Thick set

In cosmological perturbation theory, the scalar-vector-tensor decomposition is a decomposition of the most general linearized perturbations of the Friedmann–Lemaître–Robertson–Walker metric into components according to their transformations under spatial rotations. It was first discovered by E. M. Lifshitz in 1946. The general metric perturbation has ten degrees of freedom. The decomposition states that the evolution equations for the most general linearized perturbations of the Friedmann–Lemaître–Robertson–Walker metric can be decomposed into four scalars, two divergence-free spatial vector fields (that is, with a spatial index running from 1 to 3), and a traceless, symmetric spatial tensor field with vanishing doubly and singly longitudinal components. The vector and tensor fields each have two independent components, so this decomposition encodes all ten degrees of freedom in the general metric perturbation. Using gauge invariance four of these components (two scalars and a vector field) may be set to zero.

If the perturbed metric $g{\text{'}}_{\mu \nu }=g_{\mu \nu }+h_{\mu \nu }$ where $h_{\mu \nu }$ is the perturbation, then the decomposition is as follows,

$h_{00}=-2\psi$

$h_{0i}=w_i$

$h_{ij}=2(\varphi g_{ij}+S_{ij})$

where the Latin indices i and j run over spatial components (1,…,3). The tensor field $S_{ij}$ is traceless under the spatial part of the background metric $g_{ij}$ (i.e. $g^{ij}{S}_{ij}=0$). The spatial vector $w_i$ and tensor $S_{ij}$ undergo further decomposition. The vector is written

$w_i=w^{\left|\right|}{}_i+w^{\perp }{}_i,$

where $\mathrm{\nabla }\times {\mathbf{w}}^{\left|\right|}=\mathbf{0}$ and $\mathrm{\nabla }\cdot {\mathbf{w}}^{\perp }=0$ (${\mathrm{\nabla }}_i$ is the covariant derivative defined with respect to the spatial metric $g_{ij}$). The notation is used because in Fourier space, these equations indicate that the vector points parallel and perpendicular to the direction of the wavevector, respectively. The parallel component can be expressed as the gradient of a scalar, $w^{\left|\right|}{}_i={\mathrm{\nabla }}_{i}A$. Thus $\mathbf{w}$ can be written as a combination of a scalar and a divergenceless, two-component vector.

Finally, an analogous decomposition can be performed on the traceless tensor field $S_{ij}$. It can be written

$S_{ij}=S^{\left|\right|}{}_{ij}+S_{ij}^{\perp }+S^T{}_{ij},$

where

$S^{\left|\right|}{}_{ij}=({\mathrm{\nabla }}_i{\mathrm{\nabla }}_j-\frac{1}{3}{g}_{ij}{\mathrm{\nabla }}^2)B$,

where $B$ is a scalar (the combination of derivatives is set by the condition that $S$ be traceless), and

$S^{\perp }{}_{ij}={\mathrm{\nabla }}_i{S}^{\perp }{}_j+{\mathrm{\nabla }}_j{S}^{\perp }{}_i$,

where $S^{\perp }{}_i$ is a divergenceless spatial vector. This leaves only two independent components of $S^T{}_{ij}$, corresponding to the two polarizations of gravitational waves. (Since the graviton is massless, the two polarizations are orthogonal to the direction of propagation, just like the photon.)

The advantage of this formulation is that the scalar, vector and tensor evolution equations are decoupled. In representation theory, this corresponds to decomposing perturbations under the group of spatial rotations. Two scalar components and one vector component can further be eliminated by gauge transformations. However, the vector components are generally ignored, as there are few known physical processes in which they can be generated. As indicated above, the tensor components correspond to gravitational waves. The tensor $S^T{}_{ij}$ is gauge invariant: it does not change under infinitesimal coordinate transformations.

References

• Template:Cite arxiv
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  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
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  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