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Half 2: Mathematical Particulars
Most individuals who’ve spent any time in any respect learning GR are acquainted with the Schwarzschild resolution. (A collection of Insights articles discusses the important thing properties of that resolution.) A lot of that familiarity most likely derives from the truth that the Schwarzschild resolution describes a black gap. Nevertheless, for some cause, a lot much less consideration is paid to the best resolution that truly describes the collapse of a large object to a black gap. This resolution was found by J. Robert Oppenheimer and Hartland Snyder in 1939 and is known as the Oppenheimer-Snyder mannequin. On this article, we’ll briefly sketch how this mannequin is constructed, and study its key properties.
Our start line is an idealized huge object which is completely spherically symmetric, with fixed density in its inside, of finite extent, and surrounded by vacuum. After all, such an object is extremely unrealistic. But it surely makes the maths tractable, within the sense that we are able to truly discover closed-form options for the entire equations of curiosity, as an alternative of getting to resolve them numerically. When Oppenheimer and Snyder have been growing their mannequin, they may solely hope that the drastic idealizations they have been making wouldn’t make their mannequin’s predictions irrelevant to actual gravitational collapses. However right this moment, we’ve got loads of laptop simulations of extra sensible collapse eventualities, and we all know that actually, the entire key properties of the Oppenheimer-Snyder mannequin are nonetheless there in sensible fashions.
We assume that our huge object begins out static: in different phrases, each a part of the article is at relaxation relative to each different half. Which means that we are able to outline a typical relaxation body for all components of the article. We then assume that, at some instantaneous of time on this widespread relaxation body, the strain in every single place inside this object is zero, and that it stays zero for all occasions after that instantaneous. (Word that we don’t make any assumptions about how the strain turned zero, or what occurred earlier than that, besides that the article was static to the previous of the moment of time when the strain is zero. Clearly, that is one other extremely idealized assumption, however as above, it nonetheless preserves the important thing properties of the mannequin.) Because the object is spherically symmetric, there could be no shear stresses, and since it’s at relaxation at this instantaneous of time, it may possibly haven’t any momentum or power circulation. Due to this fact, if the strain can be zero, the stress-energy tensor inside the article at this instantaneous of time consists of its power density, and nothing else.
We already know that, since we’ve got assumed precise spherical symmetry, the spacetime geometry of the vacuum area of our mannequin (i.e., from the floor of the large object out to infinity) is the Schwarzschild geometry. We all know this due to Birkhoff’s Theorem (a brief proof of which you’ll be able to see within the Insights article on that subject). So to acquire the preliminary circumstances for our mannequin, all we’d like is the spacetime geometry inside the article on the instantaneous of time, within the object’s relaxation body, at which the strain is zero. We get hold of this by noting that, because the geometry inside the article is spherically symmetric, and because the density is fixed, we’ve got an apparent candidate: a portion of a closed matter-dominated FLRW universe. We received’t attempt to show right here that that is the one risk (although as a matter of reality this may certainly be proved); we’ll simply undertake it as our assumed preliminary situation because it satisfies the entire necessities, and that’s all that’s needed for constructing a mannequin.
Moreover, because the object is at relaxation on the instantaneous of time described above, we all know one thing else about its inside: it isn’t only a portion of a closed matter-dominated FLRW universe, however such a portion on the instantaneous of most enlargement. This have to be the case since that’s the solely instantaneous at which all components of the universe are at relaxation relative to one another.
So we now have an outline of the geometry of the spacelike hypersurface on the instantaneous of time described above: it’s a portion of a 3-sphere bounded by a 2-sphere of some finite space, and out of doors that it’s a Flamm paraboloid, i.e., a floor of fixed Schwarzschild coordinate time within the Schwarzschild geometry.
With out doing a single line of math, we are able to now see, qualitatively, what the remainder of the spacetime geometry, to the way forward for this spacelike hypersurface, seems to be like. The FLRW area will collapse in a finite time (by which we imply a finite correct time for an observer comoving with the matter within the area) to a singularity as a result of that’s what a closed matter-dominated FLRW universe does from the purpose of most enlargement; and the vacuum area exterior will probably be Schwarzschild as a result of a spherically symmetric vacuum area must be. On the instantaneous at which the floor space of the FLRW area is the same as ##16 pi M^2##, the place ##M## is the whole mass of the matter within the area, the floor of the collapsing matter will intersect the occasion horizon, which stays at that floor space in every single place to the way forward for that instantaneous. (To the previous of that instantaneous, the floor space of the horizon decreases, till at some earlier instantaneous it’s zero; that instantaneous corresponds to the occasion on the worldline on the middle of the matter area at which a radially outgoing mild ray could be emitted that can attain the floor of the collapsing matter on the similar instantaneous that floor intersects the horizon. In different phrases, all such outgoing mild rays type the turbines of the horizon.)
After all, Oppenheimer and Snyder didn’t have it this straightforward. They didn’t have the familiarity with the properties of the Schwarzschild and FLRW geometries that we’ve got right this moment. They needed to crank via the maths, and whereas the answer they obtained does certainly have the above properties, it nonetheless takes a good bit of labor to see them in the event you take a look at their paper instantly. In a follow-up article to this one, I’ll stroll via the identical normal line of reasoning they did, however making use of all that we’ve got discovered since 1939 concerning the Schwarzschild and FLRW geometries, as summarized within the qualitative description I gave above. However even with out going via the small print of that, we are able to see in broad define that there have to be a mathematical description of the answer that has the next properties:
(1) There’s a timelike coordinate ##tau## such that the ##tau = 0## spacelike hypersurface has the geometry described above (a portion of a 3-sphere joined to a Flamm paraboloid at a 2-sphere of finite space);
(2) The ##tau## coordinate provides the correct time of comoving observers, i.e., observers who fall radially together with the collapsing matter if they’re inside it, or alongside radial infalling geodesics of the Schwarzschild geometry ranging from relaxation at some finite altitude if they’re exterior the matter;
(3) There’s a curvature singularity similar to areal radius ##r = 0##, and each comoving worldline ends on this singularity, at a time coordinate ##tau## that’s the similar for all worldlines contained in the collapsing matter, after which will increase with growing altitude exterior the matter;
(4) There’s a spacelike coordinate ##R##, with ##0 le R < infty##, such that every comoving worldline has a novel worth of ##R## (which could be considered the areal radius at which that worldline is at ##tau = 0##);
(5) The opposite two coordinates are the usual angular coordinates on a 2-sphere.
For now, I’ll go away it to the reader to confirm {that a} coordinate chart should exist for this resolution with the above properties. The small print of what the metric seems to be like in these coordinates will likely be given within the follow-up article:
Oppenheimer-Snyder Mannequin of Gravitational Collapse: Mathematical Particulars
References:
On Continued Gravitational Contraction
J. R. Oppenheimer and H. Snyder
Phys. Rev. 56, 455 – Revealed 1 September 1939
https://journals.aps.org/pr/summary/10.1103/PhysRev.56.455
Misner, Thorne & Wheeler (1973), Part 32.4
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