This is a translation of the paper ¨Uber das Gravitations-
feld eines Massenpunktes nach der Einsteinschen Theorie by Karl
Schwarzschild, where he obtained the metric of a space due to the
gravitational field of a point-mass. The paper was originally published
in 1916, in Sitzungsberichte der K¨oniglich Preussischen Akademie der
Wissenschaften, S. 189–196. Translated from the German in 2008 by
Larissa Borissova and Dmitri Rabounski.
Crack problems concerning boundaries of convex lens like formsijtsrd
The singular stress problem of aperipheral edge crack around a cavity of spherical portion in an infinite elastic medium whenthe crack is subjected to a known pressure is investigated. The problem is solved byusing integral transforms and is reduced to the solution of a singularintegral equation of the first kind. The solution of this equation is obtainednumerically by the method due to Erdogan, Gupta , and Cook, and thestress intensity factors are displayed graphically.Also investigated in this paper is the penny-shaped crack situated symmetrically on the central plane of a convex lens shaped elastic material. Doo-Sung Lee"Crack problems concerning boundaries of convex lens like forms" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-3 , April 2018, URL: http://www.ijtsrd.com/papers/ijtsrd11106.pdf http://www.ijtsrd.com/mathemetics/applied-mathamatics/11106/crack-problems-concerning-boundaries-of-convex-lens-like-forms/doo-sung-lee
A Proof of Hypothesis Riemann and it is proven that apply only for equation ζ(z)=0.Also here it turns out that it does not apply in General Case.(DOI:10.13140/RG.2.2.10888.34563/5)
Abstract: The Riemann zeta function is one of the most Leonhard Euler important and fascinating functions in mathematics. Analyzing the matter of conjecture of Riemann divide our analysis in the zeta function and in the proof of conjecture, which has consequences on the distribution of prime numbers.
1) Two methods are provided to find the shape of a minimal surface between two equal radius rings. Both methods result in the solution y(x) = (1/b)cosh(bx), where b is determined from the boundary conditions.
2) There is a maximum ratio of ring separation / radius (=η0) above which no solution exists. This critical value is found to be η0 ≈ 0.663.
3) The area of the limiting surface at the critical ratio is approximately 20% greater than the area of two separate circles.
It turns out that an equation x^p+y^q=z^w has a solution in Z+ if and only if at least one exponent p or q or w equals 2.DOI:10.13140/RG.2.2.15459.22567/7
The Euler-Mascheroni constant is calculated by three novel representations over these sets respectively: 1) Turán moments, 2) coefficients of Jensen polynomials for the Taylor series of the Riemann Xi function at s = 1/2 + i.t and 3) even coefficients of the Riemann Xi function around s = 1/2.
Presentation of the work on Prime Numbers.
intended for mathematics loving people.
Please send comments and suggestions for improvement to solo.hermelin@gmail.com.
More presentations can be found in my website at http://solohermelin.com.
Crack problems concerning boundaries of convex lens like formsijtsrd
The singular stress problem of aperipheral edge crack around a cavity of spherical portion in an infinite elastic medium whenthe crack is subjected to a known pressure is investigated. The problem is solved byusing integral transforms and is reduced to the solution of a singularintegral equation of the first kind. The solution of this equation is obtainednumerically by the method due to Erdogan, Gupta , and Cook, and thestress intensity factors are displayed graphically.Also investigated in this paper is the penny-shaped crack situated symmetrically on the central plane of a convex lens shaped elastic material. Doo-Sung Lee"Crack problems concerning boundaries of convex lens like forms" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-2 | Issue-3 , April 2018, URL: http://www.ijtsrd.com/papers/ijtsrd11106.pdf http://www.ijtsrd.com/mathemetics/applied-mathamatics/11106/crack-problems-concerning-boundaries-of-convex-lens-like-forms/doo-sung-lee
A Proof of Hypothesis Riemann and it is proven that apply only for equation ζ(z)=0.Also here it turns out that it does not apply in General Case.(DOI:10.13140/RG.2.2.10888.34563/5)
Abstract: The Riemann zeta function is one of the most Leonhard Euler important and fascinating functions in mathematics. Analyzing the matter of conjecture of Riemann divide our analysis in the zeta function and in the proof of conjecture, which has consequences on the distribution of prime numbers.
1) Two methods are provided to find the shape of a minimal surface between two equal radius rings. Both methods result in the solution y(x) = (1/b)cosh(bx), where b is determined from the boundary conditions.
2) There is a maximum ratio of ring separation / radius (=η0) above which no solution exists. This critical value is found to be η0 ≈ 0.663.
3) The area of the limiting surface at the critical ratio is approximately 20% greater than the area of two separate circles.
It turns out that an equation x^p+y^q=z^w has a solution in Z+ if and only if at least one exponent p or q or w equals 2.DOI:10.13140/RG.2.2.15459.22567/7
The Euler-Mascheroni constant is calculated by three novel representations over these sets respectively: 1) Turán moments, 2) coefficients of Jensen polynomials for the Taylor series of the Riemann Xi function at s = 1/2 + i.t and 3) even coefficients of the Riemann Xi function around s = 1/2.
Presentation of the work on Prime Numbers.
intended for mathematics loving people.
Please send comments and suggestions for improvement to solo.hermelin@gmail.com.
More presentations can be found in my website at http://solohermelin.com.
The document summarizes an analytic approach to solving the Collatz conjecture proposed by Berg and Meinardus. It introduces two linear operators, U and V, acting on the space of holomorphic functions on the open unit disk. The kernel K of U and V, defined as the set of functions where both operators equal 0, is of main interest. If it can be shown that K equals the two-dimensional space Δ2, then the Collatz conjecture would be proven true. The individual kernel KV of V is already known from prior work. The paper aims to compute U[h] for h in KV and show that K = Δ2, which would imply the truth of the Collatz conjecture.
1) Index notation is a useful tool for performing vector algebra using subscript notation. Vectors can be expressed as sums of their coordinate components multiplied by basis vectors.
2) Key concepts include: the Kronecker delta symbol δij, Einstein notation where repeated indices imply summation, and the Levi-Civita symbol ijk which determines the sign of cross products.
3) Using these tools, vector operations like the dot product and cross product can be written concisely in index notation. For example, the dot product of two vectors ~a and ~b is aibi and their cross product is ijkaibjêk.
Signals and transforms in linear systems analysisSpringer
This document introduces the Z-transform and its properties for analyzing discrete signals and systems. It begins by defining the Z-transform as an extension of the discrete Fourier transform with a convergence factor, allowing it to represent a wider class of sequences. Properties discussed include the unilateral and bilateral Z-transforms, time shifting, and the region of convergence. Several examples of direct Z-transforms are provided, such as for exponential, unit step, and impulse sequences.
The document discusses several theorems related to twin prime conjectures:
- Theorem 1 states that a prime p can be written in the form 3k+1 or 3k-1, with k being even.
- Theorem 4 characterizes twin primes as pairs where n(n+2) satisfies a modular condition.
- Theorems aim to prove there are infinitely many twin primes, relying on the ratio of primes increasing without bound as n increases without bound.
This document discusses the use of Airy stress functions to solve problems in two-dimensional elasticity. It provides the equations relating stresses to the Airy stress function in plane stress, plane strain, and polar coordinates. Examples are given to show how specific Airy stress functions can be used to solve problems like bending of beams, stresses in curved beams with end moments, and stresses in a quarter-circle beam under an end load.
This document summarizes the numerical solution of the time-independent Schrodinger equation for particles in chaotic stadium and Sinai billiard potentials using the finite difference method. Scars, or regions of high probability density around unstable periodic orbits, were observed in particular eigenstates of both systems, consistent with previous studies. The method was validated by reproducing analytical solutions for 1D and circular wells and showing convergence of eigenvalues with increasing numerical resolution.
I. Antoniadis - "Introduction to Supersymmetry" 2/2SEENET-MTP
1) The Supersymmetric Standard Model (SSM) extends the particle content of the Standard Model by introducing supersymmetric partners for each particle, called sparticles. This includes gluinos, winos, binos, squarks, sleptons, and higgsinos as sparticle counterparts to gluons, W/Z bosons, photons, quarks, leptons, and Higgs bosons.
2) The SSM Lagrangian contains additional terms beyond the Standard Model including gauge interactions between fermions and gauginos, quartic scalar interactions, and a superpotential with Yukawa-like couplings.
3) Spontaneous supersymmetry breaking is required to make sparticles heavy enough to have evaded detection
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
I. Antoniadis - "Introduction to Supersymmetry" 1/2SEENET-MTP
Supersymmetry (SUSY) is a symmetry that relates bosonic and fermionic degrees of freedom. It extends the Poincaré algebra by including spinorial generators (supercharges) that transform bosonic fields into fermionic fields and vice versa. SUSY provides motivations like natural elementary scalars, gauge coupling unification, and a dark matter candidate. SUSY is formulated using superspace, which extends spacetime by Grassmann coordinates. Chiral and vector superfields contain the bosonic and fermionic components of supermultiplets and allow constructing SUSY invariant actions.
First part of description of Matrix Calculus at Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com.
For more presentations please visit my website at
http://www.solohermelin.com.
The document discusses permutations and the symmetric group S3. It defines what a permutation is and introduces the six permutations that make up S3: the identity E, a 120 degree rotation R, a 240 degree rotation R2, a vertical reflection V, and reflections RV and R2V. It explains that S3 forms a group under composition of permutations. It also introduces the alternating group A3, which is the subgroup of S3 made up of the even permutations E, R, and R2.
This document provides an overview of vector algebra concepts including scalars, vectors, unit vectors, position vectors, vector operations, gradients of scalars, divergence of vectors, and del operators. It defines these concepts, provides examples of their use in Cartesian, cylindrical and spherical coordinate systems, and works through examples of calculating gradients and divergences. The key topics covered are the definitions and calculations of gradients, divergences, and how to apply these concepts to vector fields in different coordinate systems.
On a Generalized 휷푹 − Birecurrent Affinely Connected Spaceinventionjournals
In the present paper, we introduced a Finsler space whose Cartan's third curvature tensor 푅푗푘 ℎ 푖 satisfies the generalized 훽푅 −birecurrence property which posses the properties of affinely connected space will be characterized by ℬ푚 ℬ푛푅푗푘 ℎ 푖 = 푎푚푛 푅푗푘 ℎ 푖 +푏푚푛 훿푘 푖 푔푗ℎ − 훿ℎ 푖 푔푗푘 , where 푎푚푛 and 푏푚푛 are non-zero covariant tensors field of second order called recurrence tensors field, such space is called as a generalized 훽푅 −birecurrent affinely connected space. Ricci tensors 퐻푗푘 and 푅푗푘 , the curvature vector 퐻푘 and the curvature scalars 퐻 and 푅 of such space are non-vanishing. Some conditions have been pointed out which reduce a generalized 훽푅 −birecurrent affinely connected space 퐹푛 (푛 > 2) into Finsler space of scalar curvature.
1) The document discusses directional derivatives and the gradient of functions of several variables. It defines the directional derivative Duf(c) as the slope of the function f in the direction of the unit vector u at the point c.
2) It shows that the partial derivatives of f can be computed by treating all but one variable as a constant. The gradient of f is defined as the vector of its partial derivatives.
3) It derives an expression for the directional derivative Duf(c) in terms of the partial derivatives of f and the components of the unit vector u, showing the relationship between directional derivatives and the gradient.
This document discusses Poisson's and Laplace's equations in the context of a class on field theory taught by Prof. Supraja. It provides definitions and derivations of Poisson's and Laplace's equations in different coordinate systems. It gives examples of using these equations to find electric potential and field between charged surfaces, as well as the capacitance between them. It also states the uniqueness theorem, which guarantees a unique solution to Laplace's/Poisson's equation given the boundary conditions.
This document discusses orthogonal trajectories and provides examples of finding orthogonal trajectories for different families of curves. It begins by defining orthogonal trajectories as curves that intersect each other at right angles. It then provides a method for finding the differential equation that describes the orthogonal trajectories for a given family of curves. Several examples are worked out, such as finding the orthogonal trajectories of the family of parabolas with equation y = x^2. Applications to equipotential lines and electric fields and electromagnetic waves are also mentioned.
(Neamen)solution manual for semiconductor physics and devices 3edKadu Brito
This document contains solutions to problems from Chapter 1 of Semiconductor Physics and Devices: Basic Principles, 3rd edition. The problems calculate properties of semiconductor unit cells such as number of atoms, density, and volume percentage occupied by atoms. Lattice structures including simple cubic, body-centered cubic, face-centered cubic and diamond are considered. Properties of common semiconductors such as silicon and gallium arsenide are also calculated.
NUMERICAL METHODS -Iterative methods(indirect method)krishnapriya R
The document discusses two iterative methods for solving systems of linear equations: Gauss-Jacobi and Gauss-Seidel. Gauss-Jacobi solves each equation separately using the most recent approximations for the other variables. Gauss-Seidel updates each variable with the most recent values available. The document provides an example applying both methods to solve a system of three equations. Gauss-Seidel converges faster, requiring fewer iterations than Gauss-Jacobi to achieve the same accuracy. Both methods are useful alternatives to direct methods like Gaussian elimination when round-off errors are a concern.
This document summarizes research on Casimir torque in the weak coupling approximation. It examines manifestations of Casimir torque between planar objects characterized by delta function potentials. The key findings are:
1) An exact calculation of the Casimir torque between a finite rectangular plate above a semi-infinite plate is presented and agrees well with the proximity force approximation when the plate separation is small compared to their sizes.
2) Cusps in the torque arise when the corners of the finite plate pass over the edge of the semi-infinite plate.
3) A similar calculation is done for a disk above a semi-infinite plate, again finding good agreement with the proximity force approximation.
1) The document discusses the problem of a particle sliding off a moving hemisphere, with the goal of finding the angle at which the particle's horizontal velocity is maximum.
2) Conservation of momentum and energy equations are used to derive an expression for the particle's horizontal velocity in terms of the angle, which reaches a maximum at a certain angle.
3) Setting the derivative of this expression to zero results in a cubic equation that determines the maximum angle, which can be solved numerically for different ratios of particle to hemisphere mass.
1) The document discusses the problem of a particle sliding off a moving hemisphere, using conservation of momentum and energy equations to derive an expression for the particle's horizontal velocity vx as a function of the angle θ.
2) Setting the derivative of vx equal to zero yields a cubic equation that determines the angle θ at which vx is maximized, corresponding to the particle losing contact with the hemisphere.
3) For the special case where the particle and hemisphere masses are equal (ratio r = 1), the cubic equation can be solved to find θ ≈ 42.9 degrees.
The document summarizes an analytic approach to solving the Collatz conjecture proposed by Berg and Meinardus. It introduces two linear operators, U and V, acting on the space of holomorphic functions on the open unit disk. The kernel K of U and V, defined as the set of functions where both operators equal 0, is of main interest. If it can be shown that K equals the two-dimensional space Δ2, then the Collatz conjecture would be proven true. The individual kernel KV of V is already known from prior work. The paper aims to compute U[h] for h in KV and show that K = Δ2, which would imply the truth of the Collatz conjecture.
1) Index notation is a useful tool for performing vector algebra using subscript notation. Vectors can be expressed as sums of their coordinate components multiplied by basis vectors.
2) Key concepts include: the Kronecker delta symbol δij, Einstein notation where repeated indices imply summation, and the Levi-Civita symbol ijk which determines the sign of cross products.
3) Using these tools, vector operations like the dot product and cross product can be written concisely in index notation. For example, the dot product of two vectors ~a and ~b is aibi and their cross product is ijkaibjêk.
Signals and transforms in linear systems analysisSpringer
This document introduces the Z-transform and its properties for analyzing discrete signals and systems. It begins by defining the Z-transform as an extension of the discrete Fourier transform with a convergence factor, allowing it to represent a wider class of sequences. Properties discussed include the unilateral and bilateral Z-transforms, time shifting, and the region of convergence. Several examples of direct Z-transforms are provided, such as for exponential, unit step, and impulse sequences.
The document discusses several theorems related to twin prime conjectures:
- Theorem 1 states that a prime p can be written in the form 3k+1 or 3k-1, with k being even.
- Theorem 4 characterizes twin primes as pairs where n(n+2) satisfies a modular condition.
- Theorems aim to prove there are infinitely many twin primes, relying on the ratio of primes increasing without bound as n increases without bound.
This document discusses the use of Airy stress functions to solve problems in two-dimensional elasticity. It provides the equations relating stresses to the Airy stress function in plane stress, plane strain, and polar coordinates. Examples are given to show how specific Airy stress functions can be used to solve problems like bending of beams, stresses in curved beams with end moments, and stresses in a quarter-circle beam under an end load.
This document summarizes the numerical solution of the time-independent Schrodinger equation for particles in chaotic stadium and Sinai billiard potentials using the finite difference method. Scars, or regions of high probability density around unstable periodic orbits, were observed in particular eigenstates of both systems, consistent with previous studies. The method was validated by reproducing analytical solutions for 1D and circular wells and showing convergence of eigenvalues with increasing numerical resolution.
I. Antoniadis - "Introduction to Supersymmetry" 2/2SEENET-MTP
1) The Supersymmetric Standard Model (SSM) extends the particle content of the Standard Model by introducing supersymmetric partners for each particle, called sparticles. This includes gluinos, winos, binos, squarks, sleptons, and higgsinos as sparticle counterparts to gluons, W/Z bosons, photons, quarks, leptons, and Higgs bosons.
2) The SSM Lagrangian contains additional terms beyond the Standard Model including gauge interactions between fermions and gauginos, quartic scalar interactions, and a superpotential with Yukawa-like couplings.
3) Spontaneous supersymmetry breaking is required to make sparticles heavy enough to have evaded detection
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
I. Antoniadis - "Introduction to Supersymmetry" 1/2SEENET-MTP
Supersymmetry (SUSY) is a symmetry that relates bosonic and fermionic degrees of freedom. It extends the Poincaré algebra by including spinorial generators (supercharges) that transform bosonic fields into fermionic fields and vice versa. SUSY provides motivations like natural elementary scalars, gauge coupling unification, and a dark matter candidate. SUSY is formulated using superspace, which extends spacetime by Grassmann coordinates. Chiral and vector superfields contain the bosonic and fermionic components of supermultiplets and allow constructing SUSY invariant actions.
First part of description of Matrix Calculus at Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com.
For more presentations please visit my website at
http://www.solohermelin.com.
The document discusses permutations and the symmetric group S3. It defines what a permutation is and introduces the six permutations that make up S3: the identity E, a 120 degree rotation R, a 240 degree rotation R2, a vertical reflection V, and reflections RV and R2V. It explains that S3 forms a group under composition of permutations. It also introduces the alternating group A3, which is the subgroup of S3 made up of the even permutations E, R, and R2.
This document provides an overview of vector algebra concepts including scalars, vectors, unit vectors, position vectors, vector operations, gradients of scalars, divergence of vectors, and del operators. It defines these concepts, provides examples of their use in Cartesian, cylindrical and spherical coordinate systems, and works through examples of calculating gradients and divergences. The key topics covered are the definitions and calculations of gradients, divergences, and how to apply these concepts to vector fields in different coordinate systems.
On a Generalized 휷푹 − Birecurrent Affinely Connected Spaceinventionjournals
In the present paper, we introduced a Finsler space whose Cartan's third curvature tensor 푅푗푘 ℎ 푖 satisfies the generalized 훽푅 −birecurrence property which posses the properties of affinely connected space will be characterized by ℬ푚 ℬ푛푅푗푘 ℎ 푖 = 푎푚푛 푅푗푘 ℎ 푖 +푏푚푛 훿푘 푖 푔푗ℎ − 훿ℎ 푖 푔푗푘 , where 푎푚푛 and 푏푚푛 are non-zero covariant tensors field of second order called recurrence tensors field, such space is called as a generalized 훽푅 −birecurrent affinely connected space. Ricci tensors 퐻푗푘 and 푅푗푘 , the curvature vector 퐻푘 and the curvature scalars 퐻 and 푅 of such space are non-vanishing. Some conditions have been pointed out which reduce a generalized 훽푅 −birecurrent affinely connected space 퐹푛 (푛 > 2) into Finsler space of scalar curvature.
1) The document discusses directional derivatives and the gradient of functions of several variables. It defines the directional derivative Duf(c) as the slope of the function f in the direction of the unit vector u at the point c.
2) It shows that the partial derivatives of f can be computed by treating all but one variable as a constant. The gradient of f is defined as the vector of its partial derivatives.
3) It derives an expression for the directional derivative Duf(c) in terms of the partial derivatives of f and the components of the unit vector u, showing the relationship between directional derivatives and the gradient.
This document discusses Poisson's and Laplace's equations in the context of a class on field theory taught by Prof. Supraja. It provides definitions and derivations of Poisson's and Laplace's equations in different coordinate systems. It gives examples of using these equations to find electric potential and field between charged surfaces, as well as the capacitance between them. It also states the uniqueness theorem, which guarantees a unique solution to Laplace's/Poisson's equation given the boundary conditions.
This document discusses orthogonal trajectories and provides examples of finding orthogonal trajectories for different families of curves. It begins by defining orthogonal trajectories as curves that intersect each other at right angles. It then provides a method for finding the differential equation that describes the orthogonal trajectories for a given family of curves. Several examples are worked out, such as finding the orthogonal trajectories of the family of parabolas with equation y = x^2. Applications to equipotential lines and electric fields and electromagnetic waves are also mentioned.
(Neamen)solution manual for semiconductor physics and devices 3edKadu Brito
This document contains solutions to problems from Chapter 1 of Semiconductor Physics and Devices: Basic Principles, 3rd edition. The problems calculate properties of semiconductor unit cells such as number of atoms, density, and volume percentage occupied by atoms. Lattice structures including simple cubic, body-centered cubic, face-centered cubic and diamond are considered. Properties of common semiconductors such as silicon and gallium arsenide are also calculated.
NUMERICAL METHODS -Iterative methods(indirect method)krishnapriya R
The document discusses two iterative methods for solving systems of linear equations: Gauss-Jacobi and Gauss-Seidel. Gauss-Jacobi solves each equation separately using the most recent approximations for the other variables. Gauss-Seidel updates each variable with the most recent values available. The document provides an example applying both methods to solve a system of three equations. Gauss-Seidel converges faster, requiring fewer iterations than Gauss-Jacobi to achieve the same accuracy. Both methods are useful alternatives to direct methods like Gaussian elimination when round-off errors are a concern.
This document summarizes research on Casimir torque in the weak coupling approximation. It examines manifestations of Casimir torque between planar objects characterized by delta function potentials. The key findings are:
1) An exact calculation of the Casimir torque between a finite rectangular plate above a semi-infinite plate is presented and agrees well with the proximity force approximation when the plate separation is small compared to their sizes.
2) Cusps in the torque arise when the corners of the finite plate pass over the edge of the semi-infinite plate.
3) A similar calculation is done for a disk above a semi-infinite plate, again finding good agreement with the proximity force approximation.
1) The document discusses the problem of a particle sliding off a moving hemisphere, with the goal of finding the angle at which the particle's horizontal velocity is maximum.
2) Conservation of momentum and energy equations are used to derive an expression for the particle's horizontal velocity in terms of the angle, which reaches a maximum at a certain angle.
3) Setting the derivative of this expression to zero results in a cubic equation that determines the maximum angle, which can be solved numerically for different ratios of particle to hemisphere mass.
1) The document discusses the problem of a particle sliding off a moving hemisphere, using conservation of momentum and energy equations to derive an expression for the particle's horizontal velocity vx as a function of the angle θ.
2) Setting the derivative of vx equal to zero yields a cubic equation that determines the angle θ at which vx is maximized, corresponding to the particle losing contact with the hemisphere.
3) For the special case where the particle and hemisphere masses are equal (ratio r = 1), the cubic equation can be solved to find θ ≈ 42.9 degrees.
The document discusses membrane harmonics and the Helmholtz equation. It begins by considering the one-dimensional Helmholtz equation on an interval, finding the eigenfunctions and eigenvalues. It then extends this to the two-dimensional case on a rectangle using separation of variables, obtaining eigenfunctions that are products of sine waves and eigenvalues that are sums of the one-dimensional eigenvalues.
Geometric properties for parabolic and elliptic pdeSpringer
This document discusses recent advances in fractional Laplacian operators and related problems in partial differential equations and geometric measure theory. Specifically, it addresses three key topics:
1. Symmetry problems for solutions of the fractional Allen-Cahn equation and whether solutions only depend on one variable like in the classical case. The answer is known to be positive for some dimensions and fractional exponents but remains open in general.
2. The Γ-convergence of functionals involving the fractional Laplacian as the small parameter ε approaches zero. This characterizes the asymptotic behavior and relates to fractional notions of perimeter.
3. Regularity of interfaces as the fractional exponent s approaches 1/2 from above, which corresponds to a critical threshold
Mathematical formulation of inverse scattering and korteweg de vries equationAlexander Decker
This document summarizes the mathematical formulation of inverse scattering and the Korteweg-de Vries (KdV) equation. It begins by defining inverse scattering as determining solutions to differential equations based on known asymptotic solutions, specifically by solving the Marchenko equation. It then discusses how the KdV equation describes shallow water waves and solitons, and how the inverse scattering transform method can be used to determine soliton solutions from arbitrary initial conditions. The document outlines the procedure, including deriving the scattering data from an initial potential function and using its time evolution to reconstruct solutions to the KdV equation at later times. It provides examples using reflectionless potentials, specifically obtaining the single-soliton solution from an initial sech^2
The document discusses solving multiple non-linear equations using the multidimensional Newton-Raphson method. It provides an example of solving for the equilibrium conversion of two coupled chemical reactions. The key steps are: (1) writing the reaction equations in root-finding form as two non-linear equations f1 and f2; (2) defining the Jacobian matrix J with the partial derivatives of f1 and f2; and (3) using the multidimensional Taylor series expansion and Jacobian matrix to linearize the system and iteratively solve for the root where f1 and f2 equal zero.
The document summarizes key concepts of quantum mechanics from chapters 3 and 4 of McQuarrie, including:
1) The Schrodinger equation and its solutions in 1D and 3D.
2) Solving the time-independent Schrodinger equation involves finding the general solution, applying boundary conditions, and normalizing the wavefunction.
3) Wavefunctions represent probability distributions, and the probability of finding a particle in a region is calculated by integrating the wavefunction.
This document discusses convexity properties of the gamma function. It begins with an introduction to the gamma function and its history. It then revisits earlier work by Krull on functional equations of the form f(x+1)-f(x)=g(x). Several lemmas are presented that characterize properties of convex functions satisfying certain conditions. These results are then applied to derive classical and new representations and characterizations of the gamma function based on its convexity.
The document discusses work and line integrals. It defines work as the force component parallel to the distance moved, and how line integrals are used to calculate work when the force varies over the distance. It provides examples of evaluating line integrals, including along a straight line path and broken into multiple path segments. For a given force function F(x,y,z), it shows how to set up and evaluate the line integral to find the work done along different paths between two points.
This document discusses counterexamples to classical calculus theorems like the Inverse and Implicit Function Theorems in the context of Scale Calculus. Scale Calculus generalizes multivariable calculus to infinite-dimensional spaces and is foundational to Polyfold Theory. The authors construct examples of scale-differentiable and scale-Fredholm maps whose differentials are discontinuous when the base point changes, disproving naive extensions of the classical theorems. However, they show the scale-Fredholm notion in Polyfold Theory ensures continuity of differentials in specific coordinates, justifying its technical complexity and allowing for a perturbation theory without the classical theorems.
This document provides an introduction to the finite element method by first discussing the calculus of variations. It explains that the finite element formulation can be derived from a variational principle rather than an energy functional. It then presents three examples that illustrate functionals - the brachistochrone problem, geodesic problem, and isoperimetric problem. The document defines the concepts of extremal paths, varied paths, first variation, and the delta operator to derive the Euler-Lagrange equation, which provides the necessary condition for a functional to be extremized.
The document provides the steps to solve a multi-part calculus problem. It involves finding the derivative of two functions, determining the equations of tangent lines to those functions at given points, finding the intersection points of those tangent lines, and using those intersection points to write the equations of three circles with a radius of 5.
M. Visinescu: Hidden Symmetries of the Five-dimensional Sasaki - Einstein SpacesSEENET-MTP
Balkan Workshop BW2013
Beyond the Standard Models
25 – 29 April, 2013, Vrnjačka Banja, Serbia
M. Visinescu: Hidden Symmetries of the Five-dimensional Sasaki - Einstein Spaces
MA 243 Calculus III Fall 2015 Dr. E. JacobsAssignmentsTh.docxinfantsuk
MA 243 Calculus III Fall 2015 Dr. E. Jacobs
Assignments
These assignments are keyed to Edition 7E of James Stewart’s “Calculus” (Early Transcendentals)
Assignment 1. Spheres and Other Surfaces
Read 12.1 - 12.2 and 12.6
You should be able to do the following problems:
Section 12.1/Problems 11 - 18, 20 - 22 Section 12.6/Problems 1 - 48
Hand in the following problems:
1. The following equation describes a sphere. Find the radius and the coordinates of the center.
x2 + y2 + z2 = 2(x + y + z) + 1
2. A particular sphere with center (−3, 2, 2) is tangent to both the xy-plane and the xz-plane.
It intersects the xy-plane at the point (−3, 2, 0). Find the equation of this sphere.
3. Suppose (0, 0, 0) and (0, 0, −4) are the endpoints of the diameter of a sphere. Find the
equation of this sphere.
4. Find the equation of the sphere centered around (0, 0, 4) if the sphere passes through the
origin.
5. Describe the graph of the given equation in geometric terms, using plain, clear language:
z =
√
1 − x2 − y2
Sketch each of the following surfaces
6. z = 2 − 2
√
x2 + y2
7. z = 1 − y2
8. z = 4 − x − y
9. z = 4 − x2 − y2
10. x2 + z2 = 16
Assignment 2. Dot and Cross Products
Read 12.3 and 12.4
You should be able to do the following problems:
Section 12.3/Problems 1 - 28 Section 12.4/Problems 1 - 32
Hand in the following problems:
1. Let u⃗ =
⟨
0, 1
2
,
√
3
2
⟩
and v⃗ =
⟨√
2,
√
3
2
, 1
2
⟩
a) Find the dot product b) Find the cross product
2. Let u⃗ = j⃗ + k⃗ and v⃗ = i⃗ +
√
2 j⃗.
a) Calculate the length of the projection of v⃗ in the u⃗ direction.
b) Calculate the cosine of the angle between u⃗ and v⃗
3. Consider the parallelogram with the following vertices:
(0, 0, 0) (0, 1, 1) (1, 0, 2) (1, 1, 3)
a) Find a vector perpendicular to this parallelogram.
b) Use vector methods to find the area of this parallelogram.
4. Use the dot product to find the cosine of the angle between the diagonal of a cube and one of
its edges.
5. Let L be the line that passes through the points (0, −
√
3 , −1) and (0,
√
3 , 1). Let θ be the
angle between L and the vector u⃗ = 1√
2
⟨0, 1, 1⟩. Calculate θ (to the nearest degree).
Assignment 3. Lines and Planes
Read 12.5
You should be able to do the following problems:
Section 12.5/Problems 1 - 58
Hand in the following problems:
1a. Find the equation of the line that passes through (0, 0, 1) and (1, 0, 2).
b. Find the equation of the plane that passes through (1, 0, 0) and is perpendicular to the line in
part (a).
2. The following equation describes a straight line:
r⃗(t) = ⟨1, 1, 0⟩ + t⟨0, 2, 1⟩
a. Find the angle between the given line and the vector u⃗ = ⟨1, −1, 2⟩.
b. Find the equation of the plane that passes through the point (0, 0, 4) and is perpendicular to
the given line.
3. The following two lines intersect at the point (1, 4, 4)
r⃗ = ⟨1, 4, 4⟩ + t⟨0, 1, 0⟩ r⃗ = ⟨1, 4, 4⟩ + t⟨3, 5, 4⟩
a. Find the angle between the two lines.
b. Find the equation of the plane that contains every point o ...
The document provides the steps to solve a multi-part calculus problem involving derivatives, tangent lines, and circles. It determines the derivative of two functions f(x) and g(x), finds the equations of the tangent lines at specific x-values, identifies the intersection points of the tangent lines, and uses those intersection points as the centers of three circles with a radius of 5 to write the equations of the circles.
A general approach is presented to describing nonlinear classical Maxwell electrodynamics with conformal symmetry. We introduce generalized nonlinear constitutive equations, expressed in terms of constitutive tensors dependent on conformal-invariant functionals of the field strengths. This allows a characterization of Lagrangian and non-Lagrangian theories. We obtain a general formula for possible Lagrangian densities in nonlinear conformal-invariant electrodynamics. This generalizes the standard Lagrangian of classical linear electrodynamics so as to preserve the conformal symmetry.
Maxwell's formulation - differential forms on euclidean spacegreentask
One of the greatest advances in theoretical physics of the nineteenth
century was Maxwell’s formulation of the the equations of electromag-
netism. This article uses differential forms to solve a problem related
to Maxwell’s formulation. The notion of differential form encompasses
such ideas as elements of surface area and volume elements, the work
exerted by a force, the flow of a fluid, and the curvature of a surface,
space or hyperspace. An important operation on differential forms is
exterior differentiation, which generalizes the operators div, grad, curl
of vector calculus. the study of differential forms, which was initiated
by E.Cartan in the years around 1900, is often termed the exterior
differential calculus.However, Maxwells equations have many very im-
portant implications in the life of a modern person, so much so that
people use devices that function off the principles in Maxwells equa-
tions every day without even knowing it
Published by:
Wang Jing
School of Physical and Mathematical Sciences
Nanyang Technological University
jwang14@e.ntu.edu.sg
N. Bilic - "Hamiltonian Method in the Braneworld" 2/3SEENET-MTP
This document discusses braneworld models and the Randall-Sundrum model. It begins by introducing the relativistic particle and string actions used to describe dynamics in higher dimensions. It then summarizes the two Randall-Sundrum models: RS I contains two branes separated in a fifth dimension to address the hierarchy problem, while RS II has the negative tension brane sent to infinity and observers on a single positive tension brane. Finally, it derives the RS II model solution, using Gaussian normal coordinates and imposing junction conditions at the brane.
Candidate young stellar objects in the S-cluster: Kinematic analysis of a sub...Sérgio Sacani
Context. The observation of several L-band emission sources in the S cluster has led to a rich discussion of their nature. However, a definitive answer to the classification of the dusty objects requires an explanation for the detection of compact Doppler-shifted Brγ emission. The ionized hydrogen in combination with the observation of mid-infrared L-band continuum emission suggests that most of these sources are embedded in a dusty envelope. These embedded sources are part of the S-cluster, and their relationship to the S-stars is still under debate. To date, the question of the origin of these two populations has been vague, although all explanations favor migration processes for the individual cluster members. Aims. This work revisits the S-cluster and its dusty members orbiting the supermassive black hole SgrA* on bound Keplerian orbits from a kinematic perspective. The aim is to explore the Keplerian parameters for patterns that might imply a nonrandom distribution of the sample. Additionally, various analytical aspects are considered to address the nature of the dusty sources. Methods. Based on the photometric analysis, we estimated the individual H−K and K−L colors for the source sample and compared the results to known cluster members. The classification revealed a noticeable contrast between the S-stars and the dusty sources. To fit the flux-density distribution, we utilized the radiative transfer code HYPERION and implemented a young stellar object Class I model. We obtained the position angle from the Keplerian fit results; additionally, we analyzed the distribution of the inclinations and the longitudes of the ascending node. Results. The colors of the dusty sources suggest a stellar nature consistent with the spectral energy distribution in the near and midinfrared domains. Furthermore, the evaporation timescales of dusty and gaseous clumps in the vicinity of SgrA* are much shorter ( 2yr) than the epochs covered by the observations (≈15yr). In addition to the strong evidence for the stellar classification of the D-sources, we also find a clear disk-like pattern following the arrangements of S-stars proposed in the literature. Furthermore, we find a global intrinsic inclination for all dusty sources of 60 ± 20◦, implying a common formation process. Conclusions. The pattern of the dusty sources manifested in the distribution of the position angles, inclinations, and longitudes of the ascending node strongly suggests two different scenarios: the main-sequence stars and the dusty stellar S-cluster sources share a common formation history or migrated with a similar formation channel in the vicinity of SgrA*. Alternatively, the gravitational influence of SgrA* in combination with a massive perturber, such as a putative intermediate mass black hole in the IRS 13 cluster, forces the dusty objects and S-stars to follow a particular orbital arrangement. Key words. stars: black holes– stars: formation– Galaxy: center– galaxies: star formation
JAMES WEBB STUDY THE MASSIVE BLACK HOLE SEEDSSérgio Sacani
The pathway(s) to seeding the massive black holes (MBHs) that exist at the heart of galaxies in the present and distant Universe remains an unsolved problem. Here we categorise, describe and quantitatively discuss the formation pathways of both light and heavy seeds. We emphasise that the most recent computational models suggest that rather than a bimodal-like mass spectrum between light and heavy seeds with light at one end and heavy at the other that instead a continuum exists. Light seeds being more ubiquitous and the heavier seeds becoming less and less abundant due the rarer environmental conditions required for their formation. We therefore examine the different mechanisms that give rise to different seed mass spectrums. We show how and why the mechanisms that produce the heaviest seeds are also among the rarest events in the Universe and are hence extremely unlikely to be the seeds for the vast majority of the MBH population. We quantify, within the limits of the current large uncertainties in the seeding processes, the expected number densities of the seed mass spectrum. We argue that light seeds must be at least 103 to 105 times more numerous than heavy seeds to explain the MBH population as a whole. Based on our current understanding of the seed population this makes heavy seeds (Mseed > 103 M⊙) a significantly more likely pathway given that heavy seeds have an abundance pattern than is close to and likely in excess of 10−4 compared to light seeds. Finally, we examine the current state-of-the-art in numerical calculations and recent observations and plot a path forward for near-future advances in both domains.
Anti-Universe And Emergent Gravity and the Dark UniverseSérgio Sacani
Recent theoretical progress indicates that spacetime and gravity emerge together from the entanglement structure of an underlying microscopic theory. These ideas are best understood in Anti-de Sitter space, where they rely on the area law for entanglement entropy. The extension to de Sitter space requires taking into account the entropy and temperature associated with the cosmological horizon. Using insights from string theory, black hole physics and quantum information theory we argue that the positive dark energy leads to a thermal volume law contribution to the entropy that overtakes the area law precisely at the cosmological horizon. Due to the competition between area and volume law entanglement the microscopic de Sitter states do not thermalise at sub-Hubble scales: they exhibit memory effects in the form of an entropy displacement caused by matter. The emergent laws of gravity contain an additional ‘dark’ gravitational force describing the ‘elastic’ response due to the entropy displacement. We derive an estimate of the strength of this extra force in terms of the baryonic mass, Newton’s constant and the Hubble acceleration scale a0 = cH0, and provide evidence for the fact that this additional ‘dark gravity force’ explains the observed phenomena in galaxies and clusters currently attributed to dark matter.
The binding of cosmological structures by massless topological defectsSérgio Sacani
Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field
equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational
field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin
spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling
concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect
light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is
mitigated, at least in part.
EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended Weste...Sérgio Sacani
Context. With a mass exceeding several 104 M⊙ and a rich and dense population of massive stars, supermassive young star clusters
represent the most massive star-forming environment that is dominated by the feedback from massive stars and gravitational interactions
among stars.
Aims. In this paper we present the Extended Westerlund 1 and 2 Open Clusters Survey (EWOCS) project, which aims to investigate
the influence of the starburst environment on the formation of stars and planets, and on the evolution of both low and high mass stars.
The primary targets of this project are Westerlund 1 and 2, the closest supermassive star clusters to the Sun.
Methods. The project is based primarily on recent observations conducted with the Chandra and JWST observatories. Specifically,
the Chandra survey of Westerlund 1 consists of 36 new ACIS-I observations, nearly co-pointed, for a total exposure time of 1 Msec.
Additionally, we included 8 archival Chandra/ACIS-S observations. This paper presents the resulting catalog of X-ray sources within
and around Westerlund 1. Sources were detected by combining various existing methods, and photon extraction and source validation
were carried out using the ACIS-Extract software.
Results. The EWOCS X-ray catalog comprises 5963 validated sources out of the 9420 initially provided to ACIS-Extract, reaching a
photon flux threshold of approximately 2 × 10−8 photons cm−2
s
−1
. The X-ray sources exhibit a highly concentrated spatial distribution,
with 1075 sources located within the central 1 arcmin. We have successfully detected X-ray emissions from 126 out of the 166 known
massive stars of the cluster, and we have collected over 71 000 photons from the magnetar CXO J164710.20-455217.
The debris of the ‘last major merger’ is dynamically youngSérgio Sacani
The Milky Way’s (MW) inner stellar halo contains an [Fe/H]-rich component with highly eccentric orbits, often referred to as the
‘last major merger.’ Hypotheses for the origin of this component include Gaia-Sausage/Enceladus (GSE), where the progenitor
collided with the MW proto-disc 8–11 Gyr ago, and the Virgo Radial Merger (VRM), where the progenitor collided with the
MW disc within the last 3 Gyr. These two scenarios make different predictions about observable structure in local phase space,
because the morphology of debris depends on how long it has had to phase mix. The recently identified phase-space folds in Gaia
DR3 have positive caustic velocities, making them fundamentally different than the phase-mixed chevrons found in simulations
at late times. Roughly 20 per cent of the stars in the prograde local stellar halo are associated with the observed caustics. Based
on a simple phase-mixing model, the observed number of caustics are consistent with a merger that occurred 1–2 Gyr ago.
We also compare the observed phase-space distribution to FIRE-2 Latte simulations of GSE-like mergers, using a quantitative
measurement of phase mixing (2D causticality). The observed local phase-space distribution best matches the simulated data
1–2 Gyr after collision, and certainly not later than 3 Gyr. This is further evidence that the progenitor of the ‘last major merger’
did not collide with the MW proto-disc at early times, as is thought for the GSE, but instead collided with the MW disc within
the last few Gyr, consistent with the body of work surrounding the VRM.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
Gliese 12 b: A Temperate Earth-sized Planet at 12 pc Ideal for Atmospheric Tr...Sérgio Sacani
Recent discoveries of Earth-sized planets transiting nearby M dwarfs have made it possible to characterize the
atmospheres of terrestrial planets via follow-up spectroscopic observations. However, the number of such planets
receiving low insolation is still small, limiting our ability to understand the diversity of the atmospheric
composition and climates of temperate terrestrial planets. We report the discovery of an Earth-sized planet
transiting the nearby (12 pc) inactive M3.0 dwarf Gliese 12 (TOI-6251) with an orbital period (Porb) of 12.76 days.
The planet, Gliese 12 b, was initially identified as a candidate with an ambiguous Porb from TESS data. We
confirmed the transit signal and Porb using ground-based photometry with MuSCAT2 and MuSCAT3, and
validated the planetary nature of the signal using high-resolution images from Gemini/NIRI and Keck/NIRC2 as
well as radial velocity (RV) measurements from the InfraRed Doppler instrument on the Subaru 8.2 m telescope
and from CARMENES on the CAHA 3.5 m telescope. X-ray observations with XMM-Newton showed the host
star is inactive, with an X-ray-to-bolometric luminosity ratio of log 5.7 L L X bol » - . Joint analysis of the light
curves and RV measurements revealed that Gliese 12 b has a radius of 0.96 ± 0.05 R⊕,a3σ mass upper limit of
3.9 M⊕, and an equilibrium temperature of 315 ± 6 K assuming zero albedo. The transmission spectroscopy metric
(TSM) value of Gliese 12 b is close to the TSM values of the TRAPPIST-1 planets, adding Gliese 12 b to the small
list of potentially terrestrial, temperate planets amenable to atmospheric characterization with JWST.
Gliese 12 b, a temperate Earth-sized planet at 12 parsecs discovered with TES...Sérgio Sacani
We report on the discovery of Gliese 12 b, the nearest transiting temperate, Earth-sized planet found to date. Gliese 12 is a
bright (V = 12.6 mag, K = 7.8 mag) metal-poor M4V star only 12.162 ± 0.005 pc away from the Solar system with one of the
lowest stellar activity levels known for M-dwarfs. A planet candidate was detected by TESS based on only 3 transits in sectors
42, 43, and 57, with an ambiguity in the orbital period due to observational gaps. We performed follow-up transit observations
with CHEOPS and ground-based photometry with MINERVA-Australis, SPECULOOS, and Purple Mountain Observatory,
as well as further TESS observations in sector 70. We statistically validate Gliese 12 b as a planet with an orbital period of
12.76144 ± 0.00006 d and a radius of 1.0 ± 0.1 R⊕, resulting in an equilibrium temperature of ∼315 K. Gliese 12 b has excellent
future prospects for precise mass measurement, which may inform how planetary internal structure is affected by the stellar
compositional environment. Gliese 12 b also represents one of the best targets to study whether Earth-like planets orbiting cool
stars can retain their atmospheres, a crucial step to advance our understanding of habitability on Earth and across the galaxy.
The importance of continents, oceans and plate tectonics for the evolution of...Sérgio Sacani
Within the uncertainties of involved astronomical and biological parameters, the Drake Equation
typically predicts that there should be many exoplanets in our galaxy hosting active, communicative
civilizations (ACCs). These optimistic calculations are however not supported by evidence, which is
often referred to as the Fermi Paradox. Here, we elaborate on this long-standing enigma by showing
the importance of planetary tectonic style for biological evolution. We summarize growing evidence
that a prolonged transition from Mesoproterozoic active single lid tectonics (1.6 to 1.0 Ga) to modern
plate tectonics occurred in the Neoproterozoic Era (1.0 to 0.541 Ga), which dramatically accelerated
emergence and evolution of complex species. We further suggest that both continents and oceans
are required for ACCs because early evolution of simple life must happen in water but late evolution
of advanced life capable of creating technology must happen on land. We resolve the Fermi Paradox
(1) by adding two additional terms to the Drake Equation: foc
(the fraction of habitable exoplanets
with significant continents and oceans) and fpt
(the fraction of habitable exoplanets with significant
continents and oceans that have had plate tectonics operating for at least 0.5 Ga); and (2) by
demonstrating that the product of foc
and fpt
is very small (< 0.00003–0.002). We propose that the lack
of evidence for ACCs reflects the scarcity of long-lived plate tectonics and/or continents and oceans on
exoplanets with primitive life.
A Giant Impact Origin for the First Subduction on EarthSérgio Sacani
Hadean zircons provide a potential record of Earth's earliest subduction 4.3 billion years ago. Itremains enigmatic how subduction could be initiated so soon after the presumably Moon‐forming giant impact(MGI). Earlier studies found an increase in Earth's core‐mantle boundary (CMB) temperature due to theaccumulation of the impactor's core, and our recent work shows Earth's lower mantle remains largely solid, withsome of the impactor's mantle potentially surviving as the large low‐shear velocity provinces (LLSVPs). Here,we show that a hot post‐impact CMB drives the initiation of strong mantle plumes that can induce subductioninitiation ∼200 Myr after the MGI. 2D and 3D thermomechanical computations show that a high CMBtemperature is the primary factor triggering early subduction, with enrichment of heat‐producing elements inLLSVPs as another potential factor. The models link the earliest subduction to the MGI with implications forunderstanding the diverse tectonic regimes of rocky planets.
Climate extremes likely to drive land mammal extinction during next supercont...Sérgio Sacani
Mammals have dominated Earth for approximately 55 Myr thanks to their
adaptations and resilience to warming and cooling during the Cenozoic. All
life will eventually perish in a runaway greenhouse once absorbed solar
radiation exceeds the emission of thermal radiation in several billions of
years. However, conditions rendering the Earth naturally inhospitable to
mammals may develop sooner because of long-term processes linked to
plate tectonics (short-term perturbations are not considered here). In
~250 Myr, all continents will converge to form Earth’s next supercontinent,
Pangea Ultima. A natural consequence of the creation and decay of Pangea
Ultima will be extremes in pCO2 due to changes in volcanic rifting and
outgassing. Here we show that increased pCO2, solar energy (F⨀;
approximately +2.5% W m−2 greater than today) and continentality (larger
range in temperatures away from the ocean) lead to increasing warming
hostile to mammalian life. We assess their impact on mammalian
physiological limits (dry bulb, wet bulb and Humidex heat stress indicators)
as well as a planetary habitability index. Given mammals’ continued survival,
predicted background pCO2 levels of 410–816 ppm combined with increased
F⨀ will probably lead to a climate tipping point and their mass extinction.
The results also highlight how global landmass configuration, pCO2 and F⨀
play a critical role in planetary habitability.
Constraints on Neutrino Natal Kicks from Black-Hole Binary VFTS 243Sérgio Sacani
The recently reported observation of VFTS 243 is the first example of a massive black-hole binary
system with negligible binary interaction following black-hole formation. The black-hole mass (≈10M⊙)
and near-circular orbit (e ≈ 0.02) of VFTS 243 suggest that the progenitor star experienced complete
collapse, with energy-momentum being lost predominantly through neutrinos. VFTS 243 enables us to
constrain the natal kick and neutrino-emission asymmetry during black-hole formation. At 68% confidence
level, the natal kick velocity (mass decrement) is ≲10 km=s (≲1.0M⊙), with a full probability distribution
that peaks when ≈0.3M⊙ were ejected, presumably in neutrinos, and the black hole experienced a natal
kick of 4 km=s. The neutrino-emission asymmetry is ≲4%, with best fit values of ∼0–0.2%. Such a small
neutrino natal kick accompanying black-hole formation is in agreement with theoretical predictions.
Detectability of Solar Panels as a TechnosignatureSérgio Sacani
In this work, we assess the potential detectability of solar panels made of silicon on an Earth-like
exoplanet as a potential technosignature. Silicon-based photovoltaic cells have high reflectance in the
UV-VIS and in the near-IR, within the wavelength range of a space-based flagship mission concept
like the Habitable Worlds Observatory (HWO). Assuming that only solar energy is used to provide
the 2022 human energy needs with a land cover of ∼ 2.4%, and projecting the future energy demand
assuming various growth-rate scenarios, we assess the detectability with an 8 m HWO-like telescope.
Assuming the most favorable viewing orientation, and focusing on the strong absorption edge in the
ultraviolet-to-visible (0.34 − 0.52 µm), we find that several 100s of hours of observation time is needed
to reach a SNR of 5 for an Earth-like planet around a Sun-like star at 10pc, even with a solar panel
coverage of ∼ 23% land coverage of a future Earth. We discuss the necessity of concepts like Kardeshev
Type I/II civilizations and Dyson spheres, which would aim to harness vast amounts of energy. Even
with much larger populations than today, the total energy use of human civilization would be orders of
magnitude below the threshold for causing direct thermal heating or reaching the scale of a Kardashev
Type I civilization. Any extraterrrestrial civilization that likewise achieves sustainable population
levels may also find a limit on its need to expand, which suggests that a galaxy-spanning civilization
as imagined in the Fermi paradox may not exist.
Jet reorientation in central galaxies of clusters and groups: insights from V...Sérgio Sacani
Recent observations of galaxy clusters and groups with misalignments between their central AGN jets
and X-ray cavities, or with multiple misaligned cavities, have raised concerns about the jet – bubble
connection in cooling cores, and the processes responsible for jet realignment. To investigate the
frequency and causes of such misalignments, we construct a sample of 16 cool core galaxy clusters and
groups. Using VLBA radio data we measure the parsec-scale position angle of the jets, and compare
it with the position angle of the X-ray cavities detected in Chandra data. Using the overall sample
and selected subsets, we consistently find that there is a 30% – 38% chance to find a misalignment
larger than ∆Ψ = 45◦ when observing a cluster/group with a detected jet and at least one cavity. We
determine that projection may account for an apparently large ∆Ψ only in a fraction of objects (∼35%),
and given that gas dynamical disturbances (as sloshing) are found in both aligned and misaligned
systems, we exclude environmental perturbation as the main driver of cavity – jet misalignment.
Moreover, we find that large misalignments (up to ∼ 90◦
) are favored over smaller ones (45◦ ≤ ∆Ψ ≤
70◦
), and that the change in jet direction can occur on timescales between one and a few tens of Myr.
We conclude that misalignments are more likely related to actual reorientation of the jet axis, and we
discuss several engine-based mechanisms that may cause these dramatic changes.
The solar dynamo begins near the surfaceSérgio Sacani
The magnetic dynamo cycle of the Sun features a distinct pattern: a propagating
region of sunspot emergence appears around 30° latitude and vanishes near the
equator every 11 years (ref. 1). Moreover, longitudinal flows called torsional oscillations
closely shadow sunspot migration, undoubtedly sharing a common cause2. Contrary
to theories suggesting deep origins of these phenomena, helioseismology pinpoints
low-latitude torsional oscillations to the outer 5–10% of the Sun, the near-surface
shear layer3,4. Within this zone, inwardly increasing differential rotation coupled with
a poloidal magnetic field strongly implicates the magneto-rotational instability5,6,
prominent in accretion-disk theory and observed in laboratory experiments7.
Together, these two facts prompt the general question: whether the solar dynamo is
possibly a near-surface instability. Here we report strong affirmative evidence in stark
contrast to traditional models8 focusing on the deeper tachocline. Simple analytic
estimates show that the near-surface magneto-rotational instability better explains
the spatiotemporal scales of the torsional oscillations and inferred subsurface
magnetic field amplitudes9. State-of-the-art numerical simulations corroborate these
estimates and reproduce hemispherical magnetic current helicity laws10. The dynamo
resulting from a well-understood near-surface phenomenon improves prospects
for accurate predictions of full magnetic cycles and space weather, affecting the
electromagnetic infrastructure of Earth.
Extensive Pollution of Uranus and Neptune’s Atmospheres by Upsweep of Icy Mat...Sérgio Sacani
In the Nice model of solar system formation, Uranus and Neptune undergo an orbital upheaval,
sweeping through a planetesimal disk. The region of the disk from which material is accreted by
the ice giants during this phase of their evolution has not previously been identified. We perform
direct N-body orbital simulations of the four giant planets to determine the amount and origin of solid
accretion during this orbital upheaval. We find that the ice giants undergo an extreme bombardment
event, with collision rates as much as ∼3 per hour assuming km-sized planetesimals, increasing the
total planet mass by up to ∼0.35%. In all cases, the initially outermost ice giant experiences the
largest total enhancement. We determine that for some plausible planetesimal properties, the resulting
atmospheric enrichment could potentially produce sufficient latent heat to alter the planetary cooling
timescale according to existing models. Our findings suggest that substantial accretion during this
phase of planetary evolution may have been sufficient to impact the atmospheric composition and
thermal evolution of the ice giants, motivating future work on the fate of deposited solid material.
Mending Clothing to Support Sustainable Fashion_CIMaR 2024.pdfSelcen Ozturkcan
Ozturkcan, S., Berndt, A., & Angelakis, A. (2024). Mending clothing to support sustainable fashion. Presented at the 31st Annual Conference by the Consortium for International Marketing Research (CIMaR), 10-13 Jun 2024, University of Gävle, Sweden.
Or: Beyond linear.
Abstract: Equivariant neural networks are neural networks that incorporate symmetries. The nonlinear activation functions in these networks result in interesting nonlinear equivariant maps between simple representations, and motivate the key player of this talk: piecewise linear representation theory.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Corrected slides: dtubbenhauer.com/talks.html
When I was asked to give a companion lecture in support of ‘The Philosophy of Science’ (https://shorturl.at/4pUXz) I decided not to walk through the detail of the many methodologies in order of use. Instead, I chose to employ a long standing, and ongoing, scientific development as an exemplar. And so, I chose the ever evolving story of Thermodynamics as a scientific investigation at its best.
Conducted over a period of >200 years, Thermodynamics R&D, and application, benefitted from the highest levels of professionalism, collaboration, and technical thoroughness. New layers of application, methodology, and practice were made possible by the progressive advance of technology. In turn, this has seen measurement and modelling accuracy continually improved at a micro and macro level.
Perhaps most importantly, Thermodynamics rapidly became a primary tool in the advance of applied science/engineering/technology, spanning micro-tech, to aerospace and cosmology. I can think of no better a story to illustrate the breadth of scientific methodologies and applications at their best.
PPT on Direct Seeded Rice presented at the three-day 'Training and Validation Workshop on Modules of Climate Smart Agriculture (CSA) Technologies in South Asia' workshop on April 22, 2024.
The cost of acquiring information by natural selectionCarl Bergstrom
This is a short talk that I gave at the Banff International Research Station workshop on Modeling and Theory in Population Biology. The idea is to try to understand how the burden of natural selection relates to the amount of information that selection puts into the genome.
It's based on the first part of this research paper:
The cost of information acquisition by natural selection
Ryan Seamus McGee, Olivia Kosterlitz, Artem Kaznatcheev, Benjamin Kerr, Carl T. Bergstrom
bioRxiv 2022.07.02.498577; doi: https://doi.org/10.1101/2022.07.02.498577
ESR spectroscopy in liquid food and beverages.pptxPRIYANKA PATEL
With increasing population, people need to rely on packaged food stuffs. Packaging of food materials requires the preservation of food. There are various methods for the treatment of food to preserve them and irradiation treatment of food is one of them. It is the most common and the most harmless method for the food preservation as it does not alter the necessary micronutrients of food materials. Although irradiated food doesn’t cause any harm to the human health but still the quality assessment of food is required to provide consumers with necessary information about the food. ESR spectroscopy is the most sophisticated way to investigate the quality of the food and the free radicals induced during the processing of the food. ESR spin trapping technique is useful for the detection of highly unstable radicals in the food. The antioxidant capability of liquid food and beverages in mainly performed by spin trapping technique.
Describing and Interpreting an Immersive Learning Case with the Immersion Cub...Leonel Morgado
Current descriptions of immersive learning cases are often difficult or impossible to compare. This is due to a myriad of different options on what details to include, which aspects are relevant, and on the descriptive approaches employed. Also, these aspects often combine very specific details with more general guidelines or indicate intents and rationales without clarifying their implementation. In this paper we provide a method to describe immersive learning cases that is structured to enable comparisons, yet flexible enough to allow researchers and practitioners to decide which aspects to include. This method leverages a taxonomy that classifies educational aspects at three levels (uses, practices, and strategies) and then utilizes two frameworks, the Immersive Learning Brain and the Immersion Cube, to enable a structured description and interpretation of immersive learning cases. The method is then demonstrated on a published immersive learning case on training for wind turbine maintenance using virtual reality. Applying the method results in a structured artifact, the Immersive Learning Case Sheet, that tags the case with its proximal uses, practices, and strategies, and refines the free text case description to ensure that matching details are included. This contribution is thus a case description method in support of future comparative research of immersive learning cases. We then discuss how the resulting description and interpretation can be leveraged to change immersion learning cases, by enriching them (considering low-effort changes or additions) or innovating (exploring more challenging avenues of transformation). The method holds significant promise to support better-grounded research in immersive learning.
Authoring a personal GPT for your research and practice: How we created the Q...Leonel Morgado
Thematic analysis in qualitative research is a time-consuming and systematic task, typically done using teams. Team members must ground their activities on common understandings of the major concepts underlying the thematic analysis, and define criteria for its development. However, conceptual misunderstandings, equivocations, and lack of adherence to criteria are challenges to the quality and speed of this process. Given the distributed and uncertain nature of this process, we wondered if the tasks in thematic analysis could be supported by readily available artificial intelligence chatbots. Our early efforts point to potential benefits: not just saving time in the coding process but better adherence to criteria and grounding, by increasing triangulation between humans and artificial intelligence. This tutorial will provide a description and demonstration of the process we followed, as two academic researchers, to develop a custom ChatGPT to assist with qualitative coding in the thematic data analysis process of immersive learning accounts in a survey of the academic literature: QUAL-E Immersive Learning Thematic Analysis Helper. In the hands-on time, participants will try out QUAL-E and develop their ideas for their own qualitative coding ChatGPT. Participants that have the paid ChatGPT Plus subscription can create a draft of their assistants. The organizers will provide course materials and slide deck that participants will be able to utilize to continue development of their custom GPT. The paid subscription to ChatGPT Plus is not required to participate in this workshop, just for trying out personal GPTs during it.
ESA/ACT Science Coffee: Diego Blas - Gravitational wave detection with orbita...Advanced-Concepts-Team
Presentation in the Science Coffee of the Advanced Concepts Team of the European Space Agency on the 07.06.2024.
Speaker: Diego Blas (IFAE/ICREA)
Title: Gravitational wave detection with orbital motion of Moon and artificial
Abstract:
In this talk I will describe some recent ideas to find gravitational waves from supermassive black holes or of primordial origin by studying their secular effect on the orbital motion of the Moon or satellites that are laser ranged.
ESA/ACT Science Coffee: Diego Blas - Gravitational wave detection with orbita...
On the gravitational_field_of_a_point_mass_according_to_einstein_theory
1. On the Gravitational Field of a Point-Mass,
According to Einstein’s Theory
Karl Schwarzschild
Submitted on January 13, 1916
Abstract: This is a translation of the paper ¨Uber das Gravitations-
feld eines Massenpunktes nach der Einsteinschen Theorie by Karl
Schwarzschild, where he obtained the metric of a space due to the
gravitational field of a point-mass. The paper was originally published
in 1916, in Sitzungsberichte der K¨oniglich Preussischen Akademie der
Wissenschaften, S. 189–196. Translated from the German in 2008 by
Larissa Borissova and Dmitri Rabounski.
§1. In his study on the motion of the perihelion of Mercury (see his
presentation given on November 18, 1915∗
) Einstein set up the following
problem: a point moves according to the requirement
δ ds = 0,
where
ds = Σ gµν dxµdxν , µ, ν = 1, 2, 3, 4
, (1)
where gµν are functions of the variables x, and, in the framework of this
variation, these variables are fixed in the start and the end of the path
of integration. Hence, in short, this point moves along a geodesic line,
where the manifold is characterized by the line-element ds.
Taking this variation gives the equations of this point
d2
xα
ds2
=
µ,ν
Γα
µν
dxµ
ds
dxν
ds
, α, β = 1, 2, 3, 4, (2)
where
Γα
µν = −
1
2
β
gαβ ∂gµβ
∂xν
+
∂gνβ
∂xµ
−
∂gµν
∂xβ
, (3)
while gαβ
, which are introduced and normed with respect to gαβ, mean
the reciprocal determinant†
to the determinant |gµν|.
∗Schwarzschild means the article: Einstein A. Erkl¨arung der Perihelbewegung
der Merkur aus der allgemeinen Relativit¨atstheorie. Sitzungsberichte der K¨oniglich
Preussischen Akademie der Wissenschaften, 1915, S. 831–839. — Editor’s com-
ment. D.R.
†This is the determinant of the reciprocal matrix, i.e. a matrix whose indices
are raised to the given matrix. One referred to the reciprocal matrix as the sub-
determinant, in those years. — Editor’s comment. D.R.
2. Karl Schwarzschild 11
Commencing now and so forth, according to Einstein’s theory, a test-
particle moves in the gravitational field of the mass located at the point
x1
= x2
= x3
= 0, if the “components of the gravitational field” Γ satisfy
the “field equations”
α
∂Γα
µν
∂xα
+
αβ
Γα
µβ Γβ
να = 0 (4)
everywhere except the point x1
= x2
= x3
= 0 itself, and also if the deter-
minant equation
|gµν| = −1 (5)
is satisfied.
These field equations in common with the determinant equation pos-
sess the fundamental property, according to which their form remains
unchanged in the framework of substitution of any other variables in-
stead of x1
, x2
, x3
, x4
, if the substitution of the determinant equals 1.
Assume the curvilinear coordinates x1
, x2
, x3
, while x4
is time. We
assume that the mass located at the origin of the coordinates remains
unchanged with time, and also the motion is uniform and linear up to
infinity. In such a case, according to the calculation by Einstein (see
page 833∗
) the following requirements should be satisfied:
1. All the components should be independent of the time coord-
inate x4
;
2. The equalities gρ4 = g4ρ = 0 are satisfied exactly for ρ = 1, 2, 3;
3. The solution is spatially symmetric at the origin of the coordinate
frame in that sense that it comes to the same solution after the
orthogonal transformation (rotation) of x1
, x2
, x3
;
4. These gµν vanish at infinity, except the next four boundary con-
ditions, which are nonzero
g44 = 1 , g11 = g22 = g33 = −1 .
The task is to find such a line-element, possessing such coefficients,
that the field equations, the determinant equation, and these four re-
quirements would be satisfied.
§2. Einstein showed that this problem in the framework of the first
order approximation leads to Newton’s law, and also that the second
order approximation covers the anomaly in the motion of the perihelion
∗Schwarzschild means page 833 in the aforementioned Einstein paper of 1915
published in Sitzungsberichte der K¨oniglich Preussischen Akademie der Wissen-
schaften. — Editor’s comment. D.R.
3. 12 The Abraham Zelmanov Journal — Vol. 1, 2008
of Mercury. The following calculation provides an exact solution of
this problem. As supposed, in any case, an exact solution should have
a simple form. It is important that the resulting calculation shows the
uniqueness of this solution, while Einstein’s approach gives ambiguity,
and also that the method shown below gives (with some difficulty) the
same good approximation. The following text leads to the representa-
tion of Einstein’s result with increasing precision.
§3. We denote time t, while the rectangular coordinates∗
are denoted
x, y, z. Thus the well-known line-element, satisfying the requirements
1–3, has the obvious form
ds2
= Fdt2
− G dx2
+ dy2
+ dz2
− H (xdx + ydy + zdz)
2
,
where F, G, H are functions of r = x2 + y2 + x2 .
The condition (4) requires, at r = ∞: F = G = 1, H = 0.
Moving to the spherical coordinates†
x=r sin ϑ cos ϕ, y=r sin ϑ sin ϕ,
z = r cos ϑ, the same line-element is
ds2
= Fdt2
− G dr2
+ r2
dϑ2
+ r2
sin2
ϑdϕ2
− Hr2
dr2
=
= Fdt2
− G + Hr2
dr2
− Gr2
dϑ2
+ sin2
ϑdϕ2
. (6)
In the spherical coordinates the volume element is r2
sin ϑdr dϑdϕ,
the determinant of transformation from the old coordinates to the new
ones r2
sin ϑ differs from 1; the field equations are still to be unchanged
and, with use the spherical coordinates, we need to process complicated
transformations. However the following simple method allows us to
avoid this difficulty. Assume
x1
=
r3
3
, x2
= − cos ϑ , x3
= ϕ , (7)
then the equality r2
dr sin ϑdϑdϕ = dx1
dx2
dx3
is true in the whole vol-
ume element. These new variables also represent spherical coordinates
in the framework of this unit determinant. They have obvious advan-
tages to the old spherical coordinates in this problem, and, at the same
time, they still remain valid in the framework of the considerations. In
addition to these, assuming t = x4
, the field equations and the deter-
minant equation remain unchanged in form.
∗The Cartesian coordinates. — Editor’s comment. D.R.
†In the original — “polar coordinates”. However it is obvious that Schwarzschild
means the three-dimensional spherical coordinates. — Editor’s comment. D.R.
4. Karl Schwarzschild 13
In these new spherical coordinates the line-element has the form
ds2
= F(dx4
)2
−
G
r4
+
H
r2
(dx1
)2
−
− Gr2 (dx2
)2
1 − (x2)2
+ (dx3
)2
1 − (x2
)2
, (8)
on the basis of which we write
ds2
= f4 (dx4
)2
− f1 (dx1
)2
− f2
(dx2
)2
1 − (x2)2
− f3 (dx3
)2
1 − (x2
)2
. (9)
In such a case f1, f2 = f3, f4 are three functions of x1
, which satisfy
the following conditions
1. At x1
= ∞: f1 = 1
r4
= 3x1 − 4
3
, f2 = f3 = r2
= 3x1
2
3
, f4 = 1;
2. The determinant equation f1·f2 ·f3 ·f4 = 1;
3. The field equations;
4. The function f is continuous everywhere except x1
= 0.
§4. To obtain the field equations we need first to construct the compo-
nents of the gravitational field according to the line-element (9). The
simplest way to do this is by directly taking the variation, which gives
the differential equations of the geodesic line, then the components will
be seen from the equations. The differential equations of the geodesic
line along the line-element (9) are obtained by directly taking this vari-
ation in the form
f1
d2
x1
ds2
+
1
2
∂f4
∂x1
dx4
ds
2
+
1
2
∂f1
∂x1
dx1
ds
2
−
−
1
2
∂f2
∂x1
1
1 − (x2)2
dx2
ds
2
+ 1 − (x2
)2 dx3
ds
2
= 0 ,
f2
1 − (x2)2
d2
x2
ds2
+
∂f2
∂x1
1
1 − (x2)2
dx1
ds
dx2
ds
+
+
f2 x2
[1 − (x2)2]2
dx2
ds
2
+ f2 x2 dx3
ds
2
= 0 ,
f2 1 − (x2
)2 d2
x3
ds2
+
∂f2
∂x1
1 − (x2
)2 dx1
ds
dx3
ds
− 2f2 x2
dx2
ds
dx3
ds
= 0 ,
f4
d2
x4
ds2
+
∂f4
∂x1
dx1
ds
dx4
ds
= 0 .
5. 14 The Abraham Zelmanov Journal — Vol. 1, 2008
Comparing these equations to (2) gives the components of the grav-
itational field
Γ1
11 = −
1
2
1
f1
∂f1
∂x1
, Γ1
22 = +
1
2
1
f1
∂f2
∂x1
1
1 − (x2)2
,
Γ1
33 = +
1
2
1
f1
∂f2
∂x1
1 − (x2
)2
,
Γ1
44 = −
1
2
1
f1
∂f4
∂x1
,
Γ2
21 = −
1
2
1
f2
∂f2
∂x1
, Γ2
22 = −
x2
1 − (x2)2
, Γ2
33 = − x2
1 − (x2
)2
,
Γ3
31 = −
1
2
1
f2
∂f2
∂x1
, Γ3
32 = +
x2
1 − (x2)2
,
Γ4
41 = −
1
2
1
f4
∂f4
∂x1
,
while the rest of the components of it are zero.
Due to the symmetry of rotation around the origin of the coordin-
ates, it is sufficient to construct the field equations at only the equator
(x2
= 0): once they are differentiated, we can substitute 1 instead of
1 − (x2
)2
everywhere into the above obtained formulae. Thus, after this
algebra, we obtain the field equations
a)
∂
∂x1
1
f1
∂f1
∂x1
=
1
2
1
f1
∂f1
∂x1
2
+
1
f2
∂f2
∂x1
2
+
1
2
1
f4
∂f4
∂x1
2
,
b)
∂
∂x1
1
f1
∂f2
∂x1
= 2 +
1
f1f2
∂f2
∂x1
2
,
c)
∂
∂x1
1
f1
∂f4
∂x1
=
1
f1f4
∂f4
∂x1
2
.
Besides these three equations, the functions f1, f2, f3 should satisfy
the determinant equation
d) f1(f2)2
f4 = 1 or
1
f1
∂f1
∂x1
+
2
f2
∂f2
∂x1
+
1
f4
∂f4
∂f1
= 0 .
First of all I remove b). So three functions f1, f2, f4 of a), c), and
d) still remain. The equation c) takes the form
c′
)
∂
∂x1
1
f4
∂f4
∂x1
=
1
f1f4
∂f1
∂x1
∂f4
∂x1
.
6. Karl Schwarzschild 15
Integration of it gives
c′′
)
1
f4
∂f4
∂x1
= αf1 ,
where α is the constant of integration. Summation of a) and c′
) gives
∂
∂x1
1
f1
∂f1
∂x1
+
1
f4
∂f4
∂x1
=
1
f2
∂f2
∂x1
2
+
1
2
1
f1
∂f1
∂x1
+
1
f4
∂f4
∂x1
2
.
With taking d) into account, it follows that
− 2
∂
∂x1
1
f2
∂f2
∂x1
= 3
1
f2
∂f2
∂x1
2
.
After integration, we obtain
1
1
f2
∂f2
∂x1
=
3
2
x1
+
ρ
2
,
where ρ is the constant of integration. Or
1
f2
∂f2
∂x1
=
2
3x1 + ρ
.
We integrate it once again:
f2 = λ 3x1
+ ρ
2
3
,
where λ is the constant of integration. The condition at infinity requires:
λ = 1. Hence
f2 = (3x1 + ρ)
2
3
. (10)
Next, it follows from c′′
) and d) that
∂f4
∂x1
= αf1f4 =
α
(f2)2
=
α
(3x1 + ρ)
4
3
.
We integrate it, taking the condition at infinity into account:
f4 = 1 − α 3x1
+ ρ
− 1
3
. (11)
Finally, it follows from d) that
f1 =
3x1
+ ρ
− 4
3
1 − α (3x1 + ρ)
− 1
3
. (12)
7. 16 The Abraham Zelmanov Journal — Vol. 1, 2008
As easy to check, the equation b) corresponds to the found formulae
for f1 and f2.
This satisfies all the requirements up to the continuity condition.
The function f1 remains continuous, if
1 = α 3x1
+ ρ
− 1
3
, 3x1
= α3
− ρ .
In order to break the continuity at the origin of the coordinates,
there should be
ρ = α3
. (13)
The continuity condition connects, by the the same method, both
constants of integration ρ and α.
Now, the complete solution of our problem has the form
f1 =
1
R4
1
1 − α
R
, f2 = f3 = R2
, f4 = 1 −
α
R
,
where the auxiliary quantity R has been introduced
R = 3x1
+ ρ
1
3
= r3
+ α3
1
3
.
If substituting the formulae of these functions f into the formula of
the line-element (9), and coming back to the regular spherical coordi-
nates, we arrive at such a formula for the line-element
ds2
= 1 −
α
R
dt2
−
dR2
1 − α
R
− R2
dϑ2
+ sin2
ϑdϕ2
R = r3
+ α3
1
3
, (14)
which is the exact solution of the Einstein problem.
This formula contains the sole constant of integration α, which is
dependent on the numerical value of the mass located at the origin of
the coordinates.
§5. The uniqueness of this solution follows from the aforementioned
calculations. For one who is troubled with the uniqueness of Einstein’s
method, followed from this, we consider the following example. There
above, from the continuity condition, the formula
f1 =
3x1
+ ρ
− 4
3
1 − α (3x1 + ρ)
− 1
3
=
r3
+ ρ
− 4
3
1 − α (r3 + ρ)
− 1
3
8. Karl Schwarzschild 17
was obtained. In the case where α and ρ are small values, the second
order term and the higher order terms vanish from the series so that
f1 =
1
r4
1 +
α
r
−
4
3
ρ
r3
.
This exception, in common with the respective exceptions for f1, f2,
f4 taken to within the same precision, satisfies all the requirements of
this problem. The continuity requirement added nothing in the frame-
work of this approximation, but only a break at the point of the origin of
the coordinates. Both constants α and ρ are arbitrarily determined, so
the physical side of this problem in not determined. The exact solution
of this problem manifests that in a real case, with the approximations, a
break appears not at the point of the origin of the coordinates, but in the
region r = α3
− ρ
1
3 , and we should suppose ρ = α3
in order to move
the break to the origin of the coordinates. In the framework of such
an approximation through the exponents of α and ρ, we need to know
very well the law which rules these coefficients, and also be masters in
the whole situation, in order to understand the necessity of connexion
between α and ρ.
§6. In the end, we are looking for the equation of a point moving along
the geodesic line in the gravitational field related to the line-element
(14). Proceeding from the three circumstances according to which the
line-element is homogeneous, differentiable, and its coefficients are inde-
pendent of t and ρ, we take the variation so we obtain three intermediate
integrals. Because the motion is limited to the equatorial plane (ϑ = 90◦
,
dϑ = 0), these intermediate integrals have the form
1 −
α
R
dt
ds
2
−
1
1 − α
R
dR
ds
2
− R2 dϕ
ds
2
= const = h , (15)
R2 dϕ
ds
= const = c , (16)
1 −
α
R
dt
ds
= const = 1 , (17)
where the third integral means definition of the unit of time.
From here it follows that
dR
dϕ
2
+ R2
1 −
α
R
=
R4
c2
1 − h 1 −
α
R
9. 18 The Abraham Zelmanov Journal — Vol. 1, 2008
or, for 1
R
= x,
dx
dϕ
2
=
1 − h
c2
+
hα
c2
x − x2
+ αx3
. (18)
We denote c2
h
= B, 1−h
h
= 2A that is identical to Einstein’s equations
(11) in the cited presentation∗
, and gives the observed anomaly of the
perihelion of Mercury.
In a general case Einstein’s approximation for a curved trajectory
meets the exact solution, only if we introduce
R = r3
+ α3
1
3
= r 1 +
α3
r3
1
3
(19)
instead of r. Because α
r is close to twice the square of the velocity of the
planet (the velocity of light is 1), the expression within the brackets, in
the case of Mercury, is different from 1 by a value of the order 10−12
. The
quantities R and r are actually identical, so Einstein’s approximation
satisfies the practical requirements of even very distant future.
In the end it is required to obtain the exact form of Kepler’s third law
for circular trajectories. Given an angular velocity n = dϕ
dt
, according to
(16) and (17), and introducing x = 1
R
, we have
n = cx2
(1 − αx) .
In a circle both dx
dϕ
and d2
x
dϕ2
should be zero. This gives, according to
(18), that
1 − h
c2
+
hα
c2
x − x2
+ αx3
= 0 ,
hα
c2
− 2x + 3αx2
= 0 .
Removing h from both circles gives
α = 2c2
x (1 − αx)
2
.
From here it follows that
n2
=
α
2
x3
=
α
2R3
=
α
2(r3 + α3)
.
Deviation of this formula from Kepler’s third law is absolutely invis-
ible up to the surface of the Sun. However given an ideal point-mass, the
∗Einstein A. Erkl¨arung der Perihelbewegung der Merkur aus der allgemeinen
Relativit¨atstheorie. Sitzungsberichte der K¨oniglich Preussischen Akademie der Wis-
senschaften, 1915, S. 831–839. — Editor’s comment. D.R.
10. Karl Schwarzschild 19
angular velocity does not experience unbounded increase with lowering
of the orbital radius (such an unbounded increase should be experienced
according to Newton’s law), but approximates to a finite limit
n0 =
1
α
√
2
.
(For a mass which is in the order of the mass of the Sun, this boundary
frequency should be about 104
per second.) This circumstance should
be interesting in the case where a similar law rules the molecular forces.
11. Vol. 1, 2008 ISSN 1654-9163
THE
ABRAHAM ZELMANOV
JOURNAL
The journal for General Relativity,
gravitation and cosmology
TIDSKRIFTEN
ABRAHAM ZELMANOV
Den tidskrift f¨or allm¨anna relativitetsteorin,
gravitation och kosmologi
Editor (redakt¨or): Dmitri Rabounski
Secretary (sekreterare): Indranu Suhendro
The Abraham Zelmanov Journal is a non-commercial, academic journal registered
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