What are Black Holes made of? | The Essence of the Black Hole
What is the gravitational singularity of Black Holes composed of? String Theorists and Loop Quantum Theorists have similar, but different, answers.
“The black hole teaches us that space can be crumpled like a piece of paper into an infinitesimal dot, that time can be extinguished like a blown-out flame, and that the laws of physics that we regard as ‘sacred,’ as immutable, are anything but.”
John Wheeler
The year was 1915. “The war to end all war” was reaching the ruinous heights necessary to justify its name; Einstein had recently published his groundbreaking theory of General Relativity, a project in formulation for 8 years. Only a month after the theory's publishing, a 42-year-old German Lieutenant with a physics background expeditiously completed Relativity’s first non-trivial solutions. His name was Karl Schwarzschild, a child prodigy with an early interest in astronomy; Schwarzchild had published two papers on the celestial mechanics of double star orbitals in Astronomische Nachrichten in 1890, before the age of 16.
News of the war was met with exultation and enthusiastic nationalistic support. These spirited cheers were a source of inspiration for Schwarzschild, who suspended his Professorial pursuits at the University of Göttingen and began serving the Imperial German Army in 1914, at the age of 41, as an artillery and missile trajectory specialist. Whilst on the Russian front, Schwarzschild got hold of Einstein’s freshly printed publication intent on replacing Newton’s age-old formulations on Gravity.
The paper, Annalen der Physik, presented Einstein’s Field Equations for the first time, which described the oneness of space and time, unified into a 4-dimensional manifold aptly called ‘spacetime.’ Like an elastic membrane, this manifold can bend and curve when under the influence of mass; gravity emerges from these distortions. With an increase in an object’s mass, we observe a proportional increase in local spacetime curvature, and consequently, a higher gravitational force at points around the object. Mathematical objects known as metric tensors capture this structure and act as the basis of General Relativity; in fact, differential geometry (the field which birthed tensors) was the very language Einstein used to express General Relativity.
Schwarzschild’s pen feverishly worked as seared cordite rained from above, and soon he obtained the first analytically exact solutions to Einstein’s Field Equations. Schwarzschild had obtained the structure of spacetime under the influence of a massive, non-rotating, spherically symmetric object: a solution now coined “The Schwarzschild Metric.”
Schwarzchild’s paper was received by Einstein, who responded with the commendation: “I had not expected that one could formulate the exact solution of the problem in such a simple way.” This work provided the theoretical foundation for formulating a yet confirmed astronomical entity: the Black Hole. Using the Schwarzschild metric to test a radius r = 0 (a dimensionless point), one obtains a solution with infinite spacetime curvature and density, a gravitational singularity. To this extent does spacetime become ill-defined, a result expectedly regarded as non-physical by the contemporary scientific community. A spherical surface with r = R, R being the “Schwarzschild Radius” constitutes a boundary called the “Event Horizon,” a partition between a reality well known and one wholly unknown. Induced by this infinitely dense dimensionless point does spacetime curve in likeness to a “Flamm’s paraboloid,” such curvature can be considered conceptually equivalent to the gravitational field.
Further investigation led to the realization that the gravitational singularity derived from Schwarschild’s solutions was, in fact, physically relevant. Considering their implication on reality, accepting this fact is no small task for intuition. Beyond the Event Horizon, light itself can no longer seek a path that leads away from the center of the Black Hole. Events and information beyond this spherical boundary remain inaccessible for all time. Time closer to a Black Hole would tick slower to a faraway observer. This effect, called gravitational time dilation, posits that an object falling towards a Black Hole would seem too slow as it closes in on the Event Horizon. The object would take an infinite time to reach the Event Horizon. Simultaneously, light emitted by the object will appear redder and dimmer, an effect called gravitational redshift. Thus, the falling object fades away until it can no longer be seen as visible light redshifts into nonvisible wavelengths: first infrared, next microwave, and last, a fading radio emission.
If one were to trace Flamm’s Paraboloid down to its minimum, they would reach the singularity. But what is the physical significance of this singularity, this point of infinite density and zero volume? And what is the “thing” from which Black Holes arise? Countless theories attempt an answer. Some border pseudoscience; others approach with substantial experimental evidence bodies. There are two competing theories in this latter category, String Theory, and Loop Quantum Gravity — both have garnered much consideration from professionals.
Momentarily, let us take a detour and outline the underpinnings of these theories before investigating their description of the interior of Black Holes.
String Theory (ST) and Loop Quantum Gravity (LQG) seek to complete our understanding of reality by resolving the incompatibilities between the theory of the macroscopic Universe: General Relativity and the microscopic Universe: Quantum Mechanics. Fundamentally, between relativity and quantum mechanics, there is something that grates. As currently formulated, both cannot be true before contradictions arise. The gravitational field was written without considering quantum mechanics, without bearing in mind that fields exist as quantum fields. Quantum mechanics was written with no consideration for relativity, that spacetime bends, and curves, as Einstein’s equations described.
To resolve the incompatibility, ST states that the fundamental particles composing our reality, such as electrons, photons, neutrinos, quarks, and gluons, exist not as particles but as incomprehensibly small (millionth of a billionth of a billionth of a billionth of a centimeter long) 1-dimensional vibrating strings. These strings have tensions, and their energy level determines the pattern and frequency of their vibrations. The properties of these particles, such as their charge, mass, and spin, are determined by the qualities (pattern/frequency) of the string’s vibration.
ST was theoretically expanded to Superstring Theory and now falls under an even larger umbrella called M-Theory. To most, the M in M-Theory stands for ‘mystery’, as the theory proposes the existence of a currently unknown, unifying web of relations that connect the concepts of Supergravity and Superstring Theory. As it pertains to the interests of this article, Layman’s summation goes: our Universe contains 10 dimensions, 4 can be accounted for from common experience (3 spatial and 1 time, Minkowski space), and the 6 additional dimensions arise from 6-dimensional Calabi-Yau manifolds, rather than 1-dimensional strings; variations in the vibrational quality of these manifolds provides us with the array of diverse particles our Universe currently entertains. These extra dimensions are necessary for ST to describe our Universe in a mathematically self-consistent way; in a way that agrees with experimental measurements and observations of phenomena from the quantum to the cosmological level.
ST is hailed for its elegance and beauty: our most irreducible reality harmonizes, resonates, and plays a cosmic tune that echoes now and forever into eternity. Particle collisions are considered to be stochastic symphonies, life itself, even sentience, a methodized scherzo. But it is not without criticisms. Many experimentalists and theorists alike have expressed their dissatisfaction with ST due to propositions that violate Popper's falsifiability. Because of this, some have gone as far as to question the scientific status of ST, with the common argument: that which is unfalsifiable is unscientific and no different than the claims of the ambitious street preacher. Regardless, String Theorists continue onwards and chip away at manifolds, branes, and the meaning of it all; we too will move on.
Whereas ST attempts to bring our entire understanding of reality together, Loop Quantum Gravity’s main goal is to provide a sensible theory of quantum gravity; that is, its aim is to solely resolve the incompatibilities which exist between General Relativity and Quantum Mechanics. Simply put, LQG states that the fabric of spacetime is composed of quantized geometries. It takes the pseudo-Riemannian syntax of General Relativity and replaces it with a quantum Riemannian geometry. Space itself can be imagined as a beach, macroscopically smooth, with rolling dunes, but composed of small grains of sand, and spin foam, as its individual units. It is background independent, meaning, its formulation does not require space to exist, and it makes no preliminary assumptions on the nature of space. The discretized nature of reality proposed by LQG results in lengths no longer than 10 trillionths of a trillionth of a trillionth of a meter (decimal with 35 zeros), and the shortest time no shorter than 100 quadrillionths of a quadrillionth of a quadrillionth of a second(decimal with 43 zeros). In the LQG framework, it is impossible to assume a space less than said length, and impossible for a duration to be shorter than said time. In these ways, spacetime continuity exists as an emergent phenomenon as opposed to a fundamental one.
Below is my attempt at depicting the theory: Intersections of links (red lines) are indicated by nodes (black dots), and form discrete loops of space (blue polygons). Information about the size and shape of space is contained within this network: one can come to understand the volume and area of space through an evaluation of the number of intersections (nodes), and the number of threads (links) contained in quanta of space. Such discretization is elegant indeed. The spin network shown below is composed of loops that, in summation, act as the constituents of gravity itself.
ST is broader in scope: an attempt at a grand unification between relativity, the standard model, and the fundamental forces, with the multi-dimensional Universe acting as its toolkit. On the other hand, LQG is really attempting just one thing, a well-defined description of quantum gravity. Neither has been experimentally verified or ruled out, so both continue to be worked on and advanced by theoretical practitioners across the globe. It is important to note that these are just two candidates among many others who all hope to define the undefined. Emergent gravity (gravity as an emergent thermodynamic phenomenon), causal sets, causal dynamical triangulations, and a most bizarre investigation into a holographic Universe (which was defined in the string theory frame by Leonard Susskind) are all workable Theory of Everything candidates.
With this rough and ready understanding of ST and LQG, we can delineate their respective claims regarding the essence of Black Holes.
Black Holes present two major physical dilemmas: the Information Paradox, and the singularity. To understand where ST and LQG are coming from in how they describe the Black Hole interior, we must first evaluate each of these dilemmas and then see how ST and LQG attempt to resolve them.
The Information Paradox
To no surprise, matter that falls into a Black Hole will add to the Black Hole’s mass. Although irretrievable beyond the Event Horizon, this matter, or more generally, information, enters the Black Hole and still persists within. Before Stephen Hawking, it was thought that Black Holes would only grow: they would swallow up the matter, increase in size, and continue on indefinitely. Hawking came to reveal that, in reality, Black Holes radiate, and gradually evaporate, to nonexistence as antiparticle—particle interactions on their boundaries excrete their mass over time. Hawking found that the quality of this radiation depended solely on the mass of the Black Hole, independent of the contents being swallowed.
When you eat a piece of bread, enzymes reduce the complex array of carbohydrates into their fundamental constituents. In essence, molecular information about the bread, even after enzymatic degradation, simply morphs and changes; it does not disappear (adhering to the conservation of mass). As the bread reaches one's bowel, it no longer exists in its usual form; but there was a determined, reversible path that rearranged the ‘bread’ information. Thanks to this reversibility, one could, in theory, follow the breadcrumb trail, and walk backward to obtain the original piece of bread. The information was changed, not destroyed, and is still tangibly linked to its original state. Adding more technicality to this analogy: if we were to follow a carbon atom in glucose throughout this process, there would still be quantum information about the carbon's past.
A Black Hole eats bread differently; the information crosses the event horizon and is permanently lost. Inside, it is ripped apart at the quantum level and contributes to the arbitrary energy and mass structure of the Black Hole. This would not be a problem if the Black Hole persisted forever and was a one-way door, as one could say that the information was preserved in the mass structure. But Black Holes decay, and during that decay, they destroy any trace of the bread’s existence. The bread information is released in the form of dissipative particles, which have no traceable origination beyond their emission on the event horizon. Black Holes violate a fundamental tenet: that quantum information cannot be lost; a problem that Hawking committed the rest of his life trying to resolve.
The Singularity
As discussed, Einstein’s Field Equations indicate that the center of the Black Hole is a gravitational singularity, a point of infinite spacetime curvature, zero volume, and infinite density. A Black Hole tears apart the fabric of reality, spacetime, such that ‘where’ and ‘when’ become ill-defined and inapplicable. Throughout the history of physics, mathematical singularities which arise in physical theories have acted as indications of unfinished theory and led to scientific breakthroughs upon resolution. Such was the case for the Rayleigh-Jeans ultraviolet catastrophe, a singularity that birthed quantum mechanics when it was resolved by Max Planck.
What then, does contemporary theory say? Here we will evaluate the resolutions posited by the String Theorist, and another by the Loop Quantum Theorist.
String Theory places a boundary condition on the smallness. In ST, Black Holes are, like everything else in the Universe, made of a ball of strings of finite size. More specifically, Black Holes are specific variations of these strings, 6-dimensional Calabi-Yau manifolds. Since these are discrete objects with dimension, finite density, and finite volume, the singularity problem is solved. If this resolution seems unsatisfying or unbelievable, consider a neutron star, an incredibly dense bundle of neutrons, an object composed of degenerate matter. Neutron stars form from collapsing stars, just as Black Holes do. Black Holes are stars whose mass surpasses a threshold; these stars collapse a little further than those that result in neutron stars and form a more extreme version of degenerate matter. The degenerate matter composing neutron stars are neutrons, but for Black Holes compressed neutrons are not compressed enough, and so they squeeze until they reach the very fundamental building block at which they can squeeze no further: the string, or more precisely, the Calabi-Yau manifold.
Physicists aptly named this thing a fuzzball, not because the 6-dimensional manifold looked cute, but because the event horizon of the manifold would have a misty “fuzz” to it, rather than a smooth surface. But what is a 6-dimensional manifold?
Manifolds are frequently studied in topology and are defined as topological spaces which look like Euclidean spaces locally. A manifold of dimension “n” is said to have homeomorphism with the Euclidean space of dimension “n.” For example, one-dimensional manifolds include the familiar line, circle, and even open-ended curves which extend outwards to infinity. If you were to zoom in on any one of these curves to a point of infinitesimal size, they would locally look like a straight line or 1-dimensional Euclidean space. These regions of Euclidean similarity are neighborhoods. 2-dimensional manifolds include the sphere, the torus, the double torus, the Klein bottle, and so on. In this case, you are looking for 2-dimensional Euclidean space, that is, a plane, such as the familiar x-y plane. All of those objects increasingly resemble planes as you approach their surfaces, mathematically we say: their curvature approaches zero. With regards to manifolds, one can stretch their imagination as far as possible, and still be limited to understanding 3-dimensional manifolds.
LQG expectedly prescribes a quantum discreteness to the interior structure of the Black Hole, thereby foregoing the existence of a singularity. This is similar in form to String Theory. A particular agreement has formed according to which the Black Hole singularity be replaced by a non-singular spacetime structure holding a density and curvature that is not infinite but large.
Recent LQG proposals on the substructure of Black Holes have been contentious, at least in comparison to ST’s fuzzball theory which is widely agreed upon among String Theorists. For example, a speculative LQG idea proposes that when the energy density of Black Hole matter reaches the Planckian density, quantum gravity begins presenting a repulsive force so strong as to result in a ‘bounce back’ as referred to above. In this way, the end of Black Holes constitutes the start of White Holes, namely the time reverse of a Black Hole, in which matter is violently expelled from a spherically symmetric White Horizon. This event is called a “Black Hole to White Hole Transition” or the “Quantum Transition” by LQG practitioners. The quantum component of this “Quantum Transition” is rooted in the quantum tunneling event that must occur to ensure Black Holes evolve into White Holes. Quantum tunneling consists of a “hop” across an energetic barrier in a nonclassical way.
To reiterate, the solutions posed by ST and LQG are akin in some ways. Both propose very small structures at the ‘center’ of Black Holes. Where they differ is in the composition of these structures. For ST, it is an ensemble of energetic, higher-dimensional manifolds with exterior fluctuations so rapid they appear fuzzy. For LQG, it is an incomprehensibly dense and small quanta of space containing degeneracy corresponding to the energy and mass of the Black Hole.
An Aside on Singularities and Tensors
The term “singularity” is a mathematical one: a point at which a mathematical object, such as a function, becomes undefined or pathological (undifferentiable or nonanalytic). Take, for example, the function:
As we approach x = 0 (x values are listed on the horizontal axis, per tradition), the function explodes to +∞, as we are dividing 1 by numbers that approach 0. The Schwarzschild Metric contains a term that behaves similarly, so as we approach a radius of 0 for our massive object (r = 0 case discussed above), we generate a point of infinite density (nonzero mass / zero volume).
The concept of tensors is attributable to the German mathematician Bernhard Riemann. In fact, much of mathematical physics is founded on Riemannian geometry, a subsection of differential geometry. Differential geometry, in short, involves the mathematical analysis of curves and surfaces. In General Relativity, the metric tensor can be thought of as conceptually equivalent to the gravitational potential invoked in the Newtonian theory of gravity.
In Cartesian (or Euclidean n-space), the metric tensor can be expressed as the Kronecker delta function, so we only keep terms that have indices mu= nu, in which the metric tensor = 1.
For those with an understanding of calculus, the metric tensor may be easier to understand as follows:
Imagine if one were to perform these double integrals on an object, say a disk; the disk would define our bounds and specify our functions f(x,y) and f(r,θ). We would expect the result from these integrals to be invariant with regards to any change in coordinate systems; after all, it is the same disk. You may notice that, highlighted in red, is a multiplicative factor, r, which exists in the Polar case but not in the Cartesian: this factor corresponds to the metric tensor, thus, we see more explicitly that the tensor is a measurement of transformations to geometric entities undergoing changes in their coordinate basis. In relativity, the metric tensor (referred to as the ‘metric’) describes the local curvature of spacetime, and fits into Einstein’s field equations as the ‘little g’ term:
In this context, the metric tensor informs of the local curvature of spacetime imparted by a massive object, and, subsequently, the local causal structure of spacetime. Overall, the field equations form complicated partial differential equations, with exact solutions few and far between. It never fails to amaze me that discovering the most mysterious objects in our sky came from a consideration of the simplest exact solution one can obtain.
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