Quantum Gravity & Black Holes

Black holes are a concept supposedly following from the General
Theory of Relativity and it has generated much popular and scientific attention, even a Nobel Prize in 2020! Other examples are the movie Interstellar and Stephen Hawking dedicating a great part of his scientific career to the topic. So, much research has been done on them, also in efforts to try to arrive at a Theory of Quantum-Gravity. That is hoped to solve singularity problems, at first in the Standard Model of Quantum Field Theory but now that those seem to be solved, for resolving the singularities (and also the information problem, see further on) in General Relativity itself i.e. of Black Holes and of the Big Bang.¶

§ However, it is straightforward to argue that a Theory of Quantum-Gravity would have no scientific value.*  For that, let’s compare individual ’quanta’ of gravity to the photons of electro-magnetism:

✓one photon can be produced by a fluctuating electromagnetic

dipole, while gravity only has positive charge and so a (much weaker) fluctuating quadrupole is the best one could use to produce one of its ’quanta’.

✓the transport of one photon is unproblematic, because it does not carry charge itself; one ’graviton’, though, does interact with the omnipresent gravity field and so wouldn’t go far before it is ’washed out’.

✓to detect one photon it has to couple to an electron and this coupling is relatively strong; gravity, on the other hand, is extremely weak: we need a whole planet to be able to sense it!

>>>In conclusion, individual gravitons can’t be produced, transported or detected and so any Theory of Quantum-Gravity (if it could be formulated at all) cannot but reduce to General Relativity.

§ It is also quite easy to see that Black Holes cannot be formed and so deserve to be disregarded as well:°

•Black Holes are characterized by their Horizon, through which nothing can pass; so neither the matter that should have made up the Black Hole in the first place?!™ Yet it is acknowledged that matter ‘appears’ to stay frozen just outside the ‘Black Hole’ (due to time dilation, a well known relativistic effect)...

•Alternatively, General Relativity is a time-invariant theory (like practically all physical theories) and (the formation of) its Horizon makes the Black Hole break this invariance. This is also called the information paradox because the conservation of information is directly linked to time invariance, and scientists have no clue how to deal with objects supposedly disappearing into Black Holes. 

So instead of Black Holes we'll have Frozen Stars. How does their formation look mathematically?

✓the Schwarzschild radius of a mass M is defined as:

R_S = 2G · M / c^2 with G Newton’s Gravity constant and c the speed of light

✓would this mass be located within R_S, it would produce the metric: 

ds^2 = (c·dt)^2 · (1 − R_S / r) − dr^2 / (1 − R_S / r) which has a Horizon where it becomes singular

✓a (spherically symmetric neutron) star in the final stage of its collapse would become free falling matter in the Scharzschild metric of its core n

✓free falling matter more and more resembles propagating light: ds → 0 

Its speed is then given by:  v(r) = −c · (1 − R^n_S / r)

✓it follows that for a mass distribution M(r) ∝ r its speed is homogeneous: v(r, t) = v(t)

but if δM ≶ δr → δv ≷ 0 and so a homogeneous speed is actually the limit for any mass distribution!

✓the limiting density distribution for t → ∞ becomes:

ρ(r) = c^2 / (8π G · r^2) 

so M = c^2 · R / (2G) → R = R_S, but no singularity is ever produced?!

✓the time of collapse from r0 to r1 is found from integrating its speed: 

ct(r1) = −\int^r1_r0 r / (r − R^n_S) dr = r0 − r1 − R^n_S · ln{(r1 − R^n_S) / (r0 − R^n_S)} 

✓for r1 → R^n_S + δr the expression can be inverted to give:

δr(t) = ∆r0 · exp{(∆r0 − ct − δr) / R^n_S} ≈ ∆r0 · exp{∆r0 / R^n_S} · exp{−ct / R^n_S} with ∆r0 ≡ r0 − R^n_S

✓writing out all implicit dependances on r0:

r(t) = r0 · R_S / R_0 + r0 · (1 − R_S / R_0) · exp{R_0 / R_S − 1} · exp{−ct / r_0 · R_0 / R_S}

✓and so, finally, the combined space and time dependence of the density distribution (for large times) is given by:

ρ(r, t) = M / (4π R(t) · r^2) = c^2 / (8πG · r^2) / [1 + (R_0 / R_S − 1) · exp{R_0 / R_S − 1} · exp{−ct / R_S}]

§ Finally, cosmology, a real consequence of General Relativity, exists in a state of crisis as is being acknowledged more and more:  blogs.scientificamerican.com/observations/cosmology-has-some-big-problems . First, Dark Matter was introduced to explain some anomaly (which is probably nothing spectacular, and now there are even anomalies in the dark matter hypothesis); then came Dark Energy to explain the acceleration of the cosmic expansion; then Inflation to explain the surprising thermal equilibrium of the universe (although that hypothetical process does not qualitatively differ from normal expansion, it's just a sudden and temporary increase of the so-called cosmological constant in GR and so doesn't really explain anything); then stars as old as the universe were found and finally inconsistencies in the age of the universe showed up... But at least this crisis is being acknowledged:-) 

¶another argument for the need of Quantum Gravity is the inability of formulating a Quantum Field Theory in (globally) curved space-time; yet isn't that so surprising since in such a space-time no concept of (conserved) energy can be formulated?!

*and anyhow arguing that improving one theory would solve problems of another seems rather strange...

°sorry Stephen;-) And also Sir Roger Penrose who got the 2020 Nobel Prize in physics for his work on Black Holes; seems pretty embarrassing to me...

™immediately the argument comes to mind that for a free falling observer the singularity at the Horizon disappears and so the region could be entered in finite time; but why would we need a coordinate transform to describe our reality? The Principle of Relativity states that for different observers the world isn’t just seen differently, it is different!