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 singularity theorem and for the analysis of exact solutions in general relativity . It also confirms in a simple way our intuitive notion that the local effect of gravity in general relativity corresponds to Newton's law of gravity : a general attraction between pairs of 'particles' (now understood as mass and energy ).

Mathematical formulation

The world lines of the observed particles are described by a time-like and normalized four-dimensional vector field . The description by a vector field implies that the world lines do not intersect, so the particles do not collide. The world lines of the particles do not necessarily have to be geodesics , so that the equation is also valid in the case of external force fields . ${\displaystyle X}$

From the metric tensor is the vector field and tensor ${\displaystyle g_{ab}}$

${\displaystyle h_{ab}=g_{ab}-X_{a}\,X_{b}}$

constructed, which can be understood as a metric tensor on the hypersurfaces orthogonal to the vector field .

The fundamental object of investigation of the Raychaudhuri equation is now the projection of the covariant derivative of the vector field onto the orthogonal hypersurfaces:

${\displaystyle {h^{m}}_{a}\,{h^{n}}_{b}X_{m;n}}$

This tensor is split up into its symmetrical part, the expansion tensor

${\displaystyle \theta _{ab}={h^{m}}_{a}\,{h^{n}}_{b}X_{(m;n)},}$

and its antisymmetric part, the vorticity tensor

${\displaystyle \omega _{ab}={h^{m}}_{a}\,{h^{n}}_{b}X_{[m;n]}}$

Derived quantities are

• ${\displaystyle \theta =g^{ab}\theta _{ab}}$(Expansion scalar )
• ${\displaystyle \sigma _{ab}=\theta _{ab}-{\frac {1}{3}}\,\theta \,h_{ab}}$( Shear tensor)
• ${\displaystyle \sigma ^{2}=\sigma _{mn}\,\sigma ^{mn}}$
• ${\displaystyle \omega ^{2}=\omega _{mn}\,\omega ^{mn}}$

Using these quantities, the Raychaudhuri equation reads

${\displaystyle {\dot {\theta }}=\omega ^{2}-\sigma ^{2}-{\frac {\theta ^{2}}{3}}-R_{mn}\,X^{m}\,X^{n}+{{\dot {X}}^{a}}_{;a}}$

A point above a quantity denotes the derivative according to proper time , i.e. H. denotes the acceleration field of the particles. ${\displaystyle {\dot {X}}}$

${\displaystyle R_{mn}\,X^{m}\,X^{n}}$is the trace of Gezeitentensors ( "tidal tensor"), it is also Raychaudhuri scalar called.

Physical interpretation

The Raychaudhuri equation is the dynamic equation of the expansion of the vector field. The expansion scalar describes the rate of change in the volume of a small ball of matter with respect to the time of a moving observer in the center of the ball:

• if the derivative of the expansion scalar according to proper time along a world line is positive, this corresponds to an accelerated expansion or a slowing down collapse . ${\displaystyle {\dot {\theta }}}$
• if the derivative is negative, on the other hand, this means that any expansion of a dust cloud will slow down and possibly transition into an accelerated collapse, while the collapse of an already collapsing cloud is accelerated.

The shear tensor describes the deformation of a spherical cloud to an ellipsoidal shape.

The vorticity tensor describes a twisting of near world lines, which can be clearly understood as a rotation of the cloud.

The signs clearly show which terms accelerate an expansion and which terms cause a collapse:

1. expansion
   • A rotation of the cloud accelerates the expansion, analogous to the centrifugal force of classical mechanics .
   • A positive divergence of the acceleration vector caused by the action of force, e.g. B. an explosion , accelerates the expansion. ${\displaystyle {{\dot {X}}^{a}}_{;a}>0}$
2. collapse
   • A high shear, i.e. an elliptical deformation, accelerates a collapse or slows down an expansion.
   • An initial expansion is slowed down by the term , while an initial collapse is accelerated because it is quadratic. ${\displaystyle -{\frac {\theta ^{2}}{3}}}$ ${\displaystyle \theta }$
   • Positivity of . This behavior is forced by the strong energy condition that is met for most forms of classical matter. ${\displaystyle R_{mn}\,X^{m}\,X^{n}}$
   • A negative divergence of the acceleration vector that can be caused by the application of force. ${\displaystyle {{\dot {X}}^{a}}_{;a}>0}$

In most cases the solution to the equation is an eternal expansion or total collapse of the cloud. However, stable or unstable equilibrium states can also exist:

• An example of a stable equilibrium is a cloud of perfect fluid in hydrodynamic equilibrium . Expansion, shear, and vorticity disappear, and a radial divergence of the acceleration vector compensates for the Raychaudhuri scalar that takes shape for a perfect fluid . ${\displaystyle R_{mn}\,X^{m}\,X^{n}=4\pi (\mu +3p)}$
• An example of an unstable equilibrium is the Gödel metric . In this case, the shear, expansion and acceleration disappear, while a constant vorticity is the same as the constant Raychaudhuri scalar, which comes from a cosmological constant .

Focus set

Assume that the strong energy condition holds in a spacetime region and is a time-like, geodetic (i.e. ) normalized vector field with vanishing vorticity (i.e. ). This describes, for example, the world lines of dust particles in cosmological models in which space-time does not rotate, such as the dust-filled Friedmann universe . ${\displaystyle X}$ ${\displaystyle {\dot {X}}=0}$ ${\displaystyle \omega =0}$

Then the Raychaudhuri equation is

${\displaystyle {\dot {\theta }}=-{\frac {\theta ^{2}}{3}}-\sigma ^{2}-R_{mn}\,X^{m}\,X^{n}}$.

The term is greater than or equal to zero due to the strong energy condition, so the entire right-hand side is always negative or zero, which is why the expansion scalar cannot increase over time. ${\displaystyle R_{mn}\,X^{m}\,X^{n}}$

Since the last two terms are nonnegative, the following applies:

${\displaystyle {\dot {\theta }}\leq -{\frac {\theta ^{2}}{3}}}$.

If you integrate this inequality, you get

${\displaystyle {\frac {1}{\theta }}\geq {\frac {1}{\theta _{0}}}+{\frac {\tau }{3}}}$.

If the initial value of the expansion scalar is negative, the geodesics converge after a proper time of at most in a caustic (i.e. approaches minus infinity). This need not indicate a strong singularity of curvature , but it does mean that the model is unsuitable for describing the dust cloud. In some cases the singularity will prove to be of little physical severity in appropriate coordinates. ${\displaystyle \theta _{0}}$ ${\displaystyle -3/\theta _{0}}$ ${\displaystyle \theta }$

Optical equations

There is also an optical version of the Raychaudhuri equation for families of light-like geodesics, so-called zero geodesics , which are described by a light-like vector field : ${\displaystyle U}$

${\displaystyle {\dot {\widehat {\theta }}}=-{\frac {1}{2}}{\widehat {\theta }}^{2}-{\widehat {\sigma }}^{2}-T_{\mu \nu }U^{\mu }U^{\nu }}$

Where is the energy momentum tensor . The hats above the symbols mean that the sizes are only viewed in the transverse direction. ${\displaystyle T_{\mu \nu }}$

If one assumes the zero-energy condition , caustics form before the affine parameter reaches the geodesics . ${\displaystyle T_{\mu \nu }U^{\mu }U^{\nu }\geq 0}$ ${\displaystyle 2/{\widehat {\theta }}_{0}}$

literature

• Poisson, Eric: A Relativist's Toolkit: The Mathematics of Black Hole Mechanics . Cambridge University Press, Cambridge 2004, ISBN 0-521-83091-5 . See chapter 2.
• Carroll, Sean M .: Spacetime and Geometry: An Introduction to General Relativity . Addison-Wesley, San Francisco 2004, ISBN 0-8053-8732-3 . See Appendix F.
• Stephani, Hans; Kramer, Dietrich; MacCallum, Malcom; Hoenselaers, Cornelius; Hertl, Eduard: Exact Solutions to Einstein's Field Equations (2nd ed.) . Cambridge University Press, Cambridge 2003, ISBN 0-521-46136-7 . See chapter 6.
• Hawking, Stephen; and Ellis, GFR: The Large Scale Structure of Space-Time . Cambridge University Press, Cambridge 1973, ISBN 0-521-09906-4 . See chapter 4.1
• Raychaudhuri, AK: Relativistic cosmology I. . In: Phys. Rev. . 98, 1955, p. 1123. doi : 10.1103 / PhysRev.98.1123 . Raychaudhuri's original article.
• Dasgupta, Anirvan; Nandan, Hemwati; and Kar, Sayan: Kinematics of geodesic flows in stringy black hole backgrounds. . In: Phys. Rev. D . 79, 2009, p. 124004. doi : 10.1103 / PhysRevD.79.124004 . See Chapter IV.
• Kar, Sayan; and SenGupta, Soumitra: The Raychaudhuri equations: A Brief review. . In: Pramana . 69, 2007, p. 49. doi : 10.1007 / s12043-007-0110-9 .

Web links

• John C. Baez, Emory F. Bunn: The Meaning of Einstein's Field Equation . (The Raychaudhuri equation is at the heart of this well-known (and highly recommended) semi-technical representation of what the Einstein equation says.)