What are the intricacies of time dilation

Astro-Lexicon A 3


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One of the characteristic, dimensionless numbers Magnetohydrodynamics (MHD), which are used to describe the system or a corresponding computer simulation.
It is defined as the root of the quotient from the square of the Alfvén velocity VA. over the square of a characteristic velocity V0 in the system.

general theory of relativity

The general theory of relativity, in short ART (engl. General Relativity, GR), is aTheory of gravitywho have favourited the classical theory of the English physicist and mathematician Isaac Newton (1643 - 1727) replaced. The ART offers completely new insights into the nature of gravity and must therefore be classified as revolutionary: Quantum theory and relativity theory are the most important physical theories of the 20th century, and their significance is central to modern physics to this day!

Redefinition of elementary terms & basic statements of ART

Both theories have that scientific worldview decisively modified and shaped: our ideas of space, time, matter and energy have been redefined. The essential statement of the GTR is that every form of energy (including matter) bends space-time. The space-time is a four-dimensional manifold, which is composed of the three space dimensions (length, width, height) and the time dimension. This geometric shapes is clearly defined by the metric or the line element. Its morphological properties are changed by energy and matter. Space-time is four-dimensional; To simplify matters, it can be imagined in two dimensions like a thin, stretchable skin that gets dents from the masses on it ('rubber mat model'). Without masses, the skin has no dents, it is flat. Such spacetimes are called flat. A test body, which is now placed on a skin dented by masses, must inevitably follow the curve. Therefore the test body moves in a curved space-time or in other words: he falls in free fall. The lines of motion are called geodesics and are different, depending on whether the test body has a Rest mass has (matter) or not (light). These geometric interpretation of gravity thus replaced Newton's concept of force.

Albert Einstein - physics pop icon

The theory of relativity is based on the German-born physicist Albert Einstein (1879-1955) back. He first developed the Special Theory of Relativity (SRT), which he published in 1905. This is a theory of reference systems (inertial systems) moving uniformly in a straight line with respect to one another, which already revealed the continuum of space and time (see e.g. Lorentz transformation). The central aspects of the SRT are Equivalence of mass and energy (E = mc2) and the Constancy of the speed of light c in all Inertial systems. Einstein postulated this constancy and deduced astonishing effects on space and time, such as length contraction and time dilation, which could actually be confirmed experimentally.
In the following years, from 1907 to 1915, Einstein generalized the uniformity of motion to accelerations. This approach resulted in a new theory of gravity, the General Relativitywhich he published in 1916. Generally it is in the sense because the relative movements between the reference systems generalized were and can be anything. The SRT, on the other hand, only applies to very specific relative movements, namely uniformly straight movements; in that sense she is special.
An essential basis of the theory of relativity is the principle of relativity. According to this, all reference systems that are uniformly moved in a straight line are completely equal. Identical physical experiments that are carried out in systems that move uniformly relative to one another produce the same results. In particular, systems that are relatively at rest are indistinguishable from systems that are moving in a straight line. The principle of relativity was generalized to accelerated systems in the GTR. Then it is called the equivalence principle and says that it is not possible in principle, sluggish of heavy mass to distinguish. This means that one cannot decide whether a mass is accelerated by a constant force or whether it falls freely in a gravitational field. In addition, after the Principle of general relativitythat all observers are equal.

Demanding mathematics of ART

Mathematically, ART makes use of tensors, which replaced Newton's concise vector notation and further generalized it. You are on geometrical formations Manifolds, defined and can be interpreted physically (see the tensors in the glossary). The tensors are embedded in the formalism of Differential geometry. According to the principle of general covariance, all equations in physics should tensor form to have. In addition to these principles, Einstein also directed the correspondence principle, Mach's principle, and the principle of minimal gravitational coupling. The general theory of relativity is rightly attributed to Albert Einstein, but one has to give credit to many other physicists and mathematicians that their thinking had a lasting influence on Einstein. The tensors as mandatory objects of the GTR, which first made a coordinate-independent access possible, had already been found by the mathematicians. Philosophical aspects of movement and inertia, which led to the ART, were already discussed by the Austrian physicist Seriously do (1838-1916) anticipated. In this respect, the time was ripe for a general theory of relativity.

The field equation of gravity

The most important tensors of the GTR are the Einstein tensor, which contains the information about the curved space-time, and the energy-momentum tensor, which contains the physical information about the forms of energy (as well as matter). Both tensors are in the fundamental Einstein's field equations of the ART linked together. The physical content of this connection is that, on the one hand, every energy bends the metric and, on the other hand, the metric defines the geometric structure on which the energy is localized. Formulated laconically: 'Geometry tells matter how to move, and matter dictates geometry how to curve.'.

The field equations have a very simple tensor form, as the equation above shows (here withoutΛ term). The compact notation is also justified because it reveals the essence of the symbiosis of curved space-time and energy. But the apparent mathematical simplicity is deceptive: the field equations of the GTR are in themselves 16 equations, six of which do not have to be considered due to the symmetry of the tensors. Einstein tensor and energy momentum tensor are second order tensors. Each of them can be represented as a 4 × 4 matrix. The tensor symmetry reduces the resulting 4 × 4 = 16 equations to only 10 equations. The remaining ten equations are coupled together. In addition, the equations are partial differential equations that are also not yet linear. The Non-linearity of the equations is an expression of the backward interrelationship of space-time and energy set out above. The Coupling constant Einstein's field equations (see equation above, numerical value of 8π) can be derived from a Correspondence principle derive: in the limit case of weak gravitational fields and lower speeds compared to the vacuum speed of light c the GTR must pass into Newton's theory. The coupling constant then follows from a comparison of Einstein's field equations and Poisson's equation. For the type of equation as shown in the field equations of the GTR, mathematics offers no patent remedies for writing down the general solution. The equations are therefore simplified and different sectors of the field equations are considered. Sometimes you set the energy-momentum tensor to zero and only consider vacuum solutions, sometimes you assume certain symmetries of the solutions (spherical or axial symmetry), sometimes you only consider spacetime of constant curvature - but even then you only find special solutions. The general theory of relativity certainly still contains many curved manifolds that occur in nature, but which no one knows to this day.
The Einstein field equations are therefore a Complicated system of ten partial, non-linear, coupled differential equations. It was all the more astonishing that just one year after the ART was published, in 1916, the German astrophysicist Karl Schwarzschild found a first solution. The (outer) named after him Schwarzschild metric solves the vacuum field equations. It describes spherically symmetrical spacetime of relativistic stars and in particular non-rotating, electrically uncharged black holes.

Consequences of the ART

  • Relativity of time and length, as already anticipated by the special theory of relativity. The passage of time depends quite generally in the theory of relativity on the reference system. In particular, the GTR is followed by a lengthening of time in the presence of gravitational fields. Time passes more slowly in the vicinity of masses, i.e. stronger gravitational fields. This stretching of time is called gravitational time dilation.
  • Loss of energy from radiation in the gravitational field. The radiation has to work against the gravitational field and therefore loses radiation energy. Because red radiation is lower in energy than blue and the cause of the energy loss is gravitation, this effect is called gravitational redshift or Gravitational redshift. In terms of effect, this corresponds in principle to the gravitational time dilation if one goes from the frequency to the time representation.
  • Deflection of radiation in the gravitational field. This phenomenon is called Gravitational lensing or gravitational aberration gravitational lensing).
  • relativistic generalization of the Binet's equation. The ART provides the correct and observed numerical value for the Perihelion of the innermost planet Mercury. Perihelion describes the rotation of the apsidal line, i.e. the line connecting the point closest to the sun (perihelion) with the point furthest from the sun (aphelion) of Mercury's orbit in space. The ellipse of Mercury's orbit is not closed, but rotates in space, so that a rosette shape of the orbit movement rotates. The cause of this phenomenon is the gravitational interaction of Mercury with the heavy sun. The perihelion rotation also exists in Newtonian gravitational physics, but only the GTR explains the exact, measured amount. The rotation of the perihelion is only relevant for Mercury because it is closest to the sun and is therefore the best indicator of relativistic gravity.
  • Gravitational wave emission of accelerated masses. In this way, changes in the gravitational field / curvatures in space-time propagate at the speed of light. Gravitational waves were previously not directly observed.

Multiple experimental confirmations

All of these phenomena have been verified experimentally and are considered brilliant confirmations of GTR. The ART was able to achieve its first experimental successes as early as 1919 during a solar eclipse in Africa, because it really did Light deflection in the sun predicted. The Perihelion of Mercury calculated correctly. The observed discrepancy of about 43 arc seconds per century for the displacement of the elliptical planetary orbit was well explained by the ART. 1993 was the indirect confirmation of the issue of Gravitational waves Awarded the Nobel Prize at the binary pulsar PSR 1913 + 16: Hulse and Taylor were able to prove experimentally that the pulsars gradually approach each other because the binary star system loses rotational energy through the emission of gravitational waves.

Einstein's cosmos

The GTR has a cosmological relevance because it is to be regarded as the first physical theory of the universe. The world is then four-dimensional and locally generally not Euclidean. Whether the universe is globally Euclidean or not depends on the Friedmann world model. This question is still the subject of modern cosmology. The exact energy content of the universe has to be measured (see also Missing-Mass Problem). Currently, an infinite, open, expanding and flat universe is indexed and favored (measurements from balloon experiments and the microwave satellite WMAP). Its dynamics are dominated by dark energy, which is the most important form of energy alongside baryonic matter and dark matter. Mathematically, this four-dimensional universe is described globally using the Robertson-Walker metric. The content of matter is described with a relativistic, ideal fluid. The dark energy is realized in the cosmological lambda term in the field equations. It is considered in modern cosmology that the cosmological constant is not constant, but can vary over time. These models are called quintessences. The lambda cosmology and the search for a world formula were among Einstein's last work. Einstein justified his lambda by stating that it enabled a static universe that was favored at the time. However, when the expanding universe was observed (Hubble effect), Einstein withdrew his lambda and called it 'the greatest donkey of his life'. Modern cosmological models need it because it is an important parameter to explain observations. The lambda term is clearly interpreted in such a way that the quantum vacuum already provides energy (Vacuum polarizations) that bends spacetime. However, this interpretation is not yet consolidated and will be examined in the context of the quintessence.

Extreme gravity: compact stars

The ART provides a mathematical description for black holes, which in this context provides a solution to the Vacuum field equations or Einstein-Maxwell field equations. The gravitational fields of these compact objects are so strong that Newton's theory fails. Other compact objects such as neutron stars, magnetars, quark stars and gravastars can also only be correctly described with the ART. The relativistic corrections for white dwarfs, on the other hand, are marginal: Here, astrophysicists often still use Newton's theory, to which the structural equations (Lane-Emden equation) of the compact star. But the Stability of the white dwarf can only be explained relativistically. Now is relativistic in the sense of the specially relativistic quantum mechanics (and not the general theory of relativity) to understand. Because: the spin of the electrons, an essential species of particles in the interior of white dwarfs (in addition to carbon), is responsible for the stabilizing degenerative pressure.

Beyond the limits of Einstein's greatest throw

ART is one classic theory, classic now in the sense of not quantized to understand because quantum properties, such as spatial impulse uncertainty or the quantum vacuum Not enter. There are areas of nature or physics where GTR fails. One can outline the parameter space where this happens with strong gravitational fields in connection with very small, atomic and subatomic space scales. That leads to the Planck scale.
The occurrence of singularities could also be interpreted as an indication that the theory has to be modified or replaced by a superordinate theory, quantum gravity. According to the Singularity theorems of Hawking and Penrose singularities are something 'natural' and independent of the GTR. The example of the gravastars shows, however, that at least the singularity of the Schwarzschild hole can be represented by a regular alternative can replace. For many physicists, regularity is extremely attractive. In this sense, the Singularity question not yet clarified and needs to be further discussed through experimental and theoretical research.This could be designed in such a way that astronomers would actually succeed in confirming singularities or gravastars, or that theorists could succeed in formulating robust quantum gravity. For a long time, string theories were seen as the only way to achieve quantum gravity. Current research has found another alternative in the form of loop quantum gravity. The loop quantum gravity can be seen as a direct further development of the general theory of relativity, which tries to do justice to the concepts of quantum mechanics. In the description of nature, neither string theory nor loop quantum gravity has proven itself so far. The development of tests has already begun and will certainly be pursued more intensively in the next few years.

No place for doubters

One thing is clear, however: claims such as'The theory of relativity is wrong!'or'Einstein was wrong!'are to be dismissed, dubious and absolutely anachronistic. The ART has been well verified by many experiments and is a (in Popper's sense of the philosophy of science) proven theory. As with Newton's theory or, more generally, with physical theories, there is also one with GTR Validity framework, which leads to a failure of the theory with certain parameters (strong fields, small spatial scales). Failure is signaled by divergent quantities, such as curvature or density, and possibly even by the occurrence of singularities. Any theory that is superordinate to the GTR must contain general relativity as a borderline case, just as the GTR contains Newton's theory. Therefore, even after a robust quantum gravity has been found, the GTR will retain its raison d'être!

Reading note on deepening

Outside the lexicon you will find a detailed article on important objects of general relativity and astrophysics: Black Holes - The Darkest Secret of Gravity.

One of the three forms of radioactivity besides beta and gamma decay. In the case of radioactivity, certain atomic nuclei (technical term: Radionuclides) certain matter particles (electrons, positrons, helium atomic nuclei, also neutrons) or high-energy, electromagnetic radiation. Radioactivity is dangerous to life due to its strong ionizing effect! In some cases, radioactivity can be shielded and thus 'defused' with simple means.

What exactly is α-decay?

The radionuclides in α-decay are so-called α emitters, i.e. they send out helium atomic nuclei, a combination of two protons and two neutrons. These are then called He nuclei α-particle (not to be confused with the alpha male in wolves). The reaction equation is noted in general form for any atomic nucleus X with the number of protons Z (atomic number of the element) and the atomic mass A (sum of the number of protons and neutrons bound in the atomic nucleus) at the top right. If the nucleus X sends out an alpha particle, its atomic mass is reduced by four and the atomic number by two, i.e. a nuclear reaction has taken place in the form of a conversion of the element. The corresponding shifts in the periodic table of the elements or on the nuclide map regulate the so-called Soddy-Fajans Displacement Theorems. As always in physics, these reactions apply Conservation Laws (e.g. for mass, energy and particle type), so that what is on the left must correspond to the sum of the components on the right. The mass of an alpha particle is 3.7274 GeV (take into account 'mass defect' due to the binding of the four nucleons).

The cause can only be understood in terms of quantum theory

Only through quantum theory it was possible to explain this form of radioactivity (the other two as well). The Tunnel effect enables the alpha particle to 'tunnel' through the Coulomb barrier of the atomic nucleus and exit the nucleus: the nucleus emits alpha radiation.
This radiation is the most dangerous of all radioactive radiation for life because the biological damage caused by the heavy alpha particles is enormous. Fortunately, due to the high mass (and charge) of the He atomic nuclei, alpha radiation is short-range and can be effectively shielded with a sheet of paper.

The equations of the are used in many areas of theoretical astrophysics and in fluid mechanics in general Hydrodynamics (HD) and Magnetohydrodynamics (Best before). They have proven themselves in astrophysics to simulate the dynamics of numerous cosmic objects on the computer. The equations can be used on very different length scales, depending on how big the bodies are.
Sometimes the objects are very extended and we are interested in the dynamics on many scales at the same time - in one and the same simulation. An example is the accretion to a supermassive black hole in an Active Galactic Core (AGN) or the expansion of a large-scale jet emitted by the AGN. Then you have to find methods that can map the dynamics on many size scales without exceeding the hardware requirements. These techniques are called Adaptive grid methods (engl. adaptive mesh refinement, AMR). To understand this, it must first be explained what a numerical Grid is.

This is how it works in practice

In HD / MHD simulations, the area to be examined is broken down. solution domain) into smaller cells. It is a process of Discretizationwhich is necessary to be able to do numerics at all. A physical function assumes certain values ​​on every cell, which is uniquely fixed with location coordinates in the area. Typical functions in hydrodynamics are pressure, density and temperature, in magnetohydrodynamics magnetic field, magnetic pressure and Alfv n speed. They vary spatially over the area under consideration, but also in time, if, for example, one takes out a certain cell and studies its development over time separately. The dynamic lies in the time dependence. The smaller cells, in the simplest case squares (2D) or cubes (3D), form a (here equidistant) Grid. The example is a regular, structured grid. You can also turn the area into a unstructured lattice disassemble what is commonly found at Finite element methods finds. AMR now fits the Delicacy of the lattice, i.e. the size of the grid cells, is different in each area of ​​the grid. The criterion is whether the size in question changes significantly in a certain region of the grid or remains more or less constant. Only where it changes strongly does it have to be resolved more strongly, i.e. the grid has to be refined. These grids are called adaptive. A measure of the spatial variation of a size is gradient. It can be used as a control parameter for the AMR. AMR is a numerically efficient method because it bundles the hardware resources only where structures occur, i.e. where 'something interesting' happens.

This principle, named after the ancient Greek word anthropos For human, is used in cosmology and says in short:

We see the universe as it is because we are here to see it.

The starting point of the anthropic principle is the question of the nature of the universe. Why is it just the way we watch it and not different? According to the interpretation of the anthropic principle, the answer lies in our existence: physically other realizations of the universe are entirely conceivable. But because only certain, possible universes allow the existence of humans, the universe must be as we observe it, because we are hereto watch it.

Two formulations of the anthropic principle

  • weak anthropic principle: The conditions for the development of life are found only in certain areas of the universe.
  • strong anthropic principle: The conditions for the development of life are found only in a few universes.

The weak anthropic principle states that the universe must first undergo development before life can arise. The development goes through the formation of particle species, atoms and molecules, the clumping of intergalactic matter into galaxies from gravitational instabilities, the formation of stars in galaxies, the formation of planets around a few stars to the formation of life on selected planets. This process takes time, apparently as long as it took our universe: about 13.7 billion years. The local universe (with a cosmological redshift of z = 0) therefore only fulfills the conditions for life. As everyone knows: this is supported by observation.
The strong anthropic principle is a modification made with the Many worlds theory came up. As part of a Quantum cosmology one can transfer the mechanisms of quantum theory to cosmological models: the creation and annihilation of particles has an analogy in the creation and annihilation of universes! This leads accordingly Everett and Wheeler to the possibility of interpretation that not only one universe can arise, but many, a cosmic coming and going. But also in this one Multiverse only certain universes could meet the conditions for life, hence the above formulation of the strong anthropic principle.

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© Andreas Müller, August 2007