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Half 2: Mathematical Particulars

In a earlier article, I described typically phrases the mannequin of gravitational collapse of a spherically symmetric large object, first revealed by Oppenheimer and Snyder of their traditional 1939 paper. On this follow-up article, I’ll give additional mathematical particulars concerning the mannequin, utilizing an strategy considerably completely different from their authentic paper (and impressed by the strategy described in MTW and Landau & Lifschitz).

(Observe: Weinberg takes a unique strategy within the vacuum area exterior the collapsing matter. As a substitute of discovering an expression for the outside vacuum metric in comoving coordinates, he finds an expression for the inside metric in coordinates just like normal Schwarzschild coordinates. We is not going to focus on that strategy right here, however it’s instructive to check the 2. The latter strategy, which additionally is analogous to the strategy taken within the authentic Oppenheimer-Snyder paper, has the plain limitation of getting a coordinate singularity on the horizon, in addition to different extra technical points; however since these sources are targeted primarily on how the collapse seems to a distant observer, these limitations are much less of a problem than they might be for us right here since we would like an outline that covers the complete collapse and contains each distant observers *and* the collapsing matter all the best way right down to the singularity. The comoving coordinate strategy we use right here is a lot better fitted to that.)

We’ll begin with the spacelike hypersurface that we labeled with ##tau = 0## within the earlier article, i.e., on which the collapsing object is momentarily at relaxation. As we famous, this hypersurface has the geometry of a 3-sphere out to some finite areal radius that we’ll name ##R_b## (“b” for “boundary” since that is the boundary of the matter area), and a Flamm paraboloid exterior this radius. We are able to specific this as follows: the 3-metric of this hypersurface is given by

$$

dSigma^2 = frac{dR^2}{1 – okay R^2} + R^2 dOmega^2

$$

for ##R le R_b##, and by

$$

dSigma^2 = frac{dR^2}{1 – frac{2M}{R}} + R^2 dOmega^2

$$

for ##R ge R_b##. Right here ##dOmega^2## is the usual metric on a unit 2-sphere by way of the angular coordinates, and ##M## is the full mass of the matter.

Since these two metrics should match at ##R = R_b##, we will get hold of an equation for ##okay##:

$$

okay = frac{2M}{R_b^3}

$$

which tells us that ##okay## is expounded to the density of the matter at ##tau = 0##. We is not going to be discussing density on this article so we received’t discover that side any additional. This equation for ##okay## accommodates ##2M##, so it permits us to rewrite the 3-metric above within the following helpful type, legitimate for all values of ##R##:

$$

dSigma^2 = frac{dR^2}{1 – 2M frac{R_-^2}{R_b^2} frac{1}{R_+}} + R^2 dOmega^2

$$

the place we’ve outlined the features ##R_- = min(R, R_b)## and ##R_+ = max(R_b, R)##. We are able to consider ##R_-## as capturing radial variation contained in the matter solely, and ##R_+## as capturing radial variation exterior the matter solely.

Our technique is to make use of the coordinate ##R## on the ##tau = 0## hypersurface to label the geodesics each inside and outdoors the collapsing matter. This strategy matches normal FLRW coordinates for a closed universe contained in the matter and is considerably just like Novikov coordinates exterior the matter; nonetheless, we might want to look rigorously on the latter case to make sure that we’re appropriately describing the vacuum area since normal Novikov coordinates don’t use the areal radius on the ##tau = 0## hypersurface straight, however outline a brand new radial coordinate, referred to as ##R^*## in MTW, and specific the metric by way of this coordinate. We’ll return to this beneath.

We now make use of the truth that the geodesic movement each inside and outdoors the matter may be described utilizing a cycloidal time parameter ##eta##, which ranges from ##0## at ##tau = 0## to ##pi## on the on the spot when every geodesic hits the singularity at ##r = 0##. We notice that contained in the collapsing matter, the moment ##eta = pi## corresponds to the *similar* ##tau## in all places; this follows from the usual FRW metric. Nonetheless, *exterior* the matter, it seems that the moment ##eta = pi## corresponds to a price of ##tau## that will increase with ##R##. We are able to specific all this within the following pair of equations:

$$

r(eta, R) = frac{1}{2} R left( 1 + cos eta proper)

$$

$$

tau(eta, R) = frac{1}{2} sqrt{frac{R_+^3}{2M}} left( eta + sin eta proper)

$$

We received’t show these intimately right here, however trying on the referenced sections in MTW and Landau & Lifschitz ought to make it clear the place they arrive from. Observe the ##R_+## within the second method; that is what captures the truth that the connection between ##tau## and ##eta## is fixed contained in the matter, however varies with ##R## exterior the matter. Observe additionally that the primary method is identical for all values of ##R##, i.e., each inside *and* exterior the matter. In different phrases, we’ve boiled down the variations inside and outdoors the matter to simply two issues: the ##dR^2## time period within the 3-metric above, and the connection between ##tau## and ##eta##. These are the one locations the place radial variation adjustments at ##R_b##.

All of this implies that we must always be capable of write the complete metric in our chosen coordinates within the type:

$$

ds^2 = – dtau^2 + A^2 left( eta proper) d Sigma^2

$$

the place ##A left( eta proper) = left( 1 + cos eta proper) / 2##. Observe that, whereas ##A## is a operate of ##eta## solely, ##eta## will not be a coordinate, and if we use the above equation for ##tau## as a operate of ##eta## and ##R## to implicitly outline ##eta## as a operate of ##tau## and ##R##, we’ll discover that ##A## will then be a operate of ##tau## and ##R##. Extra exactly, ##A## will probably be a operate of ##tau## and ##R## for ##R > R_b##, i.e., exterior the collapsing matter; however contained in the collapsing matter, ##A## will probably be a operate of ##tau## solely (which is what we count on from the usual FRW metric). This modification in dependence at ##R_b## is the worth we pay for having the correct time ##tau## of comoving observers as our time coordinate.

(We may rewrite the metric to make use of ##eta## because the time coordinate, but when we did, whereas we might get a cleaner separation of time and radial dependence within the spatial half, we might then pay a unique value: the metric would now not be diagonal. It is a consequence of the truth that, whereas surfaces of fixed ##tau## are orthogonal to our comoving worldlines (the radial geodesics), surfaces of fixed ##eta## usually are not–extra exactly, they aren’t within the vacuum area exterior the collapsing matter. We received’t pursue this additional right here, nevertheless it guarantees to be instructive if any reader desires to sort out it.)

We’ll go away these issues as an train for the reader and return to our ansatz for the metric above. For the area contained in the collapsing matter, we already know that it’s right, as a result of, as above, we all know that ##A## is a operate of ##tau## solely and we all know that ##d Sigma^2## on this area has the usual FRW type. So all we have to confirm is that our ansatz is right for the vacuum area exterior the collapsing matter. We’ll try this by rewriting the same old type of the metric in Novikov coordinates by way of ##R## as an alternative of ##R^*##.

The metric within the standard Novikov coordinates, utilizing ##R^*##, is:

$$

ds^2 = – dtau^2 + frac{{R^*}^2 + 1}{{R^*}^2} left( frac{partial r}{partial R^*} proper)^2 d{R^*}^2 + r^2 dOmega^2

$$

the place

$$

R^* = sqrt{ frac{R}{2M} – 1 }

$$

We now notice the next:

$$

frac{partial r}{partial R^*} dR^* = frac{partial r}{partial R} frac{partial R}{partial R^*} dR^* = frac{partial r}{partial R} dR

$$

$$

frac{partial r}{partial R} = frac{r}{R}

$$

When you’re unhappy with the informal use of the chain rule within the first of those, you possibly can confirm it by specific computation from the above equation for ##R^*## by way of ##R##, as is finished in this PF thread. The second is clear from the above equation for ##r## by way of ##R##.

Utilizing these and the truth that ##r^2 = R^2 left( r / R proper)^2##, we will rewrite the metric for the vacuum area within the type we would like:

$$

ds^2 = – dtau^2 + left( frac{r}{R} proper)^2 left( frac{1}{1 – frac{2M}{R}} dR^2 + R^2 dOmega^2 proper)

$$

Right here ##r / R## is identical because the operate ##A left( eta proper)## that we outlined above, as may be seen from the equation for ##r## by way of ##eta## that we gave above, and the issue contained in the parentheses within the spatial half is ##d Sigma^2## that we noticed above for the area ##R > R_b##. So, placing the whole lot collectively, we’ve our metric for the complete Oppenheimer-Snyder collapse, together with each the inside of the collapsing matter and the outside vacuum area, in comoving coordinates:

$$

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)

$$

In a follow-up article, we’ll take a look at what this metric tells us concerning the physics concerned.

**References:**

Landau & Lifschitz (Fourth Version), Quantity 2, Sections 102, 103

Misner, Thorne & Wheeler (1973), Sections 31.4, 32.4

Weinberg, Gravitation & Cosmology (1972), Part 11.9

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