EARTH'S IONOSPHERIC ELECTRIC FIELD
THE STRENGTH OF PLANTS
OHM'S LAW and QUANTUM HALL EFFECT
ROSSBY WAVES - ATMOSPHERE, OCEAN, EARTH CORE?
THE EVIDENCE FOR QUARKS
IS ZERO AN EVEN OR ODD NUMBER?
LEARNING MATHEMATICS
HOW BIG IS BIG AND HOW SMALL IS SMALL
ENTROPY AS DISORDER: HISTORY OF A MISCONCEPTION
BIOLUMINESCENCE IN SEA WATER WAVES
SPEED OF SOUND ON PLANET MARS
LOOKING FOR BLACK HOLES
Aurora over Aberdeen, Scotland September 2024 (courtesy of BBC news).
EARTH'S IONOSPHERIC ELECTRIC FIELD
In the ionosphere, where high energy solar photons ionize the atmospheric atoms to produce electrons and positive ions, and by virtue of statistical mechanical considerations of the differing gravitational masses and thermal energies of the electrons and positive ions, an "ambipolar" electric field has been postulated that provides an electric potential, helping to accelerate ions to escape from the atmosphere. As reported by Glyn A. Collinson, et al, Nature 2024; 632: 1021, this field may now have been discovered by the NASA Endurance rocket mission, launched from Norway on 11 May 2022 for the aim of examining the existence of this field near the polar regions.
The craft used a photoelectron spectrometer to measure the vertical attitude profile drop in the electron kinetic energies to provide an indicator of the change in electric potential energy, as the mission traversed the atmosphere from the earth's surface to beyond 700 km. It noted a fairly linear drop of 1.09 ± 0.17 μV m¯¹ of electric field potential between the attitude of 250 to 768 km. The authors note: "The measurements support the hypothesis that the ambipolar electric field is the primary driver of ionospheric H+ outflow, and of the supersonic polar wind of light ions escaping from the polar caps. The 1.09 μV m¯¹ field measured over the sunlit polar region is sufficient to provide an outward force on ionospheric H+ of 10.6 times that of gravity. This value is close to eight times the gravity expected from theoretical calculations, assuming an isothermal O+-dominated ionosphere with an electric field driven purely by the electron pressure gradient".
THE STRENGTH OF PLANTS
Plants like trees can exert huge pressures. Examine a tree trunk plastic ring protector, sometimes placed to protect a young plant and consisting of a short segment of a polyvinyl chloride cylindrical tube, which splits apart once the tree trunk has grown in size and stretched it to its ultimate stress strength. One can estimate the pressure exerted by the tree trunk on such a tube, using the cylindrical circumferential stress formula for a thin-walled cylinder: σ = Pr/t where σ is the circumferential stress, P is the net internal pressure, r radius and t thickness of the wall.
Approximating the ultimate stress for polyvinyl chloride cylinder to be around 10 MPa, the ring radius to be about 6 cm and the thickness 0.2 cm, we can get an estimated value of the net internal pressure that a tree trunk can generate to be around 3.5 atmospheres.
OHM'S LAW AND QUANTUM HALL EFFECT
We are taught in junior school that Ohm's law states that voltage is linear to current (ie. the resistance of the wire is constant). In high school, we are taught about the Hall effect, which is the production of a transverse voltage difference across a conductor carrying a current, when placed in a magnetic field perpendicular to the conductor.
Interestingly, when the Hall effect is performed in a very thin 2-dimensional material under low temperatures and reasonably high magnetic field B, it is found that the graph of the transverse resistivity of the material has plateaus as magnetic field B is increased, rather than demonstrate a straight line (ie transverse resistivity is quantized). Each of these plateau resistivity is in (1/n) units of h/e², where h is the Planck's constant and e is the elementary charge. This phenomenon is called the quantum Hall effect and is partly related to the quantization of the orbital motion of free electrons in a magnetic field, giving rise to discrete energy levels called Landau levels (after the well-known Russian physicist Davidovich Lev Landau). Associated with this quantized transverse resistivity, the longitudinal resistivity vanishes, and electrons can be transported longitudinally without dissipation, energy consumption or heat production.
Recently, investigators (including one in Singapore) found that this quantum Hall effect can also be seen in 3-dimensional materials. The likely first report of this experimental finding was published in Tang, F., Ren, Y., Wang, P. et al. Three-dimensional quantum Hall effect and metal–insulator transition in ZrTe5. Nature 569, 537–541 (2019). The authors wrote:
"The discovery of the quantum Hall effect (QHE) in two-dimensional electronic systems has given topology a central role in condensed matter physics. Although the possibility of generalizing the QHE to three-dimensional (3D) electronic systems was proposed decades ago, it has not been demonstrated experimentally. Here we report the experimental realization of the 3D QHE in bulk zirconium pentatelluride (ZrTe5) crystals. We perform low-temperature electric-transport measurements on bulk ZrTe5 crystals under a magnetic field and achieve the extreme quantum limit, where only the lowest Landau level is occupied, at relatively low magnetic fields. In this regime, we observe a dissipationless longitudinal resistivity close to zero, accompanied by a well-developed Hall resistivity plateau proportional to half of the Fermi wavelength along the field direction. This response is the signature of the 3D QHE and strongly suggests a Fermi surface instability driven by enhanced interaction effects in the extreme quantum limit. By further increasing the magnetic field, both the longitudinal and Hall resistivity increase considerably and display a metal–insulator transition, which represents another magnetic-field-driven quantum phase transition. Our findings provide experimental evidence of the 3D QHE and a promising platform for further exploration of exotic quantum phases and transitions in 3D systems."
As shown in this diagram, the red line shows the transverse resistivity flattening out at about 1.5 Tesla in a 3-dimensional material zirconium pentatelluride.
[taken from Galeski, S., Ehmcke, T., Wawrzyńczak, R. et al. Origin of the quasi-quantized Hall effect in ZrTe5. Nat Commun 12, 3197 (2021).]
There is another aspect of the quantum Hall effect. The quantum Hall effect is generally seen at very low temperatures of a few milliKelvins and under strong magnetic fields of several Teslas. In certain materials however, the quantum Hall effect can be demonstrated even without an external magnetic field, and this is called the "anomalous quantum Hall effect". The physics underlying this is fairly elaborate and complex, and much research is now devoted to this area of condensed matter physics, particularly in view of the quest towards superconducting quantum computing.
ROSSBY WAVES - ATMOSPHERE, OCEAN, EARTH CORE?
First described by Carl-Gustaf Arvid Rossby in 1939, these are slowly travelling waveforms in the atmosphere of a rotating planet, at the boundary of the polar vortex and the more warmer lower latitudes atmosphere, that causes the boundary jet stream to follow a meandering path (see diagram). In planetary atmospheres, including Earth, Rossby waves are due to the variation in the Coriolis effect with latitude, along with pressure gradient and other landmass contributions. In theory, a cylindrical earth will not have Rossby waves.
The Rossby waves are like a superposition of differentials of Coriolis and vorticity effects over the latitudes causing a superimposed waveform to move with a slow group velocity eastward.
Occasionally with deep indentations and amplification of these Rossby waves, pockets of high and low pressure may form, and these can develop into cold polar zones moving south from the polar regions, leading to significant unexpected climate changes from heat waves to cold waves, extreme storms and cyclones.
In the ocean, such Rossby waves can also occur in the thermocline zone (between the surface warm ocean and the deeper colder ocean). A more interesting idea has been proposed by Bardsley (2018) for the origin of the westward drift of the Earth’s magnetic field, based upon the propagation of hydrodynamic Rossby waves in the Earth's liquid outer core (Proc. R. Soc. A 474: 20180119).
THE EVIDENCE FOR QUARKS
Quarks are so incredibly small and trapped within hadrons. How do we know if they really exist?
There are 3 main lines of evidence (as well explained in Allday's book).
1.Theoretical
Gell-Mann and George Zweig in 1964 independently proposed an explanation for the observed 'eightfold way' pattern of arranging hadrons using 3 basic sets of properties in combination. Bjorken published a seminal paper in 1966 on probing the inner world of protons and in a conference in 1967 he gave a talk in which he imagined the proton to be built of quarks. Feynman himself was also toying with the idea that the proton was made up of smaller parts which he called "partons" (Frank Close tells the story as an insider during this exciting time of discovery).
2.Deep inelastic scattering of electrons by protons (SLAC Experiment)
Physicists Friedman, Taylor and Kendal in the late 1960s performed an experiment where electrons were fired at protons, in a similar way to the Rutherford-Marsden experiment for the nuclear atomic model. What was found was just as surprising: at low energy, the electrons were deflected slightly by the protons. However, as the electron energy increased over a threshold, suddenly the elections were deflected over much greater angles and the protons fragmented in showers of particles. This experiment suggested that the electrons were interacting directly with the quarks within the protons.
3.Electron-positron annihilation reactions
The energy released from these annihilation reactions can produce lepton-antilepton pairs such as μ+μ- via the electromagnetic interaction (photon) or weak interaction (Z⁰). Occasionally, instead of 2 leptonic particles moving away in opposite directions, 2 showers of particles called jets emerge from the reaction point. The fact that jets occur suggests that particles were being formed from more basic constituents (in this case probably quarks):
The results of such reactions can be plotted as a composite graph of R versus the energy of the reaction (as shown above). R is the ratio of the cross-section of the e+e- to hadrons reaction to the cross-section of e+e- to muon pair reaction. As Perkins explains, the constancy of R over large energy ranges with sudden steps is evidence of pointlike constituents of hadrons (relative to the muon reaction which is pointlike).
References:
Jonathan Allday, Quarks, Leptons and the Big Bang, CRC Press 2017, chapter 9.
Frank Close, The Infinity Puzzle, Basic Books 2011, chapter 12.
Donald Perkins, Introduction to High Energy Physics, Cambridge University Press 2000, chapter 5.
IS ZERO AN EVEN OR ODD NUMBER? The question is surprisingly relevant today. To deal with fuel shortages after Superstorm Sandy, NY Mayor Bloomberg introduced fuel rationing on 8 Nov 2012. Drivers who have licence plates that end in an odd number or some character will be allowed to gas on odd numbered days, while those ending with an even number or zero will gas on even-numbered days.
Fast forward 9 years to 2021, during the second year of the Covid-19 pandemic, to encourage safe distancing in 2 large popular malls, odd-even weekend entry restrictions, based on the last digit of the national identity card, were imposed by the South-East Asian country Singapore. So, is zero odd or even?
From a certain mathematical perspective, zero is an even number, because it belongs to the ideal 2Z of the ring Z of integers. But sometimes it is not so clear. A survey of primary school children in the 1990s showed that 50% thought zero is even, about 20% thought it odd and 30% thought it neither or both or not sure, reported BBC.
The early Greeks such as Pythagoras and Euclid apparently had no concept of zero. The early Chinese left blanks in the place of zeros. The Babylonians apparently were the first to introduce a sign to fill the space for zero. Early Indian-Arabic writings popularized the symbol "o" for zero. Zero became more widely known through the writings of Renaissance mathematicians such as Fibonacci. A crucial question debated was "is zero a symbol representing nothing or a number?" What is so special about zero as a number? Or how would a person define zero? The idea of emptyness is vague. Mathematically, zero is a number which results when a number and its inverse (under addition) are combined, or alternatively zero is a number which when added to a number, does not change it. This idea of zero is a modern algebraic concept and zero can thus be identified abstractly as the unique identity under the operation of addition. For an interesting account of zero, one may read Charles Seife's Zero, The Biography of a Dangerous Idea (Penguin Books 2000) or refer to a book on the history of mathematics such as Merzbach and Boyer's A History of Mathematics (Wiley 2011).
LEARNING MATHEMATICS
One facet of learning mathematics is like learning how to paint a piece of art, even an abstract one, creative and yet demanding a high level of rigor and fine sophistication. If this is an apt illustration, how do you learn or teach mathematics? What underlying motivations would drive this passion to learn or teach mathematics?
There are several views of why anyone should do or learn mathematics. Michael Harris, in his book Mathematics without Apologies, lists 3 typical reasons - mathematics is good, true, and beautiful. The last was eloquently argued by G.H. Hardy.
However, Harris adds 3 additional reasons. One was tradition, the second pleasure and third the quest for a romanticized idealized intellectual authority which he called "charisma". "The word charisma colloquially means a kind of personal magnetism, often mixed with glamour, but Weber chose the word to designate the quality endowing its bearer with authority (Herrschaft, also translated domination) that is neither traditional nor rational (legally prescribed)." (p 15).
On the other hand, Sabine Hossenfelder, in the book Lost in Math, argues that beauty in mathematics and science may not necessarily lead to discovery. She gave the example of supersymmetry which postulates the existence of supersymmetric particles, but so far there has been no experimental evidence of such particles even after the Large Hadron Collider has produced quadrillions of particle collisions.
Roberto Unger and Lee Smolin attribute this inefficacy of beautiful science or mathematics to what they called the "selective realism" of mathematics (in the book The Singular Universe and the Reality of Time).
The following books, among many, explore what mathematics is and how to teach or learn it:
1. Shai Simonson, Rediscovering Mathematics, Mathematical Association of America 2012.
2. Robert Kaplan and Ellen Kaplan, Out Of The Labyrinth, Bloomsbury Press 2007.
3. Mark Kac and Stanislaw M Ulam, Mathematics And Logic, Dover 1992.
4. Philip J Davis and Reuben Hersh, The Mathematical Experience, Penguin 1990.
5. Richard Courant and Herbert Robbins, What is Mathematics? Oxford University Press 1996.
6. G Polya, How To Solve It, Princeton University Press 1985.
7. Ian Stewart, Concepts of Modern Mathematics, Dover 1995.
8. Keith Devlin, Mathematics: The New Golden Age, Penguin 1990.
9. Felix Klein, Elementary Mathematics from a Higher Standpoint: Arithmetic, Algebra, Analysis Volume 1, Springer 2016.
To see the human side of mathematics, there is a nicely written book by Donald Albers and Gerald Alexanderson on contemporary mathematicians such as Lars Ahlfors and Tom Apostol, titled Fascinating Mathematical People, Interviews and Memoirs. Most of the mathematicians interviewed in this book recalled being inspired towards mathematics by an excellent teacher when they were young.
HOW BIG IS BIG AND HOW SMALL IS SMALL The sizes of everything and why by Timothy Paul Smith Oxford University Press 2013. "If the meter is the measure of humans, then we are closer to quarks than we are to quasars. However, if we take the second as the heartbeat of our lives, then we are closer to the age of the universe than to the lifetime of elementary particles. There are well over forty-five orders of magnitude between the largest things we have ever measured - the grand breadth of the universe itself - and our smallest measurement, the probing of those iotas of matter, quarks, electrons and gluons ... the cubit is one of those ancient measurements that is very convenient for humans. It is the distance from the elbow to the outstretched fingertips ... " (quote from the first chapter of the book).
ENTROPY AS DISORDER: HISTORY OF A MISCONCEPTION (an excerpt)
Dan Styer, The Physics Teacher 57, 454 (2019)
"More telling than any specific passage is the overall tone of Henry Adams’s Education that things are going downhill and that nature is responsible. That tone comes through even more clearly in other writings by Adams, which draw an explicit connection between “entropy” and “bad things.” His 1909 essay “The Rule of Phase Applied to History” claims to use Gibbs’s phase rule to calculate that “nature’s power” acting through the “sharp curve” of history would “bring Thought to the limit of its possibilities in the year 1921.” His 1910 “Letter to American Teachers of History” asserts that “physicists cease to be physicists unless they hold that the law of Entropy includes Gods and men as well as universes” and that “the law of Entropy imposes a servitude on all energies, including the mental.”
In conclusion, I hope I’ve convinced you that entropy is not a synonym for disorder or uniformity. The association between entropy and disorder was started by scientists like Boltzmann and Helmholtz in connection with gases, where it’s appropriate. The association was inappropriately championed into the popular imagination by Henry Adams. There might have been earlier champions, but Adams certainly was one such champion and I suspect that he was the keystone .... "
"In the physics sense, the word “entropy” doesn’t mean “undesirability.” Yet you can find published statements like, “Entropy is why ice cream melts and people die. It is why cars rust and forests burn.” In the physics sense, the word “entropy” doesn’t mean “immorality” or “wickedness.”
Yet you can find published statements like:
"DEAR ABBY: … it’s a sad fact that when good and bad associate, it isn’t the rotten guy who gets good, it’s the good guy who gets rotten. In scientific language, each entity seeks the lowest energy level. It is related to the concept of entropy, which is fact, not theory. –S.A.S."
Entropy and Heat
My comments: dS = dQ/T can be imaginatively read as if entropy dS is a kind of "thick spread" of heat energy dQ, like pouring honey onto a piece of bread; the greater the heat energy introduced into the system, the thicker the "spread", but the spread is also dependent on the temperature (represented by the area of the bread), and the change in the thickness of the spread is relatively more pronounced when it is cold (small piece of bread and assuming that the honey does not flow beyond the margins) than when it is hot.
One could imagine that this 'spread' or entropy S is somewhat akin to the amount of variety or permutations of how heat energy can be distributed in the system. Permutations at the microscopic energy levels are referred to as "microstates". One might therefore think that the thickness or variety of 'spread' or entropy S should be linearly related to the number of microstates; that is not so, rather, it is related to the (natural) logarithm of the number of microstates.
Julian Barbour makes a similar point from a different perspective: "Reflecting that, the increment of entropy dS in Clausius's definition measures the quality of the added heat dQ. If it is added at a low temperature T, then the entropy increment dS = dQ/T is large; the heat is, as it were, 'spread out' or diffuse, as indeed it is in the large room you would not choose to enter. If T is large, dS is small, the heat is more 'concentrated' and it has higher quality. You can achieve more with it" (The Janus Point, A New Theory of Time, Penguin Random House 2020, chapter 2).
Heat is a kind of energy that causes 'small-scale molecular motion' quantitated by entropy S (alluded to by Shang-Keng Ma in his book Statistical Mechanics). When heat is added to a system, it causes shifts in the position and momentum of the particles comprising the system, ie the "changes in microstates" and hence, entropy S. Temperature T is the "force" against which these small-scale molecular motion dS takes place, and thus TdS is the "work done" equivalent to the energy dQ or heat. The internal energy U of a body necessarily consists of small-scale molecular motion energy encoded by the permutations (ie. entropy S) while the rest are large scale motion energy (ie. work due to change in macroscopic thermodynamic variables like volume V) - this dual nature of internal energy U is clearly illustrated in Kittel's Thermal Physics.
From a statistical mechanics viewpoint, if U is the internal energy of a system, p being the probability of occupation of the energy level E, then:
From the second equation, the first term on the right represents change in the energy levels without any change in the occupational numbers of these levels, suggesting an organized change in energy ie. work done. The second term represents a change or permutation in the occupational numbers of the energy levels, due to small-scale molecular motion or heat (please refer to Donald McQuarrie and John Simon, Molecular Thermodynamics, University Science Books 1999, p 198 or Franz Schwabl, Statistical Mechanics, Springer 2006, p 61 for more information).
This second equation corresponds to the first law of thermodynamics. This means that the first term on the right corresponds to dW and the second term to dQ.
References
Thomas Engel, Philip Reid. Thermodynamics, Statistical Thermodynamics & Kinetics, Pearson 2013.
BIOLUMINESCENCE IN SEA WATER WAVES
A bluish-green glow observed in the incoming waves along the beach of north-eastern Singapore on 22 March 2022, due to dinoflagellates (see below). Dinoflagellates are eukaryotic protists that are often capable of producing cold light.
The production of light occurs in organelles termed scintillons, which contain the substrate luciferin, the enzyme luciferase and sometimes, a luciferin binding protein. The oxidation of luciferin (a substance with structural similarity to chorophyll) produces light, either directly or through a luciferin binding protein. This reaction is catalysed by the enzyme called luciferase.
A theory of bioluminescence postulates that mechanical shear stress over the cell membrane activates a G-protein coupled receptor, which allows ionic calcium to accumulate in the cytoplasm, which then produces an action potential across the vacuole membrane causing proton channels to open and to allow the intra-vacuolar pH to drop and activate the luciferin reaction, as shown in the diagram on the left.
(Diagram taken from Valiadi M, Iglesias-Rodriguez D. Understanding Bioluminescence in Dinoflagellates - How Far Have We Come? Microorganisms. 2013;1(1):3-25).
SPEED OF SOUND ON PLANET MARS
As announced at the 53rd Lunar and Planetary Science Conference by planetary scientist Baptiste Chide of the Los Alamos National Laboratory, the speed of sound on the surface of planet Mars as determined by the NASA rover Perseverance is about 240 m/s. Chide and his team measured the time between the laser firing and the sound reaching the SuperCam microphone. This speed value is much lower than that on the Earth's surface (340 m/s).
We can do a simple estimation of the speed of sound on planet Mars.
The speed of sound is given by the formula shown, where 𝛾 is the adiabatic constant of the gas, R is the gas constant, T is the temperature in Kelvins, and M is the molar mass of the gas:
As Mars atmosphere is 95% carbon dioxide, substituting 𝛾 = 1.28 for carbon dioxide, R = 8.31 J/mol, T = 243 K ( -30 C ) and M = 0.044 for carbon dioxide, we get a speed of sound of approximately 240 m/s, which is consistent with what was measured on planet Mars by the Perseverance rover.
A.P. French derived the above formula for velocity of sound as if it was due to a "spring of air" (refer to the book Vibrations and Waves, Norton & Company 1971, p 57). That sound obeys the second-order linear partial differential equation for a wave is well explained in Demtroder, Mechanics and Thermodynamics, Springer (2017) p 347.
LOOKING FOR BLACK HOLES
The James Webb Space Telescope was launched on 25 December 2021 from French Guiana, and arrived at the Sun–Earth L2 Lagrange point in January 2022. This is the largest space telescope to date, working mainly in the optical infrared spectrum and its primary mirror is about 6 times larger in terms of light collecting area than that of the Hubble Telescope. One of the striking images captured was the Stephan's quintet, where the 4 right-most galaxies are close together and of interest due to their potential interactions with each other (picture from NASA):
In 2022, the James Webb space telescope will devote part of its time to investigating the supermassive compact object (black hole) Sagittarius A* at the centre of the Milky Way, captured previously by radiotelescopes as shown here (image from NASA):
The ring shown in the image is likely due to the emission of electromagnetic radiation from the hot accretion ring around the horizon of the black hole. The horizon itself is the "surface" of the black hole (assumed to be stationary and non-rotating), that occurs when the metric ɡᵣᵣ in the Schwarzschild's solution to Einstein's equation approaches infinity, and this horizon occurs at the radial distance of (2GM/c²), where G is the gravitational constant, M is the mass of the compact body in the black hole and c is the speed of light.
Hawking's radiation from blackholes
Within the horizon, light cannot escape, but Stephen Hawking theorized that a certain kind of low-energy blackbody radiation can emerge from the black hole horizon (called Hawking radiation). Virtual particle-antiparticle creation at the horizon from a Dirac quantum field "vacuum state" ⟨0⎟ T (ψψ†𝛾⁰)⎟0⟩ (ie. at zero point energy) can allow each of the 2 virtual particles to go on separate worldlines, one inside the horizon and one outside, without breaking the conservation of energy.
"The result is that, to a distant observer, the black hole emits a particle with positive energy. This is known as the Hawking effect or Hawking radiation. Clearly, the above discussion is applicable for either particle to reach infinity. Thus, we expect the Hawking radiation to be composed of an equal number of particles and antiparticles. This radiation can take place because of an addition to the black hole of particles of negative energy ie. a decrease of the black hole's positive energy and mass" (taken from Ta-Pei Cheng, Relativity, Gravitation and Cosmology, A Basic Introduction, Oxford University Press (2010), p 173).
Temperature of a blackhole
Interestingly, the temperature of the black hole is calculated to be inversely proportional to its mass (that is, T ∝ 1/M). Strange and counterintuitive as it might be, as the black hole radiates, it loses energy and mass, yet becomes hotter, radiates faster, and eventually evaporates completely, as Hartle explains (James B Hartle, Gravity An Introduction to Einstein's General Relativity, Addison Wesley (2003), p 293).
Anthony Zee, in his delightful book Fly by Night Physics, gave an intuitive proof of the above relation by dimensional analysis that the temperature of a black hole is inversely proportional to its mass.
Entropy of a blackhole
Since mass falling into the black hole is energy, and using the relation dQ = TdS, this means that dM = (1/M)dS and, hence by integration, the entropy S of a black hole should be proportional to M², which is unusual since if mass is distributed uniformly, this means that the entropy of the black hole is proportional to its surface area (not volume)!
Precession of the perihelion of the planet Mercury
A strong experimental support for Einstein's theory of General Relativity was the prediction of 43 seconds of precession of the perihelion (nearest location of the planet to the Sun) of the orbit of planet Mercury per century. This was previously unaccounted for, despite correcting for perturbations from adjacent planets.
From the Schwarzschild's metric for spacetime outside a gravitating mass body (Sun), the geodesic (ie. shortest/straightest path) travelled by the planet Mercury has an additional term of the order of (1/r³) in the dr/dt equation. Translated into the orbit equation, there is an extra term of the order (1/r²) in the dr/d𝜃 equation.
Solving the latter non-linear differential equation by various approximate methods [eg. perturbation (Tevian Dray, Rindler, Bergmann, Ludvigsen, Sean Carroll, D'Inverno), integration (Sidney Coleman) or by using the effective potential function (Carlo Rovelli, Wald)] gives rise to a small extra angle per revolution (approximately 6πGM/c²r) for the planet to reach its next perihelion.
A simple guided calculation based on the effective potential function is found in the book, Exploring Black Holes by Edwin F Taylor and John Archibald Wheeler.
This precession is best observed in the planet nearest the Sun with the smallest radius of orbit, that is Mercury (image courtesy of Professor Kenneth Lang @ NASA Cosmos).
Gravitational Lensing
Astronomer have observed possibly the most distant star ever detected, using the James Webb Space Telescope. This massive B-type star, named Earendel, is found in the so-called Sunrise Arc which arose through gravitational lensing by an intervening galaxy WHL0137-08 into an Einstein ring. It is estimated to be in existence within the first 1 billion years of the universe.
(courtesy of NASA and ESA).
For other examples of Einstein rings due to gravitational lensing:
References:
Tevian Dray, Differential Forms and the Geometry of General Relativity, CRC Press (2015), p 122-123.
SIdney Coleman, (eds. David Griffiths, David Derbes, Rischard Sohn), Sidney Coleman's Lectures on Relativity, Cambridge University Press (2022), p 171-176.
Carlo Rovelli, General Relativity: The Essentials, Cambridge University Press (2021), p 108-113.
Robert M Wald, General Relativity, University of Chicago Press (1984), p 142.
Wolfgang Rindler, Relativity special, general and cosmological, Oxford University Press (2006), p 243.
Peter Gabriel Bergmann, Introduction to the Theory of Relativity, Dover (1976), p 215.
Anthony Zee, Fly by Night Physics, Princeton University Press (2020), p 146.
Edwin F Taylor and John Archibald Wheeler, Exploring Black Holes, Addison Wesley Longman (2000), p C1-C13.
Malcolm Ludvigsen, General Relativity, A Geometric Approach, Cambridge University Press (1999), p 144.
Sean M Carroll, Spacetime and Geometry, Addison Wesley (2004), p 214-5.
Ray D'Inverno, James Vickers, Introducing Einstein's Relativity, Oxford University Press (2022), p 301.
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