Can the Oppenheimer-Snyder Collapse Model Account for Variable Mass Inflow?

[tom.stoer]

I am looking for a mathematical solution of the following "paradox" regarding the Oppenheimer-Snyder collapse of a finite sphere of dust.

Suppose the sphere of mass M has just collapsed to its Schwarzschild radius

$$R_S(M) = \frac{2GM}{c^2}$$

1) Suppose there is another thin shell of dust just outside the Schwarzschild radius at $[R, R + dr], \, R > R_S$ which is falling into the black hole. Using the Schwarzschild time coordinate t it is clear that this shell never reaches the horizon at finite Schwarzschild time and therefore the mass flow through the event horizon is zero.

2) Suppose there is a stationary observer just outside the Schwarzschild radius at constant $R > R_S$ who sees the free falling dust. This shell has some mass dm. The shell will not fall through the Schwarzschild radius, but it will certainly fall through the radius R of the stationary observer and will reach $[R^\prime, R^\prime + dr^\prime]$ with $R_S < R^\prime < R^\prime + dr^\prime < R$ in finite Schwarzschild time.

(b/c the observer is stationary we can simply rescale Schwarzschild time to his proper time using a finite constant)

From the mass m one can derive the new Schwarzschild radius

$$R_S(M+dm) = \frac{2G(M+dm)}{c^2} = R_S(M) + R_S(dm) = R_S + dR_S$$

Of course for large enough mass dm we find for this new Schwarzschild radius $R_S < R^\prime < R^\prime + dr^\prime < R_S + dR_S$. Therefore the mass flow through the event horizon at $R_S^\prime$ is not zero - which contradicts the conclusion drawn from (1) that the mass flow is zero.

My question is whether there is a realistic solution for infalling matter of finite mass from which the growth of the event horizon can be derived analytically as a function of Schwarzschild time as some function $R_S(t)$

 

[PeterDonis]

Using the Schwarzschild time coordinate t it is clear that this shell never reaches the horizon at finite Schwarzschild time and therefore the mass flow through the event horizon is zero.

This argument is not valid. If it were, it would prove too much: it would prove that the original shell could not have collapsed inside its Schwarzschild radius either. The correct resolution to this is to realize that Schwarzschild coordinates are singular at the horizon, so you can't use them to analyze this problem (or at least, you can't use them "naively" the way you are implicitly using them here). You either have to pick better coordinates, or you have to carefully distinguish in your analysis between coordinate quantities and actual physical quantities, like the proper time it takes a dust particle to fall to the horizon. The original Oppenheimer-Snyder paper used the latter method; more modern analyses use the former.

My question is whether there is a realistic solution for infalling matter of finite mass from which the growth of the event horizon can be derived analytically as a function of Schwarzschild time as some function $R_S(t)$

No, there isn't. There can't be, because, as I noted above, Schwarzschild coordinates are singular at the horizon. The key aspect of that for your question here is that Schwarzschild coordinates assign the same time coordinate (plus infinity) to *any* event on the horizon; i.e., Schwarzschild coordinates can't distinguish between distinct events on the horizon, so of course you can't construct a function $R_S(t)$ where $t$ is Schwarzschild time. You need to pick better coordinates.

The simplest solution I know of that describes mass-energy falling into a "black hole" is the ingoing Vaidya null dust, which is described here:

http://en.wikipedia.org/wiki/Vaidya_metric#Ingoing_Vaidya_with_pure_absorbing_field

This gives a solution $M(v)$ for the mass of the black hole as a function of "advanced time" $v$, which is basically the same as the ingoing Eddington-Finkelstein null coordinate $v$. The key point here, again, is that $v$ *can* be used as a valid coordinate on the horizon, i.e., each distinct event on the horizon is labeled by a distinct, finite value of $v$. So the function $M(v)$ naturally yields a function $R_S(v) = 2M(v)$ for the horizon radius as a function of $v$.

The infalling mass-energy in the Vaidya solution is null dust, not timelike dust, i.e., it's ingoing null radiation like photons. I have not seen an analogous solution using continuously infalling timelike dust, but it seems like there ought to be one, perhaps using ingoing Painleve coordinate time $T$ to label events on the horizon (which basically works the same as $v$ does, each distinct event on the horizon has a distinct, finite value of $T$).

 

[George Jones]

The infalling mass-energy in the Vaidya solution is null dust, not timelike dust, i.e., it's ingoing null radiation like photons. I have not seen an analogous solution using continuously infalling timelike dust, but it seems like there ought to be one, perhaps using ingoing Painleve coordinate time $T$ to label events on the horizon (which basically works the same as $v$ does, each distinct event on the horizon has a distinct, finite value of $T$).

There might me exact solutions, but I am not aware of any, e.g., I don't see this type of exact solution in "Exact Space-Times in Einstein's General Relativity" by Griffiths and Podolsky. All the realistic work of which I am aware involve numerical codes, e.g. Chapter 8 "Spherically symmetric spacetimes" from the book "Numerical Relativity: Solving Einstein's Equations on the Computer" by Baumgarte and Shapiro.

 

[tom.stoer]

Thanks for you hints.

That means that - unfortunately - it is hardly possible to describe this from the perspective of a Schwarzschild observer (or a static observer hovering at constant r).

I am aware of the Vaidya metric; anyway, thanks for your answers.

 

[sardar]

Thank you for your question. The Oppenheimer-Snyder collapse is a classical solution to Einstein's field equations that describes the collapse of a spherical object made of dust into a black hole. This solution is often used as a simplification to study the dynamics of black hole formation.

In this solution, the dust particles are assumed to be pressureless and non-interacting, which is not a realistic scenario for infalling matter. However, it can provide insights into the general behavior of gravitational collapse.

In your question, you have raised a paradox regarding the mass flow through the event horizon during the collapse. This paradox arises because in the Oppenheimer-Snyder collapse, the mass of the collapsing object is assumed to be constant. However, in reality, the mass of the collapsing object will increase as more matter falls into it.

To address this paradox, we need to consider the mass-energy conservation law in general relativity. In this case, the mass of the collapsing object can be expressed as a function of the Schwarzschild time, M(t). As more matter falls into the black hole, the mass M(t) will increase, and the Schwarzschild radius will also increase accordingly.

Therefore, the correct expression for the Schwarzschild radius in this scenario would be R_S(M(t)) = 2GM(t)/c^2. This means that the event horizon will continue to grow as more matter falls into the black hole, and the mass flow through the event horizon will not be zero.

In summary, the paradox you have raised is a result of the simplifications made in the Oppenheimer-Snyder collapse solution. In reality, the mass of the collapsing object will increase, and the Schwarzschild radius will also increase accordingly. This can be derived analytically as a function of Schwarzschild time, taking into account the mass-energy conservation law.