### Articles

Physics Articles

- 11d Gravity From Just the Torsion Constraint
- 15 of the Best PF 2017 Eclipse Photos
- It Was 20 Years Ago Today — the M-theory Conjecture
- A First Idea of Quantum Field Theory – 20 Part Series
- A Problem From “Incandescence”
- Acoustic ‘beats’ from Mismatched Musical Frequencies
- All about Earth’s Gravity
- Why Road Capacity Is Almost Independent of the Speed Limit
- Anyon Demystified
- Approximate LCDM Expansion in Simplified Math
- Approximate LCDM Expansion in Simplified Math (Part 2)
- Approximate LCDM Expansion in Simplified Math (Part 4)
- Assessing the Physics in Edmond Hamilton’s “The Second Satellite”
- The Balloon Analogy … the Good, the Bad, and the Ugly
- The Basics of Positron Emission Tomography (PET)
- Intro to Big Bang and Infinity Concepts
- The Block Universe – Refuting a Common Argument
- Blockworld and Its Foundational Implications Part 1: Time Dilation and Length Contraction
- Blockworld and Its Foundational Implications Part 2: The Relativity of Simultaneity and Blockworld
- Blockworld and its Foundational Implications Part 3: General Relativity and the Big Bang
- Blockworld and its Foundational Implications Part 4: General Relativity and Closed Timelike Curves
- Blockworld and its Foundational Implications Part 5: Delayed Choice and No Counterfactual Definiteness
- A Brief on the Expansion of the Universe
- Can We See an Atom?
- Causal Perturbation Theory
- “Classical Physics Is Wrong” Fallacy
- The Complexity of Modern Science
- Coordinate Dependent Statements in an Expanding Universe
- Damped Motion in Classical and Quantum Mechanics
- The Diffraction Limited Spot Size with Perfect Focusing
- Either the Sun Is Getting Smaller or Gravity Is Getting Weaker
- Emergence from the Superpoint
- Entanglement Entropy – Part 2: Quantum Field Theory
- Entanglement Entropy – Part 1: Quantum Mechanics
- Examples of Prequantum Field Theories I: Gauge Fields
- Examples of Prequantum Field Theories II: Higher Gauge Fields
- Examples of Prequantum Field Theories III: Chern-Simons-type Theories
- Examples of Prequantum Field Theories IV: Wess-Zumino-Witten-type Theories
- Exploring the Spectral Paradox
- Explosion-Generated Collapsing Vacuum Bubbles Reach 20,000 Kelvin
- Frames of Reference: A Skateboarder’s View
- Frenet Equations in 2-D Which Result in the Cornu Spiral
- Fun with Self-Avoiding Walks
- The Fundamental Difference in Interpretations of Quantum Mechanics
- General Relativity as a Gauge Theory
- Does Gravity Gravitate? Part 2
- Does Gravity Gravitate? Part 3 – The Wave
- The Greenberger-Horne-Zeilinger Experiment
- A Hand-Wavy Discussion of the Planck Length
- High Temperature Low Temperature Duality for the Ising Model on an Infinite Regular Tree
- Why Higher Category Theory in Physics?
- Higher Prequantum Geometry II: The Principle of Extremal Action – Comonadically
- Higher Prequantum Geometry III: The Global Action Functional – Cohomologically
- Higher Prequantum Geometry IV: The Covariant Phase Space – Transgressively
- Higher Prequantum Geometry I: The Need for Prequantum Geometry
- Higher Prequantum Geometry V: The Local Observables – Lie Theoretically
- Homotopy Lie-n Algebras in Supergravity
- How to Study Mathematics
- Inflationary Misconceptions and the Basics of Cosmological Horizons
- Initiating Research Work
- The Interaction of Sound and Light
- A Journey Into the Cosmos – FLRW Metric and The Friedmann Equation
- A Journey Into the Cosmos – The Friedmann Equation
- Knut Lundmark and the Prehistory of Dark Matter
- The LCDM Cosmological Model in Simplified Math (Part 3)
- Lessons From the Bizzaro Universe
- LHC Part 2: Commissioning
- LHC Part 3: Protons as Large as a Barn
- LHC Part 4: Searching for New Particles and Decays
- Measuring How Many Days Are in a Year
- Measuring Quantum Entanglement?
- Misconception of the Heisenberg Uncertainty Principle
- Misconceptions about Virtual Particles
- The Need of Infinity in Physics
- Negative absolute temperatures
- Neutrino Masses and Speed
- Mathematical Quantum Field Theory – Fields
- Mathematical Quantum Field Theory – Field Variations
- Mathematical Quantum Field Theory – Free Quantum Fields
- Mathematical Quantum Field Theory – Gauge Fixing
- Mathematical Quantum Field Theory – Gauge Symmetries
- Mathematical Quantum Field Theory – Geometry
- Mathematical Quantum Field Theory – Interacting Quantum Fields
- Mathematical Quantum Field Theory – Lagrangians
- Mathematical Quantum Field Theory – Observables
- Mathematical Quantum Field Theory – Phase Space
- Mathematical Quantum Field Theory – Propagators
- Mathematical Quantum Field Theory – Quantization
- Mathematical Quantum Field Theory – Reduced Phase Space
- Mathematical Quantum Field Theory – Renormalization
- Mathematical Quantum Field Theory – Spacetime
- Mathematical Quantum Field Theory – Symmetries
- Introduction to Perturbative Quantum Field Theory
- Permanent Magnets Explained by Magnetic Surface Currents
- The Physics of Virtual Particles
- Plus/minus What? How to Interpret Error Bars
- A Poor Man’s CMB Primer. Part 1: Orientation
- A Poor Man’s CMB Primer. Part 2: The Birth of a Cosmic Background Radiation
- A Poor Man’s CMB Primer. Part 3: Bumps on a Blackbody
- A Poor Man’s CMB Primer. Part 4: Cosmic Acoustics
- A Poor Man’s CMB Primer. Part 5: Quantum Seeds
- Quantum Amplitudes, Probabilities and EPR
- The Quantum Liar Experiment: An Instantiation of the Mermin Device
- Questions About Infinity
- Rainbows are not Vampires
- Relativistic Work-Kinetic Energy Theorem
- How Representative is the Night Sky?
- Results Shown at EPS Conference
- Understanding Retrocausality
- Retrocausality: How Fundamental is the Arrow of Time?
- The Revival of Newton-Cartan Theory
- Roger Babson’s Anti-Gravity Contest
- Scaling Laws and the Speed of Animals
- The Schwarzschild Geometry: Part 3
- The Schwarzschild Metric: Part 1, GPS Satellites
- The Schwarzschild Metric: Part 2, The Photon Sphere
- The Schwarzschild Metric: Part 3, A Newtonian Comparison
- Science for Love or for Money
- Scientific Inference P1: Logical induction ain’t logical
- Scientific Inference P2: We don’t need no stinkin’ induction
- Secondary Forces Explored
- See an Electron Lately?
- How to Self-Study Basic High School Mathematics
- Some Sins in Physics Didactics
- Spacetime Diagrams of Light Clocks
- Spectral Standard Model and String Compactifications
- The Speed of Light and Galilean Relativity
- Statistical Mechanics Part II: The Ideal Gas
- Steering: How the EPR-Paradox Fits Between Entanglement and Nonlocality
- How I Stopped Worrying and Learned to Love Orthodox Quantum Mechanics
- Struggles with the Continuum – Conclusion
- Struggles With the Continuum – Part 1
- Struggles With The Continuum – Part 2
- Struggles With the Continuum – Part 3
- Struggles with the Continuum – Part 4
- Struggles with the Continuum – Part 5
- Struggles with the Continuum – Part 6
- Struggles with the Continuum – Part 7
- Super p-Brane Theory Emerging from Super Homotopy Theory
- My Superparticle Bet with Frank Wilczek
- Why Supersymmetry? Because of Deligne’s theorem.
- Time Dilation and Redshift for a Static Black Hole
- Tools to Enrich our Quantum Mechanics Interpretations Discourse
- Understanding Buoyancy: The Limits of Archimedes
- Understanding the General Relativity View of Gravity on Earth
- Some Useful Integrals In Quantum Field Theory
- Using the Spaghetti-Twist to Align DNA
- The Vacuum Fluctuation Myth
- Vacuum Fluctuations in Experimental Practice
- “Violating” Einstein’s Photoelectric Effect Model
- Weak Values Part 1: “Asking Photons Where They Have Been”
- Weak Values Part 2: The Quantum Cheshire Cat Experiment
- What is Energy?

Math Articles

- Abstract Algebra: The Natural Numbers
- Axioms for the Natural Numbers
- A Brachistochrone Subway Is Not a Cost-effective Idea
- Can Angles be Assigned a Dimension?
- The Case for Learning Complex Math
- A Forward on Real Numbers and Real Analysis
- Friends, Strangers, 7825 and Computers
- Groups and Geometry
- Hilbert Spaces and Their Relatives Part I
- Hilbert Spaces And Their Relatives – Part II
- Intransitive Dice with a Twist
- A Journey to The Manifold SU(2) – Part I
- Math and the Obvious
- The Millennium Prize Problems: Part I
- The Monographic Substitution Cipher: From Julius Caesar to the KGB
- Perfect Secrecy and the Unbreakable Cipher
- Representations and Why Precision is Important
- Scientific Inference P3: Balancing Predictive Success with Falsifiability
- Simple Geometry, Deep Math
- How to Tell Operations, Operators, Functionals and Representations Apart
- Things Which Can Go Wrong with Complex Numbers
- Trials and Tribulations of a Physicist who Became a Math Geek
- What is a Property Formally

Engineering Articles

- AC Power Analysis: Part 1, Basics
- AC Power Analysis: Part 2, Network Analysis
- AC Power Analysis: Part 3, Cyber Resilience
- Administering TCP/IP In Automation or Measurement Networks
- The What and Why of Circuit Analysis Assumptions
- Modulation vs. Beating Confusion
- Ohm’s Law Mellow
- Reversible Vs Irreversible Gas Compression/Expansion Work
- Time Synchronization Across Switched Ethernet
- What Happens When You Flip the Light Switch?

Other Science Articles

- Atomic Positioning with DNA Hinges
- Can Gene Editing Eliminate Alzheimer’s Disease?
- Why Can’t My Computer Do Simple Arithmetic?
- What Causes Cancer: Bad Luck or Bad Lifestyles?
- Don’t Fear the CRISPR: Why New Gene Editing Technologies Won’t Lead to Designer Babies
- History’s First Science Experiments
- Intro to Algorithms for Programming
- New High Performance Ultra-wide Angle Lenses
- New Research on Untying Knots in Polymers
- Your Software is Never Perfect
- Polymer Physics and Genetic Sequencing
- When did Mitochondria Evolve?

Education Articles

- Ask the Advisors: What’s Your Most Memorable Textbook?
- Blaming Government for Teacher and Scientist Failures in Integrity
- Fake News and Science Reporting
- False Dichotomy: Theorist or Experimentalist?
- Imagination Without Knowledge Is Ignorance Waiting to Happen
- What Is The Most Important Thing That You Learned?
- Very Little Excuse To Ask A Question Cold
- Niches for Publishable Undergraduate Research
- Why Is Quantum Mechanics So Difficult?
- Real Research in Unexpected Places – Backyards, Barns, Bayous
- Real Research in Unexpected Places – Boat Ramps
- Revamping Undergraduate Introductory Physics Laboratory
- Why Study Physics? A Bit Goes a Long Way!
- The Most Important Thing You Can Learn from Physics Forums
- Why You Should Not Use Wikipedia As Your Primary Source
- Work Ethic Is a Key to Success for Science Majors

### FAQs

Physics FAQs

- How Does an Airplane Wing Work: a Primer on Lift
- Where Did the Big Bang Happen?
- Do Black Holes Really Exist?
- Can I Send a Signal Faster than Light by Pushing a Rigid Rod?
- Can We Tell Whether the Universe Is Rotating?
- Do Photons have Mass?
- Do Photons Move Slower in a Solid Medium?
- Does Gravity Gravitate?
- Does the Heisenberg Uncertainty Principle Imply Energy Nonconservation?
- Why Don’t Electrons Crash into the Nucleus in Atoms?
- Was the Early Universe in a Disordered State?
- Energy Gained by Charge in an Electrostatic Field
- A Geometrical View of Time Dilation and the Twin Paradox
- How Fast Do Changes in the Gravitational Field Propagate?
- Lenses and Pinholes: What Does “In Focus” Mean?
- PF’s policy on Lorentz Ether Theory and Block Universe
- Why is the Radius of the Observable Universe in Light-years Greater than Its Age?
- Could Redshifts Be Intrinsic Rather than Cosmological?
- Why Is the Speed of Light the Same in All Frames of Reference?
- What Is the Steady-state Model, and Why Is It No Longer Viable?
- Is the Universe a Black Hole?
- At What Velocity Does the Universe Expand? Can It Be Faster than Light?
- Building a Definition for Heat
- What is relativistic mass and why it is not used much?
- What Is the Bell Spaceship Paradox, and How Is It Resolved?
- Why Does C Have a Particular Value, and Can It Change?
- Why Is Acceleration Due to Gravity a Constant?
- Will All Matter Be Converted to Photons?
- Why Won’t You Look at My New Theory?
- You Will Not Tunnel Through a Wall

### Guides

Math Guides

- 10 Math Tips to Save Time and Avoid Mistakes
- 5 Common MATLAB Error Messages and How to Fix Them
- In High School and Want to Do Advanced Mathematics?
- Resources for High School Math at Home
- How to Self-study Algebra: Linear Algebra
- How to Self-Study Algebra. Part II: Abstract Algebra
- How to Self-study Analysis. Part II: Intermediate Analysis
- How to Self Study Analysis. Part I: Intro Analysis
- How to Self-Study Calculus
- How to Self Study Geometry. Part I: Pure Geometry

Education Guides

- 3 New Years Tips For Cooling Off Debates
- Alternative Careers for a Physics Graduate
- Applying for Physics Graduate School
- How to Become a Medical Physicist in 3653 Easy Steps
- Choosing a Research Area and an Advisor
- Am I Cut Out for Mathematics or Sciences?
- Early Physics Education in High Schools
- Entering Physics Graduate School From Another Major
- First years of Graduate School from Being a TA to the Graduate Exams
- Getting a Physics Job!
- Guide to Publishing in Peer Reviewed Journals
- So I Am Your Intro Physics Instructor
- Mathematical Preparations
- Oral Presentations
- Oral Presentations – Addendum
- How I Overcame Learning Challenges That I Faced Studying STEM
- Postdoctoral Physics Position
- Problems with Self-Studying
- Publishing in a Physics Journal
- Publishing in a Physics Journal (Addendum)
- Research Work and The Lab Book
- Resources for High School Physics at Home
- Secrets of Successful Science Projects
- Surviving the First Year of College
- The Birth of a Textbook
- The Life of a Physics Major
- The US Graduate School System
- So You Want to Go Back to School
- What to Expect from Graduate School Before You Get There
- Writing a Master Degree Thesis
- Writing Your Doctoral Thesis/Desertation
- Your Curriculum Vitae
- Your Physics Thesis Defense

Other Science Guides

### Tutorials

Physics Tutorials

- Basic Kinematics
- Centrifugal Force Reversal Near A Black Hole
- Elementary Construction of the Angular Velocity
- Explaining Rolling Motion
- Fabry-Perot and Michelson Interferometry: A Fundamental Approach
- Frames of Reference: Linear Acceleration View
- Frequently Made Errors in Equation Handling
- Frequently Made Errors in Heat – Elementary Level
- Frequently Made Errors in Mechanics: Forces
- Frequently Made Errors in Mechanics – Friction
- Frequently Made Errors in Mechanics – Hydrostatics
- Frequently Made Errors in Mechanics – Kinematics
- Frequently Made Errors in Mechanics – Moments
- Frequently Made Errors in Mechanics – Momentum and Impacts
- Frequently Made Errors in Mechanics – Springs
- Frequently Made Errors – Pseudo and Resultant Forces
- Frequently Made Errors in Vectors – Elementary Use
- Grandpa Chet’s Entropy Recipe
- The Homopolar Generator: An Analytical Example
- An Integral Result from Parseval’s Theorem
- LightCone7 Tutorial Part III – How Things are Computed
- LightCone7Combo Tutorial Part I
- LightCone7Combo Tutorial Part II – Charts
- Misconceiving Mutual Inductance Coefficients
- Orbital Precession in the Schwarzschild and Kerr Metrics
- A Partial “Derivation” of Gauss’s Law
- Precession in Special and General Relativity
- Rare Kinematic Formula
- Relativity on Rotated Graph Paper
- Relativity Using the Bondi K-calculus
- Relativity Variables: Velocity, Doppler-Bondi k, and Rapidity
- Renormalisation Made Easy
- Why Renormalisation Needs a Cutoff
- Rindler Motion in Special Relativity, Part 1: Hyperbolic Trajectories
- Rindler Motion in Special Relativity, Part 2: Rindler Coordinates
- The Schwarzschild Geometry: Part 1
- The Schwarzschild Geometry: Part 2
- The Schwarzschild Geometry: Part 4
- A Short Proof of Birkhoff’s Theorem
- Solving Einstein’s Field Equations in Maxima
- Statistical Mechanics Part I: Equilibrium Systems
- Tetrad Fields and Spacetime
- Understanding Entropy and the 2nd Law of Thermodynamics
- Visualizing the 2-D Particle in a Box

Math Tutorials

- Complex and Irrational Exponents for the Layman
- Conceptual Difficulties in the Roles of Variables
- A Continuous, Nowhere Differentiable Function: Part 1
- A Continuous, Nowhere Differentiable Function: Part 2
- Demystifying the Chain Rule in Calculus
- Frequently Made Errors in Probability – Conditionals in Natural Language
- General Brachistochrone Problem
- Informal Introduction to Cardinal Numbers
- Irrationality for Dummies
- Is There a Rigorous Proof Of 1 = 0.999…?
- A Journey to The Manifold SU(2) – Part II
- Make Units Work for You
- Matrix Representations of Linear Transformations
- Some Misconceptions on Indefinite Integrals
- Omissions in Mathematics Education: Gauge Integration
- The Pantheon of Derivatives – Part II
- The Pantheon of Derivatives – Part III
- The Pantheon of Derivatives – Part IV
- The Pantheon of Derivatives – Part V
- Partial Differentiation Without Tears
- Partial Fractions Decomposition
- Frequently Made Errors in Probability – Continuous and Discrete Distributions
- Trick to Solving Integrals Involving Tangent and Secant
- Solving the Cubic Equation for Dummies
- Solving Homogeneous Linear ODEs using Annihilators
- Solving Nonhomogeneous Linear ODEs using Annihilators
- Further Sums Found Through Fourier Series
- The Pantheon of Derivatives – 5 Part Series
- Using the Fourier Series To Find Some Interesting Sums
- What Are Eigenvectors and Eigenvalues?

### Interviews

Interviews

- MathJax Developer: Peter Krautzberger
- Interview with Astronomer Pamela Gay
- Interview with Biologist Jim McNamara
- Interview with a Biophysicist: Andy Resnick
- Interview with a Chem-Physicist: DrClaude
- Interview with a Chemist: Borek
- Interview with an Electrical Engineer: Berkeman
- Interview with an Engineer: Russ Watters
- Interview with Instrument Engineer Jim Hardy
- Interview with a Mathematical Physicist: John Baez Part 1
- Interview with a Mathematical Physicist: John Baez Part 2
- Interview with a Physicist: David Hestenes
- Interview with Mathematician Fresh_42
- Interview with Mathematician and Physicist Arnold Neumaier
- Interview with a Mentor: Drakkith
- Interview with a Physicist: David J. Griffiths
- Interview with a Physicist: Doc Al
- Interview with a Physicist: Sean Carroll
- Interview with a Physicist: ZapperZ
- Interview with a Physics Mentor: Nugatory
- Interview with a Physics Mentor: PeterDonis
- Interview with Physics Mentor: Stevendaryl
- Interview with Science Advisor DrChinese
- Interview with a Theoretical Physicist: Carlo Rovelli
- Interview with Theoretical Physicist Clifford V. Johnson
- Interview with a Theoretical Physicist: Garrett Lisi
- Interview with a Theoretical Physicist: Sabine Hossenfelder
- Interview with Astrophysicist Adam Becker
- Q&A With Universe Today’s Fraser Cain

### Quizzes

Quizzes

- All About Ice – Trivia
- How Well Do You Know Albert Einstein?
- It’s Elemental! The Periodic Table Quiz
- Name the Science Book Author Quiz
- Name the Science Photo
- Name the Scientist
- PF Insight Scavenger Hunt 2
- PF Insights Scavenger Hunt
- Test Your NASA Knowledge
- The Isaac Newton Quiz
- The Moon Quiz
- Think You Know Richard Feynman?
- Think You Know Stephen Hawking
- Think You Know about the LHC?
- Think You Know Pluto?
- Which Scientist Said What