Wightman axioms
In general relativity, Eddington–Finkelstein coordinates are a pair of coordinate systems for a Schwarzschild geometry (i.e. a black hole) which are adapted to radial null geodesics. Null geodesics are the worldlines of photons that are moving directly towards or away from the central mass. They are named for Arthur Stanley Eddington[1] and David Finkelstein,[2] even though neither ever wrote down these coordinates or the metric in these coordinates. They seem to have been given this name by Misner, Thorne, and Wheeler in their book Gravitation.
They are sometimes also called tortoise coordinate - a name that comes from one of Zeno of Elea's paradoxes on an imaginary footrace between "swift-footed" Achilles and a tortoise.
Outward (inward) traveling radial light rays (such as a null geodesic) define the surfaces of constant "time", while the radial coordinate is the usual area coordinate so that the surfaces of rotation symmetry have an area of . One advantage of this coordinate system is that it shows that the apparent singularity at the Schwarzschild radius is only a coordinate singularity and is not a true physical singularity. While this fact was recognized by Finkelstein, it was not recognized (or at least not commented on) by Eddington, whose primary purpose was to compare and contrast the spherically symmetric solutions in Whitehead's theory of gravitation and Einstein's.
Schwarzschild Metric
The Schwarzschild coordinates are , and the Schwarzschild metric is well known:
where
Note the conventions being used here are the metric signature of (− + + +) and the natural units where c = 1 (although the gravitational constant G will be kept explicit, and M will denote the characteristic mass of the Schwarzschild geometry).
Tortoise coordinate
Eddington–Finkelstein coordinates are founded upon the tortoise coordinate.
The tortoise coordinate is defined:
so as to satisfy:
The tortoise coordinate approaches −∞ as r approaches the Schwarzschild radius r = 2GM.
When some probe (such as a light ray or an observer) approaches a black hole event horizon, its Schwarzschild time coordinate grows infinite. The outgoing null rays in this coordinate system have an infinite change in t on travelling out from the horizon. The tortoise coordinate is intended to grow infinite at the appropriate rate such as to cancel out this singular behaviour in coordinate systems constructed from it.
The increase in the time coordinate to infinity as one approaches the event horizon is why information could never be received back from any probe that is sent sufficiently close to such an event horizon. This is despite the fact that the probe itself can nonetheless travel past the horizon. It is also why the space-time metric of the black hole, when expressed in Schwarzschild coordinates, becomes singular at the horizon - and thereby fails to be able to fully chart the trajectory of an infalling probe.
Metric
The ingoing Eddington–Finkelstein coordinates are obtained by replacing the coordinate t with the new coordinate . The metric in these coordinates can be written
where is the standard metric on a unit radius two sphere.
Likewise, the outgoing Eddington–Finkelstein coordinates are obtained by replacing t with the null coordinate . The metric is then given by
In both these coordinate systems the metric is explicitly non-singular at the Schwarzschild radius (even though one component vanishes at this radius, the determinant of the metric is still non-vanishing and the inverse metric has no terms which diverge there.)
Note that for radial null rays, v=const or =const or equivalently =const or u=const we have dv/dr and du/dr approach 0 and ±2 at large r, not ±1 as one might expect if one regarded u or v as "time". When plotting Eddington-Finkelstein diagrams, surfaces of constant u or v are usually drawn as cones, with u or v constant lines drawn as sloping at 45 degree rather than as planes (see for instance Box 31.2 of MTW). Some sources instead take , corresponding to planar surfaces in such diagrams. In terms of this the metric becomes
which is Minkowskian at large r. (This was the coordinate time and metric that both Eddington and Finkelstein presented in their papers.)
The Eddington–Finkelstein coordinates are still incomplete and can be extended. For example, the outward traveling timelike geodesics defined by (with τ the proper time)
has v(τ)-> -∞ as τ->2GM. Ie, this timelike geodesic has a finite proper length into the past where it comes out of the horizon (r=2GM) when v becomes minus infinity. The regions for finite v and r<2GM is a different region from finite u and r<2GM. The horizons r=2GM and finite v is a different horizon (the black hole horizon) from that with r=2M and finite u (the white hole horizon) .
The metric in Kruskal-Szekeres coordinates covers all of the extended Schwarzschild spacetime in a single coordinate system. It's chief disadvantage is that in those coordinates the metric depends on both the time and space coordinates. In Eddington-Finkelstein, as in Schwartzschild coordinates, the metric is independent of the "time" (either t in Schwartzschild, or "u" or "v" in the various Eddington–Finkelstein coordinates), but none of these cover the complete spacetime.
The Eddington–Finkelstein coordinates have some similarity to the Gullstrand–Painlevé coordinates in that both are time independent, and penetrate (are regular across) either the future (black hole) or the past (white hole) horizons. Both are not diagonal (the hypersurfaces of constant "time" are not orthogonal to the hypersurfaces of constant r.) The latter have a flat spatial metric, while the former's spatial ("time" constant) hypersurfaces are null and have the same metric as that of a null cone in Minkowski space ( in flat spacetime).
See also
- Schwarzschild coordinates
- Kruskal–Szekeres coordinates
- Lemaître coordinates
- Gullstrand–Painlevé coordinates
- Vaidya metric
References
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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