### Black Holes, the Information Paradox, and Spaghettification

Featured in the first image ever of a black hole is M87*, the supermassive black hole at the center of the galaxy Messier 87. Credit: Event Horizon Telescope via Wikimedia Commons. |

Recently, I stumbled upon an article entitled, “There Is One Way Humans Could 'Safely' Enter a Black Hole, Physicists Say” by ScienceAlert; the article, which was written by two physicists, explained how spaghettification is less pronounced in supermassive black holes due to their size. The extremes of gravity, density and mass, black holes are ultra-dense objects that stupify all who attempt to understand them. In April 2019, astronomers from 20 different countries utilized their radio telescopes to image the black hole in the center of Messier 87, the largest object in the Messier catalog. The consortium utilized many of the world’s most advanced telescopes, atomic clocks, and supercomputers, and several new techniques in image acquisition, to acquire the image, present at the beginning of this entry. M87* itself is the round, dark area in the center of the image, and the surrounding region is the black hole’s accretion disk. This image is certainly one of the most important images ever taken, as it is also our first direct observation of a black hole itself. Let us, today, consider black holes.

A rocky start

The black hole was first introduced by the English astronomer John Michell in 1784, when he postulated that certain objects could exist with such extreme gravity that even light could not escape it. He argued that an object that has the same density as the sun but with five-hundred times the diameter of the sun would require a surface escape velocity faster than that of the then-accepted value for the speed of light. Michell’s equations were simplistic, highly theoretical and could not be extrapolated to in nature. Later, as Thomas Young discovered that light is also a wave, enthusiasm dampened for Mitchell’s hypothesis, as the effects of gravity on light waves were unknown at the time. Even so, Michell was the first to propose that a large body like this can be observed through their gravitational effects on surrounding objects.

Over a century later, Albert Einstein proved that gravity does affect photons, and later developed his theory of general relativity; the equations of general relativity–known as Einstein’s field equations–describe gravitational fields of a point mass (a dimensionless point of mass that exerts a gravitational force)** and spherical masses.

Later, an odd behavior was discovered at a set of values known as the Schwarzschild radii. For any mass, there was a certain value for r (the radius) where some values in the field equations became infinite. Such an anomaly led to the discovery that aggregate densities of the largest stars must be much less than the that of the sun (thereby rendering Michell’s postulations to be false), for similar densities would lead to core collapse–and, depending on the star’s original mass, collapse straight to a black hole.

After this set of radii was discovered, the physicist Subrahmanyan Chandrasekhar used special relativity to postulate that a nonrotating body of electron-degenerate matter–a fermionic gas whose electrons are stripped from the core of the atom, and exist in an odd, superdense goo of electrons–could not have a mass above 1.4 solar masses without yielding unstable solutions. It appeared then that such an object would ultimately become a black hole.

Chandrasekhar, as brilliant as he was, was not correct: an example of a nonrotating body of electron-degenerate matter is a white dwarf; a white dwarf that reaches the Chandrasekhar limit will become a neutron star, not a black hole. A revision to the Chandrasekhar limit was made when three physicists discovered the Tolman–Oppenheimer–Volkoff limit, which established an upper limit to the mass of a cold, non-rotating neutron star. The new limit is now approximately 2.2 solar masses. Soon after, physicists identified the Schwarzschild surface of the Schwarzschild radius as the event horizon of the black hole, the boundary at which nothing, not even light, can escape–and at which time “stops” according to outside observers.

A cohesive theory on black holes began to take shape in the 1960s; before this period, black holes and neutron stars were theoretical anomalies, but after the discovery of the first pulsar, black holes entered the mainstream. As this happened, Stephen Hawking, as well as other physicists, investigated the thermodynamics of black holes, and later discovered Hawking radiation–a constant yet agonizingly slow form of blackbody radiation that black holes emit. Soon, black holes had been indirectly observed many times, and were known almost certainly to exist; it was not until 2019, however, that a black hole was directly imaged, and therefore directly observed for the first time.

The physics of black holes

As the laws of physics break down as one closes in on the Big Bang, so does physics break down at the event horizon of a black hole, particularly as one approaches the singularity, that arbitrarily small “point” that occupies the center of mass of the black hole. Because physics breaks down within black holes, many wacky things happen within and around them.

Black holes are regions of spacetime whose gravity is so strong that nothing–not even light–can escape them. Stable black holes consist of only three physical properties: mass, charge, and angular momentum.***

Black holes absorb information essentially forever. Upon crossing the event horizon, escape is impossible. The strength of the gravitational field is exactly 299,792,458 N/kg, equivalent to the dimensionless magnitude of c.

In addition to a typical black hole, there are also static, also known as Schwarzschild black holes. Schwarzschild black holes lack two of the three defining characteristics of rotating black holes: charge and angular momentum. As such, its only physical characteristic is mass. Because angular velocity causes ellipticity in rotating black holes, the lack of such angular velocity in a Schwarzschild black hole means that it is perfectly spherical. The Schwarzschild is the only known object in the universe that is truly spherical.

At the center of mass of the black hole is a dimensionless point known as the singularity, where the curve in space time becomes infinite.**** We have all seen the images of Earth bending spacetime; the more massive the object, the greater the curvature of spacetime. At the point of the singularity, the curvature of spacetime is infinite (the gravitational field, g, has an infinite value), and cannot be measured. As such, the singularity can be said to have infinite mass and density. The singularity, an infinitesimal dimensionless point with an immensely large mass, has a density that converges to infinity as the units used to define density grow smaller.

Surrounding rotating black holes is the ergosphere, the region near the event horizon of a black hole. General relativity states that a rotating body will always drag the spacetime surrounding it, forcing unmoving bodies to move. Near the event horizon of a black hole, the drag of spacetime is so pronounced that an object must be traveling faster than the speed of light in the opposite direction of the black hole’s rotation in order to remain stationary. As such, movement is essentially a requirement.

The many quirks and oddities of black holes

Within black hole astrophysics there are, unsurprisingly, many oddities and paradoxes. The information paradox is an apparent mathematical puzzle that results when general relativity and quantum mechanics are combined in black hole astrophysics. Mathematical results suggest that information is permanently lost upon entering the event horizon of the black hole, yet that idea violates the Schrodinger wave equation, which determines that the value of a quantum system at any particular time should determine its value at any other time. This contradiction existed for decades until a unified theory of gravity, string theory, was first hypothesized. String theory argues that all particles are one-dimensional, string-like structures that vibrate; these vibrations determine properties of the particles, such as mass or charge. String theory helped determine that information is, indeed, conserved upon entering a black hole. The new explanation does now exist independently of string theory (string theory, while the leading unified theory of gravity, is heavily disputed, and even viewed in many circles as unfalsifiable). Information that enters a black hole will eventually leave it, and as the black hole gets older, the exodus accelerates.

In addition to the information paradox, another concept–spaghettification–is the center of much speculation, including whether it is possible to survive within a black hole or not. Black holes with lower masses are very small (the event horizon of a black hole with mass equivalent to that of the sun would have a radius of only two miles). Being so close to the center of gravity, the gravity at your feet would be one trillion times greater than the gravity at your head if you were falling feet first. As a result, you would be spaghettified into a mere strand of particles. In a supermassive black hole, upon entering the event horizon, you would be significantly further from the center of gravity, the singularity, and as a result, the discrepancy between the gravitational field at your feet and at your head would be far less pronounced. As such, the spaghettification effect, too, would be less pronounced in supermassive black holes. You’ll probably learn quite a lot about black holes down there, but don’t count on being able to share that knowledge.

Wrapping it up

In many ways, black holes are the future: they will last far longer than the oldest stars and galaxies, and will be the final objects to die off. When the final black holes die out, the final aggregations will die out: our universe will again become a desolate abyss of darkness and nothingness. Are you excited? As always, take care and stay curious, everyone.

* Michell’s equations refer to escape velocity, not the strength of the gravitational field; current models suggest that no object other than a black hole can have values for escape velocity greater than the speed of light, and even the most extreme objects in the universe other than black holes–neutron stars–have escape velocity values around eighty percent of the speed of light. Michell’s equations were correct, as according to my calculations the value on the surface of such an object would be 3.08108 m/s, slightly greater than the speed of light. Its gravitational field would, indeed, be much stronger than that on the surface of the sun; per my Newtonian calculations, the gravitational field strength is approximately 1.37105 N/kg, but that is still approximately 2200 times less than the maximum possible gravitational field strength of 2.998108 N/kg.

** A point mass is a type of “idealization” in physics, whereby an object is assumed to exist under certain constraints in order to simplify complicated concepts.

*** A Schwarzschild black hole (which is covered later in the entry) has only one physical property: mass. As a Schwarzschild does not spin, it has no angular momentum, and it also lacks a charge. Such black holes result from an exact solution to the field equations.

**** Technically, the gravitational field strength of any massive object approaches infinity as the radius infinitesimally approaches zero.

If you have any questions, comments, or corrections, please comment on this post or email learningbywilliam@gmail.com with your concerns. Thank you.

References

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Rodriguez, L., & Rodriguez, S. (2021, February 3). There Is One Way Humans Could 'Safely' Enter a Black Hole, Physicists Say. ScienceAlert. Retrieved January 21, 2023, from https://www.sciencealert.com/what-would-happen-if-a-human-tried-to-enter-a-black-hole-to-study-it

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Sutter, P. (2019, June 6). What Is the Black Hole Information Paradox? Space.com. Retrieved January 21, 2023, from https://www.space.com/black-hole-information-paradox-mystery.html

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