Is the four-dimensional space a Hilbert space

Astro-Lexicon R 2


Spacetime space-time) is an artificial word that appropriately expresses an essential result of the theory of relativity: space and time are no longer dimensions that are independent of each other. As the Lorentz transformation of the special theory of relativity (SRT) shows, space and time form a continuum! The time and the three space coordinates are closely interwoven and form one four-dimensional spacetime.

Inventor of the term Spacetime

The term Spacetime was founded in 1908 by mathematician Hermann Minkowski (1864 - 1909) who invented a new representation of the theory of relativity using vectors of four. This elegant formulation is still in use today. One funny anecdote is that Minkowski Albert Einstein Was a mathematics teacher at the Zurich Polytechnic. Despite his talent, Einstein did not necessarily excel in Minkowski's classes. Minkowski was all the more astonished when he learned that this Einstein had invented the theory of relativity. So he responded with the following comment:

'I really wouldn't have believed Einstein to do that.'

(Source: biography Albert Einstein of Thomas Bührke, dtv 2004)

dynamic and curved spacetime

Space and time no longer exist as absolute quantities as in classical physics, but are themselves dynamic object, physical quantity, a complex one Tensor field, a metric. This spacetime can occur in the absence of matter and energyflat then it is the Minkowski metric. However, space-time is generally curved by masses or, more generally, forms of energy. Then the general theory of relativity (GTR) has to be used to describe it. The curvature of spacetime is particularly pronounced in black holes, which are described by the Schwarzschild metric (static) or Kerr metric (rotating). The curvature only becomes particularly high close to the black hole and disappears at large distances. The relativists say: there the metric is asymptotically flat.
It is by no means a trivial task how one has to imagine space-time. In principle we are used to a four-dimensional world: In a room we fill three dimensions of space - one more, the other less - namely length, width and height. At a later point in time, we may find ourselves in a different place in the room: we have moved through time. But how do these four dimensions form a continuum? To illustrate this, you have to use a few tricks and e.g. suppress a space dimension and the time dimensions. A fairly simple idea of ​​spacetime is then an analogy to a stretchable one Rubber skin. In this simple 2D model, the information about the time dimension and one spatial dimension is lost. A rubber skin stretched in a frame forms a flat surface (= flat space-time without the presence of masses). If you put a mass, e.g. a heavy metal ball on the rubber skin, it will be curved. You get a 'dent' in spacetime, a curved spacetime (see figure above right). The depth of the dent is a measure of the curvature of space-time.
The dent is a result of an extended mass. You can continue this game and let the mass collapse in gravitational collapse. The result is a black hole. Illustrative representations of this are diagrams from Gravity funnelsas presented in the entry lapse function. The funnel does not close because there is a singularity of curvature here.

mathematical representation of spacetime

Mathematically, spacetime becomes unambiguous through the metric tensor (Metric) or alternatively that Line elementds2 described. The components of the metric tensor are in the line element as coefficients. Because the metric tensor is a second order tensor, it is 4 × 4 = 16 coefficients. Such a tensor can be written down as a matrix with four columns and four rows. The symmetry of the tensor reduces the 16 to only 4 + 3 + 2 + 1 = 10 independent components. Now it depends on the complexity and the symmetry properties (see also isometries) of the respective spacetime how many of the ten independent coefficients are different from zero. In the case of high symmetries, such as in the Minkowski geometry or the Schwarzschild geometry, the metric tensor is diagonal: then there are only four independent coefficients other than zero on the matrix diagonal - the remaining entries are zero.

Time in Einstein's theory

The concept of time is generalized to relative time in the theory of relativity: time depends on the observer. When studying dynamic, relativistic phenomena in numerical GTR, one therefore encounters the question of which observer to choose to investigate the temporal course of processes. It has proven itself to split up the symmetry of space and time again. The method is called 3 + 1 split or ADM formalism. The space-time then scrolls into space-like, three-dimensional hypersurfaces, on each of which time is constant. This makes possible within the framework of the numerical relativity theory the simulation of dynamic processes from the point of view of a special observer, for example the ZAMO.

Space-time tremors

Spacetime is very rigid structures and are difficult to deform by large amplitudes. At accelerated masses will be in principle alwaysGravitational waves emitted that convey the dynamic curvature of space-time. Gravitational waves are nothing more than vibrations in space-time that propagate at the speed of light in a vacuum of almost 300,000 km / s.

Move your hand. Did you notice? They just emitted a gravitational wave and deformed spacetime!

Of course, a hand is far too light for a large - let alone measurable - effect to be expected here. Only strongly accelerated and compact, heavy masses can cause a significant deformation of space-time by means of gravitational waves, e.g. when orbiting neutron stars or stellar black holes in binary systems. compact binaries), in binary from supermassive black holes, in supernova explosions or in gamma ray bursts.
The successive convergence of the components in the binary pulsarPSR1913 + 16proves indirectlythat it emits gravitational waves. This discovery was of great importance in underpinning ART and was awarded the 1993 Nobel Prize.
The stiffness of spacetime is a godsend for humanity because it favors the development and maintenance of life by ensuring a relatively stable environment. If spacetime were more flexible than observed, we would certainly not be here. This aspect can be taken into account in the anthropic principle.

Rayleigh Jeans Radiation Formula

See in connection under Planck's radiator.

Ray tracing is a method to visualize objects in 2D or 3D. The propagation of light is simulated and - ideally - all interactions of the radiation (reflection, refraction, scattering) with the objects in the area under consideration are taken into account.

Light on straight lines

Commercial ray tracer software, such as 3D computer games or CAD software, works in flat space-time, as relativists would put it. That means it is the usual borderline case of geometric optics before: light moves along from Straight lines. However, this is actually a special case.

Light goes into the curve

Generally speaking, the General Theory of Relativity (GTR) states that Albert Einsteinthat mass and energy bend space-time. In order to follow the propagation of light, the geodesics of the light particles have to be calculated. These 'light paths' are called zero geodesics in ART. Generally they are Curved light pathsbecause the 'space is crooked' too. To put it a little more stylishly: zero geodesics follow curved spacetime.

How do you get to the light paths in ART?

First you have to know in which space-time the light spreads. Is it the gravitational field of the sun or the environment of a neutron star, or is it even intended to show how a black hole swallows light? As soon as this has been clarified, the so-called metric of the gravitational source is formulated using the now known metric Zero geodesic equation: that is nothing else than the fully relativistic equation of motion for light particles (which have zero rest mass). In the form of a Geodesic equation The particle trajectories for particles with rest mass are also given away. In any case, the geodesic equation is, mathematically speaking, one Second order differential equation.

A particle of light does not make a picture ...

... at least not an exciting one. The zero geodesic equation must therefore be solved for many light particles that have different starting conditions (starting position, beam direction). This would be a laborious procedure 'by hand', because only about one million light particles (a picture with 1000 × 1000 pixels) produce a meaningful and exciting picture. So of course you use computer to calculate the image - experts call this (both in flat and curved spacetime) Rendering.
With this calculation it must also be clear where the observer of the scenery is. Because the geodesic equation links a light beam from its starting position with a point of impact. Where the point of impact is dictated on the one hand by the light beam through its starting conditions and on the other hand by the metric with its curvature.
The whole process of calculating the propagation of light in curved spacetime is now general relativistic ray tracing (engl. general relativistic ray tracing) called. There are also special relativistic ray tracingthat calculates what the environment looks like when you move through a scene at almost the speed of light.

Light propagation near a black hole

Let's look at an extreme case straight away, namely the movement of light particles in the vicinity of a black hole. The spacetime that must be used is either the Schwarzschild metric (hole does not rotate) or the somewhat more complicated Kerr metric (hole rotates). The following figure outlines what is to be calculated: a light beam (engl. ray) starts from a thin slice thin disk) around a black hole black hole) rotate (e.g. a standard disk) and move through curved spacetime (engl. curved space-time). The observer may sit far away from the hole, where the space-time curvature of the hole is negligible (asymptotically flat space-time). In this case, the observer looks at a screen) on which the image is to be displayed.

The geodesic equation has to be solved for each pixel of the image. Here one makes use of a useful property of light for numerical reasons: light paths are reversible. It would be unnecessary to calculate the paths of all the light particles that start from the disk, because only a fraction of them reach the screen. We're only interested in the picture on the screen. So we'd rather do it the other way around and 'calculate backwards' (back tracking): the calculation starts on the screen and depending on whether the particle hits the disk, the hole or nothing, it is colored. Whoops, the picture is ready.

two calculation methods

  • The geodesic equation can firstly by direct integration be solved, which is numerically more complex and requires higher computing power. This method has the advantage that it works for any spacetime.
  • The geodesic equation can second by Integrals of motion (Conservation quantities), which is numerically efficient and fast. The advantage here is that significantly less computer power is required.

Solution of the geodesic equation with conservation quantities

In the case of a Schwarzschild or Kerr hole, one uses the knowledge of four conservation quantities of the system. In addition to mass, energy and angular momentum, the essential quantity in Kerr geometry (which, of course, contains Schwarzschild as a special case) is Carter's constant. Brandon Carter 1968 derived this new constant of motion from the separability of the Hamilton-Jacobi equation. This fourth conserved quantity is a exclusive feature of the Kerr metric and is absent in other axially symmetric spacetime, such as in neutron stars.

Einstein's warped world

The calculated course of the zero geodesics is used for production full of relativistic images: On the screen, the viewer sees images that were painted using Einstein's theory as a method and a brush made of light. And what you see there is quite amazing! The relativistic world looks completely different, asymmetrical and distorted:

  • The object that glows is generally distorted and bent, sometimes even multiple times. This is a result of the Gravitational lensing.
  • If the light source moves towards the observer, there are blue shift effects; if the light source moves away from the observer, it is a redshift. Both are summarized in theDoppler effect.
  • Of course, the hole itself also appears: black, as it should be. The physical reason is that light particles that start at the event horizon or come too close to it are swallowed up by the hole. 'Swallowed Light' is black. This effect is called Gravitational redshift.

The figure on the right shows the result of a relativistic ray tracing. In principle, it is a false color image in which the brightness of a gas disk has been color-coded: low brightness is shown in black and high brightness is shown in white; Intermediate values ​​are yellow. This is what a thin, counter-clockwise rotating disk looks like, rotating around a black hole that is also rotating. The disc was assumed to be looking almost at its edge: the inclination is 70 °. The disc looks distorted, as if it was bent from the back up. That is precisely the gravitational lensing effect. As you can see, the radiation coming from the left part of the disk is brighter than the right part. This is the Doppler effect mentioned above, which is not only noticeable in the light color (= radiation energy), but also in the brightness (= radiation flux). The hole itself has been shown here in white to make it easier to see: however, not a single light particle comes from this area. This is the black hole's event horizon.

Use for astrophysics

The rendered images have enormous information content and offer plenty of opportunity for scientific discussion. In nature, the surroundings of a black hole have unfortunately not yet been photographed in this way. This is because the resolution of the telescope is not (yet!) Sufficient. The cosmic black holes are too compact and too far away for these photos to be taken at the moment - the radio astronomers will probably be able to do this in about five years Interferometry.
Nevertheless, these calculations have been useful to astronomers for years: In a further numerical step, one can enter from the picture spectrum calculate. They are very much observable! Since the matter is very hot so close to a black hole, there is a mixture of ions and electrons here, a plasma. It's so hot that it typically radiates in the X-ray range. Correspondingly, relativistic ray tracing is used to X-ray spectra to simulate, in particular thermal radiation from the pane. relativistic multi-color black body) and iron fluorescence lines (Fe Kα). A comparison with radio spectra is also possible if one assumes that synchrotron radiation is emitted by relativistic electrons in the vicinity of the hole.
The comparison of simulation and observation allows feedback on properties of the internal accretion flow and even on the black hole, e.g. whether it rotates.

More on this in the knowledge portal

The range is a key parameter when looking at the four fundamental forces of nature in particle physics:

The difficult ones don't get that far

The messenger particles that transmit these interactions are called Calibration bosons, which in the context of a calibration theory (engl. gauge theory) to be discribed. In the quantum field theories it is shown that the Rest masses of the gauge bosons decide on the ranges: the heavier the calibration boson, the shorter the range is the associated force.

four calibration bosons

The calibration boson of electromagnetism orin quantum electrodynamics is the photon, those of weak interaction are called W and Z particles (weakons), those of strong interaction or quantum chromodynamics are called gluons and the (not yet proven and therefore hypothetical) gauge boson of a quantized theory of gravity or quantum gravity is graviton.

And so far they come

Because photon and graviton have no rest mass, the related interactions, i.e. gravitation and electromagnetism, are in principle of infinite range. The two remaining forces of the subatomic realm are, however extremely short-range. The strong interaction has a slightly greater range than the weak interaction. The gluons are also massless (in relation to the rest mass), but they carry a colored charge. The strong carrier particles themselves interact with the quarks and hadronic matter, which shortens their range (detailed explanation under the entry gluons). The range of the weak force is so short because the weak zones are so massive: 81 or 91 GeV!

Calculate the range

Mathematically, the range follows from the equation on the right by inserting the mass of the exchange particle on the right (H: Planck's quantum of action, c: Vacuum speed of light). The equation can be quickly found using the Heisenberg's uncertainty principle estimate or exactly from the Klein-Gordon equation derive within the framework of quantum field theories.

In general, one understands in physics under Reionization a re-ionization of a material, e.g. by electromagnetic radiation. Ionization refers to the process that removes charges from a neutral structure (e.g. atom).

Cosmology: ionizing the first sources

In cosmology, the term reionization refers to an entire epoch, namely the one when the first radiation sources in the universe that resulted from the recombination era (z ~ 1100) re-ionized intergalactic medium (IGM) that had become neutral. The first, formed stars and the first generation of active galaxy nuclei, essentially quasars, come into consideration as the first radiation sources. The process of reionization took place over several phases with the following names: pre-overlap (engl. pre-overlap), Overlap overlap), Reionization reionization), Post overlap post-overlap). Each phase is associated with a specific expansion of the ionization fronts around the ionizing sources. This is illustrated in the figure on the right (also as an animation, 2.2 MB, approx. 900 × 700 pixels).

He ionization at a smaller z than H ionization

The essential chemical elements in the IGM that have already formed primordially (see primordial nucleosynthesis) are hydrogen (element symbol H) and helium (element symbol Hey). Before the reionization era, they were essentially between the first cosmic objects in neutral form in front.
As soon as a radiation source ionizes its neutral environment, it forms Strömgren Spheres made of spherical structures of ionized material. The threshold value for the ionization of neutral hydrogen is 13.6 eV. The ionizer has to generate this energy in order to 'HII-Bubbles' (Strömgren spheres simply ionized hydrogen) to form.
The ionization of neutral helium only takes place at higher threshold values: the ionization of neutral helium (HeI) to simply ionized helium (HeII) occurs from an energy of 24.6 eV, the double ionization to HeIII only at 54.4 eV. That means that theHelium reionization epoch occurred at a later point in time in the development of the universe and must therefore be more easily observable for astronomers because it is smallerRedshiftsz must lie. In a sense, helium denotes the Hydrogen reionization epoch at (H preview).

neutral material observed before reionization!

The age of reionization is on the redshift scale z ~ 10. z ~ 7 denotes hydrogen reionization and accordingly z less than 7 the helium reionization. Observations (Fan et al. 2000), namely those of the Quasar SDSS 1044-0125, prove that the reionization epoch at z = 5.8 was already fully completed: there are no so-called in the spectrumGunn Peterson Troughs in front of the Lyman-Alpha edge, which is an indication of neutral, highly absorbent material. The reionization epoch can already be seen in many distant sources (HZ sources, HZ for high redshift) and is considered proven.

21cm tomography

As indicators (in technical jargon so-called Tracer) for the pre-reionization epoch, i.e. the phase with neutral IGM, is used by the 21cm line neutral hydrogen (HI). It is a hyperfine structure transition, a spin flip from the triplet to the singlet state in the hydrogen atom. Radio observations are used in 21 cm tomography to scour space for neutral hydrogen.

At the end of the dark age

The first radiation sources or the first elementary building blocks, gaseous objects, which are natural in front the reionization must have formed, one settles at redshifts of z = 15 to 30 on. That ended with their creation Dark ages (engl.dark ages) of cosmology. The first, created stars are counted Population III to. They could have enriched the ISM and IGM with metals through pair instability supernovae.




© Andreas Müller, August 2007