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Half 2: Mathematical Particulars

Within the final article on this sequence, 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 take a look at a few of the implications of this metric.

First, let’s evaluate what we already know: ##tau## is the right time of our comoving observers, who observe 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 outdoors the collapsing matter, ##eta## is a perform of each ##tau## and ##R##.

Now let’s take 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 would initially suppose that that very same geometry applies to all surfaces of fixed ##tau##.

Sadly, nonetheless, 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 growing spatial curvature and bounded by 2-spheres of reducing areal radius (the system for the lower, however when it comes to ##eta##, not ##tau##, may be discovered by plugging ##R = R_b## into the system for ##r## when it comes to ##R## within the earlier article). However outdoors 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 are able to see from the system for ##tau## within the earlier article, ##eta## decreases as ##R## will increase. That signifies that the dimensions 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 *quicker* with ##R## than they might in a Flamm paraboloid geometry. I’m undecided if there’s a easy description of this geometry; it is likely to be that it may be described as a paraboloid with a special “raise” perform than the usual Flamm paraboloid.

Subsequent, let’s take 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 system tells us that this worth of ##tau## is fixed all over the place within the collapsing matter, however will increase with ##R## within the vacuum area. Or, to place it one other means, all the comoving observers contained in the collapsing matter take the identical correct time to succeed in the singularity; however outdoors the collapsing matter, comoving observers take longer to succeed in it the additional away they’re, with the right time growing because the ##3/2## energy of ##R##.

Subsequent, let’s take into account a query that you just might 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 method to inform from the road ingredient instantly 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 system for ##r## offers

$$

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

$$

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

$$

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

$$

and due to this fact

$$

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]

$$

Notice that this system exhibits 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, after all, 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 are able to use the truth that the horizon is generated by radially outgoing null geodesics. Setting ##ds = 0## in our line ingredient and profiting from the truth that this portion of the horizon is solely inside the collapsing matter, we’ve

$$

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

$$

Contained in the collapsing matter, we’ve ##dtau = sqrt{R_b^3 / 2M} A(eta) d eta##, so we are able to rewrite this as

$$

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

$$

the place we’ve 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 in search of 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 get hold of 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 received’t attempt to develop this since it will contain some tedious algebra involving trigonometric identities. Nevertheless, we are able to learn off the qualitative conduct simply sufficient. As ##R_b to infty##, we’ve ##eta_0 to eta_H##. This might sound counterintuitive, however in reality, it simply signifies 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, nonetheless, will increase as ##R_b## will increase.

The extra fascinating case is ##R_b to 2M##, for which we’ve ##eta_0 to eta_H – pi / 2##. Since we’ve ##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 truly begins! Once more, this appears counterintuitive, however it’s merely a consequence of the truth that the occasion horizon is globally outlined; you need to already know your entire way forward for the spacetime to know the place it’s. And in our mannequin, we do know that: we’ve declared by fiat that the thing will begin collapsing at ##tau = 0## (or ##eta = 0##).

To place this one other means, the definition of ##eta_0## is that it’s the time at which mild alerts should be emitted from ##r = 0## with the intention 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 after all mild alerts *should* be emitted from ##r = 0## *earlier than* ##eta = 0## with the intention 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 are able to see that the mathematical particulars verify what we initially got here up with based mostly on common bodily rules. We now have a collapsing matter area that appears like a portion of an FRW closed universe, joined at its boundary to a Schwarzschild vacuum area, and we’ve 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. The entire collapsing matter reaches ##r = 0## on the identical prompt, however freely falling objects outdoors 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 could have nonzero strain and the collapse is probably not spherically symmetric, nevertheless it doesn’t change the fundamental options of the mannequin. So this mannequin is certainly a superb one to make use of to know the fundamental options of gravitational collapse.

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