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In physics, curvature invariants in general relativity are a set of scalars called curvature invariants that arise in general relativity. They are formed from the Riemann, Weyl and Ricci tensors - which represent curvature - and possibly operations on them such as contraction, covariant differentiation and dualisation.

Certain invariants formed from these curvature tensors play an important role in classifying spacetimes. Invariants are actually less powerful for distinguishing locally non-isometric Lorentzian manifolds than they are for distinguishing Riemannian manifolds. This means that they are more limited in their applications than for manifolds endowed with a positive definite metric tensor.

Principal invariants

The principal invariants of the Riemann and Weyl tensors are certain quadratic polynomial invariants (i.e., sums of squares of components).

The principal invariants of the Riemann tensor of a four-dimensional Lorentzian manifold are

1. the Kretschmann scalar $K_1=R_{abcd}{R}^{abcd}$
2. the Chern-Pontryagin scalar $K_2={{}^{\star }R}_{abcd}{R}^{abcd}$
3. the Euler scalar $K_3={{}^{\star }{R}^{\star }}_{abcd}{R}^{abcd}$

These are quadratic polynomial invariants (sums of squares of components). (Some authors define the Chern-Pontryagin scalar using the right dual instead of the left dual.)

The first of these was introduced by Erich Kretschmann. The second two names are somewhat anachronistic, but since the integrals of the last two are related to the instanton number and Euler characteristic respectively, they have some justification.

The principal invariants of the Weyl tensor are

1. $I_1=C_{abcd}{C}^{abcd}$
2. $I_2={{}^{\star }C}_{abcd}{C}^{abcd}$

(Because ${{}^{\star }{C}^{\star }}_{abcd}=-C_{abcd}$, there is no need to define a third principal invariant for the Weyl tensor.)

Relation with Ricci decomposition

As one might expect from the Ricci decomposition of the Riemann tensor into the Weyl tensor plus a sum of fourth-rank tensors constructed from the second rank Ricci tensor and from the Ricci scalar, these two sets of invariants are related:

$K_1=I_1+2R_{ab}{R}^{ab}-\frac{1}{3}{R}^2$

$K_3=-I_1+2R_{ab}{R}^{ab}-\frac{2}{3}{R}^2$

Relation with Bel decomposition

Recall that (in four dimensions), the Bel decomposition of the Riemann tensor, with respect to a timelike unit vector field $\overrightarrow{X}$, not necessarily geodesic or hypersurface orthogonal, consists of three pieces

1. the electrogravitic tensor $E[\overrightarrow{X}{]}_{ab}=R_{ambn}{X}^m{X}^n$
2. the magnetogravitic tensor $B[\overrightarrow{X}{]}_{ab}={{}^{\star }R}_{ambn}{X}^m{X}^n$
3. the topogravitic tensor $L[\overrightarrow{X}{]}_{ab}={{}^{\star }{R}^{\star }}_{ambn}{X}^m{X}^n$

Because these are all transverse (i.e. projected to the spatial hyperplane elements orthogonal to our timelike unit vector field), they can be represented as linear operators on three dimensional vectors, or as three by three real matrices. They are respectively symmetric, traceless, and symmetric (6,8,6 linearly independent components, for a total of 20). If we write these operators as E, B, L respectively, the principal invariants of the Riemann tensor are obtained as follows:

• $K_1/4$ is the trace of E² + L² - 2 B B^T,
• $-K_2/8$ is the trace of B ( E - L ),
• $K_3/8$ is the trace of E L - B².

Expression in Newman-Penrose formalism

In terms of the Weyl scalars in the Newman-Penrose formalism, the principal invariants of the Weyl tensor may be obtained by taking the real and imaginary parts of the expression

$I_1-i{I}_2=16(3{\mathrm{\Psi }}_2^2+{\mathrm{\Psi }}_0{\mathrm{\Psi }}_4-4{\mathrm{\Psi }}_1{\mathrm{\Psi }}_3)$

(But note the minus sign!)

The principal quadratic invariant of the Ricci tensor, $R_{ab}{R}^{ab}$, may be obtained as a more complicated expression involving the Ricci scalars (see the paper by Cherubini et al. cited below).

Distinguishing Lorentzian manifolds

An important question related to Curvature invariants is when the set of polynomial curvature invariants can be used to (locally) distinguish manifolds. To be able to do this is necessary to include higher-order invariants including derivatives of the Riemann tensor but in the Lorentzian case, it is known that there are spacetimes which cannot be distinguished; e.g., the VSI spacetimes for which all such curvature invariants vanish and thus cannot be distinguished from flat space. This failure of being able to distinguishing Lorentzian manifolds is related to the fact that the Lorentz group is non-compact.

There are still examples of cases when we can distinguish Lorentzian manifolds using their invariants. Examples of such are fully general Petrov type I spacetimes with no Killing vectors, see Coley et al. below. Indeed, it was here found that the spacetimes failing to be distinguished by their set of curvature invariants are all Kundt spacetimes.

See also

• Bach tensor, for a sometimes useful tensor generated by $I_2$ via a variational principle.
• Carminati-McLenaghan invariants, for a set of polynomial invariants of the Riemann tensor of a four-dimensional Lorentzian manifold which is known to be complete under some circumstances.
• Curvature invariant, for curvature invariants in a more general context.

References

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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
  
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