October 2025

Relativity

Deriving Einstein’s Gravity Equations From Thermodynamics

Is Gravity Just an Average of the Behavior of Unknown “Atoms” of Spacetime?

Gödel

Gödel’s Constitutional Quarrel

“The examiner was intelligent enough to quickly quieten Gödel and say ‘Oh god, let’s not go into this’ and broke off the examination at this point, greatly to our relief” — Oskar Morgenstern Kurt Gödel (1906–1978) was the greatest logician who ever lived. At the age of 24, he

Relativity

Why Time Is Encoded in the Geometry of Space

An Introduction to Geometrodynamics

Relativity

A Gentle Explanation of E= mc²

Deriving Einstein’s Famous Mass-Energy Equivalence Formula

Einstein

Length Contraction in Einstein’s Theory of Relativity

Why Moving Objects are Shortened

Gödel

Kurt Gödel’s Brilliant Madness

Hungarian polymath John von Neumann (1903–1957) once wrote that Kurt Gödel was “absolutely irreplaceable” and “in a class by himself”.

Physics

The Einstein-Szilárd Letter (1939)

The now famous Einstein-Szilárd letter was written at the initiative of Hungarian nuclear physicist Leó Szilárd with help from Edward Teller and Eugene Wigner in 1939.

Relativity

Thinking Relativistically

Problem-solving in special relativity.

Physics

When Wiener met Einstein (1925)

Mathematician and later father of cybernetics Norbert Wiener (1894-1964) crossed paths with many great minds in his life, from Bertrand Russell and G.H. Hardy, to Max Born, John F. Nash Jr. and John von Neumann.

Physics

Extending Einstein’s Field Equations

Wave Properties of Matter in Gravitational Fields

History

The Golden Age of Quantum Physics (1927)

The “most intelligent photograph ever taken”, as it is sometimes known, was captured during the Fifth Solvay International Conference on Electrons and Photons held in 1927 in Brussels, Belgium.

Physics

Einstein’s Theory of Relativity

Explained as simply as possible, but no simpler.

The architectural symmetry of Cubic polynomials supports many root approximation methods, helping intuitive understanding and with a little work, can be transposed to higher orders to minimise more complex math. This post expands on an earlier post, Cubic Polynomials — Using Similar Triangles to Approximate Roots, which promoted the use of Similar Triangles derived from Turning and Inflection Points.

Both methods are very simple and it is difficult to quantify pros and cons in a short post. The latter, being very easily formularised and not requiring calculus beyond finding the Inflection Point Ip is the quicker and simpler of the 2, is best suited when Ip(x)=B/3A

The significance of prime numbers, in both everyday applications & as a subtopic pertinent to all branches of math, cannot be overstated. We quietly rely on their special properties to carry the backbone of countless parts of our society — all because they are an irreducible part of the very fabric of nature. Resistant to any further factorization, prime numbers are often referred to as the “atoms” of the math world. As Carl Sagan so eloquently describes them:

There’s a certain importance to prime numbers’ status as the most fundamental building blocks of all numbers, which are themselves the building blocks of our understanding of the universe.

The use of prime numbers in nature & in our lives are everywhere: cicadas time their life cycles by them, clock-makers use them to calculate ticks, & aeronautical engines use them to balance frequency of air pulses. However, all of these use-cases pale in comparison to the one fact every cryptographer is familiar with: prime numbers are at the very heart of modern computational security, which means prime numbers are directly responsible for securing pretty much everything. See that lock in the URL bar? Yeap, a two-key handshake powered by primes. How is your credit card protected on purchases? Again, encryption powered by primes.

Yet for our consistent reliance on their unique properties, prime numbers have remained infamously elusive. Throughout the history of math, the greatest minds have attempted to prove a theorem for predicting which numbers are prime, or, how apart successive primes are in placement. In fact, a handful of unsolved problems such as Twin Primes, Goldbach Conjecture, Palindromic Primes, & The Riemann Hypothesis all revolve around this general unpredictability & uncertainty in prime numbers as they approach infinity. Granted, since the early days of Euclid we’ve found a handful algorithms that predict some placement, but general theorems haven’t been accepted nor did previous attempts have the tools to test large numbers. 21st-century technology, however, does allow researchers to test proposals with extremely large numbers, but that method alone invites controversy as brute-force testing isn’t quite globally accepted as a solid proof. In other words, primes have resisted any universal formula or equation, their appearance in nature remaining a status of seemingly-random.

As It Turns Out However, A Simple Scribble Proves That They’re At Least Not Completely Random…

Scribbling To A Clue — The Ulam Spiral

One of the greatest proofs we have that the appearance of prime numbers is no mere coincidence came from one of the most unlikeliest places: the effortless & accidental doodles of one bored lecture attendee.

Ulam Spiral Setup

As the story goes, one Polish mathematician, Stanislaw Ulam, stumbled upon a visual pattern in 1963 during a seminar. While drawing a grid of lines, he decided to number the intersections according to a square-spiral pattern, & then began circling the numbers in the spiral that were primes. Surprisingly, the circled primes appeared to fall along a number of diagonal straight lines or, in Ulam’s slightly more formal prose, it:

Appears To Exhibit A Strongly Nonrandom Appearance

The Ulam Spiral, or prime spiral, is the resultant graphical depiction from marking the set of prime numbers in a square-spiral. It was originally published & reached the mainstream through Martin Gardner’s Mathematical Gamescolumn in Scientific American.

377×377 (~142K) Ulam Spiral

The visualization above clearly highlights significant patterns, especially diagonally. But maybe we’re deceiving ourselves? A common counter to the Ulam spiral is that perhaps our brain is simply tricking us into assigning these patterns in randomness. Luckily, we can take two different approaches to confirm that this isn’t the case. Both a visual comparison & a logical walk-through will convince one that they’re certainly not random. First, we’ll compare an Ulam spiral made by a matrix of NxN dimensions, to an equally-sized NxN matrix containing randomly-assigned dots. Second, we’ll flex our knowledge of polynomials to reason through exactly why we should expect some pattern in the visual layout of primes.

As mentioned, the likely most intuitive way to confirm a non-random pattern with a Ulam spiral is through a direct comparison. This is done by creating both a Ulam spiral & a spiral with random placements of the same size. Below are two different 200×200 matrices that represent numerical spirals:

Originally Published On https://www.setzeus.com/

By visual comparison it’s fairly obvious that the Ulam spiral contains striking patterns, particularly along certain diagonal axes; additionally, there is little, if any clustering. On the other hand, the random placement of dots does not yield any immediately-obvious patterns — rightly resulting in multi-directional clustering. Incontrovertibly, this lacks the rigor of traditional proofs; however, there’s something pure in visualizing prime spirals, a method inattentively stumbled upon, that yields a diagram that is both logically stimulating & aesthetically captivating.

Approaching the nature of primes in a more logical & traditional manner, it’s actually reasonable to expect patterns in these visualizations. As noted above, lines, in diagonal, horizontal, & vertical directions seem to contain a clue. A few of these lines that aren’t primes can be explained by simple quadratic polynomials that inherently exclude primes — for example, one of the diagonal lines will represent y = x² which clearly excludes primes. On the other side, a handful of quadratic polynomials, known as Prime Formulas (we’ll be seeing these again), are known to produce a high density of primes, such as Euler’s prime-generating polynomial: x²- x – 41, another line that appears as a pattern in the spiral (thought it’s hard to identify & non-continuous in the diagram above).

Visual comparison hints at patterns, while a logical walk-through confirms the existence of expected patterns through mapping out primes. Again, a far-cry from the universal formula for finding all primes, but the Ulam Spiral is undeniably beautiful as both a marker of knowledge & a piece of natural art.

Sack’s Spiral

Like many topics in math, the moment an original idea arrives, a trailing army of colleagues follow to take their crack at contributing to a burgeoning field. Reasonably, the Ulam Spiral inspired generations of mathematicians that sought to build on-top of it’s fascinating findings. In 1994 one Robert Sacks, a software engineer by trade, aimed at leveraging his programming skills to visualize primes in different ways.

Much like the Ulam spiral, Sacks decided to structure his diagram using another spiral plane. Similarly to the square-spiral above, spiral planes forgo a traditional, Cartesian number system to identify a point since it is a unipolar positioningsystem. Simply knowing the number reveals its location in the spiral, its position relative to every other number in the spiral, & its distance from the previous & the next perfect square. Instead of a square-spiral, however, Sacks attempted to find patterns with integers plotted on an Archimedean Spiral with the following polar coordinates:

Polar Coordinates For Archimedean/Sacks Spiral

In this method, an Archimedean spiral is centered on zero with the squares of all natural numbers (1,4,9,16,25) plotted on the intersections of the spiral & the polar axis (directly East of the origin).

Setting Up Our Archimedean/Sacks Spiral

From this setup, we then fill in the points between squares along the spiral (counterclockwise), drawing them equidistant from one another. Finally, like the Ulam example above, we highlight the primes contained within the resultant spiral.

First published online in 2003, the Sacks Number Spiral is both visually arresting & intellectually compelling. Additionally, as we’ll demonstrate shortly, it also yields deeper insights into prime number patterns than the well-known Ulam spiral because, in effect, it joins together the broken lines of Ulam’s pseudo-spiral:

Archimedean Spiral With Primes Highlighted — A.K.A Sacks Spiral

The resultant diagram once again highlights obvious patterns. Almost immediately, it’s clear that there is a purely white line originating from the center & stretching horizontally to the east. Referring back to our setup, we can confirm that this is simply the line that contains all the perfect squares (r = n^(.5)). The second observation that jumps out is that the pattern of markings, in contrast to the straight lines seen in the Ulam Spiral, appear this time to more appropriately mimic curved lines. As it turns out, these curved lines, also known as product curves, circle back to the polynomial intuition explaining the patterns that emerged in the previous spiral. Before we jump into those, however, let’s take a quick second, for consistency, to compare the Sack’s Spiral against a randomly-plotted spiral:

Originally Published On https://www.setzeus.com/

Polynomials & Product Curves

Robert Sacks’ work following his initial finding focused extensively on these product curves, lines that originate from the spiral’s center, or near to it & traverse the spiral’s arms at varying angles. Curves are almost straight but, more typically, they perform partial, complete, or multiple clockwise revolutions — counter to the spiral itself — around the origin before straightening out at a particular offset from the east-west axis. One of the most striking aspects of the Sacks Number Spiral is the predominance of these product curves on the western hemisphere (opposing side from the perfect squares).

Sacks describes product curves as representing “products of factors with a fixed difference between them.” In other words, every curve can be represented by a quadratic equation (a second-degree polynomial) which, again, is no mere coincidence given the primacy of the perfect square in the structure of the Sacks spiral. Arguably, these product curves lend to the observation that the Sacks spiral is significantly more useful in our journey towards understanding primes than the Ulam spiral. While the original Ulam spiral hinted at patterns & possible equations, the Sacks spiral neatly provides starting points for prime formulas — their curvature & continuity is crystallized, & therefore, they’re much more readily identifiable. For example, the Sacks spiral below contains highlighted lines with their associated Prime Formula notated in standard form. As promised, Euler’s famous prime-generating quadratic formula returns as is shown below (lowest annotation: n² + n +41):

With his number spiral, Sacks is able to make the striking assertion that a prime number is: a positive integer that lies on only one product curve. Since the spiral can be extended outward infinitely, the product curves themselves may be considered infinite; theoretically, these product curves could possibly predict the prime placement of fairly large numbers — at the very least, they’re worthy of further introspection.

In general, it’s conclusive that the Sacks spiral nudges us towards a deeper understanding of primes through more readily-identifying Prime Formulas.

The Meaning Of It All

We’ve now analyzed both the Ulam Spiral & the Sacks Spiral. Through both examples our understanding of the nature behind prime numbers expanded. The Sacks Spiral, specifically, introduced us to product curves, which are essentially the set of quadratic equations known as prime formulas. Both diagrams, Ulam & Sacks, are unexpected, aesthetic diagrams that soothe our curiosity & shine a light on a universally-challenging problem.

So what’s the lesson to take away from here?

To never shy away from re-framing seemingly-impossible problems, even if simply out of curiosity & boredom; breakthroughs don’t discriminate & often reveal themselves through the most unexpected efforts. By shifting perspectives on a famous problem through visualization, Stanislaw Ulam brought us all one step closer to understanding prime numbers: who knows how many other unexpected discoveries we’ll stumble across?

Relativity

How to Compute the General Relativistic Correction to Time

General relativity is one of the most technically difficult physics theories, as it requires uncommon knowledge and mathematical skills. However, it is possible to calculate the relativistic effects induced by the curvature of space-time with good precision, and even to derive its formula, with a few simple arguments. We know

Chemistry

Salt on an Icy Road Lowers Its Temperature

So, how does it melt the ice?

Thermodynamics

Temperature Scales

The most widely adopted temperature scale is the Celsius or centigrade scale. Nowadays, we use these terms as synonyms, but, in fact, they were not such, originally.

Electromagnetism

An Unorthodox Lecture on Electric Currents

An introduction to the concept of an electrical current

Calculus

Two Myths of Numerical Integration, Debunked

Many programmers believe that the use of higher order integration algorithms, combined with a large number of integration interval…

Particle Physics

Understanding the Higgs Mechanism

A non-metaphoric explanation

A couple of things before we start, I’m not writing this as an attack, these are just personal view based on some facts. I’ll be using a little bit of math in the part of Game Theory nothing too crazy, I think if you have studied class 12th math you’ll understand it. And finally, please ignore the typos(yes I did use Grammarly).

What is Relative Grading?

The concept of Relative Grading is quite an old one, not sure when it was actually introduced but the papers referencing it go back to the 1970s here. There are quite some reasons why it exists, one of them you might have thought of is the growth of the student population, which correlates to either having bigger classes or having multiple sections for the same course, which would require more teachers, and teachers have different metrics and strategies for grading an answer, which leads to unfairness between the different sections of the same course, relative grading will normalize the variations and put everyone at plain-level field. Another reason is the incentives and the competition, I will go into some depth with this word “incentive ” but to explain it in simple terms, relative grading induces a competition, in the sense that to get a good grade, you’d need a “good” overall score and be better than rest of the class.

Let’s look at an example, now these data points do not represent a real scenario, but they capture what the distribution of marks would look like. Now if you know a bit of statistics, this looks like normal distribution and it is, the most common type there is. If you need a refresher on distributions this is a really good resource.

The X-axis represents the scores of a sample of 500. Mean = 15, Standard Deviation= 6

Now the reason why I personally used a normal distribution is that they sum up the situation really well, right? You have some people who score really well and some who score really bad and the rest are kind of in the middle.

The running assumption in this article is that the tests are fair, which means that there are no errors in the questions and the teacher grades the papers without any bias.

Let’s say, we were to give a grade based just based on this performance. And I will be using the Relative Grading System. The formula for calculating a relative grade is quite straight forward, it only differs in what grading system you are using.

So if you are in a Japan grading system A+ translates to S, A translate to A, and so on.

Now let’s see how our program based students performed.

I’d say pretty good right, the majority got A and B, and then we have the 20 or so super nerds who got the S, remember this is out of 500, so if you assume the class size to be 60, the people who got an S will be in the range of 3–6 and so will be the F’s. Now, this does prove that in order to get the best grade you have to be better than the rest but the data in practice isn’t as beautifully distributed, it would probably be a skew-normal distribution which would look like this because believe it or not scoring below 10 on a 30 mark exam whose average is 23, is pretty hard.

Mean: 23 and Standard Deviation: 5

And here are the grades.

So it is quite apparent that as the mean increases the number of S grades decreases and the distribution becomes more centered to B. And it’s this result which directs the sentiment behind this article.

Game Theory And Exams

Game Theory, if you are a movie fanatic, you might have watched Russel Crowe’s “A Beautiful Mind”, which good film but modern cinema has this thing that a story has to be dramatic to be good, so for the people who haven’t watched it now you know what to expect. A little about the field itself: It started with Plato when he described the Battle Of Delium in his piece Laches, the first mathematical formulation was done by John von Neuman, where non-mathematically speaking we can always find an equilibrium in a finite two-player game(I will define all these buzzwords, don’t worry) and John Nash who proved that equilibrium exists even in an n-player game. In this section, I wouldn’t dive very deep into mathematics(some), we’ll keep it philosophical and the field of Game Theory is extensive, there is no end, mathematically and definitely not philosophically, so I encourage you to read more about it, and appreciate it’s beauty in mathematically addressing human psychology.

What is a Game?

A game in a very “non-gamer” sense is a situation whose results depend on the actions of the parties involved.

Let’s break it down. The parties involved are called players, these are the entities that take actions that affect the state of the game. Now, why are players even interested in taking an action, well they’ll get a reward if they take an action that benefits himself/herself. This reward is defined by a utility function or payoff function, which is just a mapping of the actions of a player to a real number. So a finite n-person game is a tuple with (N, A,u) where N is the number of players, Ais the action set(the set of all actions that will be taken by any player, you can also call it action space) and u which is the utility function. So let’s formulate them mathematically

I have no idea how to type these here so I had to type it in LaTEX and paste them here.

Now that we have a naive understanding let’s look at the things that actually matter. We know that a player gets a reward for playing a good strategy, so the goal is to maximize the utility function by using the actions available in the action space, now if I know or have some intuition on what the other players will do, I can tailor my actions in order to get the highest utility, this is known as Best Response, which basically is the means that the reward I will get by playing the best response action is will be greater than any other possible set of actions. So, to understand this in a more mathematical sense,

I hope you are still with me, it’s all philosophy from now on I promise.

If every player plays his/her best response, we achieve a Nash Equilibria. Ok, if you paid attention till now you can probably imagine how Nash equilibria can be used to define human behavior in competitive environments. Although Nash Equilibrium has lead to tremendous success in economics specifically in competitive markets and auctions, it is not limited to that we can see the use of game theory even in biology and education which we will discuss in the next subsection.

Exams

As we saw game theory can be used in a competitive environment, so let’s start and break down exams, into the components we saw in the previous section.

1. Players: You are a player, I’m a player and other students in the class are players
2. Reward: Marks, Marks, Marks***
3. Action: Marking an answer.

Now this feels very ambiguous right? Marking answers will help me score well but how me marking an answer is affecting the other student, and where does the concept of best response fit into this. Well, first-off this will only be valid for a relative system(I will explain the absolute system as well) so if the majority of the players mark A as the answer but the correct answer is B and you mark B then this is your best response, but in this case, you don’t know what the other players are doing and thinking, so you have to play for yourself without the knowledge of the strategies of other players, and there is a term for this but if you think hard about it, you can assign a positive probability to the possible set of answers, this is also called Pure Strategy. So now that we have formalized exams that are relatively graded are games let’s see how rewards and incentives work.

In an absolute graded system, the players are independent and the actions of one player don’t affect the state of the game or any other players.

Now the goal of every player is to perform its best to reach the Nash Equilibria, in an exam if everyone performs for the maximum utility the mean of the class would by 30 and the standard deviation would be 0 and the whole class will get an F.

Of course, the utility here is Marks and not the grade

And just in case you are wondering it is impossible that a whole class gets an S, I encourage you to work it out. So one inference you can make the incentive to perform isn’t marks, its grades.

Cheater Cheater!!

Again I’m not writing this because I’m pissed about people cheating like crazy in a 30 marks MCQ test, I just want to enlighten them about the repercussions of their actions.

So let’s say you got the question paper and you start solving it and you are pretty confident about your capabilities so you know the answers you marked are correct. In the last 20min of the quiz you open telegram or WhatsApp where people have shared their answers, now you know exactly what other players are doing, and your task is the give the best response which is the correct answer. By now, one of thee things is happening:

1. Your answers exactly match with the answers of other players
2. Your answers somewhat match the answers of the other players
3. Your answers are totally different from the answers of other players

If you are in the first situation, you wouldn’t change your answers because of the Bandwagon effect. In the rest, you might. So in order to give what you saw was the best response you change your answers. Again two things happen,

1. The answers were right and you get 30 on 30
2. The answers were wrong and you lost marks

If you analyze the first situation it is pretty bad as well, almost like the fruit of a poisonous tree. And you know why because the mean would shift very close to 30 and the standard deviation would be low as well and hence you will get a graph like this,

And the curve for the grades will look like this,

Looks seeming normal right, except it’s not. To get an S you absolutely have to get a 30 and to get an A, your score should be between 28.5 and 29.5 but its MCQ right so you gotta round that, so having a score between 29 and 30 will get you an A.

If it is the other way round, you marked the wrong answers and even the other players did, the damage wouldn’t be as significant But if you marked the wrong answers and other players didn’t such that the difference between you and the average is of 5 marks, you’ll either get a C or a D.

Think about this for a second.

If you are aware, and you know if you cheat this could happen you’d never cheat right? But as humans, (it’s about to get 100% philosophical now) we would be in a situation where we score more regardless of the performance of others, and here lies that incentive to cheat. To make up for the uncertainty we believe anything presented to us. Also if you don’t cheat and everyone else is cheating, after some time even you might succumb to it. This is what the modern education system has done, there is so much focus on marks that we have taken a tangent from actually being open to learning to cramming on a constrained syllabus. But let’s say nobody cheats, the curve would look like Figure 2, and we wouldn’t have the crazy numbers as we saw in the last paragraph and the grading would actually be fair.

Conclusion

1. After discussing game theory and statistics I hope you now know what drastic impact would cheating on a test create.
2. The education system today is very “marks-centric” that is you can’t get that far if your scores aren’t on point and of course, there are exceptions. But this has created this longing of marks rather than knowledge.
3. Cheating on a test, that is relatively graded has disastrous implications, and it’s better not to do it.

All the references are added as links. The code will be provided on request.

What one would do to find the representation of a number written in one base to some another would probably be to take a look at the Euclidean algorithm: Dividing the number by a new base, writing down the integer part, then divide it again, and again, until it will not reach 0. The order formed from the remainders represents the same number but written in this new base.

As an example, the algorithm applied to:

looks like this:

By writing down the remainders, we get the same number but in the binary system, that is:

What is the problem?

Having so many successive 0s that cannot be skipped. The algorithm flows from start to end without any breaking points. That is where logarithms come in handy because they could wipe out these places from the calculation.

What is the solution?

As we all know, if the result and the base of some exponential function are well-known, then logarithms are used to find this exponent, to find the power.

We also know that each numeral system in modern mathematic is positional. The number 1027 written in the 10-base system means just that:

Now let us think about how we could write the same number but with logarithms. Looking at the formula above, we notice that the base is known, it is 10, and the result is also known, it is 1027. Thus:

By using the logarithms table or just with any calculator, we get that b is 3.0115704436. Of course, it means that if we would raise 10 to this power, we would get 1027 — that is clear. Yet, we could get the same number with a little different approach. The power could be divided into two parts: integer and decimal. We take the integer one, which is 3, and calculate 10 to this exponent. What do we get? We get 1000. This gives us information that digit at the position of 10³ will be non-zero, and, in fact, it will be equal to the integer part of the following fraction:

and the remainder,

we use in the next step in the same manner as 1027 was processed in this one:

Notice that we computed digits at 3rd and 1st positions. We did not calculate the one at 10² since there is only 0. It was omitted from the calculation.

The next step is:

There is nothing left: the algorithm ends.

Now, what converting is all about?

Simply instead of using 10 as the base, we want it to be something else: 2, 3, 4, or so. How could we do this? The solution comes from the properties of logarithms: its base could be changed without changing the result. But before getting into the how at once and risking losing the ground, we will go to it step by step.

We know what logarithms are, it has been talked already. Knowing what base and result are, we could compute the power. Well, are the following equations the same?

Yes. Though, the product of multiplication t times the fraction of 1 by t is 1. It does not change the exponent.

However, we do not want just to arise it like that. We want to use it for changing our base from a to a’. So we ask ourselves, “To what power do we have to raise a to get a’?” And what mathematic answers for such a question? Exactly, logarithms!

At this point, our equation looks like this:

There is no unknown left. We know what b is:

and what the fraction is:

Then we could present the formula for calculating the same c but with another base. That is:

Examples

Let us convert our 1027 from the-10 base system into the binary one. We know how it looks by now, so it would be easier to follow. Where do we start? We need to calculate t to get a’, which is 2 (binary system.)

We could find the fraction immediately knowing what t is:

We know all the constants, thus the algorithm could be started.

First iteration:

Undoubtedly, we are working with the binary system, so any integer part will be 1, always. But we are going to calculate everything each time for getting better feelings.

Second iteration:

Look at how all 0s disappeared; it is so beautiful.

Third iteration:

Thus we know everything: 1s are at 10th, 1st, and 0th positions. All others have 0s. Let’s check this in the same way as we checked it after applying the Euclidean algorithm. Using the same table:

Maybe one more with some huge number, say 94842243, that we would like to convert into the 3-base numeral system let us go. Again, we start by computing tto get a’.

and the fraction:

Now iterations, and there are a few. When we break through calculations, we will check if the result is right. So, thus they are:

And a verification test,

gives the identical number.

Emergent gravity is an idea in quantum gravity according to which the fabric of spacetime is not fundamental but emerges as a coarse-graining approximation of underlying (still unknown) microscopic degrees of freedom (similarly to a gas emerging from a large sampling of atoms or molecules). In the words of Huggett and Wuthrich, emergent gravity is the view that gravity arises due to the “collective action of the dynamics of more fundamental non-gravitational degrees of freedom.”

Figure 1: According to emergent gravity, the spacetime continuum can be viewed as the macroscopic limit of some fundamental microscopic structure (source). Here we will investigate a proposal by the physicist Ted Jacobson that Einstein’s gravity can be derived from thermodynamics (source).

In the present article, we will investigate a 1995 proposal by the American theoretical physicist Ted Jacobson that Einstein’s gravity equations can be derived from thermodynamics, “the branch of physics that deals with the relations between heat and other forms of energy” (see link). This implies that Einstein’s equations can be viewed as an equation of state, a thermodynamic equation relating variables describing the state of matter (such as, for example, the ideal gas law).

Figure 2: The French physicist Sadi Carnot considered the “father” of thermodynamics (source), and his 1824 book “Reflections on the Motive Power of Fire and on Machines Fitted to Develop that Power,” considered the founding work of thermodynamics (source).

Motivation: Entropy and Horizons in Spacetime

In the 70s, the Mexican-born Israeli-American theoretical physicist Jacob Bekenstein and the English theoretical physicist and cosmologist Stephen Hawking showed that black holes have a thermodynamic entropy proportional to the area of their event horizon.

Equation 1: The Bekenstein-Hawking entropy.

where G, c, h, and k denote Newton’s gravity constant, the speed of light, Planck’s constant, and the Boltzmann’s constant. Note that just by examining the constants in this expression, we infer that black holes lie at the intersection of gravity, quantum mechanics, and thermodynamics since:

• G is the gravitational constant, needed to calculate the gravitational effects in Newton’s law of universal gravitation and in Einstein’s theory of gravity.
• h is the Planck’s constant, the quantum of electromagnetic action relating a photon’s energy to its frequency (photons are particles of light).
• k is the Boltzmann constant, the proportionality factor relating the average kinetic energy of particles in a gas with its thermodynamic temperature.

Figure 3: The Bekenstein-Hawking entropy is the entropy ascribed to black holes. It is proportional to 1/4 of its horizon area (source).

Black holes are not the only spacetime configurations carrying entropy. Two other important examples are:

• Cosmological horizons in de Sitter (dS) space
• Observer-dependent horizons in Rindler spacetime (a coordinate system representing part of Minkowski spacetime that describes uniformly accelerated observers). This type of horizon, which will be central here, has entropy and a temperature (the Unruh temperature). The latter is proportional to the observers’ acceleration and its existence hints thatspacetime itself encodes thermodynamical information.

Figure 4: Jacob Bekenstein (source) and Stephen Hawking (source), the first who showed that black holes have a thermodynamic entropy proportional to the area of their event horizon.

Quoting the renowned American theoretical physicist Robert Wald, who made fundamental contributions to the study of gravitation physics, such as the discovery of the general formula for black hole entropy, and the development of a rigorous formulation of quantum field theory in curved spacetime:

“I believe that the relationship between black holes and thermodynamics provides us with the deepest insights that we currenly have concerning the nature of gravitation, thermodynamics, and quantum physics.”

— Robert Wald

We conclude that, since we can associate a temperature and entropy to regions of spacetime, it is not unreasonable to suppose that their properties may have some similarities to the properties of matter at the macroscopic scale. Quoting Jacobson:

“This perspective suggests that it may be no more appropriate to […] quantize the Einstein equation than it would be to quantize the wave equation for sound in air.”

Ted Jacobson and Einstein’s Equation of State

As described in the introduction, in a 1995 article, Ted Jacobson showed that Einstein’s equations can be obtained by applying thermodynamics laws to the so-called Rindler horizons. His proposal implies that spacetime is emergent.

This idea will be described in detail below, but to fully understand it we first need to review some preliminary concepts from Einstein’s relativity. One of my past articles may be useful as a revision of this material.

Quantum Gravity, Timelessness and Complex NumbersIs The Wave Equation of the Universe Timeless and Real?towardsdatascience.com

It should be noted that if we derive the Einstein equations using arguments from thermodynamics, we cannot interpret the equations geometrically (as it is usually done). If we assume gravity is, by nature, a thermodynamic phenomenon, the Einstein equations must be interpreted using thermodynamic concepts. In other words, gravitational dynamics must be re-expressed in terms of the thermalevolution of spacetime (see this reference).

Preliminary Concepts

We first need to define what are vectors and dual vectors. A manifold can be loosely defined as a space that resembles Euclidean (flat) space near each of its points.

Figure 5: A two-dimensional manifold M with a curve γ parametrized by λ (source).

Consider now a curve γ on the manifold parameterized by λ (see Fig. 5). It can be described by the parametric equations:

Equation 2: Coordinates of the curve γ on M, parametrized by λ.

Fig. 6, shows an example of a curve along the surface of a cylinder. It is given by the following parametric equations:

Equation 3: The λ-parametrized equations of the coordinates of a curve that spirals along the surface of a cylinder. Figure 6: The curve given by Eq. 3.

Now consider a function f defined along the curve γ on M. How does it vary along the curve in terms of the parameter λ?

Equation 4: Variation of the function f along the curve γ in terms of the parameter λ.

where we can identify the components of the tangent vector and the gradient of the function f. The gradient is referred to as a dual vector. Using the “,” notation for partial derivatives we write:

Equation 5: The components of a vector and a dual vector.

Note that in Eq. 4 the Einstein’s summation convention was used to omit the summation symbol.

The figure below shows a tangent vector u to the curve γ parametrized by λ:

Figure 7: The plane tangent to a manifold and a tangent vector at P (source).

Vector and dual vectors transform differently under coordinate transformations:

Equation 6: How the dual vectors and tangent vectors change under a coordinate transformation.

Tensors

We will now consider tensors (see Dirac, for a simple explanation). Vectors are tensors of the type (1, 0). Dual vectors are tensors of the type (0, 1). Following Dirac, to obtain general tensors of higher ranks, we first build the quantity

Equation 7: A simple example of a (2,0)-type tensor.

which is a particular kind of tensor of type (2, 0). Adding several tensors like T, one gets a general tensor of type (2, 0):

Equation 8: An example of a general (2,0)-type tensor.

Under a coordinate transformation, T transforms as:

Equation 9: How a contravariant tensor of second rank changes after a coordinate transformation.

If we have 2 lower (instead of upper) indexes, T is said to be of the type (0, 2). Tensors can also have mixed indexes such as the (1, 1) tensor below:

Equation 10: How the components of mixed tensor change after a coordinate transformation.

Lie derivatives

The Lie derivative is a concept from differential geometry, a mathematical discipline that applies calculus, linear algebra, and multilinear algebra to geometry problems. This type of differentiation, named after the Norwegian mathematician Sophus Lie, evaluates the change of a tensor field along the flowof another vector field (see Wiki).

Figure 8: The Norwegian mathematician Marius Sophus Lie and the front page of his most important treatise, “Theorie der Transformationsgruppen.”

Suppose we have a vector field A in a region of spacetime and a curve γ in the neighborhood of which A is defined. The tangent vector to γ is u = dx/dλ. We consider two points in the curve, x and x+dx.

Figure 9: A vector field A in a region of spacetime, two points x and x+dx inside that region, a curve γ containing both points, and a vector tangent to γ.

Under the infinitesimal change dx

Equation 11: Infinitesimal change of coordinates (see Fig. 9).

the vector A transforms in the following manner:

Equation 12: How the vector A transforms under the change of coordinates in Eq. 11.

Now, the value of the original vector field in x+dx can be written as:

Equation 13: Expansion of the α-component of A(x+dx).

The Lie derivative of A along the curve γ is defined by:

Equation 14: Definition of the Lie derivative of the dual vector A along the curve γ.

The tangent vector u is the direction in which we carry the Lie derivative (see this video for a detailed explanation of the construction of Lie derivatives).

We can better understand the Lie derivative with the following construction. Any vector field is defined by the congruence of curves for which it is the tangent field. We then draw a curve tangent to A at P (the form of the rest of the curve can be anything). We then parametrize the first curve by λ. We choose λ=1 at Pand use the cross curve to fix the parameterization of the other curves to be λ=1. The rate at which λ changes at each curve is fixed by its tangent vectors. We then slide the cross curve by dλ on all curves of the congruence. Note that the cross curve through Q is rigidly tied to the curve through P. There is a vector A’ tangent to the new curve at Q. Since we have two vectors at Q we can subtract them. The Lie derivative is then:

Equation 15: Geometrical definition of the Lie derivative of the dual vector A.

The construction is illustrated below:

Figure 10: The geometrical construction described above.

The Lie derivative is, contrary to its appearance, a tensorial expression since we can rewrite it as:

Equation 16: The Lie derivative written as a tensorial object.

Dual-vectors transform as:

Equation 17: The Lie derivative of the dual vector A along the curve γ.

Now consider a vector A which does not depend on some coordinate, say, x⁰ in some specific coordinate system:

Equation 18: A does not depend on x⁰ in some specific coordinate system.

The * indicates that the corresponding equality is valid in one specific coordinate system. Hence we have a set of curves along which x⁰ increases where A does not change. In this coordinate system:

Equation 19: Tangent vector to set of curves along which x⁰ increases where A does not change Figure 11: Set of curves along which x⁰ increases where A does not change.

From Eq. 19 we obtain:

We can rewrite Eq. 18 as:

These equations imply that:

Equation 20: The Lie derivative of the vector A along the curve with tangent vector U.

This equation expresses the invariance of A in the direction of U. But the Lie derivative on the left-hand side is independent of coordinate systems. Therefore, since it is zero in one coordinate system, it is zero in all coordinate systems.

The Lie derivative of a type-(0,2) tensor is:

Equation 21: The Lie derivative of a type-(0,2) tensor.

Consider some coordinate system the tensor A is independent of some coordinate x⁰. These can be expressed in two different ways:

• To say this in terms of coordinates is Eq. 18
• To say this covariantly is to say that the Lie derivative is zero when the vector field U is aligned with the coordinates where x⁰ is running.

Equation 22: The way to expressing covariantly that A is invariant as we make a translation in a certain direction (in this case U) in spacetime.

Eq. 22 expresses the invariance of the tensor A in the direction of the vector U.

Figure 5: Two ways to express tensor that A is independent of some coordinate x⁰.

This notion becomes very important if we have a symmetry in our spacetime. For example, if vectors don’t depend on time x⁰ or if vectors don’t depend on rotations along some axis, the way to express this in a covariant way is to say that the Lie derivative of that tensor is going to be zero along the appropriate direction, either at time translation or a rotation along an axis. The vector along which you have the symmetry becomes the so-called Killing vector.

Figure 12: The German mathematician Wilhelm Killing (source) and the front page of his article “Die Zusammensetzung der stetigen endlichen Transformationsgruppen” (Math. Ann.33, 1–48) which was considered by the renowned Canadian mathematician A. J. Coleman to be “the most significant mathematical paper he has read or heard about in fifty years” (source).

Killing Vectors and Symmetries

A Killing vector is a vector field ξ such that the Lie derivative of the metric along ξ is zero:

Equation 23: Definition of a Killing vector.

If in a given coordinate system the metric does not depend on the coordinate σ*, the α-component of ξ is:

We can also write ξ and its α-component as:

Eq. 23 says that we have a symmetry of the metric in the direction along which you know ξ points. Symmetries of the metric are called isometries.

Using the definition of Lie derivative we obtain:

Equation 24: Condition obeyed by the Killing vector.

We can use also Killing vectors to obtain constants of motion along geodesics. If u is tangent to a geodesic it is trivial to show that for ξ obeying Eq. 23, the following result follows:

Equation 25: This equation, which is can be easily proved using the equations above, gives us constants of motion along geodesics.

Isometries give origin to conserved quantities along geodesics. More precisely, Killing vectors ξ generate isometries, and transformations under which g is invariant are expressed infinitesimally as motions in the direction of ξ.

For clarity, let us now consider some simple examples of Killing vectors and their corresponding symmetries.

Example 1
Take for example the following metric in R³:

Equation 26: Example of metric in R³.

Note that the metric does not depend on x, y, or z. Therefore, the following three vectors are Killing vectors corresponding to translations:

Equation 27: Three Killing vectors associated with the metric in Eq. 26.

There are other symmetries in R³. Writing Eq. 26 in spherical coordinates, (illustrated below), the metric becomes:

Equation 28: Metric in Eq. 26 expressed in spherical coordinates. Figure 13: Definitions of the variables in Eq. 28.

Since the components of g are ϕ-independent,

Equation 29: The Killing vector corresponding to the ϕ-independence of the metric.

is another Killing vector of R³. In cartesian coordinates this becomes:

Equation 30: The Killing vector in Eq. 29 written in cartesian coordinates.

Rotations around the other two axes give us two other Killing vectors.

Example 2
Consider now a spherically symmetric spacetime (such as the Schwarzchild metric):

Equation 31: Spherically symmetric spacetime.

Since the metric g has no dependence on t and no dependence on ϕ, time translations, and rotations around the z-axis are examples of isometries. We have two obvious Killing vectors:

Equation 32: Two Killing vectors associated with metric Eq. 31.

The following quantities (related to energy and angular momentum per unit mass) are constant along geodesics to which a vector u is tangent:

Equation 33: Two constants along geodesics to which u is tangent.

Killing Horizons

Let us consider a null hypersurface (for example, a light cone). It is, by definition, a hypersurface whose normal vector at every point is null (it has zero length with respect to the local metric tensor g). A Killing horizon Σ is a null hypersurface where the norm of a Killing vector field vanishes. Also, since a null surface cannot have two linearly independent null tangent vectors, ξ will be normal to Σ.

In a Minkowski spacetime in inertial coordinates, the timelike Killing vector that generates boosts, for example, in the x-direction is given by:

Equation 34: The Killing vector that generates boosts in the x-direction in a Minkowski spacetime.

Its norm is given by:

Equation 35: The norm of the Killing vector given by Eq. 34.

When ξ is constant its orbits are hyperboles representing worldlines of uniformly accelerated observers with proper acceleration a = 1/ξ. As ξ → 0 the acceleration increases and the boost Killing vector field generates a bifurcate Killing horizon,

Equation 36: The null surfaces corresponding to Eq. 35.

the so-called Rindler horizon.

These null surfaces are therefore Killing horizons. Since a Killing vector ξ is normal to its Killing horizon Σ, along Σ it obeys the geodesic equation (the κ on the right-hand side of the geodesic equation accounts for the possibility that the integral curves of ξ are not affinely parametrized):

Equation 37: Definition of the surface gravity κ. Note that κ must be evaluated at Σ.

where κ is called surface gravity. For a static spacetime, κ is the acceleration of a static observer close to the horizon, measured by a static observer at ∞.

The Rindler Wedge

Consider an arbitrary spacetime point p. Locally, the spacetime around p is flat (because of the principle of equivalence). Now choose a small patch B of a spacelike 2-surface passing containing P and introduce Riemann normal coordinates (RNC). The metric in RNC is given by:

Equation 38: Metric in RNC at p.

The coordinates of the points on the patch B are:

Equation 39: Coordinates of the points on the patch B.

Adding past and future light sheets, say, in the z-direction we obtain:

Equation 40: Coordinates on the past and future light sheets in the z-direction.

Eq. 40 describes a local Rindler wedge (two null three-dimensional half-planes joined by the spacelike bifurcation, the 2-plane B at t=0) illustrated below.

Figure 14: The Rindler wedge (source).

Consider now a sheet of hyperbolic timelike observers close to the null surface. Their coordinates, velocity, and acceleration

Equation 41: Coordinates, velocity, and acceleration of hyperbolic timelike observers close to the null surface.

where a is the observer acceleration

describe approximately uniformly accelerated observers. For such observers, the light sheets of the Rindler wedge form the Rindler horizon. A causal horizon is the boundary of the spatial region consisting of points causally connected to an observer. More precisely, a causal horizon is a hypersurface that is a boundary between light rays that are directed outwards and moving outwards (dz/dt>0), and those directed outward but moving inward (dz/dt

When you think of knots, you might think of your double-knotted shoelaces or your hopelessly tangled earbuds. I personally think of garlic knots. However, these sorts of knots are different from mathematical knots. What kind of snob wouldn’t consider garlic knots valid knots? Well, the difference is that “knots” like shoelaces have loose ends, meaning we can always untie them. Nothing is locked in. For example, let’s take your “knotted” shoelaces. It is two strings tied together in a way that we can technically untie (though it may seem impossible) and retie another way.

For mathematical knots, the ends are joined together to create a continuous strand.

Let’s look at a more concrete example. Take a hair tie, or a stray rubber band lying around. In their untangled form, these are circles. It might surprise you that these are indeed considered knots! This is known as the unknot, or the trivial knot. It is simply a circle with no crossings. (This is a word that will come up later). Before we get into more complex knots, let’s discuss the recent beginning of knot theory.

Start of Knot Theory

Knots first became a subject of interest in the 1860s. At this time, scientists were trying to understand matter, and a man named William “Lord Kelvin” Thomson developed a theory claiming that atoms were made up of knots. He proposed that every element was a different kind of knot. While trying to sort this out, inevitable questions arose: What sorts of properties do knots have? What makes two knots different? How can we prove they are fundamentally distinct? Let’s try to answer some of these questions!

What sorts of properties can knots have?

We’ll think about knots in terms of simple knot diagrams. First, recall our nice simple knot from before: the unknot. Its diagram is pretty straightforward:

An Unknot

Let’s talk about a related type of knot. Take a simple hair tie. Suppose you’re fiddling with it because you’re bored in your Zoom meeting, and you’re just twisting and turning it and end up with a tough knot. Can you return it to its original circle form? Well, yes! It might take some effort, but you can theoretically just undo each of your actions and return it to a circle. But while it’s all tangled up, it may not appear to be the unknot — it’s disguised! Knots that can deform (through some set of moves) into the unknot are called culprit knots. (Yes, mathematicians do have a sense of humor!)

A Culprit Knot

Here is an example of a (truly deceitful) culprit knot.

Now, let’s take it up a (k)notch. The next standard knot is called the trefoil knot, and its knot notation is: 3_1. What is this nonsense? Let’s unpack it. First, here is a picture of the standard, simplest form of the trefoil knot:

Can you guess what the three stands for? Try to think about it from the perspective of a mathematician for a second, and see if you can find a nice reason… If you guessed that the 3 represents the number of intersections, or crossings, the knot has with itself, you would be right! Now, look carefully at the knot and guess what the 1 stands for… Okay sorry, that was a trick question. The subscripts are just an index to keep track of all the different knots that have the same number of crossings. Interestingly enough, the trefoil is the only knot with three crossings (up to deformation). For the standard knot notation of a knot, the number is something called the crossing number, which is the minimum possible number of crossings in the knot diagram (meaning there is no way to unknot it or deform it into something with fewer crossings), and the subscript is just a label counting how many we have of that type.

Here are the first several standard knots. In the diagram, 3_1 is our trefoil knot and 4_1 is known as the “figure 8” knot.

What makes knots the same?

Now you might ask, how can we be sure that some of these knots are not secretly the same? Maybe we just didn’t untangle them the right way. Could these all be devious culprits? Well, we say that two knots are the same if one can be deformed into the other without letting it pass through itself. This process of deforming a knot without it passing through itself is called ambient isotopy. So, the next time someone calls you out for tangling and detangling your hair-tie (if this ever happens), tell them you’re practicing ambient isotopy!

Okay, so now we know how to tell if knots are the same — you just have to see if you can deform one into another. So do we just have to keep trying to untangle until we get lucky? Is there some sort of method? Why, what a great question! There is indeed!

Reidemeister Moves

I – Twist, II – Poke, III – Slide

In the 1930s, Kurt Reidemeister proved that all deformations could be reduced to three moves: a twist (I), a poke (II), and a slide (III). This is a fantastic result because it means that we have a finite number of moves we can make, which really simplifies things. So, we can now say that two knots that are equivalent can be related by a sequence of these three moves. In other words, if you can transform a knot into another knot via these moves, they are equivalent (up to ambient isotopy). Remember our example culprit knot from before? This is the process using Reidemeister moves showing that it can be deformed into the unknot. See if you can figure out which of the three moves is being used at each step!

A culprit knot deforming into the unknot.

This is an amazing tool to prove that two knots are the same. But, can we use it to show that two knots are different? Say we are trying to deform a trefoil knot into an unknot with these moves. As we try and twist and turn it, you probably can convince yourself that it’s not possible, but can you know for sure? Here’s where it gets a little more complicated.

This is where something called invariants comes into play. Invariants are characteristics of knots that remain unchanged, and are unaffected by Reidemeister moves. One example of an invariant is something called tricolorability. Tricolorability of a knot is its ability to be colored with 3 different colors. In order for a knot to be tricolorable, at least two colors must be used and incident crossing strands are all either the same color or all different colors. This second rule is not the easiest to understand, so if you don’t get it, don’t worry about it!

You might ask: Why do we even care about tricolorability? Well, it creates two categories of knots: tricolorable knots and non-tricolorable knots. If our two knots fall into different categories, they cannot be equal. This gives us a really nice way to prove beyond doubt that two knots are different!

Take our example of trying to deform the trefoil into the unknot. We can color the trefoil with 3 different colors in a way that satisfies the two rules, as shown below. Think about why it might satisfy the second rule.

It satisfies the second rule because each of the crossing strands are all different colors. So, the trefoil is tricolorable! What about the unknot? It’s just a single circle so it can’t even be colored with two colors, breaking the first requirement for tricolorability. Thus, the unknot is non-tricolorable. Since this characteristic is invariant (meaning it’s an unchangeable characteristic of a knot) we’ve shown that the unknot and trefoil cannot be equal. Hooray! You can stop twisting and turning that trefoil.

We’re not going to get into other invariants, because they can be pretty complex. You’re welcome to look some up! Now, you might have noticed that it can be pretty difficult to categorize these knots. It’s still an open challenge to find nice descriptors that make knots unique.

Knots can actually be found in nature, structure in DNA, and in the sun’s corona! People are also considering using knots to make money more secure. When quantum computers are more prevalent, quantum money can be encoded with knots, and algorithms will check knots. Because of their complexity, this could be a pretty secure system.

And that’s where we’ll end it for today! Final question: can you knot? I don’t know about you, but I can knot. Now I’m going to go eat some garlic knots. Until next time!

If you found this interesting, make sure to check out the next one in a couple weeks! If you have any questions or comments, please email me at [email protected].

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This column, Gems in STEM, is a place to learn about various STEM topics that I find exciting, and that I hope will excite you too! It will always be written to be fairly accessible, so you don’t have to worry about not having background knowledge. However, it does occasionally get more advanced towards the end. Thanks for reading!

Puzzle

What numbers do they need to contain to display all days of the month?

Photo by Iga Palacz on Unsplash

Calendar Dice are a popular gadget on German office desks. Two dice can be moved in a small box to display the current day of the month. They’re decorative and it’s not even clear that it is possible to display any number between 01 and 31 using only two dice.

In fact, it’s a nice little puzzle to figure out which numbers you need to place on the dice to be able to display all dates in a month.

Since there are ten numbers (0, 1, 2, 3, 4, 5, 6, 7, 8, 9) and only six sides per die, we obviously cannot put all ten numbers on each die. So we have to make a selection. But what is the right selection to ensure that all days of the month can be displayed?

Before you scroll down to the solutions, I invite you to pause for a minute and think about it yourself.

Each die has six sides. This means that we can put up to six different numbers on each one.

Since two possible days are the 11th and the 22nd, we need to have the numbers 1 and 2 on both dice. Also, since every number needs to be paired with a 0 to display days 1–9 properly, we also need to put the number 0 on each die.

Die A: 0, 1, 2

Die B: 0, 1, 2

Now we can take the remaining numbers 3,4,5,6,7,8 and 9 and distribute them across the two dice, leading to

Die A: 0, 1, 2, 3, 4, 5

Die B: 0, 1, 2, 6, 7, 8, 9

Well, looks like we have a problem here. Die B now has seven numbers instead of only six. How can we solve the problem?

It took me a surprisingly long amount of time to figure it out, but the answer is simple once you’ve found it. We don’t need a 9 as an upside-down 6 serves as a 9!

Therefore we get

Die A: 0, 1, 2, 3, 4, 5

Die B: 0, 1, 2, 6, 7, 8

Of course, we can also choose to place the numbers {3,4,5,6,7,8} onto the remaining sides of the dice in whichever combination we like.

In fact, there is a surprising amount of

possible combinations to do so and to distribute the numbers on the two dice!

Cryptography

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Applied Mathematics

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Understanding Pierre de Fermat’s observation about prime numbers

Quantum Computing

How Quantum Computers Will Break Your Phone’s Encryption

These days all your devices are encrypted. Your texts, emails, all have …