Half 2: Mathematical Particulars

Most individuals who’ve spent any time in any respect finding out GR are acquainted with the Schwarzschild answer. (A sequence of Insights articles discusses the important thing properties of that answer.) A lot of that familiarity most likely derives from the truth that the Schwarzschild answer describes a black gap. Nevertheless, for some purpose, a lot much less consideration is paid to the best answer that truly describes the collapse of a large object to a black gap. This answer was found by J. Robert Oppenheimer and Hartland Snyder in 1939 and is known as the Oppenheimer-Snyder mannequin. On this article, we are going to briefly sketch how this mannequin is constructed, and look at its key properties.

Our place to begin is an idealized huge object which is completely spherically symmetric, with fixed density in its inside, of finite extent, and surrounded by vacuum. In fact, such an object is extremely unrealistic. But it surely makes the mathematics tractable, within the sense that we will truly discover closed-form options for all the equations of curiosity, as an alternative of getting to resolve them numerically. When Oppenheimer and Snyder had been growing their mannequin, they may solely hope that the drastic idealizations they had been making wouldn’t make their mannequin’s predictions irrelevant to actual gravitational collapses. However at the moment, now we have loads of pc simulations of extra real looking collapse eventualities, and we all know that the truth is, all the key properties of the Oppenheimer-Snyder mannequin are nonetheless there in real looking fashions.

We assume that our huge object begins out static: in different phrases, each a part of the thing is at relaxation relative to each different half. Because of this we will outline a typical relaxation body for all elements of the thing. We then assume that, at some immediate of time on this frequent relaxation body, the stress in all places inside this object is zero, and that it stays zero for all instances after that immediate. (Notice that we do not make any assumptions about *how* the stress turned zero, or what occurred earlier than that, besides that the thing was static to the previous of the moment of time when the stress 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 may be no shear stresses, and since it’s at relaxation at this immediate of time, it could possibly haven’t any momentum or power circulation. Subsequently, if the stress can be zero, the stress-energy tensor inside the thing at this immediate of time consists of its power density, and nothing else.

We already know that, since now we have assumed precise spherical symmetry, the spacetime geometry of the vacuum area of our mannequin (i.e., from the floor of the huge object out to infinity) is the Schwarzschild geometry. We all know this due to Birkhoff’s Theorem (a brief proof of which you’ll see within the Insights article on that matter). So to acquire the preliminary situations for our mannequin, all we want is the spacetime geometry inside the thing on the immediate of time, within the object’s relaxation body, at which the stress is zero. We get hold of this by noting that, for the reason that geometry inside the thing is spherically symmetric, and for the reason that density is fixed, now we have 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 could certainly be proved); we’ll simply undertake it as our assumed preliminary situation because it satisfies all the necessities, and that’s all that’s essential for constructing a mannequin.

Moreover, for the reason that object is at relaxation on the immediate 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 immediate of most enlargement. This should be the case since that’s the solely immediate at which all elements 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 immediate 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 will now see, qualitatively, what the remainder of the spacetime geometry, to the way forward for this spacelike hypersurface, appears to be like 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 immediate 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 all places to the way forward for that immediate. (To the previous of that immediate, the floor space of the horizon decreases, till at some earlier immediate it’s zero; that immediate corresponds to the occasion on the worldline on the middle of the matter area at which a radially outgoing mild ray may be emitted that can attain the floor of the collapsing matter on the identical immediate that floor intersects the horizon. In different phrases, all such outgoing mild rays kind the turbines of the horizon.)

In fact, Oppenheimer and Snyder didn’t have it this simple. They didn’t have the familiarity with the properties of the Schwarzschild and FLRW geometries that now we have at the moment. They needed to crank by the mathematics, and whereas the answer they obtained does certainly have the above properties, it nonetheless takes a good bit of labor to see them if you happen to have a look at their paper straight. In a follow-up article to this one, I’ll stroll by the identical common line of reasoning they did, however making use of all that now we have realized since 1939 concerning the Schwarzschild and FLRW geometries, as summarized within the qualitative description I gave above. However even with out going by the small print of that, we will see in broad define that there should 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 offers 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 equivalent to areal radius ##r = 0##, and each comoving worldline ends on this singularity, at a time coordinate ##tau## that’s the identical for all worldlines contained in the collapsing matter, after which will increase with rising altitude exterior the matter;

(4) There’s a spacelike coordinate ##R##, with ##0 le R < infty##, such that every comoving worldline has a singular worth of ##R## (which may 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 answer with the above properties. The small print of what the metric appears to be like like in these coordinates can 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 – Printed 1 September 1939

https://journals.aps.org/pr/summary/10.1103/PhysRev.56.455

Misner, Thorne & Wheeler (1973), Part 32.4

- Accomplished Academic Background: MIT Grasp’s

- Favourite Space of Science: Relativity