Raychaudhuri equation

The Raychaudhuri equation (or Landau - Raychaudhuri equation) is a fundamental result of general relativity and describes the motion of neighboring particles.

The equation is a fundamental lemma for the singularities theorem for the analysis and exact solutions of general relativity. It also confirms a simple way our intuitive notion that the local effect of gravity in general relativity corresponds to the Newtonian law of gravitation: a general attraction between pairs of 'particles' ( now understood as mass and energy).

Mathematical formulation

The world lines of the particles under consideration are described by a timelike and normalized four-dimensional vector field. The description by a vector field implies that do not intersect the world lines, ie the particles do not collide. The world lines of the particles need not necessarily be geodesics, so the equation in the case of external force fields is valid. The tensor from the metric tensor and the vector field

Constructed which can be interpreted as the metric tensor on the vector field orthogonal hypersurfaces. The basic object of investigation of the Raychaudhuri equation is now the projection of the covariant derivative of the vector field orthogonal to the hypersurfaces

Where this tensor in its symmetric part, the Expansionstensor ( "expansion tensor " )

And its anti -symmetric part, the Vortizitätstensor ( " vorticity tensor " )

Is split. Derived variables are

It is as Scherungstensor ( "shear tensor " ) and as Expansionsskalar ( "expansion scalar " ) referred.

By means of these variables is the Raychaudhuri equation

A point on a size denotes the derivative with respect to proper time, it means refers to the acceleration field of the particles.

Physical interpretation

The Expansionsskalar describes the rate of change of the volume of a small ball of material with respect to time of an additionally moving observer in the center of the ball ( therefore, this rate may also be negative). In other words, Raychaudhuri equation is the dynamic equation of the expansion of the vector field. If the derivative with respect to proper time along a world line is negative, it means that any expansion of a cloud of dust slows down and, if appropriate, passes into an accelerated collapse, during the collapse of an already collapsing cloud is accelerated. If the derivative is positive, this corresponds to an accelerated expansion or a slowing collapse.

The Scherungstensor describes the deformation of a Spherical cloud towards an ellipsoidal shape. The Vortizitätstensor describes a twist near world-lines, which can be clearly interpreted as rotation of the cloud.

Clearly, it can be stated with reference to the sign which terms accelerate expansion and cause a collapse which terms:

• A rotation of the cloud accelerates expansion, analogous to the centrifugal force of classical mechanics.
• A positive divergence of the acceleration vector, which may be caused by application of force, for example by explosion.

• A high shear, ie an elliptical deformation accelerates or decelerates the collapse expansion.
• An initial expansion is braked by the term, during an initial collapse is accelerated, because a quadratic response.
• Positivity, the track of the Gezeitentensors ( "tidal tensor " ), also called Raychaudhuri scalar. This behavior is enforced by the high energy condition is fulfilled for most forms of classical matter.
• A negative divergence of the acceleration vector, which may be caused by application of force.

In most cases, the solution of the equation is an eternal expansion or a total collapse of the cloud. It may, however, exist in both unstable and stable equilibrium states. An example of a stable equilibrium is a cloud of a perfect fluid in hydrodynamic equilibrium. Expansion, shear and vorticity disappear and a radial divergence of the acceleration vector compensates for the scalar Raychaudhuri, which takes the form of a perfect fluid.

An example of an unstable equilibrium is the Gödel metric. In this case, disappear shear expansion and acceleration, during a constant vorticity is the same as the constant Raychaudhuri scalar that results from a cosmological constant.

Focus set

Assuming that the strong energy condition is true in a space-time region and is a timelike, geodetic (ie ) normalized vector field with vanishing vorticity. This example describes the world lines of the dust particles in cosmological models in which the spacetime is not rotating, as the dust-filled Friedmann universe.

Then is the Raychaudhuri equation

Obviously, the right side is always negative. Even if the Expansionsskalar so is positive at the beginning, the observed dust cloud therefore initially expands, the Expansionsskalar will eventually negative and collapsed cloud of dust.

In fact applies

If we integrate this inequality, one obtains

If the initial value of Expansionsskalars is negative, the geodesics converge after a proper time of maximum in a caustic ( ie goes against minus infinity). This does not indicate a strong curvature singularity, but it does mean that the model for the description of the dust cloud is inappropriate. In some cases, the singularity will prove in suitable coordinates as physically less severe.

Optical equations

There is also an optical version of the Raychaudhuri equation, the lightlike geodesics for crowds, so-called null geodesics, which are described by a light -like vector field.

Here, the energy-momentum tensor. The hats on the symbols mean that the variables are considered only in the transverse direction. Substituting the zero-energy condition requires caustics to form before the affine parameter of the geodesics achieved.