's theory. Then one can derive the coupling constant by using Einstein's field equations with the **Poisson's equation** compares Newtonian gravitational physics.

What holds the world together inside

Since the SRT could already show that mass is equivalent to energy, mass also bends space-time. With this knowledge, the central question of the protagonist in *Goethe's***fist** answer:

What holds the world together at its core is the world itself.

Because the reason is that according to the GTR the matter of the earth itself (represented by the *Energy-momentum tensor***T**, the right hand side of the field equations) space and time (the space-time or metric, a four-dimensional manifold that is expressed in the form of derivatives in the*Einstein tensor***G** the left side of the field equations) bend around the earth in such a way that a self-gravitating object, the spherical earth mass, results. All earthly energy determines the geometry in the relativistic, but also in the direct literal sense (*geos*, Greek: earth, *metros*, Greek: measure)!

What to do with the field equation

If one wants to bring Einstein's field equations for the vacuum case into a more visible or practical form, one only needs to know the definition of the Einstein tensor: it is precisely the difference between Riemann's curvature tensor and its taper, the **scalar curvature**, the Ricci scalar.

The Riemann curvature tensor is obtained from a sum of partial derivatives of the Christoffel symbols. Here comes the link to the metric, which is clearly defined by the **metric tensor** or the line element: the Christoffel symbols are again sums of derivatives of components of the metric tensor.

The consequence is obvious: the Einstein tensor is roughly speaking a sum of partial, second derivatives of the metric tensor. Hence the metric tensor is fundamental and defines all properties of a curved, four-dimensional one **Spacetime** firmly.

Trivial spacetime of the SRT

The metric tensor in the special theory of relativity (SRT) is determined by the Minkowski metric and has a very simple form: the non-diagonal elements of the metric tensor written as a 4 × 4 matrix are all zero. Again, there are only constant numbers on the diagonal and no coordinate-dependent functions. In a possible convention (+ - - -), for example, the temporal component +1 and all spatial diagonal elements -1 (one also says the **Signature of the metric** be -2. That means: all derivatives (according to time and space coordinates) of these constant entries are zero. After the above explanations, the Christoffel symbols disappear first. Then the Riemann curvature tensor is also zero, and so is its taper, the scalar curvature. The relativists paraphrase this: **The spacetime of the SRT, the Minkowski space, is flat**.

In the case of black holes, this only applies to the asymptotic limit, i.e. if you are very far away from the black hole. The Schwarzschild solution and the Kerr solution (also the counterparts with electrical charge) are for radii towards infinity **asymptotically flat**.

In the vicinity of the black hole the curvatures become extremely strong and even diverge in the center. In the central singularity the curvature is infinite! The source of the hole's gravity is located in this singularity of curvature. But this is exactly where classical physics fails!

Black holes are relativistic objects

Let us now close the arc from the theory of relativity to the black holes: Black holes are solutions of the field equations of general relativity. As long as they have no electrical charge, they are solutions of the field equations in a vacuum. The widespread scientific doctrine is that the cosmic black holes are electrically neutral and are characterized by at most only two properties: **Mass and rotation**. The essential type of black hole in astrophysics is therefore the Kerr geometry, which we will discuss in great detail in the course of this paper. The Kerr solution is also a vacuum solution. In other words, if you put the metric of a rotating black hole in Einstein's equations, it turns out that the **Einstein tensor** (the 'left side' of the field equations) disappears.

Are black holes **general relativistic objects** and as such can only be correctly described with the ART. It is true that the horizon radius of non-rotating black holes can also be derived using the means of classical Newtonian theory, but the correct result must be assessed as a coincidence. Newton's theory fails to describe rotating holes. Newer theories that seek to further develop the GTR, such as string theories and loop quantum gravity, also allow black holes to be treated from the point of view of modern physics.

With the compact objects of astrophysics (white dwarf, neutron star, boson star, fermion star, quark star etc.) in general and the black holes in particular, matter and energy are combined in a very small space. Therefore, the curvature of spacetime is particularly high in these objects. Vividly argued, the curvature on the horizon of black holes is so great that the light rays are 'bent' onto the interior of the black hole. The trajectories of the radiation, the so-called zero geodesics, point to the central singularity. At this point (without rotation) or ring (with rotation) is the entire mass of a black hole! The key question is which equation of state the matter obeys there. In the context of classical GTR it can be stated that the curvature becomes infinite in this singular point of space-time. This is proven by the investigation of curvature invariants such as the Kretschmann scalar. The discussion suggested here thus shows that **Limits of the theory of relativity** and suggests an overarching theory that has been feverishly searched for for decades. So far without success! But with decisive advances only in the last few years!

p.s.

The relativistically correct answer to the initial question (Einstein's key question of the SRT), what one would see when riding on a ray of light is: **Nothing!** In other words: photons do not age! The Lorentz factor diverges when the speed of propagation equals the speed of light (*v* = c) becomes, and the 'elongation of the flow of time' (time dilation) approaches infinity, while the shortening of length scales (length contraction) approaches zero.

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