Oppenheimer-Snyder Mannequin of Gravitational Collapse: Mathematical Particulars
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Half 2: Mathematical Particulars
In a earlier article, I described on the whole phrases the mannequin of gravitational collapse of a spherically symmetric huge object, first printed by Oppenheimer and Snyder of their traditional 1939 paper. On this follow-up article, I’ll give additional mathematical particulars in regards to the mannequin, utilizing an method considerably totally different from their authentic paper (and impressed by the method described in MTW and Landau & Lifschitz).
(Observe: Weinberg takes a distinct method within the vacuum area exterior the collapsing matter. As an alternative of discovering an expression for the outside vacuum metric in comoving coordinates, he finds an expression for the inside metric in coordinates just like commonplace Schwarzschild coordinates. We is not going to talk about that method right here, however it’s instructive to match the 2. The latter method, which additionally is analogous to the method 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 wish an outline that covers all the collapse and consists of each distant observers and the collapsing matter all the way in which all the way down to the singularity. The comoving coordinate method we use right here is significantly 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 are going to 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 categorical this as follows: the 3-metric of this hypersurface is given by
$$
dSigma^2 = frac{dR^2}{1 – ok R^2} + R^2 dOmega^2
$$
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 overall mass of the matter.
Since these two metrics should match at ##R = R_b##, we will get hold of an equation for ##ok##:
$$
ok = frac{2M}{R_b^3}
$$
which tells us that ##ok## is said 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 ##ok## incorporates ##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 now we have outlined the capabilities ##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 method matches commonplace FLRW coordinates for a closed universe contained in the matter and is considerably just like Novikov coordinates exterior the matter; nevertheless, we might want to look rigorously on the latter case to make sure that we’re appropriately describing the vacuum area since commonplace Novikov coordinates don’t use the areal radius on the ##tau = 0## hypersurface instantly, however outline a brand new radial coordinate, referred to as ##R^*## in MTW, and categorical the metric by way of this coordinate. We are going to return to this under.
We now make use of the truth that the geodesic movement each inside and outdoors the matter will be described utilizing a cycloidal time parameter ##eta##, which ranges from ##0## at ##tau = 0## to ##pi## on the instantaneous when every geodesic hits the singularity at ##r = 0##. We observe that contained in the collapsing matter, the moment ##eta = pi## corresponds to the identical ##tau## all over the place; this follows from the usual FRW metric. Nevertheless, 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 categorical 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 wanting 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, now we have boiled down the variations inside and outdoors the matter to only 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 means that we must always have the ability to write the total 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 perform of ##eta## solely, ##eta## is just not a coordinate, and if we use the above equation for ##tau## as a perform of ##eta## and ##R## to implicitly outline ##eta## as a perform of ##tau## and ##R##, we’ll discover that ##A## will then be a perform of ##tau## and ##R##. Extra exactly, ##A## will likely be a perform of ##tau## and ##R## for ##R > R_b##, i.e., exterior the collapsing matter; however contained in the collapsing matter, ##A## will likely be a perform of ##tau## solely (which is what we anticipate from the usual FRW metric). This transformation in dependence at ##R_b## is the worth we pay for having the right 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’d get a cleaner separation of time and radial dependence within the spatial half, we’d then pay a distinct 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## will not be–extra exactly, they aren’t within the vacuum area exterior the collapsing matter. We received’t pursue this additional right here, but it surely guarantees to be instructive if any reader needs to sort out it.)
We are going to 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 perform 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 are going to 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 normal 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 observe the next:
$$
frac{partial r}{partial R^*} dR^* = frac{partial r}{partial R} frac{partial R}{partial R^*} dR^* = frac{partial r}{partial R} dR
$$
$$
frac{partial r}{partial R} = frac{r}{R}
$$
For those who’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 completed 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 wish:
$$
ds^2 = – dtau^2 + left( frac{r}{R} proper)^2 left( frac{1}{1 – frac{2M}{R}} dR^2 + R^2 dOmega^2 proper)
$$
Right here ##r / R## is identical because the perform ##A left( eta proper)## that we outlined above, as will 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 all the things collectively, now we have our metric for all the 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 have a look at what this metric tells us in regards to 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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