(How not to be afraid by wormhole) Octopussy (or Medusa)

//The title of this post has been slightly edited on 1st September 2016

Octopus wormhole (or wormhole octopussy?) 
(Olena Shmahalo/Quanta Magazine)

The ER = EPR idea posits that entangled particles inside and outside of a black hole’s event horizon are connected via wormholes.

"The story of Medusa tells you she destroys those who contemplate only her reflection, which is of course horrible, instead of contemplating her real face, which is the face of wholeness" 

Patrick Moran, Swimming in stone

Contemplating quantum gravity in its full nonlocal glory... without fearing for local burns at the black hole horizon?

The modern representation of Karl Schwarzschild’s spherically symmetric solution of Einstein’s equations reads

ds² = −(1 −2M/r) dt² + 1/(1 − 2M/r) dr² + r²(dθ² + sin²θ dϕ² ) . (1.1) 

[Note: In Schwarzschild’s original work [1], the coordinate r in Eq. (1.1) was called R, while he chose an other radial coordinate r such that the point R=2M corresponds to r=0 , since it seemed to be obvious to expect a singular mass distribution at the origin of the coordinate frame. Today, we know that this was unnecessary, for two reasons: first, one is free to choose the most convenient coordinate system anyway, and secondly, the surface r=2M does not represent a physical singularity at all, but just a coordinate singularity, much like the north pole of the Earth. It is the black hole horizon.]

As we now know very well, matter can enter the black hole through the horizon, defined by the surface r=2M , while in the standard, unquantised theory, nothing can emerge out of it. The horizon is a one way door. [Note: On some web pages, these facts are still being disputed, which we can only attribute to ignorance. Schwarzschild, who wrote his paper in less than two months after Einstein’s discovery, could be excused for not immediately realising the rather subtle features of black hole horizons, which required several years to be cleared up, but today’s experts cannot afford to make such mistakes.] In the coordinates of Eq. (1.1), the point r = 0 is a real physical singularity. 

Even though the horizon appears to be a regular region of space-time, we do have a problem with it. According to Hawking’s well-known result [2], it is due to vacuum fluctuations that a distant observer will observe particles leaving the black hole: Hawking radiation. These particles have a thermal spectrum, independent of the black hole formation process.

Hawking’s original conclusion was that this result must imply that a black hole as a physical object violates the laws of quantum mechanics: even if it originates from matter in a single quantum state, it ends up in a thermal, that is, a quantum mechanically mixed state. How could it be that a derivation that uses quantum mechanics can yield a result violating the laws of this theory? Hawking particles are now understood to be formed at the horizon, not, as was originally thought, somewhere near the r=0 singularity in its past...

Singularities, horizons, firewalls, and local conformal symmetry

Gerard 't Hooft

(Submitted on 13 Nov 2015)

Hawking particles emitted by a black hole are usually found to have thermal spectra, if not exactly, then by a very good approximation. Here, we argue differently. It was discovered that spherical partial waves of in-going and out-going matter can be described by unitary evolution operators independently, which allows for studies of space-time properties that were not possible before. Unitarity dictates space-time, as seen by a distant observer, to be topologically non-trivial. Consequently, Hawking particles are only locally thermal, but globally not: we explain why Hawking particles emerging from one hemisphere of a black hole must be 100 % entangled with the Hawking particles emerging from the other hemisphere. This produces exclusively pure quantum states evolving in a unitary manner, and removes the interior region for the outside observer, while it still completely agrees locally with the laws of general relativity. Unitarity is a starting point; no other assumptions are made. Region I and the diametrically opposite region II of the Penrose diagram represent antipodal points in a PT or CPT relation, as was suggested before. On the horizon itself, antipodal points are identified.

Black hole unitarity and antipodal entanglement

Gerard 't Hooft

(Submitted on 14 Jan 2016 (v1), last revised 14 Apr 2016 (this version, v4))

To make a long story short: 't Hooft proposes to solve the  information loss paradox of quantum evaporating black holes arguing that a pure state of collapsing matter forming a black hole evaporates indeed through the radiation of Hawking particles out of the horizon but globally this radiation is not thermal but made of entangled pairs of Hawking particles emitted from antipodal points on the two hemispheres of the horizon (thus restoring the unitary evolution of black hole evaporation).

Another speculative way to "feel" quantum gravity with just a snow ball's chance in hell against quantum fluctuations

The following work I am going to highlight now might be based on less solid grounds than (and potentially be incompatible with) the 't Hooft work but I find it a very evocative outline of tentative ideas about quantum gravity to be worth a quote here:

Here we use a thought experiment, based on a comparison of Hawking radiation with Unruh radiation, to show that these two quantum phenomenon imply a small violation of the equivalence principle. The manner in which the equivalence principle is violated by the comparison of these two effects might point toward a resolution of some of the short comings of general relativity such as the existence of singularities for certain space-times and the difficulty in formulating a quantum version of general relativity...

Hawking radiation [8] is the thermal radiation emitted by a black hole of mass, M. It occurs as a consequence of placing quantum fields in the gravitational background of a black hole. An observer who stays at a fixed distance, R, from a black hole of mass, M, will measure a temperature given by [9] 

T_Hawking= ℏc³/[8πGMk_B√(1−2GM/c²R)]. 

Normally, the Hawking temperature T_Hawking is quoted for an observer a large distance from the black hole (i.e. R → ∞) so the blue shift factor √(1−2GM/c²R) is not written down. It is kept here since it is crucial for seeing how the equivalence principle is violated. 

By the equivalence principle an observer, accelerating through flat, Minkowski spacetime, should also measure thermal radiation. Otherwise the observer could immediately tell the difference between a gravitational field and an accelerating frame – the accelerating frame would be the one in which no thermal radiation is detected. Soon after Hawking’s original paper on black hole radiation it was shown that an accelerating observer (with an acceleration of a = |a|) does detect thermal radiation with a temperature given by 

T_Unruh =  ℏa/2πck_B. 

This radiation and T_Unruh are know as Unruh radiation and the Unruh temperature respectively [10]. Thus, at least qualitatively, there is no violation of the equivalence principle – an observer in an Einstein elevator fixed at a distance, R, from a black hole will measure both a downward acceleration toward the floor of the elevator and thermal radiation at a temperature T_Hawking ; an observer in an Einstein elevator which is accelerating through flat, Minkowski space-time will also measure both a downward acceleration and thermal radiation. However, looking at this situation quantitatively uncovers a violation of the equivalence principle except in the limit as the observer approaches the event horizon... 

We now ask “What are the possible implications, for gravity, of this violation of the equivalence principle from the above thought experiment?”. In addressing this question we will assume that the strength of the gravitational effects are proportional to or connected with the Hawking temperature and the inertial effects are proportional to or connected with the Unruh temperature. In particular we assume the ratio of the gravitational and inertial masses are connected with the ratio of the Hawking temperature to the Unruh temperature. This assumption is not trivial since the violation of the equivalence principle discussed above deals with the Einstein elevator formulation of the equivalence principle while the distinction between gravitational and inertial masses is a different formulation of the equivalence principle. 

First we look n the near horizon limit R → 2GM/c2 a. Here the gravitational effects dominate the inertial effects since 

T_Hawking → ℏc³/[8πGMk_B] > T_Unruh  → ℏ/2πck_B   × (GM/R²)

For example, taking M = M_{Sun .}.. and R = 1AU ... yields T_Hawking ≈6.2×10⁻⁸K and T_Unruh≈2.4×10⁻²³K. While T_Hawking is 15 orders of magnitude larger than T_Unruh both temperatures are effectively zero when compared to something like the 2.7K cosmic microwave background. We take this as an indication that the violation of equivalence principle, implied by the difference in Hawking and Unruh temperatures for the same accelerations, is a very small effect under normal conditions i.e. low energy density, low gravitational field strength, non-relativistic velocity. Nevertheless, the implication would be that the variation in gravitational mass is slight larger than the variation in inertial mass. This might have some bearing on the rotation curves of galaxies. The anomalous velocity profile of outlying stars orbiting the galactic center is usually explained by an enhancement of the gravitational force coming from the presence of dark matter. Here, the enhancement of gravity over inertia would come from the slight dominance of gravitational mass over inertial mass at large distances from the galactic center. 

 Second we look in the near horizon limit R→2GM/c² and find

 T_Hawking →... T_Unruh

Thus, in the near horizon region the equivalence principle is restored. (Note that exactly at the horizon both temperatures diverge to the same infinite value due to the blue shift factor. This is as expected since for an observer fixed just above the horizon the local acceleration and Hawking temperature both diverge). This is surprising. One might have guessed that in a region of stronger gravitational field strength, such as near the horizon versus far from the horizon, the violation of the equivalence principle would be more pronounced; that the divergence between quantum mechanics (as represented by Hawking and Unruh radiation) and general relativity would be larger. The fact that this is not the case might be taken as an indication that general relativity and quantum mechanics are more compatible, not less, as the strength of the gravitational field increases...

In the near horizon limit R→2GM/c² our arguments pick out the event horizon as special – it is the surface where the (local) equivalence principle on which general relativity is based, becomes compatible with (non-local) quantum mechanics as represented by Hawking and Unruh radiation. 

Finally, behind the horizon (i.e. R<2GM/c²  ) the expressions for Hawking temperature and Unruh temperature break down, and we continue our journey inward based on the following conjecture: Outside the horizon (R>2GM/c²  ) gravitational effects dominate inertial effects; near the horizon ... gravitational effects and inertial effects are equivalent; thus we postulate that inside the horizon ... inertial effects dominate gravitational effects. As R→0 inertial effects will become ever more dominant over gravitational effects... one can say that ... any ... material that has fallen to this point inside the horizon, is “frozen” and non-dynamical, since the inertial mass of the material will have increased to the point where further motion is impossible. This postulated transition of gravity, under conditions of high mass-energy density, to a non-dynamical theory also might have relevance for the difficulty of consistently calculating quantum corrections to the gravitational field i.e. the long standing and unresolved problem of formulating a quantum theory of gravity...

Black holes are often described as a scientific version of Hell – a place of extreme conditions which is inhospitable to any person who falls inside. Taking into account the above postulated picture, if a black hole is to be compared to Hell then it would be the Hell of Dante’s Inferno, the center of which is frozen, rather than the traditional hot Hell of fire and brimstone

Hawking versus Unruh temperature as a measure of the health of the equivalence principle, Douglas Singleton (August 13, 2012)

(source)

A radiolarian (skeleton) with a bilateral symmetry may mimic the radiating horizon of a black hole with its antipodal entangled Hawking particles better than an octopussy ;-) Its lattice structure may also fit pretty well for the discreteness of spacetime that emerges from the quantum gravity effects at the black hole horizon but this is another story (for a later post...?)