Science and Math Articles

Here is the collection of expert STEM articles that span all the sciences and mathematics. Some examples include physics, mathematics, biology, chemistry, cosmology, and earth sciences. For math, we cover algebra to advanced calculus.  These articles are all peer-reviewed by the community.

anyon

Learn The Main Conceptual Ideas of Anyon Particles

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55 Comments
Every quantum physicist knows that all particles are either bosons or fermions. And the standard textbook argues that this so does not depend on the number…
frameofreference2

Frames of Reference: A Skateboarder’s View

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22 Comments
My essay Explaining Rolling Motion raised some commentary about frames of reference and their equivalence when solving physics problems. I wish to pursue…
relativitygauge

Exploring General Relativity as a Gauge Theory

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24 Comments
The fundamental interactions of the Standard Model are described by Yang-Mills theory. This is a gauge theory, which means the following: (1) Choose a…
govandteachers

Blaming Government for Teacher and Scientist Failures in Integrity

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46 Comments
The article, "Governmental policy is wrecking science," makes some interesting points but is fundamentally in error, because government policy is only…
quantumparticle

Spectral Standard Model and String Compactifications

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28 Comments
 The Connes-Lott-Chamseddine-Barrett model is the observation that the standard model of particle physics -- as a classical action functional,…
cmbprimer4

A Poor Man’s CMB Primer: Cosmic Acoustics

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1 Comment
 Before decoupling, photons and charged particles were in good thermal contact: though the primordial plasma might have varied in temperature…
spacetimecon

Struggles with the Continuum: Spacetime Conclusion

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4 Comments
 We've been looking at how the continuum nature of spacetime poses problems for our favorite theories of physics --- problems with infinities.…
11dgravity

11d Gravity From Just the Torsion Constraint

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17 Comments
 It is familiar that Einstein gravity may be formulated in terms of a vielbein field together with a "spin connection", subject to the constraint…
spacetime7

Struggles with the Continuum: General Relativity

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2 Comments
 Combining electromagnetism with relativity and quantum mechanics led to QED. Last time we saw the immense struggles with the continuum this…
ohmslaw

Learn Why Ohm’s Law Is Not a Law

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41 Comments
At first, I wanted this title to say “Ohm’s law is not a Law.”  But someone else used that phrase in a recent PF thread, and a storm of protest…
carcapacity

Why Road Capacity Is Almost Independent of the Speed Limit

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29 Comments
Let us start with a familiar situation. Take a familiar part of a road, for example, the one from your home to work. How long will it take you to drive…
quantumhard

Why Is Quantum Mechanics So Difficult?

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254 Comments
Quantum Mechanics Key points:Quantum mechanics (QM) is often perceived as difficult, especially by non-physicists. The difficulty lies in the…
Supersymmetry

Learn About Supersymmetry and Deligne’s Theorem

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54 Comments
 In 2002, Pierre Deligne proved a remarkable theorem on what mathematically is called Tannakian reconstruction of tensor categories. Here…
LHC4

LHC Part 4: Searching for New Particles and Decays

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17 Comments
 The LHC experiments are in full swing collecting data this year (more information in the forum), and a while ago the collaboration of the…
acusticbeats

Acoustic ‘beats’ from Mismatched Musical Frequencies

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11 Comments
Introduction In late high school physics courses and first-year university courses, the phenomenon of acoustical 'beats' (digital audio) is often covered.…
polymer

Polymer Physics and Genetic Sequencing

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4 Comments
Polymer Key PointsDNA is a good model system for polymer physics as it is small enough to observe single-molecule dynamics and large enough to be…
groupsandgeometry

Exploring the Relationship Between Group Theory and Geometry

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7 Comments
There is a very deep link between group theory and geometry. Sadly, this link is not emphasized a lot in most courses of group theory, even though it is…
computers

An Interesting Ramsey Theory Riddle

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5 Comments
Ramsey theory has its origins in a very nice riddle Consider a party of 6 people. Any two of these 6 will either be meeting each other for the first time…

Learn About Spacetime Diagrams of Light Clocks

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25 Comments
We demonstrate a method for constructing spacetime diagrams for special relativity on graph paper that has been rotated by 45 degrees. Many quantitative…

A Black Hole Problem From Greg Egan’s “Incandescence”

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8 Comments
The following problem was inspired by reading Greg Egan's "Incadescence".  Consider a body, orbiting a Schwarzschild black hole and tide locked to it,…
softwarenotperfect

Why Your Software is Never Perfect

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52 Comments
We occasionally have students ask for help on software, "My software is perfect, but it doesn't work!" Your software is never perfect. My software is never…
Photoacoustic

The Interaction of Sound and Light

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6 Comments
I recently wrote a post on my blog about a fairly esoteric idea regarding sound propagating through light. This inspired me to write an article about…
logicp3

Scientific Inference: Balancing Predictive Success with Falsifiability

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9 Comments
  Bayes' Theorem: Balancing predictive success with falsifiability Despite its murky logical pedigree, confirmation is a key part of learning.…
logicp2

Scientific Inference: Do We Really Need Induction?

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14 Comments
  Part 2. "We don't need no stinkin' induction" -- Sir Karl Popper The traditional scientific method supposedly employs induction both in the…
antigravity

Roger Babson’s Anti-Gravity Contest and Modern Physics

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13 Comments
In the early 20th century, an eccentric businessman named Roger Babson declared gravity to be mankind's greatest threat. He devoted part of his fortune…
logic1

Scientific Inference: Logical Induction Ain’t Logical

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39 Comments
 Three million years ago, the grandfathers of our genus used sharpened stones to chop wood and cut bone. Today, surgeons use fiber optic-guided…
virtualparticlesmyths

Learn the Top Misconceptions about Virtual Particles

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1 Comment
This Insight Article is a sequel to the Insight Article ''The Physics of Virtual Particles'', which contains an exposition of definitions that are physically…
acpoweranalysisp3

AC Power Analysis: The 4 Levels of Cyber Resilience

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9 Comments
 Power system security is much too big a topic for this article.  But I can squeeze in a few points about vulnerability to cyber attacks.…
nightsky

How Representative is the Night Sky?

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3 Comments
A couple of weeks back, a thread was started about whether or not most of the stars we see at night are brighter than our Sun. The consensus was yes. I'd…
encryption

Is It Possible to Design an Unbreakable Cipher?

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10 Comments
Is it possible to design an unbreakable cipher? Do methods of encryption exist that guarantee privacy from even the most capable and highly-resourced…
quantumentanglement

How the EPR Paradox Fits Between Entanglement and Nonlocality

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11 Comments
If we violate a Bell inequality, we know we have quantum entanglement, and it used to be thought that these two notions were one and the same.At least,…
acpoweranalysisp2

Intro to AC Power Analysis: Network Analysis

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26 Comments
 Let me use the terms “power grid” and “network” interchangeably. What is the Power Grid Required to Do?Deliver energy to customers,…
Newton-Cartan-Theory

The Revival of Newton-Cartan Theory in General Relativity

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12 Comments
It's a calculation that you can find in any textbook on General Relativity (GR): when gravity is weak and does not vary with time, and when particles are…
virtualparticles

Learn the Physics of Virtual Particles in Quantum Mechanics

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33 Comments
In discussions on the internet (including a number of Wikipedia pages) and in books and articles for non-experts in particle physics, there is considerable…
universeexpand

Learn about the Big Bang and the Expansion of the Universe

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29 Comments
The "ordinary Big Bang and expansion" (no inflation, no dark energy) Einstein's General Relativity allows a solution (FLRW metric) where an empty Universe…
acpoweranalysisp1

Intro to AC Power Analysis: Learn System Basics

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13 Comments
 Instantaneous PowerThe formula for power is: ##P=V\cdot I## , power= voltage*current.   We call that, instantaneous power.   Even…
cipher

The Monographic Substitution Cipher: From Julius Caesar to the KGB

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7 Comments
A monographic substitution cipher works by replacing individual characters of plaintext with corresponding characters of ciphertext. It is perhaps the…
pet_scan

Learn the Basics of Positron Emission Tomography (PET)

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7 Comments
PET Key PointsPaul Dirac first theorized positrons in 1928 and they were observed four years later by Carl Anderson. Positron Emission Tomography…
tcp_ip

How to Manage TCP/IP In Automation or Measurement Networks

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10 Comments
The 1990s saw the widespread adoption of network technology on two fronts: Office automation and factory automation. But while office automation soon standardized…
Ethernettime

Learn Time Synchronization Across Switched Ethernets

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18 Comments
Now and then you come across measurement problems that are tightly associated with the notion of synchronicity, meaning that things need to happen simultaneously.…
qminterpret

Tools to Enrich our Quantum Mechanics Interpretations Discourse

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13 Comments
IntroductionMany times, we encounter ourselves tangled in some Quantum Mechanics (QM) interpretations debates. Unfortunately, the standard…
spacetime6

Struggles with the Continuum – Freeman Dyson and QED

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24 Comments
 Last time I sketched how physicists use quantum electrodynamics, or 'QED', to compute answers to physics problems as power series in the…
Mitochondria

When did Mitochondria Evolve?

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7 Comments
Historically, life is categorized into three broad domains: bacteria, archaea, and eukaryota.  Whereas bacteria and archaea are almost all relatively…
islingmodel

High Temperature and Low Duality for the Ising Model on an Infinite Regular Tree

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1 Comment
Today I felt like sharing a computation that I thought was interesting. I'm sure I'm not the first to explore the Ising-Ising duality in this context,…
secondaryforces

Introduction to the Secondary Forces in Physics

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7 Comments
Many are familiar with the "fundamental forces" of nature: gravity, electromagnetism, and the strong and weak nuclear forces. Three of these tend to bind…
complexmath

Hear the Case for Learning Complex Math

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15 Comments
Resistance to complex math seems to never die out.  I see it frequently in PF posts.  Often it takes the form of challenges rather than questions. …
spacetime5

Struggles with the Continuum: Quantum Electrodynamics

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10 Comments
  Quantum field theory is the best method we have for describing particles and forces in a way that takes both quantum mechanics…
darkmatter

Learn About Knut Lundmark and the Prehistory of Dark Matter

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16 Comments
Look in your standard cosmology textbook (in my case, the closest one at hand is Barbara Ryden's Introduction to Cosmology) and go to the section that…
spacetime4

Struggles with the Continuum – Relativity and Quantum

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5 Comments
 In this series, we're looking at mathematical problems that arise in physics due to treating spacetime as a continuum---basically, problems…
complexnumbers

Things Which Can Go Wrong with Complex Numbers

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3 Comments
At the first sight, there are many paradoxes in complex number theory. Here are some nice examples of things that don't seem to work:Example A [itex]-1=i^2=\sqrt{-1}\cdot\sqrt{-1}=\sqrt{(-1)(-1)}=\sqrt{1}=1[/itex]Example…
theneedforinfinity

Why the Need of Infinity in Physics

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31 Comments
It is beyond doubt that the notion of infinity lies somewhere near the core of all mathematics, probably in a finite vicinity of zero. But where does it…
complexscience

Thoughts on the Complexity of Modern Science

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122 Comments
Science today is incredibly complex. Anyone who ever did scientific research knows that, but many in the general public seem to be completely unaware of…
MillenniumPrize

Intro to the Millennium Prize Problems

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13 Comments
IntroductionIn this Insight, I will go over the background information for the Millennium Prize problems and briefly describe three of them. A future…
rainbows

Exploration into the Physics of Rainbows

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88 Comments
https://www.youtube.com/watch?v=JYr0LNWwhAs For several years, I have been contemplating this beautiful picture by photographer Brian McPhee.  I have…
micro2

Learn Axioms for the Natural Numbers

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0 Comments
** Bloch Chapter 1.2The Peano system in Bloch has a special element ##1\in \mathbb{N}##. The intuitive idea here is that ##\mathbb{N} = \{1,2,3,...\}##.…
micro3

An Intro on Real Numbers and Real Analysis

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0 Comments
It is important to realize that in standard mathematics, we attempt to characterize everything in terms of sets. This means that notions such as natural…
earthgravity

Learn All About Earth’s Gravity

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18 Comments
Earth's gravitational field at the surface is approximately 9.8 Newtons/kilogram or 9.8 meters/second/second. But how does that change due to its shape,…
greenhouse

Frequently Made Errors in Climate Science – The Greenhouse Effect

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87 Comments
1.What is meant by "The Greenhouse Effect"? Many gases, such as H2O, CO2, CH4, are transparent to visible light but absorb and emit parts of the infrared…
speedoflight1

Learn About the Speed of Light and Galilean Relativity

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43 Comments
Speed of Light Key PointsEarly experiments attempted to measure the speed of light, including Galileo's lantern experiment, Ole Rømer's eclipse…
cancer

What Causes Cancer: Bad Luck or Bad Lifestyles?

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23 Comments
What causes cancer? For the most part, cancer is a disease that arises from mutations in the body that accumulate over time.  These mutations knock out…
buoyancy

Understanding Buoyancy: The Limits of Archimedes

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19 Comments
It’s a principle as old as science itself, and remarkably, remains essentially unchanged in the light of revolution after revolution in our understanding…
physicsturnedmath

Trials and Tribulations of a Physicist who Became a Math Geek

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6 Comments
How did I go from the brink of changing my major from physics to ceramics (no more math) to the Math faculty of the Air Force Academy? How did I go from…
studymath

How to Self-Study Basic High School Mathematics

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5 Comments
  IntroductionWe often get questions here from people self-studying mathematics. One of those questions is what mathematics should I study…
studymath1

The Essential Guide to Self Study Mathematics

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13 Comments
  How to self-study mathematics? People self-study mathematics for a lot of reasons. Either out of pure interest, because they want to get ahead,…
seeelectron

Have Scientists Seen an Electron?

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18 Comments
This is not about seeing an electron, but rather, the notion that seeing something with our eyes is the end-all requirement for the validity of anything.…
peterblockuniverse

The Block Universe – Refuting a Common Argument

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153 Comments
The "block universe" interpretation of SR has come up repeatedly in threads here on PF. Rather than link to them, I want to summarize a common argument…
Retrocausality

Understanding Retrocausality and Blockworld

In an Insights series “Blockworld and its Foundational Implications,” I quoted Huw Price and Ken Wharton in several places advocating a blockworld…
blockuniverse5

Blockworld and its Foundational Implications: Delayed Choice and No Counterfactual Definiteness

 In parts 1 and 2 of this 5-part Insights series, I explained the blockworld (BW) implication of special relativity. In part 3, I introduced…
blockuniverse4

Blockworld and its Foundational Implications: General Relativity and Closed Timelike Curves

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2 Comments
 In parts 1 and 2 of this 5-part Insights series, I explained the blockworld (BW) implication of special relativity (SR). In part 3, I introduced…
blockuniverse3

Blockworld and its Foundational Implications: General Relativity and the Big Bang

 In parts 1 and 2 of this 5-part Insights series, I explained the blockworld (BW) implication of special relativity (SR). Geroch sums up the…
blockuniverse2

Blockworld and Its Foundational Implications: The Relativity of Simultaneity and Blockworld

 In part 1 of this 5-part Insights series, I introduced two consequences of the second postulate of special relativity (SR): time dilation…
blockuniverse1

Blockworld and Its Foundational Implications: Time Dilation and Length Contraction

 This is the first in a 5-part series of Insights that will introduce blockworld (aka "block universe”) and use it to address a puzzle…
DNA

Atomic Positioning with DNA Hinges

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11 Comments
DNA Hinge Key PointsFunke and Dietz created a DNA hinge which is adjustable by varying the length of one of the DNA molecules, allowing positioning…
prequantum4

Examples of Prequantum Field Theories IV: Wess-Zumino-Witten-type Theories

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4 Comments
  At the end of this page, we will have come full circle back to the first article in the series 20 years ago -- The M-Theory Conjecture.We…
prequantum3

Examples of Prequantum Field Theories III: Chern-Simons-type Theories

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1 Comment
  After having constructed gauge fields and higher gauge fields in the previous article by systems of ##L_\infty##-algebroid-valued…
guagetheory

Examples of Prequantum Field Theories II: Higher Gauge Fields

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1 Comment
   After having recalled ordinary gauge fields from a dg-algebraic perspective in the previous article, here I discuss…
prequantumft

Examples of Prequantum Field Theories I: Gauge Fields

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1 Comment
  After having motivated the need for prequantum field theory and having laid out its principles (i. extremal action, ii. global…
animationwalk

Fun with Self-Avoiding Walks Simulations

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16 Comments
This post is about some simulations I did of self-avoiding random walks. These are what they sound like with each step, the position of the walk moves…
prequantumgeometry5

Higher Prequantum Geometry V: The Local Observables – Lie Theoretically

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1 Comment
 This article discusses how the previous considerations naturally follow the concepts of local observables of local field theories and of…
prequantumgeometry4

Higher Prequantum Geometry IV: The Covariant Phase Space – Transgressively

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13 Comments
  The Euler-Lagrange ##p##-gerbes discussed in the previous article are singled out as being exactly the right coherent refinement…
prequantumgeometry3

Higher Prequantum Geometry III: The Global Action Functional – Cohomologically

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9 Comments
  The previous article ended with the concept of classical locally variational field theories, of which a class of examples are…
blackbody

A Poor Man’s CMB Primer: Bumps on a Blackbody

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15 Comments
 Astronomers Arno Penzias and Robert Wilson discovered the cosmic microwave background in 1965. They were not looking for it.They were…
prequantumgeometry2

Higher Prequantum Geometry II: The Principle of Extremal Action – Comonadically

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1 Comment
  The previous article motivated the importance of considering "pre-quantum field theory" in-between classical and quantum field…
redshiftblackholes

Learn Time Dilation and Redshift for a Static Black Hole

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76 Comments
The following is an overview of the time dilation and gravitational redshift effects of a static (Schwarzschild) black hole. By general relativity, a strong…
prequantumgeometry

Higher Prequantum Geometry I: The Need for Prequantum Geometry

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18 Comments
 Before proceeding with a discussion of the super p-brane sigma models, whose emergence from the superpoint I discussed in the previous article,…
atoms

Can We See an Atom?

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12 Comments
In this article, I will discuss how these images of atoms are made, what exactly we are looking at, and what it means to see an atom. Over the decades…
gravity3

Does Gravity Gravitate: The Wave

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17 Comments
 In the first two posts in this series, we looked at different ways of interpreting the question "does gravity gravitate?" We left off at…
gravitate2

Does Gravity Gravitate? Part 2

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30 Comments
 In the first post of this series, I talked about two ways to answer the title question, one leading to the answer "no" and the other leading…
birthcmb

A Poor Man’s CMB Primer: The Birth of a Cosmic Background Radiation

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3 Comments
 The early universe was hot. So hot that nuclei boiled. The great thermal energy of the universe overwhelmed the confining efforts of the…
modulation

Understand the Acoustic Modulation vs. Beating Confusion

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36 Comments
A long time ago I read a paper in the IEEEProceedings recounting the history of the superheterodyne receiver. Overall it was a very interesting and informative…
balloonanalogy

Learn Inflation Balloon Analogy Misconceptions

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62 Comments
The Balloon Analogy is a simple-minded way to help describe (but not completely describe) two facts of cosmology that are difficult for many people to…
spacetime4

Struggles With the Continuum: Point Particles and the Electromagnetic Field

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20 Comments
 In these posts, we're seeing how our favorite theories of physics deal with the idea that space and time are a continuum, with points described…
bubbles

Explosion-Generated Collapsing Vacuum Bubbles Reach 20,000 Kelvin

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5 Comments
Key PointsJérôme Duplat and Emmanuel Villermaux published a paper in Physical Review Letters (PRL) to generate centimeter-sized vacuum bubbles…
cmb

A Poor Man’s CMB Primer: Orientation of the Universe

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4 Comments
 This is a picture of the cosmic microwave background:Fig 1. The cosmic microwave background as seen by the European Space Agency's…
planck

What Planck Length Is and It’s Common Misconceptions

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29 Comments
The Planck length is an extremely small distance constructed from physical constants. Many misconceptions generally overstate its physical significance,…
spacetime2

Struggles With The Continuum: Quantum Mechanics of Charged Particles

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39 Comments
 Last time we saw that nobody yet knows if Newtonian gravity, applied to point particles, truly succeeds in predicting the future. To be precise:…
imagination

Why Imagination Without Knowledge Is Ignorance Waiting to Happen

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15 Comments
Introduction My feelings on people who think that imagination is more important than knowledge are well-known. These people simply are parroting Einstein's…
animal speeds

Exploring a Paper on Scaling Laws and the Speed of Animals

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19 Comments
Key PointsScaling arguments are powerful tools in physics between vague descriptive arguments and rigorous formulae. Examples of scaling arguments…
infinity_faq

Learn the Top 5 Misconceptions About Infinity

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17 Comments
Introduction Understanding the behavior of infinity is one of the major accomplishments of mathematics. Sadly, the infinite is often misunderstood and…
man_on_moon

Last Man On The Moon | Look Back at Previous Missions

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0 Comments
On December 11, 1972, astronaut Eugene Cernan became the last person to walk on the lunar surface, marking the conclusion of NASA's Apollo missions—a…