Gravitational Collapse

The \(\Upsilon\)-term may be ignored for usual solar system computations, as well as for cosmological observations, except for the very early universe: \(\Upsilon\) > 0 leads to a big bounce instead of the big bang (in other words, a variant of inflation) in the very early universe. But to be compatible with observation, we have to choose a very small \(\Upsilon\), so that the \(\Upsilon\)-term becomes unobservable in solar system observations.

But this term nonetheless becomes important also at another place – near horizon formation during the gravitational collapse.

Finding the preferred coordinates

To show this, we have to find where the GR solution starts to fail. For this purpose, it is necessary to have a hypothesis about the preferred coordinates \(\mathfrak{x},\mathfrak{y},\mathfrak{z},\mathfrak{t}\). We know they have to be harmonic. The initial values should be obtained from gluing with the cosmological solution. That means, for a collapsing star we have approximately Minkowski coordinates before the collapse. But in the case of spherical symmetry we can as well use symmetry considerations and require coordinates for the stable initial star that the solution does not depend on time in these coordinates. A class of time-independent harmonic metrics is the following: \[ds^2 = \left(1-\frac{m \frac{\partial m}{\partial \mathfrak{r}} }{ \mathfrak{r}}\right) \left(\frac{\mathfrak{r}-m}{ \mathfrak{r}+m}dt^2-\frac{\mathfrak{r}+m}{ \mathfrak{r}-m}d\mathfrak{r}^2\right) - (\mathfrak{r}+m)^2 d\Omega^2 \]

Here \(m=m(\mathfrak{r}) = GM(\mathfrak{r}) c^{-2}\), where \(M(\mathfrak{r})\) is the mass inside the sphere of radius \(\mathfrak{r}=\sqrt{\mathfrak{x}^2 + \mathfrak{y}^2 + \mathfrak{z}^2}\). In particular, outside the matter \(M(\mathfrak{r})= const\), thus, the leading factor is trivial and the expression reduces to the Schwarzschild solution in harmonic coordinates.

In this solution, the coordinates \(\mathfrak{x},\mathfrak{y},\mathfrak{z}\) are harmonic even if M = \(M(\mathfrak{r},\mathfrak{t})\) depends on time. The time coordinate \(\mathfrak{t}\) is harmonic only for time-independent \(M = M(\mathfrak{r})\). Thus, to obtain the preferred coordinates for a collapsing star of this type, we have to solve only one harmonic wave equation for \(\mathfrak{t}(\mathfrak{r},t)\) with well-defined initial values for this background.

It depends on the inner structure of the star (the material equations) as well as on the equations for the gravitational field, if a star with a given distribution of mass \(M(\mathfrak{r})\) is stable or not.

Now, we have defined the preferred coordinates and are now able to look if the new terms are important or not. In the region where they become important, the GR solution is no longer a good approximation.

Frozen stars

Let's show now that in GLET exist stable stars close to horizon size. Assume \(m(\mathfrak{r})=(1-\Delta)\mathfrak{r}\) and assume matter with \(k\approx 1\). This leads to a simple constant metric inside, which models, for sufficiently small \(\Delta\), the inner part of a star very close to forming a horizon. We obtain the following set of equations: \[ ds^2 = \Delta^2 d\mathfrak{t}^2 - (2-\Delta)^2 (d\mathfrak{r}^2 + \mathfrak{r}^2d\Omega^2) \] \[ 0 = -\Upsilon\Delta^{-2} + 3 \Xi (2-\Delta)^{-2} + \Lambda + \varepsilon,\] \[ 0 \approx \Upsilon\Delta^{-2} + \Xi (2-\Delta)^{-2} + \Lambda - k \varepsilon.\]

As long as we can ignore \(\Xi \approx \Upsilon \approx \Lambda \approx 0\), we have no solutions of this type, because the energy should be always greater zero. But close to the horizon the \(\Upsilon\)-term becomes more and more important. Ignoring \(\Xi \approx \Lambda \approx 0\), we obtain a stable solution for \[ \Upsilon \Delta^{-2} = \varepsilon > 0\]

Thus, close to the horizon, before horizon formation, we obtain stable states for collapsing stars.

Once no horizon has been formed, we do not have black holes. To describe these solutions, the old notion frozen star seems much more appropriate. The surface of these stars remains, in principle, visible, but is highly redshifted (that means, much more "cold"). How much, depends on \(\Upsilon\). This, again in principle, allows to measure \(\Upsilon\).