Oppenheimer-Snyder Mannequin of Gravitational Collapse: Mathematical Particulars

In a earlier article, I described on the whole phrases the mannequin of gravitational collapse of a spherically symmetric large object, first revealed by Oppenheimer and Snyder of their basic 1939 paper. On this follow-up article, I’ll give additional mathematical particulars concerning 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 a substitute of discovering an expression for the outside vacuum metric in comoving coordinates, he finds an expression for the inside metric in coordinates much like customary Schwarzschild coordinates. We is not going to focus on that method right here, however it’s instructive to match the 2. The latter method, which additionally is similar to the method taken within the authentic Oppenheimer-Snyder paper, has the apparent limitation of getting a coordinate singularity on the horizon, in addition to different extra technical points; however since these sources are centered primarily on how the collapse seems to a distant observer, these limitations are much less of a difficulty than they’d 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 best way all the way down to the singularity. The comoving coordinate method we use right here is significantly better suited 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 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 whole mass of the matter.

Since these two metrics should match at ##R = R_b##, we will receive 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## incorporates ##2M##, so it permits us to rewrite the 3-metric above within the following helpful kind, 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 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 customary FLRW coordinates for a closed universe contained in the matter and is considerably much like Novikov coordinates exterior the matter; nevertheless, we might want to look fastidiously on the latter case to make sure that we’re accurately describing the vacuum area since customary Novikov coordinates don’t use the areal radius on the ##tau = 0## hypersurface immediately, however outline a brand new radial coordinate, known as ##R^*## in MTW, and specific the metric by way of this coordinate. We are going to return to this beneath.

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 notice that contained in the collapsing matter, the moment ##eta = pi## corresponds to the similar ##tau## in every single place; 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 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 formulation; 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 formulation is similar 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 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 should always have the ability to write the complete metric in our chosen coordinates within the kind:

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

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## can be a perform of ##tau## and ##R## for ##R > R_b##, i.e., exterior the collapsing matter; however contained in the collapsing matter, ##A## can be a perform of ##tau## solely (which is what we count on from the usual FRW metric). This variation in dependence at ##R_b## is the worth we pay for having the right time ##tau## of comoving observers as our time coordinate.

(We might 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 not be diagonal. This can be a consequence of the truth that, whereas surfaces of fixed ##tau## are orthogonal to our comoving worldlines (the radial geodesics), surfaces of fixed ##eta## are usually not–extra exactly, they don’t seem to be 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 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 kind. 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 standard type of the metric in Novikov coordinates by way of ##R## as an alternative of ##R^*##.

The metric within the regular 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:

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

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

In case 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 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 kind 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 similar 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 every thing collectively, we’ve 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 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