#### SimonsCat

**Registered Member**

**Quantum Repulsion**

Using the Eddington Finkelstein coordinates, the metric of a [Schwarzschild] black hole is

$$ds^2 = r^2d\Omega^2 + 2dvdr - F(r)du^2$$

where the two sphere is $$d\Omega^2$$ and the redshift is

$$F(r) = (1 - \frac{2m}{r})$$

The ingoing null geodesics satisfy a constant $$u$$ with $$(r > 2m)$$ while the outgoing ones $$(r< 2m)$$ satisfy

$$\frac{dr}{dv} = \frac{1}{2} (1 - \frac{2m}{r})$$

The interesting thing, is the presence of a gravitational repulsive force through a correction of the red shift

$$F(r) = 1 - \frac{2mr^2}{r^3 + 2 \alpha^2 m}$$

expanding in $$r^{-1}$$ gives

$$F(r) = 1 - \frac{2m}{r} + \frac{4 \alpha^2m^2}{r^4}$$

The new term actually represents a short-scale repulsive force due to quantum effects. Take note, that the effect is actually well-known, for certain classes of quantum gravity models and has been used in this context to create Planck Stars.

It is this quantum mechanical force acting on the short scale that interests me as it could be efficient to actually halt the collapsing universe from reaching the dreaded singularity. When Penrose and Hawking formulated the singularity theorems for the univese, the second equation of Einstein reveals that an initial singularity cannot be avoided in the standard model since $$\ddot{R} < 0$$ that is, the acceleration of a collapsing universe is always negative so that nothing can escape it from collapsing to a pure point (a system with no dimenisons but infinite energy) as long as

$$[\rho + 3(\frac{p}{c^2} + \frac{u}{3c^2})]$$ positive, which is necessary if the pressure is positive $$p > 0$$. Largely, they remained steadfast in their assertion for many years, but Penrose and Hawking eventually changed their stance from many years of believing the singularty theorems had to be the only conclusion of a universe following the rules of General Relativity. But perhaps the mistake was, that General Relativity is only one factor which may be significant, one other factor is the effects of quantum gravity, the quantum interpretation of gravity which seems so estranged to Relativity but is responsible for quantum fluctuations.

It is interesting that both Penrose and Hawking believed for many years that singularities could not be avoided in a universe. Today, they both concede that this was probably a wrong approach. Scientists are allowed to change their minds after all It would help though that scientists stop making such absolutist statements.