We are in a Schwarzschild black hole--T or F?

yuiop

Hi Marcus,

I normally hang around in the relativity forum (but I am by no means a relativity expert) and while playing around with Schwarzschild solutions I made a discovery that I think is very relevant to this thread and may provide an alternative answer to the question you pose here.

The equation for coordinate acceleration in the exterior Schwarzchild solution is:

[tex]a '=\frac{GM}{R^2}\left(1-\frac{R_s}{R}\right)[/tex]

When R is greater than the Schwarzschild radius the gravitational acceleration is positive towards the mass as you would expect. When R is less than the Schwarzchild radius the gravityational acceleration is negative and directed outwards towards the event horizon. if for example all the mass of the universe was originally confined to radius of R=Rs/10 then the outward acceleration is -900 GM/Rs^2. If the mass was confined to R=Rs/1,000 then the outward acceleration is -999,000,000 GM/Rs^2. Obviously, the outward gravitational acceleration gets considerably larger as original density increases.

Now if we look at the coordinate velocity of photon falling from infinity the equation is:

[tex]c '= c\left(1-\frac{R_s }{R}\right)[/tex]

and for R>Rs the coordinate velocity is always less than c, the velocity of light at infinity. Below the Schwarzchild radius the coordinate velocity of light get larger than c and is negative. This value for R<Rs is the speed of light falling from the centre outwards towards the event horizon. So for a universe with an extreme initial density photons (and particles with mass) move outwards towards the Schwarzchild radius at velocities much greater than c. In other words the outward expansion would very rapid until the universe reached the size of its own Schwarzchild radius. In fact the expansion would be arbitarily high and only limited by the initial density. The greater the initial density the greater the initial expansion. This would be very like the inflation that is thought to have occured early in the history of the universe. For falling particles the coordinate velocity is given by:

[tex]v ' = c\sqrt{{Rs \over R}} \left(1-\frac{R_s}{R} \right)[/tex]

One possible objection to this idea is that the coordinate velocity of the outward moving particles becomes zero at the Schwarzchild radius bring everything to a stop. I think this issue can be resolved by considering a universe with an initially flat spacetime. The rapid expansion of the particles within the Schwarzschild volume sends a gravitational shock wave that ripples outwards. Gravity waves have no difficulty passing event horizons and carry energy away with them. The loss of energy from the Schwarschild volume reduces the Schwarschild radius, releasing the particles trapped at the event horizon. The process is self destructive and the event horizon dissappears.

If dark energy is ignored this model would basically oscillate, with the universe expanding and collapsing to point and then expanding again. With dark energy it may never collapse.

I came to this conclusion while investigating the interior Schwarzschild solution that enables you examine what happens to a black hole as it forms and found that normal stable black holes do not have a singularity of infinite density at the centre but are a thin shell of matter just outside the event horizon.


For more equations and background on these ideas, see these threads:

http://www.physicsforums.com/showthr...=238839&page=2 post #19 onwards.

http://www.physicsforums.com/showpos...2&postcount=17

http://www.physicsforums.com/showthr...=223730&page=2 post#19

I hope these ideas are of interest. The nice thing about them is that they basically fall straight out of the Schwarzschild solutions. I am not saying dark energy does not exist or that the Schwarschild solutions might have to be modified a bit to allow for expanding spacetime, but I am saying that that even without those things the Schwarzschild equations do not imply the universe would be trapped in a black hole even when there technically enough mass within a given radius to be a black hole. In fact, examination of the solutions show the universe would be very different if we were inside a black hole.

yuiop

I just found a counter argument to my above statement. Damn!
http://en.wikipedia.org/wiki/Birkhof...em_(relativity)

Birkhoff's theorem states a pulsating spherical mass can not give off gravitational waves. That seems reasonable as the gravitaional filed of a sperical object always looks like a point source outside the mass of the body.

There are however any number of potential ways that the mass trapped in a shell at the Schwarzschild radius can escape. The loss of a single atom or photon by Hawking radiation or quantum tunelling would start the destruction of the event horizon. This is even more likely as there is no CMB radiation adding to the mass/energy of the Schwarzschild mass at this epoch. The other method is to observe that the escape velocity at the event horizon is c and that during the inflation period the velocities of exceed c as explained in my last post.

So for those who cherish the notion that if the universe is expanding, that it must have been smaller and denser at some time in the past, GR can cope with that. For those that dont like that notion, you can take comfort with thought of a universe that started infinite in volume and mass and then continued expanding.