# Should I learn general relativity theory

## general theory of relativity

to which the Einstein tensor apply Gαβ subject to (Bianchi identities). (The semicolon in the above equation denotes the covariant derivative, Connection). Therefore, only six equations of system (1) are independent. This reflects the invariance of general relativity under arbitrary (sufficiently often differentiable, invertible) coordinate transformations of the metric tensor, which can be used to use a certain coordinate system that is adapted to the physical situation. An example of one calibration of the coordinates is the Einstein-Hilbert or De-Donder calibration, which we will encounter in connection with gravitational waves. Another example are normal coordinates, which are defined by the condition G00 = 1, G01 = G02 = G03 = 0 are defined.

The local energy-momentum conservation follows from the Einstein equations by applying the contracted Bianchi identities

. Einstein recognized that this resulted in the equation of motion of the general theory of relativity, the geodetic equation

(2)

can be derived, where the connection Γαβγ in contrast to the special-relativistic equation in the general theory of a Riemann tensor

belongs.

The equation of motion does not have to be postulated additionally, but is contained in the field equations, as in electrodynamics. This closes an important circle: through the Einstein equation (1), matter in the universe "tells" space-time geometry how it should bend. At the same time, the space-time geometry "tells" matter how it should move: on the geodesics determined by equation (2). This shows the implementation of Mach's principle in general relativity.

For Γαβ = 0 the Einstein equations (1) describe a space-time without matter (Vacuum field equations). This includes the Minkowski space, i.e. the flat geometry of the special theory of relativity, but also, for example, gravitational waves and the outer space of celestial bodies are covered by the vacuum field equations. Because of their non-linearity, the field equations can hardly be solved exactly, i.e. without numerical methods, so exact solutions are only known for highly symmetrical geometries. The most important of these is the Schwarzschild solution, found in 1916, which describes the gravitational field of a static, spherically symmetrical body. The corresponding generalization to a stationary rotating body, the Kerr solution, was only discovered in 1963. These solutions describe the spacetime geometry of static or rotating stars, of neutron stars and also of black holes, which will be discussed in more detail below.

3. Experiments

Despite all plausibility arguments, the Einstein equations (1), like all basic equations in physics, cannot necessarily be derived. Ultimately, only the comparison with the experiment can justify whether the general theory of relativity is the "correct" theory of spacetime and gravity. It is thanks to Einstein's genius that all experimental tests have excellently confirmed the predictions of General Relativity, although the deviation from Newton's gravity is often minimal. Einstein himself proposed three effects to test his new theory: the Perihelion of Mercury, the gravitational light deflection and the Gravitational redshift. Although other, partly more meaningful experiments have been carried out in the meantime, these three "classical" tests form the core of the experimental confirmation of Einstein's theory of gravity.

The first success was the calculation of the perihelion of Mercury, which Einstein performed himself. In contrast to Newton's gravitation, the gravitational potential of a spherical body in general relativity only falls approximately like 1 /r. For this reason, a planet does not move in a closed elliptical orbit around the sun, but rather a small closure error occurs, whereby the perihelion, the point of the shortest distance to the sun, shifts a small amount with each orbit. The effect is most pronounced for planets close to the sun with the greatest possible orbital eccentricity. The best candidate is therefore Mercury, whose perihelion rotation has been known since the mid-19th century. The measured amount of 5600 ″ per century is mainly caused by effects already known at the time, such as the disturbance of Mercury's orbit by the other planets in the solar system. However, a numerical evaluation showed a deviation of this Newtonian perihelion from the observed value of around 43 "per century. Einstein calculated the additional perihelion predicted by general relativity and found the missing 43 "per century in very good agreement.

The second important test is based on an effect which, using the special-relativistic equivalence of mass and energy, can also be understood in the context of Newton's theory: the deflection of light in the gravitational field. However, the value predicted by the general theory of relativity is exactly twice as large as Newton's due to the additional deflection caused by the curvature of space-time, so measuring the strength of the gravitational light deflection can experimentally distinguish between the two theories. One possibility is to observe the deflection of the light of the fixed stars in the gravitational field of the sun. For a ray of light that is not touching the edge of the sun, the apparent shift in its position is around 1.75 ″, which should be easily measurable during a solar eclipse. The measurements of the apparent positions of the fixed stars near the Sun, made by two expeditions during the solar eclipse in May 1919, confirmed Einstein's prediction. Although only an accuracy of about 30% was achieved, this result signifies the breakthrough in general relativity and suddenly made Einstein world famous. Today, corresponding measurements are carried out with much higher accuracy on the radio emissions from quasars. In addition, the gravitational deflection of light causes the gravitational lensing effect, which has been observed many times.

The third classic effect is gravitational redshift. A photon that escapes from the earth into space, for example, loses energy when overcoming the gravitational potential, which is associated with a corresponding redshift of the frequency. This effect was first observed in 1960 and 1965 (Pound-Rebka-Snyder experiment), and the expected frequency shift was confirmed with an accuracy of 1%.

The tests of the general theory of relativity also include an extraordinary "test laboratory": the double pulsar PSR 1913 + 16 enables a whole series of general relativistic effects to be checked, the most important of which should be mentioned here. The pulsar and its invisible companion move at a maximum orbit speed of 400 km / s, which corresponds to

, at an orbit period Tb = 7.75 h. The general theory of relativity says for such a system an energy loss due to the radiation of Gravitational waves ahead. This phenomenon will be discussed in more detail later, here it should only be pointed out that the energy loss causes a decrease in the orbit period. The measured relative change

corresponds very closely to the value that general relativity predicts for this system. This measurement is the first indirect observation of gravitational waves. The discoverers of PSR 1913 + 16, John Hulse and Joseph Taylor, were awarded the Nobel Prize in Physics in 1993.

4. Black holes

One of the most spectacular predictions of general relativity is the existence of black holes. According to the theory of stellar evolution, burned-out stars collapse into white dwarfs or neutron stars. If the mass of the star exceeds an upper limit, however, then the gravity during the collapse becomes so strong that the internal pressure can no longer brake the collapse - the entire mass is concentrated in a very small space. In the context of the classical general relativity theory, space-time curvature and mass density even become singular, i.e. they grow into infinity. Characteristic of such an extreme gravitational field Space-time singularity is the event horizon, a two-dimensional space-like surface that encloses the singularity like a semi-permeable membrane. While particles outside the event horizon can still escape the black hole, the gravitational force in its vicinity becomes so strong that neither matter nor light of the singularity can escape within the horizon. Therefore, a distant observer sees a "black hole". One of the amazing properties of black holes is by name No-hair theorem known: Regardless of the details of the initial state and the collapse, a black hole in the final state is fully characterized by a maximum of three parameters, namely by the mass M., the angular momentum J and the electric charge Q.

Theoretical research into the mechanics of black holes found surprising analogies to classical thermodynamics in the 1970s. The development of the thermodynamics of black holes, especially promoted by Stephen Hawking since then, is linked with the hope of finding indications of a - so far unsuccessfully sought - Quantum theory of gravity without knowing the details of the theory.

Disregarding the possibility of observing a dying, massive star as it collapses, observing black holes can only rely on indirect effects. This includes the movement of binary systems with an invisible partner whose mass can be estimated downwards. The X-ray source Cygnus X-1, discovered in 1971, was identified as such a binary system consisting of a red supergiant and an invisible companion. By the mass of at least 9M.S. (M.S. means the solar mass), according to the current theory, only a black hole is possible as a companion, since white dwarfs or neutron stars only up to a maximum of 1.35 or 4M.S. are stable. A total of six similar X-ray binary systems with masses> 4 are known M.S. . However, it cannot always be ruled out that the system two Contains invisible companions, the mass of which is below the critical limit.

Y. Zeldowitsch and I.D. In 1964, Novikov suggested that in addition to such stellar black holes, supermassive black holes could be present in the centers of active galaxies. The cores of such galaxies have a typical extension of

km, which is roughly the size of our solar system. Continuous radiation with a typical luminosity of

W, that corresponds to the

- times the luminosity of the sun! The only convincing mechanism for the release of such energies is the model one Accretion disk (Accretion) around a rotating, 106-109M.S. heavy black hole. Similarly, all observations suggest that Quasars of supermassive, 108-1012M.S. heavy black holes are energized. But inactive galaxies may also harbor supermassive black holes: by observing the proper motion of hundreds of stars, the existence of a 10 is valid6M.S. severe black hole in the center of the Milky Way is practically proven. The latest spectroscopic investigations by the Hubble Space Telescope even indicate that the central regions of practically all galaxies harbor supermassive black holes whose mass is proportional to the mass of the respective galaxy. The existence of these objects is therefore closely linked to the question of the formation of galaxies, which has not yet been clearly clarified. A mechanism that includes the formation of central black holes during the birth of galaxies has yet to be found. Solving this riddle could provide crucial clues to the evolution of the early cosmos.

5. Gravitational waves

As indicated, General Relativity makes another exciting prediction: the existence of gravitational waves. For this purpose, the Einstein equations are linearized in the simplest approximation. For weak gravitational fields, the space-time metric can be determined by a small perturbation Hαβ 1 superimposed Minkowski metric ηαβ be approximated. With the approach

the Einstein equations (1) reduce to the wave equation

for the non-marking tensor

. The field Ψαβ is subject to the calibration condition

(Einstein-Hilbert or De-Donder calibration). The solutions to this wave equation describe transverse waves that propagate at the speed of light. However, one should not forget that this simple picture is only valid in the linear approximation carried out. In particular, the non-linearity of the complete theory has the effect that no superposition principle applies to realistic gravitational waves. In extreme cases, colliding gravitational waves can even form intrinsic space-time singularities.

Because transverse waves generally do not allow monopole radiation and dipole radiation could only occur if there were also negative mass, gravitational waves are concerned Quadrupole radiation. A source of gravitational radiation therefore needs a time-varying gravitational quadrupole moment. The energy radiated from a massive object in the form of gravitational waves is

,

where the third time derivative

of the reduced quadrupole moment of the source. This famous one, already calculated by Einstein in 1916 Quadrupole formula was confirmed very precisely by measurements on the Hulse-Taylor double pulsar PSR 1913 + 16. As already mentioned, this is taken as a first, albeit indirect, evidence for the existence of gravitational waves.

Possible sources suggest three different types of gravitational waves. Shock waves can arise from the gravitational collapse of massive stars and are therefore associated with supernova explosions and the formation of neutron stars and black holes, as well as with the collision of black holes and neutron stars and the merging of compact binary star systems. Generate rotating, deformed stars and especially binary systems such as PSR 1913 + 16 periodic gravitational waves. After all, it is to be expected that the entire universe will have a background of stochastic gravitational waves contains. Similar to the cosmic microwave background (cosmic background radiation), this "remainder" of the hot Big Bang stores information from processes in the very young universe, e.g. from the first gravitational inhomogeneities that acted as seeds of the structure of the universe observed today.

The experimental investigation of gravitational waves is therefore a new window of astrophysical observation, combined with a wealth of exciting, otherwise inaccessible phenomena. Nevertheless, even 80 years after the birth of the general theory of relativity, no direct measurement has yet been achieved. The reason lies in the enormous smallness of the predicted effects. The size of the dimensionless perturbation H = | Hαβ | can e.g. be passed through for periodic sources in the already mentioned quadrupole approximation

estimate.

is the part of the kinetic energy associated with the non-sphericity of the rotation and r the distance of the source from the earth. For merging neutron stars and stellar black holes, the order of magnitude applies

≈ M S. . So lies H in the range of 10 - 22 for the distance range of the Hubble length (r ≈ 3000 Mpc, 1 parsec (corresponds to the symbol Me) 3.26 light years) to 10 - 17 for the outer region of the Milky Way (r ≈ 20 kpc). Optimistic estimates show a rate of a few such events per year within a radius of r ≈ 200 Mpc, so when building a detector a sensitivity of H ≈ 1021-1022 can be achieved. For interferometer detectors (gravitational wave detectors) with a time-dependent difference between the two arm lengths of the interferometer of ΔL. = L.1L.2 the approximation applies H(t) = (ΔL., L.). Precision laser interferometry will allow a maximum resolution of Δ in the next few yearsL. ≈ 10 - 18 m, which corresponds to about a thousandth of an atomic nucleus! Together with the aforementioned requirement of H ≈ 1021-1022 for the sensitivity follows for the arm length L. = (ΔL., H) ≈ 1-

km. Several such detectors are under construction as major international projects.If the first direct signals of gravitational waves are registered with these detectors after the turn of the second millennium, as hoped, then this signals a new era of relativistic astrophysics and another triumph of Einstein's general theory of relativity.

literature

[1] A. Einstein: Basics of the theory of relativity, 6th edition, Vieweg, Braunschweig 1990.
[2] R. D'Inverno: Introduction to the theory of relativity, VCH, Weinheim 1995.
[3] I. Ciufolini and J.A. Wheeler: Gravitation and Inertia, Princeton University Press, Princeton 1995.
[4] R.U. Sexl and H.K. Urbantke: Gravitation and cosmology, Fourth edition, Spektrum Akademischer Verlag, Heidelberg 1995.
[5] E.W. Mielke: Sun, moon and ... black holes, Vieweg, Braunschweig 1997.

General relativity 1: Title page of a work by Albert Einstein and Marcel Grossmann on general relativity from 1913.

General theory of relativity 2: The star HDE 226868.

General relativity 3: Two-dimensional representation of the space-time solution of the Einstein equations near a double star that emits gravitational waves.

General relativity 4: The sky around the pulsar 1913 + 16. The cross marks an object that could possibly be the companion star of the pulsar. (Courtesy T. Tyson, Bell Laboratories.)