Perturbation theory (general relativity theory)

In the general theory of relativity , perturbation theory is used as a calculation method to obtain approximate solutions for the gravitational field in complicated systems.

In contrast to perturbative approaches in other areas of physics, symmetries in particular play a prominent role. In addition, there is a systematic differentiation of the disturbances according to their tensor character.

Many of the predictions of general relativity have been derived from perturbation theory. The perihelion rotation , the curvature of light and the Shapiro delay in the solar system can be calculated very precisely and easily using perturbative methods. So far, gravitational waves have only been treated with methods of perturbation theory and cosmological perturbation theory provides important predictions about the temperature fluctuations of the cosmic background radiation , which are very precisely confirmed by the measurements of the WMAP satellite .

Basics

The central equations of general relativity are the Einstein field equations

{\ displaystyle G _ {\ mu \ nu} = {\ frac {8 \ pi G} {c ^ {4}}} T _ {\ mu \ nu}}

where is the Einstein tensor , the gravitational constant , the speed of light and the energy-momentum tensor . The Einstein tensor is given by derivatives of the metric tensor . ${\ displaystyle G _ {\ mu \ nu}}$ ${\ displaystyle G}$ ${\ displaystyle c}$ ${\ displaystyle T _ {\ mu \ nu}}$ ${\ displaystyle g _ {\ mu \ nu}}$

In perturbation theory, the metric tensor is now divided into a known background part , for example the Minkowski metric , and a perturbation that is much smaller than the background metric . An analogous division is also carried out with the energy-momentum tensor. The background sizes are chosen in such a way that they describe a known universe, i.e. are a solution to the Einstein equations (idealization of the actual problem). The terms that contain perturbations are then sorted as usual according to the powers of the perturbations that occur. From the assumption that the equation must be fulfilled individually for each order, then equations for the disturbances result. ${\ displaystyle {\ overline {g}} _ {\ mu \ nu}}$ ${\ displaystyle \ delta g _ {\ mu \ nu}}$

Covariance

Einstein's field equations, like the whole general theory of relativity, are framed in a form that transforms covariantly under diffeomorphisms . This means that they do not change their shape when changing coordinates. This property has the consequence that the perturbation variables in perturbation theory change in a certain way with coordinate transformations. Since it is assumed that the disturbances are very small compared to the background sizes, infinitesimal diffeomorphisms are considered which do not destroy this relationship.

Diffeomorphisms can be understood as flows of vector fields , so that an infinitesimal diffeomorphism can be understood as the flow of an infinitesimal vector field . A disturbance variable with an associated background variable is transformed under such a calibration transformation ${\ displaystyle \ xi}$ ${\ displaystyle \ delta X}$ ${\ displaystyle {\ overline {X}}}$

{\ displaystyle \ delta X \ rightarrow \ delta X + {\ mathrm {Lie}} _ {\ xi} {\ overline {X}}}

where is the Lie derivative with respect to the vector field . ${\ displaystyle {\ mathrm {Lie}} _ {\ xi}}$ ${\ displaystyle \ xi}$

In order to separate the physical results from the effects of the coordinates, disturbance variables are often determined which do not change under calibration transformations, i.e. are calibration invariant. For this purpose, certain linear combinations of the disturbance variables and their derivatives can be formed.

Classification of disturbances

The disturbances can be broken down into scalar, vectorial and tensorial disturbances within the framework of the Hamiltonian formulation of the field equations. This decomposition facilitates the determination of calibration invariant quantities and the physical interpretation of the results. As part of this breakdown, spatial and temporal variables are separated, so the background metric is broken down into

{\ displaystyle {\ overline {g}} = - dt ^ {2} + \ gamma}

where is the spatial fraction of the metric. The disturbances can now be broken down into three types. ${\ displaystyle \ gamma}$

Scalar disorders can now be four functions , , , characterize ${\ displaystyle A}$ ${\ displaystyle B}$ ${\ displaystyle D}$ ${\ displaystyle E}$

{\ displaystyle \ delta g_ {s} = \ left ({\ begin {array} {c | c} -A & -B_ {| i} \\\ hline -B_ {| i} & D \ gamma _ {ij} + E_ {| ij} \ end {array}} \ right)}

.

The lines indicate the separation of spatial and temporal components, with the upper left quadrant being the purely temporal component, i.e. a simple number, while the lower right quadrant is to be understood as a 3x3 matrix. The other quadrants are correspondingly a row and a column vector. The notation means the covariant derivative belonging to the metric . Two of the four scalar degrees of freedom can be eliminated by calibration transformations, so that two calibration invariant degrees of freedom remain. ${\ displaystyle B_ {| i}}$ ${\ displaystyle \ gamma}$

Vector perturbations are characterized by two three-dimensional (co-) vector fields , the divergence of which disappears, i.e. where Einstein's sum convention was used in the formulas . The disorders have the form ${\ displaystyle \ beta _ {i}}$ ${\ displaystyle h_ {i}}$ ${\ displaystyle \ beta ^ {i} {} _ {| i} = h ^ {i} {} _ {| i} = 0}$

{\ displaystyle \ delta g_ {v} = \ left ({\ begin {array} {c | c} 0 & \ beta _ {i} \\\ hline \ beta _ {i} & h_ {i} {} _ {| j} + h_ {j} {} _ {| i} \ end {array}} \ right)}

.

Of the four degrees of freedom that remain due to the freedom from divergence of the fields, two can be removed by calibration, so there are two calibration-invariant degrees of freedom.

Tensor disturbances by a 3x3 tensor describes the both traceless is and divergence-free, that is . The general form of tensorial disorders is ${\ displaystyle H_ {ij}}$ ${\ displaystyle H ^ {i} {} _ {i} = H ^ {i} {} _ {j} {} _ {| i} = 0}$

{\ displaystyle \ delta g_ {t} = \ left ({\ begin {array} {c | c} 0 & 0 \\\ hline 0 & H_ {ij} \ end {array}} \ right)}

.

Both degrees of freedom, which remain due to the absence of tracks and divergence, are gauge invariant.

In total there are 6 independent gauge invariant degrees of freedom, which corresponds exactly to the number of independent Einstein equations. It can be seen from this that all possible disturbances of the metric are included in this characterization.

Applications

The perturbation theory has two main areas of application in general relativity, the approximate calculation of the gravitational fields of mass distributions and cosmological calculations.

Gravitational fields

In this context, the background spacetime is usually assumed to be flat spacetime and the gravitational field is described by the perturbations. The scalar perturbations describe the Newtonian law of gravity and additional effects such as the deflection of light by masses, the Shapiro delay and the relativistic perihelion rotation . Einstein's first calculation of light deflection and perihelion were based on such a perturbative approach, since the Schwarzschild metric had not yet been discovered at the time of its publication .

The vector disturbances can be interpreted as an expression of a rotating mass distribution and are therefore suitable for an approximate description of the Lense-Thirring-Effect . For an exact description it is possible to use the Kerr metric, which was first described in 1963 .

The tensor disturbances are interpreted as gravitational waves that can be generated by an inhomogeneous, rotating mass distribution, for example in a binary star system . Today (2009) there is no known exact solution of Einstein's field equations for two-body systems, so that the generation of gravitational waves in binary star systems can only be calculated using perturbation theory. The indirect proof of gravitational wave generation at the double pulsar PSR 1913 + 16 thus also represents a significant success for the perturbation theory of general relativity.

Cosmological perturbation theory

→ Main article: Cosmological perturbation theory

In cosmology, the methods of perturbation theory are used to describe the deviations of the observed mass distribution of the universe from a homogeneous, isotropic mass distribution. The Friedmann-Lemaître-Robertson-Walker metric is used as the background space- time, which describes a homogeneous and isotropic universe because astronomical observations show that the universe is homogeneous and isotropic to a good approximation on scales of more than 100 mega parsec .

In the cosmological perturbation theory, mainly scalar and tensor perturbations are taken into account. The disturbances are used to explain the inhomogeneity of the mass distribution of today's universe and the temperature fluctuations of the background radiation , which the satellite COBE first discovered. The successful prediction of the spectrum of the inhomogeneity of the background radiation, confirmed by the WMAP satellite , is a great achievement for cosmological perturbation theory and the theory of the inflationary universe .

literature

Norbert Straumann : General Relativity. With applications to astrophysics . In: Texts and Monographs in Physics. Springer Verlag, Berlin 2004, ISBN 3-540-21924-2
Wiatscheslaw Mukhanow , Hume Feldman , Robert Brandenberger : Theory of Cosmological Perturbations . In: Phys. Rep. 215: 203 (1992).
Norbert Straumann: From primordial quantum fluctuations to the anisotropies of the cosmic microwave background radiation . In: Annals of Physics . 15, No. 10–11, pp. 701–845 (2006) ( online version , as PDF in arXiv ).