Dark Matter (TV series), Superposition, and Parallel Universes

Posted on June 30, 2024 by Sung Lee

There is a new TV series titled Dark Matter on Apple TV+. It is based on a novel of the same title by Blake Crouch. I have not read the book yet but it is on my reading list. The show’s basic premise is the following: Jason Dessen was once a promising physicist but chose to have a family with love of his life, Daniela over his ambition in quantum physics research 15 years ago. Now he is teaching physics at a college and living an ordinary and peaceful (but somewhat boring) life with his wife Daniela and son Charlie.  One night, on his way back home from a bar, he was kidnapped and was thrown into a world that he doesn’t recognize…

The show is quite entertaining. Jason has to go through some (mostly unpleasant) adventures, which reminded me of the 1990’s Sci-Fi TV series  Sliders, until he figures out how to control superposition and to select the reality that he wants. (Wouldn’t this be the ultimate dream of all quantum computing scientists?) Contrary to the title, the show’s plots do not have anything to do with dark matter, but the main scientific idea of the show appears to be based on Hugh Everett’s many-worlds interpretation of quantum mechanics just like another recent TV series on Apple TV+, Constellation which I wrote about here. The show Constellation made less physical sense. For example, there was a lack of explanation why two particular realities matter among infinitely many choices of realities as superposition or how an observer experience two different realities against the collapse of wave function. On the other hand, in the show Dark Matter, at least some ideas (brilliant, though not so really scientific or logical) were introduced in an attempt to address the observer effect (collapse of wave function) or a manifestation of superposition.

In here, besides the measurement issue (which is critically important for a viable physical theory) with many world interpretation, I also mentioned that extending the wave nature of matter (consequently, the probabilistic nature of quantum mechanics) to the macroscopic world or to the entire universe is too far-fetched to be even remotely true. I am going to elaborate more on this. But first,

What is a Superposition?

A superposition is actually a mathematical concept. While a superposition is a common terminology in physics, in mathematics, it is usually called a linear combination. To explain a linear combination, I will have to start from vectors. As you learned in high school physics, a vector is a quantity that has both magnitude and direction. So, vectors are often represented by directed line segments i.e. arrows. On the other hand, a quantity that has only magnitude is called a scalar. (A scalar is nothing other than a number.) You also remember that one can add two vectors using a parallelogram or equivalently a triangle.  One can also multiply a vector by a scalar. It’s called scalar multiplication. One can stretch or shrink a vector, or reverse the direction of a vector by scalar multiplication. (If you want to brush up about vectors, see here.) It all made sense until higher dimensional spaces came in. For instance, Einstein’s theory of special relativity taught us that the world is not 3-dimensional (as in Newtonian physics) but actually 4-dimensional, called spacetime. Unfortunately, for being 3-dimensional beings our perception is limited to 3-dimensions, so the classical notion of a vector is not adequate to study mathematics or physics in higher dimensions. It turns out, as an alternative and a general definition of a vector, an $n$-dimensional vector can be defined by an $n$-tuple $(a_1, a_2,\cdots, a_n)$. (For details, see here.) This certainly provides a nice framework for studying mathematics and physics in higher dimensional spaces. However, as we advance our knowledge in mathematics and physics, we realized that such a general definition of a vector is still not adequate to deal with new mathematical objects arising in mathematics and physics. This time, mathematicians got smarter. They chose not to define an individual vector. Noting that the set of classical vectors with vector addition and scalar multiplication satisfies certain properties associated with the operations, they defined a vector space by any set with addition and scalar multiplication satisfying those properties as axioms. See here for details on the definition of a vector space. Once we have a vector space, any element belonging to the vector space is called a vector. This allows us to identify certain objects arising not only in mathematics but also in physics (for example, quantum mechanics) and in engineering (for example, signal processing) as vectors. From classical sense, it is inconceivable to see them as vectors as we will see later. Let $V$ be a vector space. Let $v_1, v_2\in V$, i.e. $v_1, v_2$ be vectors and $c_1, c_2$ scalars. Since $V$ is closed under vector addition and scalar multiplication, we see that $c_1v_1+c_2v_2\in V$. The expression $c_1v_1+c_2v_2$ is called a linear combination or a superposition of two vectors $v_1$ and $v_2$.

By the way, I must admit that what I described above may not be (or more likely, is probably not) how the notion of a vector has evolved in mathematics. In other words, I made things up to emphasize the need for an evolution of the notion of a vector in accordance with the progress of modern physics. So, what is the real story? I don’t know. Regrettably, I am not so privy to history of vectors. But it’s not really important here.

Ok, Then How Does a Superposition Come into Quantum Mechanics?

A quick answer is the famous Schrödinger equation $$i\hbar \frac{\partial\psi(\vec{x},t)}{\partial t}=H\psi(\vec{x},t)$$ where $H=-\frac{\hbar^2}{2m}\nabla^2+V$ is a Hamiltonian. If you have no idea what those symbols mean, that’s okay. You don’t actually have to know them. All you have to know is that Schrödinger equation tells us how particles such as electrons behave and mathematically, it is a second-order linear partial differential equation. It is quite mouthful but the most important keyword here is linear. What that means is that if $\psi_1$ and $\psi_2$ are solutions of Schrödinger equation, so is $c_1\psi_1+c_2\psi_2$, i.e. a superposition of $\psi_1$ and $\psi_2$. This means that the set of all solutions of Schrödinger equation form a vector space! Furthermore, it becomes in general an infinite dimensional vector space called a Hilbert space. So, in quantum mechanics, functions $\psi$ as solutions of Schrödinger equation are vectors. Before continue, what is $\psi$? $\psi$ is so called a state function or a wave function. It represents a certain physical state of a particle, hence called a state function. But then why is it also called a wave function? Because it is literally a (complex-valued) wave. You remember hearing about the wave-particle duality of matter even in high school physics, right? Just like light, particles such as electrons also act like waves. So, physically it makes sense to use a wave function to represent a state of a particle but it is also easier to use a wave function to model quantum mechanics. I don’t mean to delve too much into quantum mechanics here as this article is not intended to be a lecture note on quantum mechanics. But I would like to show you how a wave function naturally entered into building quantum mechanics. First, the following complex-valued plane wave \begin{equation}\label{eq:debroglie}\psi(x,t)=Ae^{i(kx-\omega t)}\end{equation} was well-known to physicists even long before the birth of quantum mechanics. For the sake of simplicity, I am considering only one-dimensional case. The plane wave has quantities that characterize a wave: the wave number $k$ and the frequency $\omega$. Reasonably enough to guess, physicists might have thought of using the wave function to model a particle like an electron. A lot of time, modeling means writing an ordinary or a partial differential equation that describes a physical phenomenon. Differentiating \eqref{eq:debroglie} with respect to $t$ yields $$i\hbar\frac{\partial\psi}{\partial t}=\hbar\omega\psi=E\psi$$ and differentiating \eqref{eq:debroglie} twice with respect to $x$ yields $$-\frac{\hbar^2}{2m}\frac{\partial^2\psi}{\partial x^2}=\frac{\hbar^2 k^2}{2m}\psi=\frac{p^2}{2m}\psi$$ For a free particle, kinetic energy is the only energy involved, hence we obtain free Schrödinger equation $$i\hbar\frac{\partial\psi}{\partial t}=-\hbar^2\frac{\partial^2\psi}{\partial x^2}$$  With the dispersion relation $\omega=ck$ for light, one can also easily show that \eqref{eq:debroglie} satisfies the wave equation $$-\frac{1}{c^2}\frac{\partial^2\psi}{\partial t^2}+\frac{\partial^2\psi}{\partial x^2}=0$$ A word of caution here. What I said above is in no way a historical account of how quantum mechanics was actually formulated. I am not privy to history of quantum mechanics either, so I don’t know. I am trying to show what one could have undertaken to mathematically model quantum mechanics in the beginning from an already well-known wave function in electromagnetism. \eqref{eq:debroglie} resembles circular water waves from a single source but apparently that is not how free particles behave, so the plane wave itself is not a mathematical manifestation of a free particle. How do we then interpret the wave function and relate it to a (free) particle? Physicists interpret it as something that represents a physical state of a particle. There can be many different physical states such as bound state and spin state, etc. depending on Hamiltonians. Since wave function itself is not observable (not something we can physically measure, i.e. numbers), it has to be connected to an observable. In general, a superposition state $\psi$ is given by $$\psi=\sum_{i=1}^\infty c_i\psi_i=c_1\psi_1+c_2\psi_2+\cdots$$ where $\psi_i$, $i=1,2,3,\cdots$ are eigenstates. Whenever a measurement is done, $\psi$ collapses to an eigenstate $\psi_i$. But again, we don’t really measure an eigenstate $\psi_i$ but its corresponding eigenvalue (energy) $E_i$ (they satisfy the Schrödinger equation $H\psi_i=E_i\psi_i$). Each eigenstate $\psi_i$ can be normalized, meaning it can be made into satisfying $|\psi_i|^2=\bar\psi_i\psi_i=1$. The quantity $$\mathrm{Pr}(\psi=\psi_i)=\frac{|c_i|^2}{\sum_{i=1}^\infty|c_i|^2}$$ in interpreted as the probability of the superposition $\psi$ being the eigenstate $\psi_i$.

Schrödinger’s Cat

Schrödinger’s cat is a thought experiment used by physicists to explain quantum superposition. Regrettably, often by misunderstanding or on purpose laypersons are misled to believe that quantum superposition is something so out of the world phenomenon. Here is what the thought experiment is about.

Consider a sealed box containing a cat, a flask of poison, a radioactive substance and a Geiger counter.

If the Geiger counter detects a radiation due to the radioactive substance decaying, the flask will be shattered, releasing toxic gas which will kill the cat. Over the course of time, the cat is simultaneously alive and dead. Once you open the box, the state of the cat collapses to either alive or dead.

Here are some problems with the conventional interpretation of Schrödinger’s cat. First, the state of a cat is not a quantum state. So applying the superposition principle of quantum state to the state of a cat is just far-fetched. Second, the cat being simultaneously alive and dead is misleading. Although we can’t observe the cat until we open the box, she is either alive or dead at any moment in time. The cat will never be half-alive and half-dead. The correct description of what’s going on is that the probability of the cat being alive (or dead) is 50% at a certain moment in time. So there is really no mystery after all regarding quantum superposition contrary to what has been portrayed in physics literature and Sci-Fi.

There Is No Universal Wavefunction

To recap, there is quantum superposition due to the wave nature of matter and the fact that the equation of motion (Schrödinger equation) for such matter waves is linear. There is no wave function for the state of the poor cat inside a sealed box and she will never be in any quantum superposition. Likewise, there is no physical justification or reason why there should be universal wavefunction, i.e. the wave function for the whole universe. Hugh Everett’s many-worlds interpretation is based on the premise (existence of universal wavefunction) that is purely speculative. This means that whatever box Jason builds he will never experience quantum superposition of the world i.e. infinite possibilities of realities. To his greatest disappointment, nothing will happen in the box. Regardless, if you don’t think about real physics, I believe you will enjoy the show.

So Does This Mean There Are No Parallel Universes?

No, many-worlds interpretation of quantum mechanics is not the only thing that implies the existence of parallel universes. For example, parallel universes can exist as a consequence of traveling back in time. Is traveling back in time possible? Yes it is theoretically possible to travel back in time using a wormhole as seen in many recent studies by physicists. Besides technical possibilities of traveling back in time, the biggest problem with traveling back in time is causality violation such as the grandfather paradox. The grandfather paradox exists based on the assumption that time is one-dimensional. If time is multi-dimensional, we no longer have grandfather paradox! How time can be multi-dimensional? It can happen due to a Chronology Protection Conjecture (CPC). As for time travel, there are currently four different CPCs.

  • The Radical Rewrite Conjecture: One can travel forward and back in time and rewrite the history. (All Hell breaks loose.)
  • Novikov’s Consistency Conjecture: The Universe is consistent, so whatever temporal transitions and trips one undertakes, events must conspire in such a way that overall result is consistent.
  • Hawking’s CPC: The cosmos works in such a way that time travel is completely and utterly forbidden. This conjecture permits spacewarps/wormholes but forbids timewarps/time machines.
  • The Boring Physics Conjecture: There are no wormholes and/or spacewarps. There are no time machines/timewaprs.

I am personally fond of Novikov’s consistency conjecture but I put forth another CPC which is a stronger version of Novikov’s consistency conjecture.

Protect History At All Cost!

Here is Lee’s CPC: It does permit timewarps/time machines but does not permit any violation of causality and any tempering with already recorded history.

A consequence of this CPC is that as soon as any tempering with recorded history occurs, a new timeline must be created in order to prevent history from being rewritten, i.e. a new reality is created separately from the previously existing reality. This new reality shares exactly the same history up until the moment a tempering with history was occurred. From that moment onward, it may turn out to be pretty similar to the previous reality or it may turn out to be completely different.

I am currently writing a book on time travel and more details about this will be discussed there.

Posted in Quantum Mechanics, Sci-Fi | Leave a comment

Three-Body, The Wandering Earth, and Is Starship Earth Possible?

Posted on June 16, 2024 by Sung Lee

There is a TV series called 3 Body Problem on Netflix. It is based on the first novel of the trilogy Remembrance of Earth’s past (the original Chinese title is 地球往事 whose literal translation is Earth’s past) by a Chinese Sci-Fi writer Liu Cixin. The first novel’s Chinese title is 三体 meaning three-body and its English title is The Three-Body Problem. The storyline of 3 Body Problem is interesting and I am planning on reading the trilogy in the near future (when I find time). I learned from my daughter that the English translation of the trilogy won a Hugo Award for Best Novel and it was the first novel by an Asian author to win a Hugo Award. The show’s storyline involves search effort of an alien civilization (like that of SETI) by the Chinese government during the time of Mao Zedong’s Cultural Revolution and a highly advanced but dying alien civilization in a triple star system due to the instability of their gravitationally bounded three stars. Is there actually a triple star system? In the Chinese TV series with the same title 三体 as that of the novel, it appears to indicate that the three star system is Alpha Centauri. Indeed, Alpha Centauri is a well-known triple star system. But in Alpha Centauri, only two of the three stars, Rigil Kentaurus (α Centuri A) and Toilman (α Centuri B) are gravitationally bound and form a binary star system. Due to their proximity to each other they look like a single star from the Earth. The other star is Proxima Centauri (α Centauri C). It is a red dwarf and is the closest star to the Sun (about 4.2 light-years). Proxima Centauri has two known planets. One of them, Proxima b is orbiting within Proxima Centrauri’s habitable zone. Also due to the closest distance from us, although its actual habitability is uncertain, Proxima b may be a good realistic candidate for Earth 2 if humans ever have to migrate themselves to another place outside of our solar system. There is also a known triple star system in which all three stars are gravitationally bound like the one described in the show 3 Body Problem or in the Chinese one 三体. It is called EZ Aquarii which is located 11.1 light-years away from the Sun. It is a part of the constellation Aquarius within the Milky Way. EZ Aquarii has two stars forming an inner binary star system and the other star orbiting around the binary system. They are red dwarfs much smaller than our Sun. The mass of each red dwarf is about only 10% of the solar mass. No planets orbiting them have been found yet.

Liu Cixin also wrote a novella The Wandering Earth. I have not read the novella but I watched a movie (available on Netflix) with the same title which is based on the novella. Its premise is that the Sun will soon become a supernova. Facing the ultimate cataclysmic extinction event, people on Earth turns their entire planet into a spaceship and attempt to relocate it to Proxima Centauri. It is going to be a long 2,500 years journey. So, at what speed the starship Earth must travel? Assuming that it makes no stops, on average, it will be 510 km/sec which is 0.19% of the speed of light. This is quite a fast speed. So far the fastest object that has ever been built is Parker Solar Probe. By 2025, it is expected to travel as fast as 191 km/sec which is 0.064% of the speed of light. Is it physically possible for the starship Earth to travel at 510 km/sec? The energy required for Earth to travel at 510 km/sec can be easily calculated using $E=\frac{1}{2}mv^2$. With $m=5.9722\times 10^{24}$ kg, the Earth’s mass and $v=510$ km/sec, the energy $E$ is calculated to be $7.767\times 10^{29}$ J=$1.856\times 10^{14}$ megatons. This is 2,000 times the energy output of the Sun per second which is $9.1\times 10^{10}$ J. The most powerful nuclear weapon that has ever been created and tested is Tsar bomb (Царь-бомба) by the Soviet Union. Interestingly, the project was overseen by the famed physicist Andrei Sakharov. Its yield was about 50 megatons. In terms of Tsar bomb, the energy is equivalent to denotating $3.712\times 10^{12}$, i.e. almost 4 trillion Tsar bombs! It seems such an enormous amount of energy is beyond our reach even in a distant future. Ultimately, the answer to the question about whether we can put giant thrusters on Earth to make it travel at 510 km/sec is determined by the famous Tsiolkovsky rocket equation \begin{equation}\label{eq:tre}\Delta v=v_e\ln\frac{m_0}{m_f}=I_{\mathrm{sp}}g_0\ln\frac{m_0}{m_f}\end{equation} where

  • $\Delta v$ is the maximum change of velocity of the vehicle;
  • $v_e=I_{\mathrm{sp}}g_0$ is the effective exhaust velocity;
  • $I_{\mathrm{sp}}$ is the specific impulse in dimension of time;
  • $g_0=9.8\ \mathrm{m}/\mathrm{sec}^2$ is the gravitational acceleration of an object in a vacuum near the surface of the Earth;
  • $m_0$, called wet mass, is the initial mass, including propellant;
  • $m_f$, called dry mass, is the final total mass without propellant.

Tsiolkovsky rocket equation is named after the Russian rocket scientist Konstantin Eduardovich Tsiolkovsky (September 5, 1857 – September 19, 1935). He is dubbed the father of Russian rocket science. For a derivation of the rocket equation, see here. From \eqref{eq:tre}, we obtain \begin{equation}\label{eq:tre2}\frac{m_0-m_f}{m_0}=1-\frac{m_f}{m_0}=1-e^{-\frac{\Delta v}{v_e}}\end{equation} \eqref{eq:tre2} gives rise to the percentage of the initial total mass which has to be propellant. This tell us how efficient the rocket engine is. The most realistic propulsion method for the starship Earth would be a nuclear-thermal rocket. As far as I know, the best performing nuclear-thermal rocket engine that has ever been built and tested was the USSR made nuclear-thermal rocket engine RD0410. It was developed in 1965-94. Its specific impulse is $I_{\mathrm{sp}}=910$ sec [1], so its effective exhaust velocity with $g_0$ is 9 km/sec. We need effective exhaust velocity using the gravitational acceleration for the Sun which is  $274\ \mathrm{m}/\mathrm{sec}^2$. The resulting effective exhaust velocity is $v_e=249$ km/sec. The orbiting speed of the Earth around the Sun can be calculated using the formula $v=\sqrt{\frac{GM}{r}}$. With $G=6.67\times 10^{-11}\ \mathrm{N}\mathrm{m}^2/\mathrm{kg}^2$, $r=1.5\times 10^{11}$ m and $M=1.99\times 10^{30}$ kg, we have $v=29.7$ km/sec. Since the escape velocity is $v_{\mathrm{escape}}=\sqrt{\frac{2GM}{r}}$, for an orbiting object its escape velocity is just $\sqrt{2}$ times its orbiting speed. So, the minimum velocity required for the Earth to break away from its orbit around the Sun is 42 km/sec. With $\Delta v=42$ km/sec and $v_e=249$ km/sec, \eqref{eq:tre2} is evaluated to be $$1-e^{\frac{-\Delta v}{v_e}}=0.155$$ This means that about 16% of the mass of Earth has to be propellant just to break away from the orbit. Since the mass of Earth is $6\times 10^{24}$ kg=$6\times 10^{21}$ tons, 16% would be $10^{21}$ tons. For a nuclear-thermal rocket, the usual propellant is liquid hydrogen. (RD0410’s propellant was also liquid hydrogen.) The basic principle is that liquid hydrogen is heated to a high temperature in a nuclear (fission) reactor and then expands through a rocket nozzle to create thrust. Earth does not even remotely have that much amount of hydrogen. While hydrogen is the most abundant element in the universe, Earth does not have it a lot. This decisively concludes that the starship Earth can’t even break away from its orbit around the Sun let alone travel at the speed of 150 km/sec. Regardless, for the sake of completion, let us calculate how much propellant the starship Earth would need just to reach the speed of 150 km/sec. Now, with $\Delta v=150$ km/sec, we calculate \eqref{eq:tre2} to be $$1-e^{\frac{-\Delta v}{v_e}}=0.453$$ That is, 45% of the mass of Earth has to be propellant!

By the way, Moving Earth is not just a science fiction. It is nothing like relocating Earth to a distant star system but scientists have been pondering how to shift Earth’s orbit farther away from the Sun in order to mitigate rising temperatures on Earth. In my opinion, such an extreme meddling in Mother Nature must not be attempted even if possible as it likely results in unintended catastrophic consequences.

References:

  1. Ядерный ракетный двигатель РД0410 (РД0411), Воронежский центр ракетного двигателестроения
Posted in Astronomy, Rocket Science, Sci-Fi | Leave a comment

What is a Tactical Nuclear Weapon?

Posted on June 6, 2024 by Sung Lee

As tensions are rising between the NATO and Russia, we hear a lot about tactical nuclear weapons lately. But people who talk about them do not appear to have a clear definition of what a tactic nuclear weapon is and they often understood it as a low yield nuclear weapon but that is not exactly so. To understand the correct meaning of a tactical nuclear weapon, I would like to point out that all nuclear weapons are categorized as strategic or non-strategic nuclear weapons. A strategic nuclear weapon is designed to be used on targets in a settled territory (a sovereign state) far away from the battle field. In general, strategic nuclear weapons have high yields which are 100 kilotons and up. 1 kiloton is approximately the energy released by the detonation of 1000 tons of TNT (equivalent to $4.184\times 10^{12}$J). Strategic nuclear weapons are delivered by intercontinental ballistic missiles (ICBM), submarine-launched ballistic missiles (SLBM), or heavy bombers. The United States is the only country that has ever used strategic nuclear weapons against a sovereign state. The United States detonated an atomic bomb called Little Boy over the Japanese city of Hiroshima on August 6th, 1945 and another atomic bomb called Fat Man over the Japanese city of Nakasaki on August 9th, 1945. The yields of Little Boy and Fat Man are, respectively, 15 kilotons and 20 kilotons. In today’s standard, they are considered low yields (for example, Russia’s non-strategic nuclear weapons have yields of 70-75 kilotons) but they were the most powerful nuclear weapons existed at that time. The two bombings resulted in the deaths of between 129,000 and 226,000 mostly civilians.  All other nuclear weapons are non-strategic nuclear weapons. They are also called tactical nuclear weapons or theater nuclear weapons. As the name theater nuclear weapons suggests, they are designed for use in the battle field. Due to a possible proximity to friendly forces, they have low yields between 10 tons and 100 kilotons. According to the reference [1], non-strategic nuclear weapons have historically include bombs delivered by dual-capable aircraft (DCA) which can be used for both nuclear and conventional missions; warheads in cruise missiles delivered by non-strategic aircraft, warheads on sea-launched cruise missiles (SLCMs); warheads on ground-launched cruise missiles (GLCMs); warheads on ground-launched ballistic missiles (GLBMs) with a maximum range that does not exceed 5,500 km, including air-defense missiles (ADMs); warheads fired from cannon artillery; and anti-submarine warfare nuclear depth bombs. But the reference [1] also mentioned that today, only air-launched cruise missiles (ALCMs) and gravity bombs delivered by DCA are in the non-strategic category. I believe this statement should be understood as the case for the US non-strategic nuclear weapons. As far as I know, the lowest yield non-strategic nuclear weapon that has ever been deployed is W48. It is an AFAP (Artillery Fired Atomic Projectile) which can be fired from any standard 155mm howitzer. It had an yield of 72 tons. In the US, it was manufactured starting in 1963 and all units were retired in 1992. For the case of Russia, officially, all nuclear artillery shells have been decommissioned by the year 2000, but rumor has it that some of Soviet 152mm nuclear artillery shells like 3BV3 were not all removed in 2000 and they can be used in 152mm self-propelled howitzer 2S19 Msta-S. Also, rumor has it that Russia has been developing new 152mm nuclear artillery shells for the artillery systems on Armata platform. Even if they were all decommissioned in 2000, nuclear artillery shells are something that can be easily produced by Russia and can be immediately put to use on currently existing delivery systems. Less risking a full brown nuclear war, nuclear artillery shells can be an ideal demonstrative nuclear deterrent against the NATO in the current theater of Russian SMO (Special Military Operation) in Ukraine when the NATO crosses what Russia considers as red lines. So, I wouldn’t be surprised if Russia already has them in stock and ready for use. Thus far, non-strategic nuclear weapons have never been used.

Physics Tidbit: From the famous Einstein’s formula $E=mc^2$, we see that 1 g of mass is equivalent to $9\times 10^{13}$J of energy. Since 1 kiloton=$4.184\times 10^{12}$J, 1 g of mass is equivalent to 20.15 kilotons of energy which is about the yield of Fat Man .

References:

  1. Chapter 4. Nuclear Weapons, The Nuclear Matters Handbook 2020, The Office of the Deputy Assistant Secretary of Defense for Nuclear Matter
Posted in Nuclear Weapons | Leave a comment

Can a toroidal black hole exist?

Posted on April 13, 2024 by Sung Lee

Many years ago, a brilliant Berkeley mathematician Richard Borcherds started a blog on math. I have no doubt he has so many interesting things to share about math. Regrettably though, for whatever reason, it didn’t last long. In one of his earliest blog articles, he posed a question as to whether a toroidal (doughnut shaped) black hole can exist. We will answer this question here but first, what do we mean by the shape of a black hole? What is called the shape of a black hole is actually the shape of its event horizon. The simplest black hole is Schwarzschild black hole. It is a spherically symmetric solution of the vacuum Einstein’s field equations. The Schwarzschild solution is given by the metric
\begin{equation}
\label{eq:bh}
ds^2=-\left(1-\frac{2MG}{c^2r}\right)c^2dt^2+\left(1-\frac{2MG}{c^2r}\right)^{-1}dr^2+r^2(d\theta^2+\sin^2\theta d\phi^2)
\end{equation}
The even horizon is the sphere of radius $r_g:=\frac{2MG}{c^2}$ which is called the Schwarzschild radius. The Schwarzschild solution \eqref{eq:bh} requires that $r>r_g$. The event horizon is what determines the inside and the outside of a black hole. \eqref{eq:bh} is a solution of the vacuum Einstein’d field equations, so it really doesn’t tell us anything happens once an object falls into a black hole past its event horizon. In order to see what happens inside of a black hole, we need information on the source of its gravity, i.e. the stress-energy tensor $T_{ij}$.

In 1923, an American mathematician George David Birkhoff proved the following theorem.

Theorem. Any spherically symmetric solution of the vacuum Einstein’s field equations must be static and asymptotically flat.

One may wonder if the converse of the Birkhoff’s theorem is also true, namely is any static asymptotically flat solution of the vacuum Einstein’s field equations spherically symmetric? The answer is affirmative and it is called Israel’s theorem named after Werner Israel (Phys. Rev. 164, 1776). So, Israel’s theorem settles it. There can’t be a doughnut shaped black hole which is static and asymptotically flat. This also can be shown directly. First the ansatz for a static toroidal black hole can be given by
\begin{equation}
\label{eq:toroidalbh}
ds^2=-A(r)dt^2+B(r)dr^2+r^2d\theta^2+(R+r\cos\theta)^2d\phi^2
\end{equation}
with asymptotically flat condition
$$\lim_{r\to\infty}A(r)=\lim_{r\to\infty}B(r)=1$$
The Ricci tensors are computed to be:
\begin{align*}
R_{tt}&=\frac{A^{\prime\prime}}{2B}-\frac{A’}{4B}\left(\frac{A’}{A}+\frac{B’}{B}\right)+\frac{A’}{2B}\left(\frac{1}{r}+\frac{\cos\theta}{R+r\cos\theta}\right)\\
R_{rr}&=-\frac{A^{\prime\prime}}{2A}+\frac{A’}{4A}\left(\frac{A’}{A}+\frac{B’}{B}\right)+\frac{B’}{2B}\left(\frac{1}{r}+\frac{\cos\theta}{R+r\cos\theta}\right)\\
R_{\theta\theta}&=-\frac{r}{2B}\frac{A’B-AB’}{AB}+\frac{r\cos\theta}{R+r\cos\theta}\left(1-\frac{1}{B}\right)\\
R_{\phi\phi}&=(R+r\cos\theta)\cos\theta\left[-\frac{1}{2}\frac{A’}{AB}+\frac{1}{2}\frac{B’}{B^2}+\frac{1}{r}\left(1-\frac{1}{B}\right)\right]
\end{align*}
$$BR_{tt}+AR_{rr}=\frac{1}{2B}\left(\frac{1}{r}+\frac{\cos\theta}{R_r\cos\theta}\right)(AB)’$$
Since $BR_{tt}+AR_{rr}=0$ regardless of what the value of $\theta$ is, $(AB)’=0$ i.e. $AB$ is a constant. The asymptotically flat condition indicates that $AB=1$. Consequently, $R_{\theta\theta}$ becomes
\begin{equation}
\label{eq:riccitheta}
R_{\theta\theta}=-rA’+\frac{r\cos\theta}{R+r\cos\theta}(1-A)
\end{equation}
$R_{\phi\phi}=0$ yields
\begin{equation}
\label{eq:ricciphi}
-\frac{1}{2}\frac{A’}{AB}+\frac{1}{2}\frac{B’}{B^2}+\frac{1}{r}-\frac{1}{rB}=0
\end{equation}
Finally, subtracting $r$ times \eqref{eq:ricciphi} from \eqref{eq:riccitheta} and then setting the result equal to 0 we obtain
$B=1$ and thereby $A=1$. Therefore, the ansatz \eqref{eq:toroidalbh} becomes the flat Minkowski space-time, so there is no toroidal black hole.

How about then for a rotating black hole? Can there be a rotating toroidal black hole? The answer is negative for this case also. Hawking and Ellis proved that the shape of a rotating black hole has to be a sphere and it can be found in Proposition 9.3.2 of S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time, Cambridge University Press, 1973. (Hat tip to David Garfinkle.)

Posted in General Relativity | Leave a comment

diff (A Linux Command)

Posted on February 24, 2024 by Sung Lee

diff is one of my favorite Linux commands, which is very useful. What does it do? It does compare two text documents and shows their differences if there are any. The syntax is as simple as

diff textfile1 textfile2

and so it is easy to use even if you are not so familiar with command lines. Here is an example.

I am currently working on my old MacBook, so you see an Apple terminal in the above image. But diff is a common Linux command for any Linux distribution, of course, including Ubuntu. I have two files named caesar.py and caesar2.py in my ~/Desktop/python folder. I ran diff command to compare these two. What the output says is that caesar.py has the following two lines:

# every possible symbol that can be encrypted
LETTERS = 'ABCDEFGHIJKLMNOPQRSTUVWXYZ'

that caesar2.py does not have. Pay attention to the direction of the arrow <. This is particularly useful when you are programming. Suppose that you have a backup of the original code say caesar.py and you are editing its copy caesar2.py. Somehow you deleted or incorrectly altered certain lines by mistake and you couldn’t identify your mistakes. In that case, you can solve the issue by simply running the command diff as shown in the example.

Posted in Linux, macOS | Leave a comment