Congruence (general relativity)

In general relativity, the Raychaudhuri equation, or Landau-Raychaudhuri equation, is a fundamental result describing the motion of nearby bits of matter.

The equation is important as a fundamental lemma for the Penrose-Hawking singularity theorems and for the study of exact solutions in general relativity, but has independent interest, since it offers a simple and general validation of our intuitive expectation that gravitation should be a universal attractive force between any two bits of mass-energy in general relativity, as it is in Newton's theory of gravitation.

The equation was discovered independently by Amal Kumar Raychaudhuri and Lev Landau.

Mathematical statement

Given a timelike unit vector field ${\vec {X}}$ (which can be interpreted as a family or congruence of nonintersecting world lines, not necessarily geodesics), Raychaudhuri's equation can be written

${\dot {\theta }}=-{\frac {\theta ^{2}}{3}}-2\sigma ^{2}+2\omega ^{2}-{E[{\vec {X}}]^{a}}_{a}+{{\dot {X}}^{a}}_{;a}$

where

$\sigma ^{2}={\frac {1}{2}}\sigma _{mn}\,\sigma ^{mn},\;\omega ^{2}={\frac {1}{2}}\omega _{mn}\,\omega ^{mn}$

are (non-negative) quadratic invariants of the shear tensor

$\sigma _{ab}=\theta _{ab}-{\frac {1}{3}}\,\theta \,h_{ab}$

and the vorticity tensor

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

respectively. Here,

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

is the expansion tensor, $\theta$ is its trace, called the expansion scalar, and

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

is the projection tensor onto the hyperplanes orthogonal to ${\vec {X}}$. Also, dot denotes differentiation with respect to proper time counted along the world lines in the congruence. Finally, the trace of the tidal tensor $E[{\vec {X}}]_{ab}$ can also be written

${E[{\vec {X}}]^{a}}_{a}=R_{mn}\,X^{m}\,X^{n}$

This quantity is sometimes called the Raychaudhuri scalar.

Intuitive significance

The expansion scalar measures the fractional rate at which the volume of a small ball of matter changes with respect to time as measured by a central comoving observer (and so it may take negative values). In other words, the above equation gives us the evolution equation for the expansion of the timelike congruence. If the derivative (with respect to proper time) of this quantity turns out to be negative along some world line (after a certain event), then any expansion of a small ball of matter (whose center of mass follows the world line in question) must be followed by recollapse. If not, continued expansion is possible.

The shear tensor measures any tendency of an initially spherical ball of matter to become distorted into an ellipsoidal shape. The vorticity tensor measures any tendency of nearby world lines to twist about one another (if this happens, our small blob of matter is rotating, as happens to fluid elements in an ordinary fluid flow which exhibits nonzero vorticity).

The right hand side of Raychaudhuri's equation consists of two types of terms:

1. terms which promote (re)-collapse
   • initially nonzero expansion scalar,
   • nonzero shearing,
   • positive trace of the tidal tensor; this is precisely the condition guaranteed by assuming the strong energy condition, which holds for the most important types of solutions, such as physically reasonable fluid solutions),
2. terms which oppose (re)-collapse
   • nonzero vorticity, corresponding to Newtonian centrifugal forces,
   • positive divergence of the acceleration vector (e.g., outward pointing acceleration due to a spherically symmetric explosion, or more prosaically, due to body forces on fluid elements in a ball of fluid held together by its own self-gravitation).

Usually one term will win out. However there are situations in which a balance can be achieved. This balance may be:

• stable: in the case of hydrostatic equilibrium of a ball of perfect fluid (e.g. in a model of a stellar interior), the expansion, shear, and vorticity all vanish, and a radial divergence in the acceleration vector (the necessary body force on each blob of fluid being provided by the pressure of surrounding fluid) counteracts the Raychaudhuri scalar, which for a perfect fluid is $E[{\vec {X}}]_{ab}=4\pi (\mu +3p)$. In Newtonian gravitation, the trace of the tidal tensor is $4\pi \mu$; in general relativity, the tendency of pressure to oppose gravity is partially offset by this term, which under certain circumstances can become important.
• unstable: for example, the world lines of the dust particles in the Gödel solution have vanishing shear, expansion, and acceleration, but constant vorticity just balancing a constant Raychuadhuri scalar due to nonzero vacuum energy ("cosmological constant").

Focusing theorem

Suppose the strong energy condition holds in some region of our spacetime, and let ${\vec {X}}$ be a timelike geodesic unit vector field with vanishing vorticity, or equivalently, which is hypersurface orthogonal. For example, this situation can arise in studying the world lines of the dust particles in cosmological models which are exact dust solutions of the Einstein field equation (provided that these world lines are not twisting about one another, in which case the congruence would have nonzero vorticity).

Then Raychaudhuri's equation becomes

${\dot {\theta }}=-{\frac {\theta ^{2}}{3}}-2\sigma ^{2}-{E[{\vec {X}}]^{a}}_{a}$

Now the right hand side is always negative, so even if the expansion scalar is initially positive (if our small ball of dust is initially increasing in volume), eventually it must become negative (our ball of dust must recollapse).

Indeed, in this situation we have

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

Integrating this inequality with respect to proper time $\tau$ gives

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

If the initial value $\theta _{0}$ of the expansion scalar is negative, this means that our geodesics must converge in a caustic ($\theta$ goes to minus infinity) within a proper time of at most $-3/\theta _{0}$ after the measurement of the initial value $\theta _{0}$ of the expansion scalar. This need not signal an encounter with a curvature singularity, but it does signal a breakdown in our mathematical description of the motion of the dust.

Optical equations

There is also an optical (or null) version of Raychaudhuri's equation for null geodesic congruences.

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

Here, the hats indicate that the expansion, shear and vorticity are only with respect to the transverse directions. When the vorticity is zero, then assuming the null energy condition, caustics will form before the affine parameter reaches $2/{\widehat {\theta }}_{0}$.

Applications

The event horizon is defined as the boundary of the causal past of null infinity. Such boundaries are generated by null geodesics. The affine parameter goes to infinity as we approach null infinity, and no caustics form until then. So, the expansion of the event horizon has to be nonnegative. As the expansion gives the rate of change of the logarithm of the area density, this means the event horizon area can never go down, at least classically, assuming the null energy condition.

See also

• congruence (general relativity), for a derivation of the kinematical decomposition and of Raychaudhuri's equation.
• Gravitational singularity
• Penrose-Hawking singularity theorems for an application of the focusing theorem.

Notes

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References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 See chapter 2 for an excellent discussion of Raychaudhuri's equation for both timelike and null geodesics, as well as the focusing theorem.

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 See appendix F.

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 See chapter 6 for a very detailed introduction to geodesic congruences, including the general form of Raychaudhuri's equation.

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 See section 4.1 for a discussion of the general form of Raychaudhuri's equation.

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  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang See section IV for derivation of the general form of Raychaudhuri equations for three kinematical quantities (namely expansion scalar, shear and rotation).

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  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang See for a review on Raychaudhuri equations.

External links

• The Meaning of Einstein's Field Equation by John C. Baez and Emory F. Bunn. Raychaudhuri's equation takes center stage in this well known (and highly recommended) semi-technical exposition of what Einstein's equation says.