# Where Does the Physics of Extreme Gravitation Buckle?

## Where Does the Physics of Extreme Gravitation Buckle?

Dr. TJ Gunn Ph.D. Applied Physics, Ph.M., Masters Philosophy

## Abstract

**:**

## 1. Introduction

## 2. Black Holes and Spacetime Singularities: The Standard GR Picture

#### 2.1. Event Horizons

We will concentrate on the simplest models of gravitational collapse, which are spherically symmetric. Eventually we will consider the effect of introducing small, non-spherical perturbations in this simple picture. As is well known, in the spherically symmetric case and for a star with mass M whose radius is larger than its Schwarzschild radius, rs=2GMc2

In those extreme situations, the strength of the gravitational interaction surpasses that of any other possible known force in the system, provoking an indefinite contraction of the stellar structure. The resulting trajectory of the surface of the collapsing star can be described by means of a function R(τ), where τ is the proper time of an observer attached to the surface. If the regime in which the strength of the gravitational interaction surpasses that of any other relevant force in the system is reached, this surface will inevitably cross the Schwarzschild radius in finite time. In terms of equations, for a given initial stellar radius at τ=τi there always exists a finite proper time interval Δτ so that: R(τi+Δτ)=rs

#### 2.2. Singularities

The last ingredient is a condition on the curvature of the manifold. This crucial geometrical condition for the theorem is motivated by its translation, through the use of the Einstein field equations, into a condition upon the matter content. The so-called null convergence condition holds by definition if, for every null vector field ua in the manifold,

## 3. Black Holes and Spacetime Singularities: Ultraviolet Effects beyond GR

#### 3.1. Trapping Horizons

When trying to build coherent scenarios in which the collapse of matter is affected by the very quantum nature of matter, the classical picture significantly changes. Robust semiclassical calculations tell us that black holes cannot be absolutely stationary, as they evaporate through the emission of Hawking quanta [9,34]. This is the last piece composing the so-called black-hole thermodynamics: a black hole of mass M has a temperature TH=ℏc38πGMkB.

#### 3.2. Preventing Singularities

These qualitative expectations have been embedded in mathematical frameworks of different nature, essentially since the first explorations in semiclassical gravity [41], and most frequently in the field of cosmology [42]. One of the most popular implementations, probably because of its fundamental flavor, is the one that results from the application of the loop quantum gravity techniques (see [43] for an introduction to the subject). Loop quantum cosmology (see, e.g., [44]) is the result of applying non-perturbative quantization techniques borrowed from the general theory and applied to some highly symmetric cosmological spacetimes. Although the results of this procedure have to be taken with a grain of salt [45,46], these models are regarded as valuable tools in understanding the implications of the wider quantization program of the canonical structure of GR. One of the robust results of this approximation is that near the cosmological singularities, there appear effective forces with a net repulsive effect. These forces are strong enough to overcome the fatal attraction of gravity which would otherwise engender a singularity, provoking the bounce of the matter distribution and connecting a classical contracting cosmology with a classical expanding cosmology [47]; the Big Bang event is identified with the moment of the bounce, so that the expanding branch corresponds to our present universe. The bounce generally takes place when a critical density of matter, of the order of the Planck density, is reached: ρc∼ρP=mPℓ3P=c5ℏG2≃5×1096 kg⋅m−3

In loop quantum cosmology, the bouncing behavior is successfully captured by an effective Friedmann equation which, in the flat (k=0) case, takes the form [48] (a˙a)2=8πG3ρ(1−ρρc)

## 4. Ultraviolet Effects: Phenomenological Considerations

#### 4.1. Horizon Predominance

One could argue that, even accepting that the gravitational collapse process itself could be useless to boost our understanding of the physics near singularities, the Big Bang singularity could be used as a substitute from which to obtain this knowledge. The first observation that comes to mind concerning this assertion is the issue of repeatability: while we expect that numerous processes of gravitational collapse are taking place now around us in the universe, and will be quite surely taking place in the future, in the currently accepted model of the universe there is only one Big Bang event. Also the Big Bang singularity lies in the past so that, while we can hopefully describe its properties in a simple way, it is questionable to what extent we are able to perform experiments in the usual sense of the word, or if the most we can hope to do is constructing some sort of cosmological “archeology”. Leaving aside these issues, which are indeed ubiquitous to the field of cosmology (see [105] for instance), there are good physical reasons to believe that the singularities associated with black holes should be different from the initial cosmological singularity. These arguments are based on the second law of thermodynamics and have been repeatedly exposed by Penrose [25]. While one may think that these singularities correspond from a mechanical point of view as the time-reversal of each other, thermodynamical considerations break this apparent time reversal. On the one hand, if the second law of thermodynamics is to be applicable to the universe, the behavior near the initial singularity is to be associated with a low amount of entropy. On the other hand, the process of gravitational collapse to black holes is generally expected to lead to high entropies. In particular, this should indeed be the case in the standard scenario, in order to fulfill the thermodynamic description of black holes and match the Bekenstein-Hawking entropy formula: S=c3A4Gℏ

#### 4.2. Preponderance of the Singularity Regularization

The possibility of measuring the ultraviolet effects that are responsible for singularity avoidance in reasonable time scales would imply that Hawking radiation would not be able to carry away from the object a significant amount of energy in the meanwhile. Indeed, to make these measurements possible the horizon barrier should be broken in some way so that some signals from the inside may reach an external observer and, as the evaporation process is extremely slow, the net evaporative effect would be negligible. Even if we wait the entire age of the universe (specifically, the Hubble time tH), the energy loss of a stellar-mass black hole would be of the order o10−4ℏc6G2tHM2⊙∼10−11J

## 5. Effective Bounces, Black-Hole to White-Hole Transitions and Shock Waves

It will prove useful to describe the metric using generalized Painlevé-Gullstrand coordinates [117]. These coordinates are adapted to observers attached to the stellar body. We will write the metric as ds2=−A2(t,r)dt2+1B2(t,r)[dr−v(t,r)dt]2+r2dΩ22

#### 5.1. Collapse from Infinity and Homogenous Thin-Layer Transition

To describe the bounce, we will glue two geometries, one corresponding to the Oppenheimer- Snyder collapse of a homogeneous ball of dust, the other one being its time-reversal. Let us consider for the moment the metric outside the star, corresponding to the Schwarzschild solution. The velocity profile v(t,r) presents a flip of sign between the black-hole and white-hole patches of the Kruskal manifold. Thus the metric we are seeking for will be characterized by a velocity profile v(t,r)=vs(r)[1−2θ(t)]=−c[1−2θ(t)]rsr−−√,R(t)≤r≤+∞

This gluing procedure by itself is nothing but a brute force exercise, and the result presents unpleasant features. However, we want to present a constructive procedure in which the details are progressively added on demand, eventually arriving to the complete picture in all its generality. For instance, using the Heaviside function to construct the geometry makes all the t=0 hyperplane singular for r>R0, in the sense that the metric is discontinuous there. Of course, this does not necessarily reflect any physical reality, as we can always introduce a regulator to make the metric continuous. Let us introduce the continuous, differentiable but non-analytic function f(t)={exp(−1/t)00≤tt≤0

Curvature tensors may not be well defined in the strict limit tR→0, as these may contain terms that are quadratic in the Dirac delta function, which are ill-defined (in general, the product of two distributions cannot be defined [118]). Let us emphasize again that we always keep in the back of our minds that the relevant physical situation corresponds to tR small, but nonzero. For tR>0 we can safely insert the metric in the Einstein field Equation (4) to obtain (using, e.g., Equations 3.2 in [117]) the following nonzero components of the effective stress-energy tensor supporting this geometry: T10=−c2π(1−2gtR)g˙tRMr2T11=−c2πg˙tRMr2(rsr)−1/2T22=−c8πg˙tRMr2(rsr)−1/2

#### 5.2. Non-Perturbative Ultraviolet Effects

The curvature scalar corresponding to the effective source (Equation (18)) is given by R=6g˙tRcrsr3

#### 5.3. Collapse from a Finite Radius and Triangular-Shaped Transition

Up to now, we have considered that the initial radius of the stellar structure is infinite. Let us look for the description of the more realistic case of gravitational collapse from a finite radius ri. The resulting metric will be written again in the form of Equation (13). Again, the collapsing and expanding classical phases geometrically represent the gluing of a homogenous internal geometry with an external Schwarzschild spacetime. We follow tightly [117], which presents a careful analysis of how to handle these matchings by using generalized Painlevé-Gullstrand coordinates. They are helpful in describing a collapsing star from a finite radius. The collapse of the star begins at rest at the initial radius ri and is represented by the trajectory of the star’s surface R(t). The arbitrary zero of time is chosen such that the collapse starts at t=−tB/2, where

A new feature in this situation is the necessary introduction of a radial dependence on the parameter tR that controls the interpolation time, which then becomes a function tR(r). This quantity enters through the function gtR(r)(t) in Equation (31). This is necessary to guarantee that curvature invariants are kept finite in the surroundings of the r=ri hypersurface. Had we taken this parameter as being constant, the leading order of the Ricci scalar in the limit r→ri from below would be given, at t=0 for instance, by

**Figure 1.**Transition from the black-hole patch to the white-hole patch by using a smooth interpolation with characteristic time scale tR. The qualitative behavior of light cones for different values of the radial coordinate is shown.

#### 5.4. Short-Lived Trapping Horizons

#### 5.5. Short Transients and the Propagation of Non-Perturbative Ultraviolet Effects

All the transients lead to a characteristic imprint as we have already discussed: the deviation of the near-horizon outer geometry, that is, the Schwarzschild geometry beyond r=rs. Such a feature is anathema in the orthodox view on the possible relevance of ultraviolet effects on classical geometries. It belongs to the conventional wisdom that appreciable deviations from the classical behavior are to be expected in regions of large curvature, when measured in Planck units. The argument is that only then the possible corrections to the Einstein field equations are expected to be non-perturbative. In the Schwarzschild solution the Ricci curvature tensor is zero but the Weyl part of the Riemann curvature tensor leads to the Kretschmann scalar. K=RabcdRabcd=12r2sr6

This is neatly illustrated by taking as working example the geometry considered in Section 5.1, though the situation is generically the same for all the geometries we have considered. The Kretschmann scalar of the transient in this case is given in Equation (26). As argued above, the short (Planckian) time tR associated with the transient implies that the modifications of the spacetime curvature in the surroundings of the stellar structure, when the latter reaches its minimum radius R0, are also of the order of the Planck curvature. Thus in geometries with short transients, and only in these cases, the trigger of non-perturbative ultraviolet effects on the metric outside the star is inextricably tied up to spacetime regions with high curvature. This region of high curvature will propagate outwards, getting diluted in this process: already when reaching the horizon, the magnitude of this curvature several orders of magnitude lower, roughly by (R0rs)3/2∼mPM

## 6. Physical and Observational Consequences

#### 6.1. Towards New Figures of Equilibrium

#### 6.2. Energetics of the Transient Phase

When considering realistic situations with dissipation, the transient phase might leave some traces, for instance in the physics of gamma-ray bursts (see, e.g., [137]). There is experimental evidence that a subset of these events, the so-called longgamma-ray bursts, are associated with the final stages in the life of very massive stars. The most widely accepted theoretical picture is known as the collapsar model [138]. It is natural to expect that a modification of the standard gravitational collapse process to a black hole that is considered here could leave clear imprints associated with a reverberant collapse. However, in the collapsar model of GRBs the emission zone is supposed to be very far from the collapsed core [137]. This means that the connection between the processes at the core and those at the external wind shells could be very far from direct. However, the general features of the model are enough in order to roughly compare its energetics to those of GRBs. This comparison may be used in order to understand whether or not the bounce process is a reasonable candidate for the mechanism behind these bursts. So let us assume that the picture discussed above is realized in nature: the occurrence of violent bouncing processes, dissipation and final stabilization in the form of a black star. We can estimate the effect of the energy loss in the entire process by using the following argument. Recall that for dust matter content, and in the absence of rotation, the differential equation for the trajectory of the surface of the star is mathematically equivalent to that of a test particle with the overall mass of the star following a radial geodesic in Schwarzschild spacetime. We can then use the conserved quantities associated with the geodesic equations in this spacetime. In particular, we will use the conserved quantity E that is associated with energy. So let ri be the initial radius and rs the Schwarzschild radius of the star, and consider the Schwarzschild effective potential for radial motion. If the structure was originally at rest, its energy is given by: (EMc2)2=1−rsri

#### 6.3. Ripples from the Transient Phase

Let us sketch the necessary steps in order to do so, taking for instance the metric described in Section 5.3 as a specific representative of the bounce process. This metric satisfies by construction the equations.

#### 6.4. Recent Detection of Gravitational Waves from Coalescing Black Holes

## 7. Conclusions

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