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Black Hole 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…
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]…
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…
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…
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…
The following problem was inspired by reading Greg Egan’s “Incadescence”. Consider a body, orbiting a Schwarzschild black hole and tide locked to it, as in figure 1 below. The body itself has negligible mass, but due to the tidal gravity around the black hole, an observer on the body will feel “weight”. The basic problem…
Introduction The purpose of this article is to discuss the title question from several different viewpoints, to show that it isn’t as simple as it looks. We will look at some common misconceptions that lead people to think the answer must be “no”, and we will look at some of the issues involved that prevent…
In the early universe, the matter was gathered together at very high density, so why wasn’t it a black hole? The first thing to understand is that the Big Bang was not an explosion that happened in one place in a preexisting, space. The Big Bang happened everywhere at once, so no location would be…
The following is an overview of the time dilation and gravitational redshift effects of a static (Schwarzschild) black hole. By general relativity, a strong gravitational field can slow downtime. The closer you get to the event horizon of a black hole (if you can survive the gravity gradients, and g-forces and have some means of…
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 hole at some constant radius [itex]r > 2M[/itex]. (Note that this is not the same as being in a stable circular orbit about the hole; we allow for…
Abstract There is so much to say about the many endeavors by professional scientists to explain to us the world. The list is long: Carl Sagan, Harald Lesch, Neil deGrasse Tyson, Sabine Hossenfelder, Michio Kaku, and I even saw Roger Penrose and Steven Hawking on TV. The list is – of course – considerably longer…
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…
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…
Introduction The theory of geodesic congruences is extensively covered in many textbooks (see References); what follows in the introduction is a brief summary. Consider a 1-parameter family of timelike geodesics ##\gamma_s(\lambda)##, where ##s## labels each geodesic in the family whilst ##\lambda## is an affine parameter along each ##\gamma_s##. Then the vector field ##\xi \equiv \partial / \partial…
The Tolman law describes how the temperature in a fixed gravitational field depends on the position (see https://arxiv.org/abs/1803.04106 for a pedagogic review). Here I present a simple general derivation of the Tolman law. It’s generally in the sense that it is applicable to any (thermalized) physical degrees of freedom for which one can define Lagrangian…
I’m sure anyone who has hung out long enough here on Physics Forums has encountered threads that go something like this (I’ll use an example based on threads I’ve seen and participated in in the relativity forum, but I’m sure similar things occur in other forums as well): Original Poster: I don’t understand how black…
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…
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…
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…
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…
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…
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,…
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….
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…
Introduction: Why Rindler Motion? When students learn relativity, it’s usually taught using inertial (constant velocity) motion. There are lots of reasons for this, but mainly it’s because it’s the easiest kind of motion for deriving the results of relativity, and historically, thinking about inertial motion is what led to Einstein’s theory. An unfortunate side-effect of…
Adam Becker is an astrophysicist and science writer whose first book “What Is Real?: The Unfinished Quest for the Meaning of Quantum Physics” just hit the bookshelves! Give us some background on how you got interested in physics and some experiences in youth/school that were formative. I don’t remember a time when I wasn’t interested…
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…
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)…
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…
This is the beginning of a series that gives an introduction to perturbative quantum field theory (pQFT) on Lorentzian spacetime backgrounds in its rigorous formulation as locally covariant perturbative algebraic quantum field theory. This includes the theories of quantum electrodynamics (QED) and electroweak dynamics, quantum chromodynamics (QCD), and perturbative quantum gravity (pQG) — hence the…
We are pleased to introduce Sabine Hossenfelder. Sabine is a theoretical physicist from the Frankfurt Institute for Advanced Studies. Sabine also authors a very popular physics blog called backreaction. Reading your blog over the past 6 months or so, one gets a feeling that your views about (particle) dark matter versus modified gravity have changed during that…
In the last article, we looked at various counterintuitive features of the Schwarzschild spacetime geometry, as illustrated in the Kruskal-Szekeres spacetime diagram. But counterintuitive, in itself, does not mean physically unreasonable or unlikely. So the obvious next question is, how much of the entire spacetime geometry we have been looking at is actually believed…
When we left off in part 2, we were looking at the metric for the Schwarzschild geometry in Kruskal-Szekeres coordinates: $$ ds^2 = \frac{32 M^3}{r} \left( – dT^2 + dX^2 \right) + r^2 \left( d\theta^2 + \sin^2 \theta d\phi^2 \right) $$ And we saw that we can draw a spacetime diagram using these coordinates…
We’ve been looking at how the continuum nature of spacetime poses problems for our favorite theories of physics — problems with infinities. Last time we saw a great example: general relativity predicts the existence of singularities, like black holes and the Big Bang. I explained exactly what these singularities really are. They’re not points…
Combining electromagnetism with relativity and quantum mechanics led to QED. Last time we saw the immense struggles with the continuum this caused. But combining gravity with relativity led Einstein to something equally remarkable: general relativity. In general relativity, infinities coming from the continuum nature of spacetime are deeply connected to its most dramatic successful…
We are pleased to introduce Sean Carroll! Sean Carroll is a theoretical physicist at Caltech, specializing in cosmology and quantum mechanics. Sean has a new book out called The Big Picture, where the topic is “On the Origins of Life, Meaning, and the Universe Itself”. Key Points Sean Carroll’s educational journey took him from suburban Philadelphia to Villanova University,…
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 to defeating it, creating the Gravity Research Foundation which sponsors a yearly essay contest focused on the understanding of gravity. This contest has been won by the likes of Stephen Hawking…
Please give us a bit of background on your life and professional experience Life: Born educated in Italy, then 10 years in the States and 15 in France. I got to physics late: before I was more into the hippies dreams or trying to overthrow the Italian government. Professional: in my third university year I…
Read part 1 of this interview. Tell us about your experience with past projects like “This Week’s Finds in Mathematical Physics”. I was hired by U.C. Riverside back in 1989. I was lonely and bored, since Lisa was back on the other coast. So, I spent a lot of evenings on the computer. We had…
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. Since then, I’ve learned more about the theory, and it turns out that Fred Hoyle was a lot cleverer than I’d assumed. The following is an…
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 general relativity (GR) and brought BW to bear on general relativistic cosmology to ‘explain’ what puzzles so many people about big bang cosmology, i.e., the origin of the universe….
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 randomly, with the constraint that it can’t visit the same spot more than once. These are mathematically somewhat interesting and crop up in a few areas of physics;…
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 of several posts on this subject. Short answer: mu. (The terms of the question are not well-defined, so it doesn’t have a well-defined answer.)…
The Planck length is an extremely small distance constructed from physical constants. There are a lot of misconceptions that generally overstate its physical significance, for example, stating that it’s the inherent pixel size of the universe. The Planck length does have physical significance, and I’ll talk about what it is, and what it isn’t. Key…
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 popularizations, which is that all the matter in the universe will first be sucked into black holes, and then recycled into photons in the distant future through…
News of the LHC progress has dazzled scientists and hobbyists alike. It’s now time to show just how much you know about the operation. Please share your results Ready for your next quiz? How well do you know about the Periodic Table of Elements? 1. What is the Large Hadron Collider (LHC)? The Large Hadron…
Star Fate 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…
Meet a Mentor is a weekly series to help you get to know your wonderful Mentors better. Constructive questions and comments are welcome! Today we meet: George Jones Give us a brief history of George Jones. On my mother’s side, my roots in Canada have been traced back to the late 1600s; on my father’s side,…
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…
In the previous insights article (How to Use Duality in Computational Electromagnetic Problems), I covered some uniqueness theorems for the Riemann-Silberstein vector, ##F=E+iB##, for time-harmonic fields. We showed that the boundary value, ##f##, completely determines the vector field throughout the volume. In this article, I’d like to step through constructing a numerical algorithm for finding…
Authorship Contributions From Anorlunda You may want to believe the article you read, or the video you saw, or you may be very proud of your PMM invention. It can be very difficult to analyze what is inside the box to find the flaw. Maybe we can’t find it. Maybe nobody can find it. If…
The electric power industry faces much turmoil in the coming decades. The business model of the electric utility company (public or private) may not survive. In addition, the power needs of high-density cities may diverge significantly from non-urban areas causing political turmoil and technical hurdles. The increasing share of renewable energy will be a major…
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…
Financial options (the right to purchase (“call“) or sell (“put“) stock (or other assets)) at a fixed price at a future date have been around for a long time. They are attractive for the option buyer since his/her total risk is the premium (price) paid for the option, as contrasted with futures contracts which are…
Give us a little background on Jim Hardy Born and raised in Miami Florida of parents with roots in Springfield Missouri . So at heart i’m a small town midwestern fellow. Miami was small in the 1950’s but it became cosmopolitan and outgrew me. I spent a career working in the nuclear plant south of…
“I’ve tried astrophotography and want to know how to improve.” Here’s where I assume you are contemplating the purchase of a tracking mount- a motorized tripod. Far from being an afterthought, “A mount must relate to the telescope tube like a clockwork to the hand on the clock.” Purchasing a tracking mount represents a…
Polymer Key Points DNA is a good model system for polymer physics as it is small enough to observe single–molecule dynamics and large enough to be thermodynamically driven When DNA is coiled in the cell nucleus, it has no direct correlation between spatial position and genetic position When stretched out in a nanochannel, there is…
Bayes’ Theorem: Balancing predictive success with falsifiability Despite its murky logical pedigree, confirmation is a key part of learning. After all, some of the greatest achievements of science are unabashed confirmations, from the discovery of acquired immunity to the gauge theory of particle physics. However because we cannot isolate a unique hypothesis from the…
We are pleased to introduce Peter Krautzberger. Peter is a manager for MathJax. MathJax is the technology behind the display of equations on Physics Forums. “The core of the MathJax project is the development of its state-of-the-art, open source, JavaScript platform for display of mathematics.” Please give us a bit background on yourself from growing up…
Part 2. “We don’t need no stinkin’ induction” — Sir Karl Popper The traditional scientific method supposedly employs induction both in the course of forming hypotheses and empirically confirming them. The examples of inductive inference from the previous note—whether the sun will rise tomorrow, or whether the sequence 1,3,5,7,9,… anticipates all odd numbers—were discussed…
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. First, this topic follows naturally from Part 2’s discussion of grid control (remember that flyball governor?). Second, I feel that’s apropos in an era where the public believes that…
Let me use the terms “power grid” and “network” interchangeably. What is the Power Grid Required to Do? Deliver energy to customers, and thereby earn money so that the utilities can profit. Deliver power wherever and whenever the customers demand it. That implies maintaining the dynamic balance between energy produced, energy lost, and energy…
A monographic substitution cipher works by replacing individual characters of plaintext with corresponding characters of ciphertext. It is perhaps the simplest encryption scheme ever devised: early monographic substitution ciphers were employed by Julius Caesar to secure private correspondence. These ciphers were low-tech, required virtually no mathematics, and encryption and decryption could be accomplished by finger…
This is a new Interview category for Insights. While I line up some great new interviews I’ll be migrating some previous mentor interviews. Doc Al is a physics mentor for Physics Forums Can you give us a brief bio? Okay. It was never easy for me. I was born a poor black child. I remember the days,…
Earth’s gravitational field at the surface is approximately 9.8 Newtons/kilogram, or equivalently, 9.8 meters/second/second. But how does that change due to its shape, rotation, and composition and various locations along its surface and within its interior? This article will answer those questions. Earth’s Gravity Key Points Earth’s gravitational field at the surface is approximately 9.8…
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 spectrum. Most of the visible light reaching the Earth’s surface gets re-emitted, eventually, as infrared. Media that pass visible light through but block infrared can act as heat…
The early universe was hot. So hot that nuclei boiled. The great thermal energy of the universe overwhelmed the confining efforts of the nuclear and electromagnetic forces, and droves of fundamental particles—quarks, gluons, leptons, photons—bounced and jostled in a tightly coupled plasma. The universe, however, has a built-in cooling mechanism: expansion. Matter particles dilute…
What is evolution? In every field of scientific endeavor, there comes now and then a quantum shift in knowledge, a grand denouement of the way things are. In physics, we have Newton’s laws of mechanics, Einstein’s theories of Relativity, and the new theories of Quantum Mechanics. In Biology, we have the discovery of DNA and…
This is a picture of the cosmic microwave background: Fig 1. The cosmic microwave background as seen by the European Space Agency’s Planck satellite, 2013. If the universe is an 80-year-old man, here it is as a zygote only a tenth of a second old. It is the oldest light ever observed, and it tells…
PF Insights Writing Guidelines Follow these guidelines when writing for PF Insights, and refer to them before submitting content. What is PF Insights? PF Insights is a community blog for science and math related content. PF Insights is a single blog where many writers contribute to it. The attitude of the blog is friendly with a coat of professionalism. It…
Meet a Mentor is a fun series to help you get to know your wonderful Mentors better. Constructive questions and comments are welcome! Today we meet: DaleSpam Give us a brief history of DaleSpam I have moved around a decent amount during my life: Arizona, Indiana, Colorado, Texas, Michigan, Utah, Ohio, and North Carolina. With all…
Digital Modeling Key Points Digital Signal Processing (DSP) was introduced in the mid-seventies, allowing musicians to simulate the acoustic properties of instruments and analog effects. Until recently, a guitarist’s sounds were limited to the gear available, requiring vintage gear for certain sounds. Recent advances in microprocessor power have allowed current amplifiers to closely match the…