Half 2: Mathematical Particulars

Within the final article on this collection, we completed up with a metric for the Oppenheimer-Snyder collapse:

$$

ds^2 = – dtau^2 + A^2 left( eta proper) left( frac{dR^2}{1 – 2M frac{R_-^2}{R_b^2} frac{1}{R_+}} + R^2 dOmega^2 proper)

$$

Now we are going to have a look at among the implications of this metric.

First, let’s assessment what we already know: ##tau## is the correct time of our comoving observers, who comply with radial timelike geodesics ranging from mutual relaxation for all values of ##R## at ##tau = 0##. ##R## labels every geodesic with its areal radius ##r## at ##tau = 0##. ##eta## is a cycloidal time parameter that ranges from ##0## to ##pi##; ##eta = 0## is the place to begin of every geodesic at ##tau = 0##, and ##eta = pi## is the purpose at which every geodesic hits the singularity at ##r = 0##. Contained in the collapsing matter, ##eta## is a perform of ##tau## solely, however within the vacuum area exterior the collapsing matter, ##eta## is a perform of each ##tau## and ##R##.

Now let’s have a look at the geometry of the hypersurfaces of fixed ##tau##, for ##tau > 0##. We noticed that, for ##tau = 0##, this geometry is what we anticipated from our preliminary dialogue: a portion of a 3-sphere joined to a Flamm paraboloid by a 2-sphere boundary at areal radius ##R_b##. That’s what the issue contained in the parentheses within the spatial a part of the metric above describes. So we’d initially assume that that very same geometry applies to all surfaces of fixed ##tau##.

Sadly, nevertheless, that’s not the case. Contained in the collapsing matter, it’s true that every floor of fixed ##tau## is a portion of a 3-sphere, however with rising spatial curvature and bounded by 2-spheres of reducing areal radius (the formulation for the lower, however when it comes to ##eta##, not ##tau##, may be discovered by plugging ##R = R_b## into the formulation for ##r## when it comes to ##R## within the earlier article). However exterior the collapsing matter, the Flamm paraboloid geometry (with the areal radius of its inside boundary 2-sphere reducing with ##eta##) is the geometry of surfaces of fixed ##eta##, *not* surfaces of fixed ##tau##. And, as we noticed within the earlier article, these surfaces usually are not the identical, as a result of, as famous above, within the vacuum area, ##eta## is a perform of each ##tau## and ##R##.

Alongside surfaces of fixed ##tau##, as we will see from the formulation for ##tau## within the earlier article, ##eta## decreases as ##R## will increase. That implies that the size issue ##A(eta)## will increase as ##R## will increase. Because of this ##r / R## will increase as ##R## will increase: in different phrases, the areas of 2-spheres enhance *sooner* with ##R## than they might in a Flamm paraboloid geometry. I’m unsure if there’s a easy description of this geometry; it may be that it may be described as a paraboloid with a distinct “carry” perform than the usual Flamm paraboloid.

Subsequent, let’s have a look at the locus of the singularity at ##r = 0##. That is at ##eta = pi##, as famous above, however when it comes to ##tau##, this turns into

$$

tau = frac{pi}{2} sqrt{frac{R_+^3}{2M}}

$$

The presence of ##R_+## on this formulation tells us that this worth of ##tau## is fixed in all places within the collapsing matter, however will increase with ##R## within the vacuum area. Or, to place it one other method, all the comoving observers contained in the collapsing matter take the identical correct time to succeed in the singularity; however exterior the collapsing matter, comoving observers take longer to succeed in it the additional away they’re, with the correct time rising because the ##3/2## energy of ##R##.

Subsequent, let’s think about a query that you simply may need been desirous to ask for a while now: the place is the occasion horizon in all this? We are able to see that the metric above is manifestly nonsingular for ##eta < pi##, so there isn’t any solution to inform from the road ingredient straight the place the horizon is. We do know that within the exterior vacuum area, the horizon is at ##r = 2M##, and plugging this into the formulation for ##r## offers

$$

R_H = frac{4M}{1 + cos eta}

$$

For the floor of the infalling matter, we will set ##R_H = R_b## within the above to acquire

$$

eta_H = cos^{-1} left( frac{4M}{R_b} – 1 proper)

$$

and subsequently

$$

tau_H = frac{1}{2} sqrt{frac{R_b^3}{2M}} left[ cos^{-1} left( frac{4M}{R_b} – 1 right) + sqrt{frac{8M}{R_b} – frac{16M^2}{R_b^2}} right]

$$

Observe that this formulation reveals that we will need to have ##R_b > 2M##, in order that the argument of the inverse cosine is lower than ##1##, and that as ##R_b to infty##, ##eta_H to pi## and ##tau_H to infty##, as we might count on.

As we transfer to the way forward for the occasion at ##eta_H##, the place the floor of the infalling matter crosses the horizon, ##R_H## will increase, so in these coordinates, the horizon is just not vertical however is inclined outward. This, in fact, simply displays the truth that geodesics with bigger and bigger ##R## take longer and longer to succeed in the horizon.

To the *previous* of the occasion at ##eta_H##, we will use the truth that the horizon is generated by radially outgoing null geodesics. Setting ##ds = 0## in our line ingredient and benefiting from the truth that this portion of the horizon is totally inside the collapsing matter, now we have

$$

dtau = A (eta) frac{1}{sqrt{1 – frac{2M R^2}{R_b^3}}} dR

$$

Contained in the collapsing matter, now we have ##dtau = sqrt{R_b^3 / 2M} A(eta) d eta##, so we will rewrite this as

$$

d eta = frac{1}{sqrt{frac{1}{ok} – R^2}} dR

$$

the place now we have returned to our earlier notation ##ok = 2M / R_b^3##. This integrates to

$$

eta = sin^{-1} left( R sqrt{ok} proper) + eta_0

$$

The worth of ##eta_0## is what we’re searching for since that is the worth of ##eta## for the horizon at ##R = 0##, i.e., the worth of ##eta## at which the horizon types on the heart of the collapsing matter and begins increasing outward. We are able to acquire it by plugging in ##R = R_b## and ##eta = eta_H##:

$$

eta_0 = eta_H – sin^{-1} left( sqrt{frac{2M}{R_b}} proper)

$$

The corresponding worth of ##tau## is

$$

tau_0 = frac{1}{2} sqrt{frac{R_b^3}{2M}} left( eta_0 + sin eta_0 proper)

$$

We gained’t attempt to increase this since it will contain some tedious algebra involving trigonometric identities. Nonetheless, we will learn off the qualitative conduct simply sufficient. As ##R_b to infty##, now we have ##eta_0 to eta_H##. This may appear counterintuitive, however the truth is, it simply implies that, for collapses of bigger and bigger objects, the time between the horizon forming on the heart, ##r = 0##, and the floor of the matter crossing the horizon at ##r = 2M##, is a smaller and smaller *fraction* of the whole time the collapse takes. The *correct* time ##tau## between these occasions, nevertheless, will increase as ##R_b## will increase.

The extra attention-grabbing case is ##R_b to 2M##, for which now we have ##eta_0 to eta_H – pi / 2##. Since now we have ##eta_H to 0## on this restrict, we see that on this case, the horizon types on the heart, ##r = 0##, at a time that’s *earlier than* the collapse really begins! Once more, this appears counterintuitive, however it’s merely a consequence of the truth that the occasion horizon is globally outlined; it’s important to already know your entire way forward for the spacetime to know the place it’s. And in our mannequin, we do know that: now we have declared by fiat that the thing will begin collapsing at ##tau = 0## (or ##eta = 0##).

To place this one other method, the definition of ##eta_0## is that it’s the time at which mild alerts should be emitted from ##r = 0## with a view to attain the floor of the collapsing matter simply because the matter reaches ##r = 2M##. And since we’re wanting on the restrict ##R_b to 2M##, the collapsing matter is *at* ##r = 2M## at ##eta = 0##, so in fact mild alerts *should* be emitted from ##r = 0## *earlier than* ##eta = 0## with a view to simply attain the floor *at* ##eta = 0##. The above equation with ##R_b = 2M## plugged in simply tells us how a lot earlier than.

In abstract, we will see that the mathematical particulars affirm what we initially got here up with based mostly on normal bodily rules. We have now a collapsing matter area that appears like a portion of an FRW closed universe, joined at its boundary to a Schwarzschild vacuum area, and now we have an occasion horizon that types on the heart of the collapsing matter, expands outward till it reaches the floor of the collapsing matter simply as that floor passes ##r = 2M##, after which stays at ##r = 2M## thereafter. All the collapsing matter reaches ##r = 0## on the similar immediate, however freely falling objects exterior the collapsing matter take longer to succeed in ##r = 0## the additional away they’re once they begin to fall. And, as we now know from numerical simulations, these qualitative options stay principally the identical even for collapses that don’t meet the extremely idealized situations of our mannequin: the matter might have nonzero stress and the collapse might not be spherically symmetric, but it surely doesn’t change the fundamental options of the mannequin. So this mannequin is certainly one to make use of to grasp the fundamental options of gravitational collapse.

- Accomplished Instructional Background: MIT Grasp’s

- Favourite Space of Science: Relativity