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Introduction The applicability of Newton’s second law in the oft-quoted “general form” $$\begin{align}\frac{d\mathbf{P}}{dt}=\mathbf{F}_{\text{ext}}\end{align}$$ was an issue in a recent thread (see post #4) in cases of systems with variable mass. The following example illustrates the kind of confusion that could arise from the (mis)application of Equation (1): A rocket is hovering in place above ground…
In this article, we will look at why there are maximum mass limits for objects that are supported against gravity by degeneracy pressure instead of kinetic pressure. We will look at the two known cases of this, white dwarfs and neutron stars; but it should be noted that similar arguments will apply to any postulated…
Corrections to MIT Open Courseware 8-01sc classical mechanics, fall 2016 Applying Newton’s Laws to systems of Varying Mass PDF Course Link My concerns with the solutions provided on the cited MIT Open Courseware web page are, mostly: Despite arriving at the correct answers, the algebraic methods are invalid. As a result, they can be utterly…
Introduction In a previous article entitled “Alternate Approach to 2D Collisions” we analyzed collisions between a moving and stationary object by defining the co-ordinate axes as being respectively parallel and perpendicular to the post-collision direction of motion of the stationary object. In this article, we will be adopting the same approach to analyze the well…
Introduction This article is essentially an addition to the previous one on (mainly) inelastic collisions to include the particular case of inelastic relativistic collisions. Reasons for writing a separate article are first that this author is not particularly well qualified to write on the topic and so may well need to request the scrutiny of…
Definition/Summary Abstract from Poisson and Israel’s 1990 paper, ‘Internal structure of black holes’- ‘The gravitational effects associated with the radiative tail produced by a gravitational collapse with rotation are investigated. It is shown that the infinite blueshift of the tail’s energy density occurring at the Cauchy horizon of the resulting black hole causes classically unbounded…
Do photons have mass? The quick answer: NO. However, this is where it gets a bit confusing for most people. This is because in physics, there are several ways to define and measure a quantity that we call “mass”. Now, it doesn’t create any confusion among physicists because we tend to know in what context…
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 to use relativistic mass to a large extent is a matter of convention, convenience, and semantics as long as it is used properly and does not have any impact…
It was hard to miss the 2011 OPERA neutrino speed measurement that indicated superluminal neutrino speeds (and turned out to be a measurement error), but measurements of neutrino masses and speeds have a long tradition. Neutrinos are very light particles that interact via the weak interaction and gravity only. There are three types of neutrinos:…
Key Points Stars are born in primordial nebulae, gaseous molecular clouds on the spiral arms of galaxies Stars smaller than 1.4 suns will settle into a white dwarf and eventually become a black dwarf Stars greater than 1.4 suns will undergo a much more violent death, as their nuclear fuel is exhausted and their core…
The slinky drop is a rather simple experiment. In its most basic form, it requires only a popular toy for children, a stable hand, and a keen eye. For a better view, using a modern smart phone to capture a video of the experiment also helps to capture the falling slinky. Apart from the commonly…
Introduction The figure on the right shows a “double-double” Atwood machine with three ideal pulleys and four masses. All pulleys are released from rest simultaneously. Which of the choices below describes the angular motion of the top pulley P after some time has elapsed and why? It rotates clockwise with increasing angular speed. It rotates…
Abstract “Mathematics is the native language of nature.” is a phrase that is often used when it comes to explaining why mathematics is all around in natural sciences, especially in physics. What does that mean? A closer look shows us that it primarily means that we describe nature by differential equations, a lot of differential…
Abstract Richard Pierce describes the intention of his book [2] about associative algebras as his attempt to prove that there is algebra after Galois theory. Whereas Galois theory might not really be on the agenda of physicists, many algebras are: from tensor algebras as the gown for infinitesimal coordinates over Graßmann and Banach algebras for…
Abstract My school teacher used to say “Everybody can differentiate, but it takes an artist to integrate.” The mathematical reason behind this phrase is, that differentiation is the calculation of a limit $$ f'(x)=\lim_{v\to 0} g(v) $$ for which we have many rules and theorems at hand. And if nothing else helps, we still can…
Abstract We explain by simple examples (one-parameter Lie groups), partly in the original language, and along the historical papers of Sophus Lie, Abraham Cohen, and Emmy Noether how Lie groups became a central topic in physics. Physics, in contrast to mathematics, didn’t experience the Bourbakian transition so the language of for example differential geometry didn’t…
Part 1: Overview Part 2: Mathematical Details Part 3: Implications In a previous article, I described in general terms the model of gravitational collapse of a spherically symmetric massive object, first published by Oppenheimer and Snyder in their classic 1939 paper. In this follow-up article, I will give further mathematical details about the model, using…
Part 1: Overview Part 2: Mathematical Details Part 3: Implications Most people who have spent any time at all studying GR are familiar with the Schwarzschild solution. (A series of Insights articles discusses the key properties of that solution.) Much of that familiarity probably derives from the fact that the Schwarzschild solution describes a black…
Problem statement (simplified) An object slides down a ramp at angle θ to encounter level ground. Both surfaces have kinetic friction: μ’ on the ramp, μ on the level. The object reaches the ground at speed u. What is its speed when first fully on the level? (Original is at https://www.physicsforums.com/threads/distance-a-block-slides-along-a-surface-with-friction-given-with-an-initial-velocity.1047556/.) There are several missing…
I asked myself why different scientists understand the same thing seemingly differently, especially the concept of a metric tensor. If we ask a topologist, a classical geometer, an algebraist, a differential geometer, and a physicist “What is a metric?” then we get five different answers. I mean it is all about distances, isn’t it? “Yes”…
Introduction In an earlier article, we examined a magnet falling through a solenoid. We argued that the point dipole model can account for the basic features of the induced emf across the solenoid ends. Here, we extend the model to a magnet falling through a conducting pipe along its axis. With the falling dipole moment…
Introduction Modeling a magnet realistically is a task best done numerically. Even the simplified model of two separated disks with uniform surface magnetization ##\pm~\sigma_M## involves elliptic integrals simplifying assumptions. As a model, the point dipole may be unrealistic to some but the math is tractable and accessible. The usefulness of the point dipole model in…
It is relatively common on Physics Forums to see arguments that are effectively similar to the following: When we jump off the ground, the ground does not move. Because of this, the force from the ground on us does zero total work. Since the force does no work, we cannot gain any kinetic energy. We…
Introduction Every secondary school student who has encountered trigonometry in his/her Math syllabus will most likely have come across the sine, cosine, and area rules which are typically used to solve triangles in which certain information is supplied and the remainder are to be calculated. Somewhat surprisingly (because it is relatively simple to derive), the…
Introduction In a previous Physics Forums article entitled “How to Master Projectile Motion Without Quadratics”, PF user @kuruman brought to our attention the vector equation ##\frac{|V_0 \times V_f|}{g} = R## and lamented the fact that: “Equally unused, untaught and apparently not even assigned as a “show that” exercise is Equation (4) that identifies the range as the…
Euler’s amazing identity The mathematician Leonard Euler developed some surprising mathematical formulas involving the number ##\pi##. The most famous equation is ##e^{i \pi} = -1##, which is one of the most important equations in modern mathematics, but unfortunately, it wasn’t invented by Euler.Something that is original with Euler is this amazing identity: Equation 1: ##1…
Like many others, I have been seeking new and fun things to do during a pandemic. I decided on the new Microsoft Flight Simulator 2020, but the Condor 2 Soaring Simulator is my intermediate step. I am certified as a pilot, including gliders, but I have not flown for many years because of the expense. …
Note: It is assumed that the reader has read part I and part II of the series. Is Mechanical Energy Conservation Free of Ambiguity? Can We Do Better Than Mechanical Energy Conservation? Preface Because of what has already been said, there seem to be three options for proceeding with the teaching of mechanical energy conservation…
Introduction Physics teachers who are either writing physics questions that deal with waves on a string or setting up equipment for a class lab or demo of standing waves on a string might find the following analysis useful. When writing questions for physics tests or homework, it is preferable to use physically plausible values for…
ref: asi.edu.au/programs/past-exams/physics-olympiad-past-exams/ Classification of issues: [1*] Quibble regarding the solution. [2*] Clearer problem description suggested. [3*] Alternative solution should be accepted [4*] Significant error, likely leading to incorrect marking. Tally matrix: Year 1* 2* 3* 4* 2019 3 0 0 0 2017 2 2 0 2 2016 1 1 2 0 2015 2 2 1…
All undergraduate physics majors are shown how the counterintuitive aspects (“mysteries”) of time dilation and length contraction in special relativity (SR) follow from the light postulate, i.e., that everyone measures the same value for the speed of light c, regardless of their motion relative to the source (see this Insight, for example). And, we can…
Introduction In this article we take as our starting point the original equations which Compton drew up and solved in his ground-breaking 1925 article: From the above equations, Compton solved for two variables namely ##\beta## the ratio of electron speed to the velocity of light, and for ##\nu_{\theta}##, the frequency of the scattered…
The “twin paradox” is often discussed in the introductory treatment of special relativity. Under “twin paradox” we understand the fact that if two twins start from the same place with synchronized clocks, traveling in an arbitrary way and then meet again at the same spacetime point, where they compare their clocks, in general, they find…
In a previous series of articles, I posed the question “Does Gravity Gravitate?” and explained how, depending on how you interpreted the terms “gravity” and “gravitate”, one could answer the question, either way, yes or no. This article will treat its title question in a similar fashion. :-) To be sure, this case is a…
This is Part 2 of a 3-part series explaining evidence for so-called “dark energy” leading to a current positive cosmological acceleration. The evidence comes from fitting the SCP Union2.1 type Ia supernova data which indicates the existence of a cosmological constant ##\Lambda## (read “Lambda”, thus ##\Lambda##CDM is sometimes written LCDM) in Einstein’s equations (EEs) of…
In the first two articles in this series, we looked at the Einstein Field Equation and Maxwell’s Equations in a static, spherically symmetric spacetime. Using formulas from those two previous articles, I now want to consider the question: what is the maximum amount of work that can be extracted by slowly lowering an object into…
Key Points Astronomers have been trying to understand space for thousands of years. Supernova Refsdal was the first known multiply imaged supernova. Multiple images of the supernova are due to strong gravitational lensing. Astronomers managed to predict the reappearance of the supernova. Multiple images of the supernova have a time delay between them. HST observations…
In this 3-part series, I want to motivate the (re)introduction of the cosmological constant ##\Lambda## into Einstein’s equations of general relativity (GR) per the Supernova Cosmology Project (SCP) Union2.1 type Ia supernova data. As you probably know, this discovery won Perlmutter, Schmitt, and Riess the 2011 Nobel Prize in Physics “for the discovery of the accelerating…
The universe was not perfectly uniform when it started, some areas had a higher density than others. During the evolution of the universe, these areas of high density contained most of the matter and started forming galaxies where there was the highest concentration of matter. This large-scale structure (‘cosmic web’) connects the observed clusters of…
Gravitational waves (GW’s) are disturbances in spacetime produced by any massive object moving asymmetrically. However, only the most massive and most relativistic objects produce large enough GW’s to be detectable. The Laser Interferometer Gravitational-Wave Observatory (LIGO) and Virgo detectors are using laser interferometry to detect tiny ripples in the fabric of spacetime. They have detected…
Uranus spins on its side. Uranus has an obliquity (tilt) of 98º, making its axis of rotation closer to the ecliptic plane than any other planet. It is not known how it got this peculiar tilt. Key Points Uranus has an obliquity (tilt) of 98º, making its axis of rotation closer to the ecliptic plane…
Key Points Black Holes (BH’s) have a gravitational pull so strong that nothing, not even light, can escape it BH’s can be divided into two main classes: Stellar–mass and Super Massive Black Holes (SMBH’s) BH’s are observed indirectly through the flow they accrete and emit, also by the supersonic collimated jets they produce and by…
Note: It is assumed that the reader has read part I of the series. Introduction The ambiguity and flaws discussed in part I can be resolved using the law of conservation of energy. In the words of Richard Feynman, There is a fact, or if you wish, a law, governing all natural phenomena that are…
Introduction This article follows on from the previous on an alternate approach to solving collision problems. In that article, we determined the equal and opposite collision impulse to have magnitude ##\mu \Delta v## for perfectly inelastic collisions, ##\mu(1+e) \Delta v## for semi-elastic collisions and ##2\mu \Delta v## for elastic collisions which will be the focus…
Introduction “Close to any question that is in the textbook, there is another question that has never been answered that is interesting.” [Stephen Wolfram, remarks to The University of Vermont physics students, September 30, 2005] Mechanical energy conservation is the assertion that the sum of kinetic and potential energies of a system (the mechanical energy)…
Introduction Collisions are very much a stock item in any school physics curriculum and students are generally taught about the use of the principles of conservation of momentum and energy for solving simple collision problems in one dimension. In this article we will be examining a very common type of collision problem: the inelastic collision….
The Google Doodle for 2 February 2020 celebrated Mary Somerville, the Scottish polymath and science writer, and with Caroline Herschel, the joint first-ever woman to be made an honorary member of the Royal Astronomical Society. Born in Jedburgh, Scotland, in 1780, Somerville received little formal education compared to her brothers. Largely self-taught, she pursued academic…
Introduction Providing accurate fluid dynamics experiments for undergraduate laboratories is challenging in several ways, including reproducibility, simplicity, and accessibility to introductory students. The data from many introductory experiments, for example, is not sufficiently accurate to test whether a linear or a quadratic relationship more appropriately models the dependence of drag force on velocity. An optimal…
As a long-time woodworker, I have an interest in being able to tell one wood from another and in the process of learning how to do that, I’ve developed some knowledge about wood anatomy that I think will be of interest to Biology students as well as woodworkers. There is an enormous variety of wood…
Introduction In previous articles relating to various transition energies in Hydrogen, Helium and Deuterium we have employed the following formula for electron energy given a particular primary quantum number n: $$ E_{n}=\mu c^2\sqrt{1-\frac{Z^2\alpha^2}{n^2}} $$ where ## \alpha ## is the fine structure constant and ## \mu ## the reduced electron mass for a single electron…
Introduction In this article, we will be revisiting a somewhat understudied (and seemingly unrepeated) experiment to measure the Deuterium Lyman Alpha line at approximately 121.5 nm. The experiment was carried out in the 1950s in the wake of the Lamb-Retherford experiment (1947) which established the tiny energy shift (Lamb shift) between the Hydrogen (and Deuterium)…
Introduction In a previous article “Calculating the Balmer Alpha Line” we mentioned how accurate predictions of the spectral lines of singly ionized Helium were of considerable importance in persuading the scientific community that Danish physicist Niels Bohr was on the right track in respect of his groundbreaking atomic model first published in 1913. In this…
Definition/Summary The four laws of black hole thermodynamics are as follows… The Zeroth Law Surface gravity [itex](\kappa)[/itex] is constant over a black holes event horizon. The First Law ‘This law deals with the mass (energy) change, dM when a black hole switches from one stationary state to another.’ The following (in natural units) applies- [tex]dM=\frac{\kappa}{8\pi}\,dA\,+\,\Omega\,dJ\,+\,\Phi\,dQ[/tex]…
We get a lot of “what if” questions here on Physics Forums. This article will explore three different types and then some suggestions for students who feel their question may fall into one of those types. “what if” questions that contradict physics as we know it “what if” questions that are themselves self-contradictory “what if”…
As a university teacher and as a PF member, I have often noted that students are largely unaware of or not using dimensional analysis to help them in their pursuit of knowledge or to check their results. A number of recent threads on PF have also highlighted this issue. The intent of this Insight is…
Preface While browsing through unanswered posts in the Classical Physics Workshop, I came across a gem at the link shown below. For the reader’s convenience, I have included (in italics) the OP’s statement of the question. https://www.physicsforums.com/threads/spacecraft-path-with-polar-coordinates.683210/ There is a circular gate rotating at a constant angular speed of ##\omega##. The circular gate has a…
Introduction Most readers acquainted with the hydrogen spectrum will be familiar with the set of lines in the visible spectrum representing transitions of electrons from energy levels 3,4,5 and 6 (H alpha, beta, gamma, and delta respectively) of atomic hydrogen to energy level 2 – the Balmer series lines. The picture below shows 3 of…
In the first article in this series, we looked at the Einstein Field Equations in a static, spherically symmetric spacetime. In this article, we are going to build on what we saw in the first article to show what Maxwell’s Equations in a static, spherically symmetric spacetime look like. The electromagnetic field tensor is given…
Definition/Summary Potential energy is simply another name for (minus) the work done by a conservative force. Since the work-energy theorem states that change in energy minus work done is constant, that means that for a conservative force, energy plus potential energy is constant. For example, a object of mass [itex]m[/itex] moving a height [itex]h[/itex] and…
Definition/Summary The moment of Inertia is a property of rigid bodies. It relates rotational force (torque) to rotational acceleration in the same way that mass relates ordinary (linear) force to ordinary acceleration. Moment of Inertia has dimensions of distance squared times mass ([itex]ML^2)[/itex]. The moment of Inertia is always relative to a given axis. The…
Introduction The student of thermodynamics, as they consider pistons and ideal gasses and such, often begin to grasp the nature of entropy only to find as they delve deeper that grasp slips away. In the deeper analysis of thermodynamic systems via statistical mechanics this grasp may slip away entirely. “How can my cup of hot…
This will be the first of several articles which will provide, for reference, useful equations for static, spherically symmetric spacetimes. This is a common special case that is studied in General Relativity, and it has the advantage of having a general solution for the Einstein Field equation that can be expressed in closed form equations….
This Insight develops equations of state that are useful in calculations about cosmology and about the insides of stars. The first calculation is for a photon gas and the second is for a ‘relativistic’ gas of particles with mass. Photon Gas Exercise 22 on p108 of Bernard Schutz’s ‘The first course in General Relativity (Second…
Commonly there is a lot of imprecision in talking about ”indistinguishable” (or ”identical”) particles, even in serious work. This Insight article clarifies the issues involved in a conceptually precise way. Classical mechanics. Historically, indistinguishable particles were introduced in order to explain the failure of the thermodynamics of a Newtonian ##N##-particle system to describe the absence…
This Insight is part of my attempt to develop a formal definition of ‘large-scale isotropy’, a concept that is fundamental to most cosmology, but that is nowhere that I have seen properly defined. The definitions of isotropy are as precise as one could wish, but the ‘large-scale’ bit is in every case I have seen…
What is a Parabola? A parabola is a U-shaped curve in mathematics that is defined by a specific set of points. It is a fundamental geometric shape that appears in various mathematical and real-world contexts. Parabolas have a distinct symmetry and are defined by a simple mathematical equation. Key characteristics of a parabola include: Symmetry:…
Introduction This article will attempt to take the mystery out of setting up surface integrals. It will explain the basic ideas underlying surface integration and show you how to parameterize surfaces to set up the corresponding integrals efficiently. You will need to have had or be taking ##3D## calculus covering multiple integrals to get the…
In the last two posts in this series, we developed some tools for looking at Fermi-Walker transport in Minkowski spacetime and then applied them in Schwarzschild spacetime. In this post, we’ll look at Kerr spacetime, which will introduce some additional complexities. The first congruence we will look at in Kerr spacetime is the hovering congruence,…
In the first post in this series, we introduced the concepts of frame field, Fermi-Walker transport, and the “Fermi derivative” of a frame field, and developed some basic machinery for dealing with them. In this post, we will use that machinery to look at two congruences in Schwarzschild spacetime, to see how things differ from…
Definition/Summary Virtual particles are a mathematical device used in perturbation expansions of the S-operator (transition matrix) of interaction in quantum field theory. No virtual particle physically appears in the interaction: all possible virtual particles, and their antiparticles, occur equally and together in the mathematics and must be removed by integration over the values of their…
This is the second installment in a continuing series of articles on Intel AVX-512 assembly programming. The first installment is An Intro to AVX-512 Assembly Programming. Problem If you have an array (Array), that contains ArrLen signed integers, how would you find the sums of the positive and negative numbers in the array? A Solution…
Learning Objectives * Execute a specific experimental procedure to test a specific hypothesis. * Use the Tracker video analysis software for a simple experiment. * Analyze the acquired data with a spreadsheet to test the hypothesis. * Explain in one’s own words whether the experimental data supported the hypothesis, and (if so), how well. *Use…
Learning Objectives Gain confidence and experimental care in making accurate measurements. Understand the relationship between force and spring stretch. Use a neat and orderly lab notebook in which data are recorded. Execute a specific experimental procedure to test a specific hypothesis. Analyze the acquired data with a spreadsheet and graphing program to test the hypothesis….
Don’t forget to read Part 1 of this interview. Professor Kaku, what do you think of “peak oil,” how serious is it? and what alternative sources of energy do you think will best provide the worlds needs in the coming decades after fossil fuels start to decline? (whether it be organic oils, nuclear breeder-types, fusion…
We are happy to have Michio Kaku answer some questions from the community. This interview was originally held in 2004. Michio Kaku is an American theoretical physicist, futurist, and popularizer of science (science communicator). He is a professor of theoretical physics in the City College of New York and CUNY Graduate Center. Kaku has written several books about physics and related topics, has made frequent…
Introduction Bernoulli’s equation is one of the most useful equations in the field of fluid mechanics, owing to its simplicity and broad applicability. That applicability does have limits, however, which has led to it being one of the most misunderstood and also misused equations in fluid mechanics. The goal of this post is to present…
We recently had a question in the relativity forums that mentioned the behavior of magnetic field lines and reminded me of my own confusion at school about what magnetic field lines actually were. You see a lot of diagrams showing the field around a magnet that looks like this: [Image source – author: Geek3, GNU…
One interesting problem when dealing with a vehicle of a certain mass is to determine what is required in order to get the maximum acceleration while going from one velocity to another. Statement If a moving vehicle has an energy source that has a variable power output, the energy source must be set to its…
Servo-constraint was invented by Henri Beghin in his Ph.D. thesis in 1922. For details see the celebrated monograph in rational mechanics by Paul Appell. To understand what this is, we consider the following example. A trolley of mass ##M## can move freely along the horizontal ground in the standard gravity field. A pendulum is placed…
Killing Vector Field The Killing vector field is a vector field on a differentiable manifold that preserves the metric over spacetime (from this I assume, in very basic terms, the Killing vector field ensures smoothness of the metric). Although time-like (c^2 dt^2 > dr^2) at infinity, it does not need to be time-like everywhere outside…
The Large Hadron Collider has produced collisions at 7 TeV. For collisions at 7 TeV, protons need to be ‘ramped’ to 3.5 TeV, the proton has a mass of 1.6726e−27 kg which, according to mass–energy equivalence (E=mc^2), is 938.272 MeV where 1 eV= 1.6022e−19 Joules. The proton will be accelerated to 0.999999964c (11,103.4 revolutions of…
From ‘Exploring Black Holes’ by John Wheeler and Edwin Taylor; can apply to any object falling radially towards a static spherical mass (where the mass of the in-falling object is much smaller than the static spherical mass). Three types of in-falling radial plunger- Drip (dropped from rest at r_o) Rain (dropped from rest at…
About Physics Forums Values and Mission Our goal is to provide a community for people (whether students, professional scientists, or hobbyists) to learn and discuss science as it is currently generally understood and practiced by the professional scientific community. As our name suggests, our main focus is on physics, but we also have forums for…
In this Insight, I’ll go over implementing basic orbital mechanics simulations in the Unity game engine as well as an approach to scaling the simulation for Augmented Reality applications. Unity is a great tool for prototyping games but also for animating physics models. The physics gets more interesting when you can watch all the interactions…
Having more than 10 years of experience in teaching vector and tensor calculus and special and general relativity, I have noted that many people have the same problems when trying to manipulate index expressions. The problem often lies in being familiar with the very basic ideas, such as repeated indices implying a sum (the summation…
When I ask questions about the conservation of information I frequently get the reply, “It depends on what you mean by information.” So, I researched how to better define information. What I found is almost more interesting than the conservation question. That sounds like the makings of a fun PF Insight article, so here goes….
This Insight takes a look at how it is possible to calculate the spin of Sagittarius A*, the supermassive black hole at the center of the Milky Way using some data and a few equations derived from Kerr metric. The following paper was used as a source- Spin and mass of the nearest supermassive black…
Physics Forums is pleased to introduce Niels Tuning. He’s a physicist working as the Run Coordinator for the LHCb experiment at CERN. Apart from the questions below you can learn more about him at his website. Niels Tuning also has an active Twitter account here. Please give us a bit of background on your life, education and…
We asked our PF Advisors to tell us what academic advice would you tell your 18 year old self. Here are their responses… Thanks to all the PF Advisors that shared their academic advice. We have more questions for them in the near future. Stay tuned for another “Ask the Advisors”!
A few months ago, pervect pointed me to a post by Chris Hillman which is an introduction to the usage of Maxima for General Relativity. Maxima is a free (both as in beer and as in speech) symbolic algebra package, and it includes a library called censor that handles tensor components and looks to have…
Paper discussion: Solar system expansion and strong equivalence principle as seen by the NASA MESSENGER mission. Antonio Genova, Erwan Mazarico, Sander Goossens, Frank G. Lemoine, Gregory A. Neumann, David E. Smith & Maria T. Zuber. Nature Communications volume 9, Article number: 289. Key Points Students of physics learn about the sun‘s gravitational influence which causes planets to…
Read Part 1: Equilibrium Systems The Ideal Gas: Boltzmann’s Approach (The Microcanonical Ensemble) Consider a monatomic gas of ##N## non-interacting particles with mass ##m## occupying the volume ##V##. Since the particles of the gas do not interact with each other, it is not difficult to explicitly calculate ##\Omega_{E}##. The position of each particle is constrained to…
One of the common questions or comments we get on PF is the claim that classical physics or classical mechanics (i.e. Newton’s laws, etc.) is wrong because it has been superseded by Special Relativity (SR) and General Relativity (GR), and/or Quantum Mechanics (QM). Such claims are typically made by either a student who barely learned…
Clifford V. Johnson is a professor in the Physics and Astronomy Department at the USC. “I mainly work on (super)string theory, gravity, gauge theory and M-theory right now, which lead me to think about things like space-time, quantum mechanics, black holes, the big bang, extra dimensions, quarks, gluons, and so forth.” Clifford V. Johnson runs a…
This is one chapter in a series on Mathematical Quantum Field Theory The previous chapter is: 15. Interacting quantum fields. 16. Renormalization In this chapter we discuss the following topics: Epstein-Glaser normalization Stückelberg-Petermann re-normalization UV-Regularization via Counterterms Wilson-Polchinski effective QFT flow Renormalization group flow Gell-Mann & Low RG Flow In the previous chapter we have…
This is one chapter in a series on Mathematical Quantum Field Theory. The previous chapter is 14. Free quantum fields. The next chapter is 16. Renormalization. 15. Interacting quantum fields In this chapter we discuss the following topics: Free field vacua Perturbative S-matrices Conceptual remarks Interacting field observables Time-ordered products (“Re”-)Normalization Feynman perturbation series Effective…
This article is part of our student writer series. The writer Arman777, is an undergraduate physics student at METU Previous Chapter: A Journey Into the Cosmos – The Friedmann Equation Chapter 2- FLRW Metric and The Friedmann Equation In this chapter, we will further investigate the Friedmann equation and we will…
A Short Proof of Birkoff’s Theorem derived the Schwarzschild metric in units of ##G = c = 1##: \begin{equation} ds^2 = -\left(1 – \frac{2M}{r}\right)dt^2 + \left(1 – \frac{2M}{r}\right)^{-1}dr^2 + r^2d\theta^2 + r^2 \sin^2\theta d\phi^2 \label{metric} \end{equation} and I used that metric in The Schwarzschild Metric: Part 1, GPS Satellites to show that Global Positioning System (GPS) clocks…
The following is one chapter in a series of Mathematical Quantum Field Theory. The previous chapter is 11. Reduced phase space. The next chapter is 13. Quantization. 12. Gauge fixing While in the previous chapter we had constructed the reduced phase space of a Lagrangian field theory, embodied by the local BV-BRST complex (example 11.21),…
A Global Positioning System (GPS) device gives your precise location by receiving light pulses from satellites with synchronized clocks then triangulating your location based on that information [1]. Since light travels at 300 million meters per second, your location will be off by about 1 meter if the clock times are off by only…