Physics, Math and Science Articles

Independent Research: Thinking Outside The Box Versus Knowing What’s In The Box

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In Memory of Dr. Thomas J. LeCompte (1964-2025), Detector Designer and Champion of Education and Science. Prologue Someone who shows interest in science is initially a welcome development. So are fresh ideas from unexpected quarters. In contrast, there is a scientific community that is meticulously organized down to the last detail, allowing little to no…

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Ice Quiz: All About Ice Trivia

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8 Comments
Winter is coming. First day of Winter is December 22nd. Time to learn a bit about a substance that the northern regions will encounter. Don't forget to…
cancer

What Causes Cancer: Bad Luck or Bad Lifestyles?

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22 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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18 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…
observableuniverse

Why the Observable Universe Radius Exceeds Its Age

The radius of the observable universe is about 46 billion light-years, which is considerably greater than its age of about 14 billion years. The radius…
heat

What Is Heat? Definition in Thermodynamics Explained

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13 Comments
One of the most frustrating misconceptions in thermodynamics centers on the actual definition of heat. Many science and engineering students refer to a…
ODE2

How to Solve Nonhomogeneous Linear ODEs using Annihilators

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3 Comments
My previous Insights article, Solving Homogeneous Linear ODEs using Annihilators, discussed several examples of homogeneous differential equations, equations…
steadystate

What Is the Steady-state Model and Why It’s No Longer Viable

Introduction to the Steady State Model Back in 2011, I wrote a FAQ entry for Physics Forums on the steady state model and why it is no longer viable.…
ODE1

Solving Homogeneous Linear ODEs using Annihilators

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11 Comments
In this Insights article we'll look at a limited class of ordinary differential equations -- homogeneous linear ODES with constant coefficients. Although…
physicsturnedmath

Trials and Tribulations of a Physicist who Became a Math Geek

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4 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…
lineartransformations

Learn About Matrix Representations of Linear Transformations

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20 Comments
Let X and Y be finite-dimensional vector spaces. Let ##T:X\to Y## be a linear transformation. Let ##A=(e_1,\dots,e_n)## and ##B=(f_1,\dots,f_m)## be ordered…
electronatom

Why Electrons Don’t Crash Into the Nucleus — Quantum View

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12 Comments
Atoms and Coulomb forces If atoms were described only by Coulomb attraction, the electron and nucleus would attract each other and, according to classical…
airplanelift

How Does an Airplane Wing Work? A Primer on Lift

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56 Comments
Many people ask how an airplane wing works, and several answers are commonly given. You might have even seen vigorous arguments between proponents…
blackholesuniverse

Why the Early Big Bang Universe Didn’t Form a Black Hole

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7 Comments
Why the Big Bang Is Not a Black Hole In the early universe, the matter was gathered together at very high density — so why wasn't it a black hole? Big…
pythondebug2

Simple Python Debugging with Pdb: Part 2

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2 Comments
This Insight article is the continuation of the first article, Simple Python Debugging with Pdb: Part 1.In this article, let's look at another important…
pythondebug1a

Learn Simple Python Debugging with Pdb

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6 Comments
I'm pretty new to Python, so I was looking around for some debugging tools. At first, I dismissed Pdb (Python debugger) as being too primitive, but after…
goingbacktoschool

A Guide to Going Back to School For Math or Science

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5 Comments
Motivations for creating the series I came to Physics Forums originally to read ZapperZ’s thread about becoming a physicist back in 2013. It was one…
electrostaticfield1

Learn About Energy Gained by Charge in an Electrostatic Field

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2 Comments
I have seen this question being asked frequently on here. A lot of students have a bit of an issue in understanding why, if an electron is placed in a…
studymath

Math Self-Study Roadmap: Topics & Book Recommendations

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5 Comments
Introduction We often get questions here from people self-studying mathematics. One common question is: what mathematics should I study, and in what…
studymath2

How to Self-Study Calculus: Topics, Order & Book Guide

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26 Comments
We often get questions here from people self-studying mathematics. One common question is: "What mathematics should I study and in what order?" To…
studymath1

The Essential Guide to Self Study Mathematics

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12 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? How We Detect Them

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18 Comments
Have Scientists Seen an Electron? Why Seeing Isn't Enough This article is not about literally seeing a single electron with your eye; it addresses the…
peterblockuniverse

The Block Universe – Refuting a Common Argument

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154 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…
twinparadox

A Geometrical View of Time Dilation and the Twin Paradox

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90 Comments
Based on the number of questions we receive on the topic at Physics Forums, there is a lot of confusion in the general public about how time dilation works…
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…
universespinning

Is the Universe Rotating? Evidence, Tests & Limits

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3 Comments
Can we tell whether the universe is rotating? If you believe wholeheartedly in Mach's principle, there is no way to test empirically for the rotation…
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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13 Comments
I was perusing the Nature Nanotechnology website today when I came across an interesting article by Funke and Dietz, called "Placing molecules with Bohr…
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…
uncertaintyenergy

Does Heisenberg Uncertainty Violate Energy Conservation?

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2 Comments
The short answer is: No — there is no violation. The longer answer Given the probabilistic aspect of quantum theory, what do we mean now by "conservation…
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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3 Comments
  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…
999equals1

Why 1 Equals 0.999… — Explanations & Rigorous Proofs

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113 Comments
Why do people say 1 and 0.999... are equal? Aren't they two different numbers? No — 1 and 0.999... really are the same number, although that can feel…
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…
photonsmedium

Do Photons Move Slower in a Solid Medium?

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5 Comments
This question appears often because it has been shown that in a normal, dispersive solid such as glass, the speed of light is slower than it is in a vacuum.…
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

Time Dilation & Redshift of Schwarzschild Black Holes

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76 Comments
Time dilation near a Schwarzschild black hole The following is an overview of the time-dilation and gravitational-redshift effects of a static (Schwarzschild)…
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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11 Comments
In this article I discuss how images of atoms are made, what exactly we are looking at, and what it means to "see" an atom. Over the decades there have…
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…
lightspeedrod

Why Rigid Rods Can’t Send Faster-Than-Light Signals

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52 Comments
Why a rigid rod can't send faster-than-light signals One common proposal for achieving faster-than-light (FTL) communication is to use a long, perfectly…
gravitate2

Does Gravity Gravitate? Part 2

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33 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…
proofs

Mathematical Proofs: How to Understand and Write Them

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28 Comments
Introduction This FAQ is about proofs. Proofs are central to mathematics, and writing proofs is a skill many people find hard to master. There are two…
blockuniverse

PF’s policy on Lorentz Ether Theory and Block Universe

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81 Comments
What is the PF's policy on Lorentz Ether Theory and Block Universe?Debates about the superiority or "truth" of modern Lorentz Ether Theory (LET) and…
doubt

Doubt, Discouragement, and Perseverance in STEM Careers

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Doubt, as odd as this may sound, can be essential to living. We all make decisions and later question whether we made the right choice; doubt helps influence…
gravitate

Does Gravity Gravitate?

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9 Comments
 The title question of this article is one that often comes up in PF threads, and I would like to give my take on it. This will be the first…
pluto

Pluto Quiz: Think You Know Pluto?

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5 Comments
Pluto has been in the news quite a bit these past several months. Let's see if you've been paying attention. Feel free to share your score in the comments…
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

Mixing vs Beating: Superheterodyne & Beat Frequency

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36 Comments
Mixing vs. Beating: a brief history A long time ago I read a paper in the IEEE Proceedings recounting the history of the superheterodyne receiver. Overall…
balloonanalogy

Balloon Analogy Explained: Cosmology, Expansion & Limits

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63 Comments
The Balloon Analogy is a straightforward way to illustrate — without being complete — two important points of modern cosmology that many people find…
gravityacceleration

Why Is Acceleration Due to Gravity a Constant?

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5 Comments
This question has popped up many times. So here is an attempt to address it.To answer this question at the elementary level, several assumptions will…
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
In a recent paper published in Physical Review Letters (PRL), one of the top physics journals, Jérôme Duplat and Emmanuel Villermaux developed a method…
what_is_evolution

What Is Evolution? How It Works — Mechanisms & Evidence

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What is evolution? In every field of scientific endeavor there sometimes comes a quantum shift in knowledge, a grand denouement that changes how we see…
cardinalnumbers

Informal Introduction to Cardinal Numbers

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1 Comment
Cardinal numbers We will now give an informal introduction to cardinal numbers. We will later formalize this by using ordinal numbers. Informally, cardinal…
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…
speedoflight

Why Does C Have a Particular Value, and Can It Change?

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53 Comments
Short answer: Because c (speed of light) has units, its value is what it is only because of our choice of units, and there is no meaningful way to test…
planck

What Planck Length Is and It’s Common Misconceptions

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33 Comments
The Planck length is an extremely small distance constructed from physical constants. Many misconceptions overstate its physical significance, claiming…
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:…
procrastination

Overcome Procrastination: Practical Student Strategies

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0 Comments
Introduction It's 6:30 in the morning. You've just woken up and feel so sleepy you think, "A few more minutes can't hurt," and drift back under your comforter.The…
gravitywaves

How Fast Do Changes in the Gravitational Field Propagate?

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20 Comments
General relativity predicts that disturbances in the gravitational field propagate as gravitational waves, and that low-amplitude gravitational waves travel…
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

Animal Speed Scaling: Body-Lengths per Second Across Sizes

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19 Comments
Introduction In a recent American Journal of Physics issue, I read an interesting paper by Nicole Meyer-Vernet and Jean-Pierre Rospars examining the top…
infinity_faq

Infinity in Mathematics: Limits and Cardinality FAQ

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

Apollo Program: Moon Missions & Last Moonwalk 1972

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0 Comments
Eugene Cernan: Last to Walk on the Moon On December 11, 1972, astronaut Eugene Cernan became the last person to walk on the lunar surface, marking the…
spacetime

Struggles With the Continuum: Is Spacetime Really a Continuum?

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29 Comments
 Is spacetime really a continuum?  That is, can points of spacetime really be described---at least locally---by lists of four real numbers…
photoelectric

“Violating” Einstein’s Photoelectric Effect Model

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4 Comments
One of the most spectacular theoretical descriptions that Einstein had ever produced is the corpuscular nature of light that he used in his 1905 photoelectric…
999

Rigorous Proof: Why 0.999… Equals 1 (Geometric Series)

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101 Comments
Yes. What 0.999... Means First, we have not addressed what 0.999... means. So it is best to first describe what the notation [tex]b_0.b_1b_2b_3...[/tex]…
CosmologicalHorizons

Inflationary Misconceptions and the Basics of Cosmological Horizons

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46 Comments
Introduction It is a common saying that during inflation "space expanded faster than the speed of light."  This statement is meant to articulate the…

Understanding Zero: History, Division, Exponents, 0!

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7 Comments
The goal of this FAQ is to clarify the concept of 0, and specifically the operations that are allowed with it.The best way to start this FAQ is to…
lightswitch

What Happens When You Flip the Light Switch: From Zero Seconds to 20 Years

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47 Comments
Introduction Few people know what happens when a customer does something like flipping a light switch. The full explanation requires descriptions of topics…
matterphoton

Will All Matter Be Converted to Photons?

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41 Comments
Will all matter be converted into black holes and then photons? No. There is a misconception that a lot of laypeople seem to be picking up recently from…
photonmass

Do Photons have Mass?

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5 Comments
Do photons have mass? The quick answer: NOHowever, this is where it gets a bit confusing for most people. This is because, in physics, there are several…
relativitytherorm

Learn the Relativistic Work-Kinetic Energy Theorem

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19 Comments
I was bothered for a long time by the reasons for the relativistic validity of the work-kinetic energy relation ##\Delta E=Fd##, which holds without any…
supergravity

Emergence From the Superpoint

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6 Comments
 In the previous article we saw that the generalization of super-Lie algebras to homotopy super-Lie n-algebras (super L-infinity algebras)…
blackholesforce

An Equation for the Centrifugal Force Reversal Near A Black Hole

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15 Comments
My goal in this article is to derive a simple equation for the proper acceleration of an observer traveling on a circular path around a Schwarzschild black…
liealgebra

Homotopy Lie-n Algebras in Supergravity

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1 Comment
 The previous article in this series claimed that the mathematics of the 21st century that had fallen into the 1970s in the form of string…
birktheo

Learn A Short Proof of Birkhoff’s Theorem

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8 Comments
Birkhoff's theorem is a very useful result in General Relativity, and pretty much any textbook has a proof of it. The one I first read was in Misner, Thorne,…
string theory

It Was 20 Years Ago Today — the M-theory Conjecture

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2 Comments
  While the world didn't end, after all, 15 years back at the turn of the millennium, in hindsight it is curious that, almost unnoticed,…

Why You Should Not Use Wikipedia As Your Primary Source

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108 Comments
It is no secret to anyone who has read my posts in this forum for a while that I do not like Wikipedia. I think that there is a fundamental flaw with the…
spaceship

Bell Spaceship Paradox: Why the String Breaks in SR

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33 Comments
Bell spaceship paradox — setup Bell describes two spaceships that start out at rest relative to each other with an elastic string between them, one…
hupmisconceptions1

Learn a Misconception of the Heisenberg Uncertainty Principle

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26 Comments
One of the common misconceptions about the Heisenberg Uncertainty Principle (HUP) is that it is the fault of our measurement accuracy.A description…
quantum_tunneling

Why You Can’t Quantum Tunnel Through a Wall

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64 Comments
Can You Quantum Tunnel Through a Wall? The short and sweet answer if a tennis ball, a bowling ball, or any other kind of ordinary macroscopic object can…
relmass

What is relativistic mass and why it is not used much?

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58 Comments
It happens that the term relativistic mass is used, in particular in the introductory text on special relativity. It should be noted that whether or not…
learnfrompf

The Most Important Thing You Can Learn from Physics Forums

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15 Comments
It is a bit pompous of me to think that I can tell you the most important thing you can learn from Physics Forums (PF). After all, each one of us here…
errorbars

Plus/minus What? How to Interpret Error Bars

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20 Comments
People sometimes find themselves staring at a number with a ± in it when a new physics result is presented. But what does it mean? This Insight aims to…
LCDM4

Approximate LCDM Expansion in Simplified Math (Part 4)

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14 Comments
  Part 4: Cosmic Recession Rates An astronomer, accompanied by his amateur relativist friend, aimed a telescope at a distant galaxy and measured…
theoex

False Physics Major Dichotomy: Theorist or Experimentalist?

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27 Comments
Many physics majors believe that pursuing a Ph.D. in physics requires them to choose between theory and experiment. Due to my weaknesses in math, I…
physics course

CV Tips for Physicists: Secure Postdoc & Research Jobs

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1 Comment
Curriculum Vitae (CV): What to include I am going to backtrack a little bit and talk about writing your Curriculum Vitae (CV) and what you should focus…
issac newton

Isaac Newton Quiz: Test Yourself on Isaac Newton

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44 Comments
One of the great heavy weights of history. How much do you really know about him? Let's find out!"To myself I am only a child playing on the beach,…
LCDM3

The LCDM Cosmological Model in Simplified Math (Part 3)

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5 Comments
  Part 3: Important Cosmological Horizons and Distances A question that often comes up is: "how big is the observable universe?"The question…
physics doctoral

Postdoctoral Positions: How to Succeed as a Physicist

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1 Comment
Postdoctoral Positions What is a postdoc? If you intend to pursue an academic or research career, you will very likely need postdoctoral experience.…