Einstein was the first to use the cosmological constant in his 1917 paper. Shortly after, Schrödinger presented a note transposing the cosmological constant term to the right side of the field equations, making it part of the stress-energy tensor. Einstein commented that if Schrödinger intended more than a mathematical convenience, he should describe its properties. Einstein then described what some of those properties might be, such as it behaving as a form of negative pressure - effectively providing the first description of dark energy.
This document summarizes a research paper on defining and analyzing the mass of asymptotically hyperbolic manifolds. The paper establishes that the total mass can be defined unambiguously for such manifolds. It proves a positive mass theorem using spinor methods under certain conditions on the scalar curvature. The paper also analyzes how the total mass is related to the geometry of the manifold near infinity and establishes that the total mass is invariant under changes of coordinates. It proposes generalizing the definition of Hawking mass to asymptotically hyperbolic manifolds and relates the total mass to the Hawking mass of outermost surfaces.
To the Issue of Reconciling Quantum Mechanics and General RelativityIOSRJAP
The notion of gravitational radiation as a radiation of the same level as the electromagnetic radiation is based on theoretically proved and experimentally confirmed fact of existence of stationary states of an electron in its gravitational field characterized by the gravitational constant K = 1042G (G is the Newtonian gravitational constant) and unrecoverable space-time curvature Λ. If the numerical values of K 5.11031 Nm2 kg-2 and =4.41029 m -2 , there is a spectrum of stationary states of the electron in its own gravitational field (0.511 MeV ... 0.681 MeV).Adjusting according to the known mechanisms of broadening does not disclose the broadening of the registered portion of the emission spectrum of the micropinch. It indicates the presence of an additional mechanism of broadening the registered portion of the spectrum of the characteristic radiation due to the contribution of the excited states of electrons in their own gravitational field. The energy spectrum of the electron in its own gravitational field and the energy spectra of multielectron atoms are such that there is a resonance of these spectra. As obvious, the consequence of such resonant interaction is appearance, including new lines, of electromagnetic transitions not associated with atomic transitions. The manuscript is the review of previously published papers cited in the references.
Vasil Penchev. Cyclic Mechanics: The Principple of CyclicityVasil Penchev
The principle of cyclicity is introduced. It allows of equating the universe with any space-time in it and implies a generalization of conservation of energy, namely conservation of action or information. Cyclic mechanics created on this base can unify quantum mechanics and general relativity.
1) The document explores the origins of complex numbers in quantum mechanics through three axioms: composability, consistency, and the separation axiom.
2) Applying the composability and consistency axioms to classical and quantum systems implies either real or complex numbers. The separation axiom eliminates real numbers, leaving complex numbers.
3) The author proposes that quantum mechanics can be derived from these axioms by defining symmetric and anti-symmetric products that satisfy certain identities for composable systems, implying the use of complex numbers in quantum mechanics.
1) A quantum particle is described by a wave function ψ(x) which is a function of position. This provides a complete description of the particle's state.
2) The wave function does not indicate a precise position for the particle. Instead, the particle is considered delocalized, meaning it does not have a well-defined position more precise than the spread of the wave function.
3) Certain properties of the wave function, like whether it is nonzero in a particular region of space, provide some information about where the particle might be found if its position were measured. But the particle does not have a well-defined position until a measurement is made.
The paper discusses using renormalization group theory to understand critical phenomena in ferromagnets. It casts the Kadanoff scaling theory for the Ising model in differential form, with the resulting equations being an example of the renormalization group differential equations. It is shown that the usual scaling laws arise naturally from the equations if the coefficients are analytic at the critical point. A generalization involving an "irrelevant" variable is also considered, where the scaling laws only result if the solution asymptotically approaches a fixed point.
Gravity as entanglement, and entanglement as gravityVasil Penchev
1) The document discusses the relationship between gravity and quantum entanglement, exploring the possibility that they are equivalent or closely connected concepts.
2) It outlines an approach to interpret gravity in terms of a generalized quantum field theory that includes entanglement, which could explain why gravity cannot be quantized.
3) The key idea is that entanglement expressed "outside" of space-time points looks like gravity "inside", and vice versa, with gravity representing a smooth constraint on the quantum behavior of entities imposed by all others.
This document discusses Born reciprocity in string theory and how it relates to the nature of spacetime. It argues that while string theory is formulated in terms of maps into spacetime, this breaks Born reciprocity. The document suggests that quasi-periodicity of strings without assuming a periodic target spacetime better respects Born reciprocity. This leads to a phase space formulation of string theory without assuming locality of spacetime at short distances.
This document summarizes a research paper on defining and analyzing the mass of asymptotically hyperbolic manifolds. The paper establishes that the total mass can be defined unambiguously for such manifolds. It proves a positive mass theorem using spinor methods under certain conditions on the scalar curvature. The paper also analyzes how the total mass is related to the geometry of the manifold near infinity and establishes that the total mass is invariant under changes of coordinates. It proposes generalizing the definition of Hawking mass to asymptotically hyperbolic manifolds and relates the total mass to the Hawking mass of outermost surfaces.
To the Issue of Reconciling Quantum Mechanics and General RelativityIOSRJAP
The notion of gravitational radiation as a radiation of the same level as the electromagnetic radiation is based on theoretically proved and experimentally confirmed fact of existence of stationary states of an electron in its gravitational field characterized by the gravitational constant K = 1042G (G is the Newtonian gravitational constant) and unrecoverable space-time curvature Λ. If the numerical values of K 5.11031 Nm2 kg-2 and =4.41029 m -2 , there is a spectrum of stationary states of the electron in its own gravitational field (0.511 MeV ... 0.681 MeV).Adjusting according to the known mechanisms of broadening does not disclose the broadening of the registered portion of the emission spectrum of the micropinch. It indicates the presence of an additional mechanism of broadening the registered portion of the spectrum of the characteristic radiation due to the contribution of the excited states of electrons in their own gravitational field. The energy spectrum of the electron in its own gravitational field and the energy spectra of multielectron atoms are such that there is a resonance of these spectra. As obvious, the consequence of such resonant interaction is appearance, including new lines, of electromagnetic transitions not associated with atomic transitions. The manuscript is the review of previously published papers cited in the references.
Vasil Penchev. Cyclic Mechanics: The Principple of CyclicityVasil Penchev
The principle of cyclicity is introduced. It allows of equating the universe with any space-time in it and implies a generalization of conservation of energy, namely conservation of action or information. Cyclic mechanics created on this base can unify quantum mechanics and general relativity.
1) The document explores the origins of complex numbers in quantum mechanics through three axioms: composability, consistency, and the separation axiom.
2) Applying the composability and consistency axioms to classical and quantum systems implies either real or complex numbers. The separation axiom eliminates real numbers, leaving complex numbers.
3) The author proposes that quantum mechanics can be derived from these axioms by defining symmetric and anti-symmetric products that satisfy certain identities for composable systems, implying the use of complex numbers in quantum mechanics.
1) A quantum particle is described by a wave function ψ(x) which is a function of position. This provides a complete description of the particle's state.
2) The wave function does not indicate a precise position for the particle. Instead, the particle is considered delocalized, meaning it does not have a well-defined position more precise than the spread of the wave function.
3) Certain properties of the wave function, like whether it is nonzero in a particular region of space, provide some information about where the particle might be found if its position were measured. But the particle does not have a well-defined position until a measurement is made.
The paper discusses using renormalization group theory to understand critical phenomena in ferromagnets. It casts the Kadanoff scaling theory for the Ising model in differential form, with the resulting equations being an example of the renormalization group differential equations. It is shown that the usual scaling laws arise naturally from the equations if the coefficients are analytic at the critical point. A generalization involving an "irrelevant" variable is also considered, where the scaling laws only result if the solution asymptotically approaches a fixed point.
Gravity as entanglement, and entanglement as gravityVasil Penchev
1) The document discusses the relationship between gravity and quantum entanglement, exploring the possibility that they are equivalent or closely connected concepts.
2) It outlines an approach to interpret gravity in terms of a generalized quantum field theory that includes entanglement, which could explain why gravity cannot be quantized.
3) The key idea is that entanglement expressed "outside" of space-time points looks like gravity "inside", and vice versa, with gravity representing a smooth constraint on the quantum behavior of entities imposed by all others.
This document discusses Born reciprocity in string theory and how it relates to the nature of spacetime. It argues that while string theory is formulated in terms of maps into spacetime, this breaks Born reciprocity. The document suggests that quasi-periodicity of strings without assuming a periodic target spacetime better respects Born reciprocity. This leads to a phase space formulation of string theory without assuming locality of spacetime at short distances.
1) John Nash presents an equation that is a 4th order covariant tensor partial differential equation applicable to the metric tensor of spacetime. This equation is formally divergence free like the Einstein vacuum equation.
2) The equation can be derived from a specific Lagrangian involving terms quadratic in the scalar curvature and Ricci tensor. Previous theorists had considered such Lagrangians in attempts at quantum gravity theories but had not focused on this specific choice.
3) The equation may allow a wider variety of gravitational waves, including compressional waves not excluded in electromagnetic theory. Standard GR only allows transverse gravitational waves.
This document discusses solutions to the Klein-Gordon equation in Schwarzschild spacetime near a black hole's event horizon. Very near the horizon, the radial Klein-Gordon equation can be approximated as an oscillatory solution in Regge-Wheeler coordinates. These solutions are then expressed in terms of outgoing and ingoing coordinates, which have distinct analytic properties inside and outside the event horizon.
This document discusses solutions to the Klein-Gordon equation in Schwarzschild spacetime near a black hole's event horizon. Very near the horizon, the radial Klein-Gordon equation can be approximated as an oscillatory solution in Regge-Wheeler coordinates. The time and radial solutions are then expressed in terms of outgoing and ingoing coordinates, which leads to outgoing and ingoing waves with different analytic properties on either side of the event horizon.
Outgoing ingoingkleingordon spvmforminit_proceedfrom12dec18foxtrot jp R
This document discusses outgoing and ingoing Klein-Gordon waves near the event horizons of a black hole. It begins by introducing the Klein-Gordon equation of motion in the background of the Schwarzschild spacetime metric. It then presents the solution to this equation in product form and discusses the radial and time component equations. Finally, it recasts the radial equation using the Regge-Wheeler coordinate and shows that very near the horizon, the equation can be approximated as a simple oscillatory solution. The goal is to obtain outgoing and ingoing wave solutions that have different properties on either side of the black hole's event horizons.
Outgoing ingoingkleingordon spvmforminit_proceedfromfoxtrot jp R
This document discusses outgoing and ingoing Klein-Gordon waves near the event horizons of a black hole. It begins by introducing the Klein-Gordon equation of motion in the background of the Schwarzschild spacetime metric. It then presents the solution to this equation in product form and discusses the radial and time component equations. Finally, it recasts the radial equation using the Regge-Wheeler coordinate and shows that very near the horizon, the equation can be approximated as a simple oscillatory solution. The goal is to obtain outgoing and ingoing wave solutions that have different properties on either side of the black hole's event horizons.
1) The document discusses the exact solution to the Klein-Gordon shutter problem, finding that the wave function does not resemble the optical expression for diffraction but the charge density does show transient oscillations resembling a diffraction pattern.
2) It presents the exact solution for the Klein-Gordon shutter problem using discontinuous initial conditions, finding the wave function solution differs from Moshinsky's approximation.
3) When the exact relativistic charge density is plotted over time, it shows transient oscillations that resemble a diffraction pattern, despite some relativistic differences, demonstrating that diffraction in time exists in relativistic scenarios.
- The Stern-Gerlach experiment in 1922 showed that a beam of silver atoms passed through an inhomogeneous magnetic field split into two beams, providing early evidence that angular momentum is quantized.
- The development of quantum mechanics helped explain phenomena like the fine structure of hydrogen emission spectra, but failed to account for observed splittings until the concept of intrinsic "spin" angular momentum was introduced.
- Angular momentum operators like Jx, Jy, and Jz are defined based on classical angular momentum expressions, with momentum terms replaced by operators involving partial derivatives. These operators obey specific commutation relationships and do not commute with one another.
Paul Dirac developed an equation to describe the dynamics of electrons in a way that was consistent with both special relativity and quantum mechanics. His equation predicted the existence of positrons - particles identical to electrons but with the opposite charge. Dirac's equation allowed for negative energy solutions, which he interpreted as occupied states in a "Dirac sea". Unoccupied positive energy states could be described as holes in the sea, behaving as positively charged particles, which came to be known as positrons. Dirac's prediction was confirmed in 1932 when Carl Anderson discovered the positron through his experiments with cosmic rays.
This document provides an introduction to dynamics and forces acting on particles moving in a straight line. It introduces Newton's second law of motion, F=ma, and defines key concepts like weight, tension, thrust, friction, and normal reaction. It explains how to resolve forces into horizontal and vertical components when multiple forces are acting. Examples show how to set up force diagrams and use Newton's second law to solve for acceleration, distance, and missing forces. Trigonometry is used to resolve forces acting at angles into their x- and y-direction components.
A quick introduction to Mach's principleAdemir Xavier
Mach's principle proposes that inertia is an interaction between a body and the rest of the mass in the universe. The document presents Mach's principle, discusses how it could explain the origin of inertia by deriving kinetic energy as an interaction energy between bodies. It then shows calculations modeling the universe as a spherical shell to derive an expression for a particle's total interaction energy with the universe that is equivalent to its inertial mass.
This document reviews research on the convergence of perturbation series in quantum field theory. It discusses Dyson's argument that perturbation series in quantum electrodynamics (QED) have zero radius of convergence due to vacuum instability when the coupling constant is negative. Large-order estimates show that perturbation series coefficients grow factorially fast in quantum mechanics and field theories. Finally, it describes the method of Borel summation, which may allow extracting the exact physical quantity from a divergent perturbation series through a unique mapping.
Conformal Nonlinear Fluid Dynamics from Gravity in Arbitrary DimensionsMichaelRabinovich
This document summarizes research on constructing solutions to Einstein's equations in arbitrary dimensions d that are dual to fluid dynamics on the boundary. The key points are:
1. Solutions are constructed perturbatively to second order in a boundary derivative expansion and are parameterized by a boundary velocity and temperature field obeying the Navier-Stokes equations.
2. The bulk metric dual to an arbitrary fluid flow on a weakly curved boundary is computed explicitly to second order.
3. The boundary stress tensor dual to these solutions is also computed and expressed in a manifestly Weyl-covariant form involving the velocity, temperature, and their derivatives.
4. Properties of the solutions like the event horizon location and an
The document summarizes key aspects of the Standard Model of particle physics. It describes how the Standard Model accounts for fundamental particles like quarks and leptons that interact via four fundamental forces - gravitation, electromagnetism, weak force, and strong force. These interactions are mediated by exchange of spin-1/2 bosons. The Standard Model has been very successful in explaining experimental observations, but questions remain like incorporating gravity and the origin of particle masses.
SOLUTIONS OF THE SCHRÖDINGER EQUATION WITH INVERSELY QUADRATIC HELLMANN PLUS ...ijrap
The solutions of the Schrödinger equation with inversely quadratic Hellmann plus Mie-type potential for
any angular momentum quantum number have been presented using the Nikiforov-Uvarov method. The
bound state energy eigenvalues and the corresponding un-normalized eigenfunctions are obtained in terms
of the Laguerre polynomials. Several cases of the potential are also considered and their eigen values obtained.
This document discusses the quantization of electromagnetic radiation fields within the framework of quantum electrodynamics (QED). It begins by introducing the classical description of radiation fields in terms of vector potentials that satisfy the transversality condition. Next, it describes quantizing the classical radiation field by treating it as a collection of independent harmonic oscillators, with each oscillator characterized by a wave vector and polarization. Finally, it discusses how the quantization of these radiation oscillators leads to treating their canonical variables as non-commuting operators, in analogy to the quantization of position and momentum in non-relativistic quantum mechanics. This lays the foundation for a quantum description of radiation phenomena using the formalism of QED.
This document provides an introduction to dynamics and forces acting on particles moving in a straight line. It introduces Newton's second law of motion, which states that force is equal to mass times acceleration (F=ma). It defines key concepts like weight, normal reaction force, friction, tension, and thrust. Examples are provided on using F=ma to calculate acceleration given force and mass, or force given mass and acceleration. The document also discusses resolving forces into perpendicular components and using this to solve problems involving multiple forces acting on an object.
The generalization of the Periodic table. The "Periodic table" of "dark matter"Vasil Penchev
The thesis is: the “periodic table” of “dark matter” is equivalent to the standard periodic table of the visible matter being entangled. Thus, it is to consist of all possible entangled states of the atoms of chemical elements as quantum systems. In other words, an atom of any chemical element and as a quantum system, i.e. as a wave function, should be represented as a non-orthogonal in general (i.e. entangled) subspace of the separable complex Hilbert space relevant to the system to which the atom at issue is related as a true part of it. The paper follows previous publications of mine stating that “dark matter” and “dark energy” are projections of arbitrarily entangled states on the cognitive “screen” of Einstein’s “Mach’s principle” in general relativity postulating that gravitational field can be generated only by mass or energy.
The document provides an outline for a course on quantum mechanics. It discusses key topics like the time-dependent Schrodinger equation, eigenvalues and eigenfunctions, boundary conditions for wave functions, and applications like the particle in a box model. Specific solutions to the Schrodinger equation are explored for stationary states with definite energy, including the wave function for a free particle and the quantization of energy for a particle confined to a one-dimensional box.
Conformal Anisotropic Mechanics And Horavas Particleguest9fa195
The document presents a model of classical mechanics that implements anisotropic scaling transformations between space and time consistent with the dispersion relation of Hořava gravity. The resulting action is a generalization of conformal mechanics and reduces to known cases in certain limits. Thermodynamic properties of a gas of free particles under the anisotropic scaling are also derived, matching results from recent literature on anisotropic black branes.
Plutino an accidental_quasi-satellite_of_plutoSérgio Sacani
Plutino 15810 (1994 JR1) is currently in a 1:1 mean motion resonance with Pluto, following a quasi-satellite path where its mean longitude circulates with a superimposed libration. N-body simulations show it will remain in this quasi-satellite state, where it moves around Pluto like a retrograde satellite but with a non-closed trajectory, for nearly 350,000 years. This makes 15810 the first known minor body in a 1:1 resonance with Pluto and the first quasi-satellite discovered in the trans-Neptunian region. The quasi-satellite behavior is triggered by the object's 2:3 mean motion resonance with Neptune.
The stelar mass_growth_of_brightest_cluster_galaxies_in_the_irac_shallow_clus...Sérgio Sacani
This document describes a study of the stellar mass growth of brightest cluster galaxies (BCGs) between z=1.5 and z=0.5 using data from the Spitzer IRAC Shallow Cluster Survey (ISCS). The researchers developed a method to select high-redshift clusters as progenitors of lower-redshift clusters to study the evolution of BCG stellar masses. They find that between z=1.5 and z=0.5, the BCGs grew in stellar mass by a factor of 2.3, matching predictions from a semi-analytic galaxy formation model. Below z=0.5, there are hints of differences between the observed BCG growth and the model predictions.
1) John Nash presents an equation that is a 4th order covariant tensor partial differential equation applicable to the metric tensor of spacetime. This equation is formally divergence free like the Einstein vacuum equation.
2) The equation can be derived from a specific Lagrangian involving terms quadratic in the scalar curvature and Ricci tensor. Previous theorists had considered such Lagrangians in attempts at quantum gravity theories but had not focused on this specific choice.
3) The equation may allow a wider variety of gravitational waves, including compressional waves not excluded in electromagnetic theory. Standard GR only allows transverse gravitational waves.
This document discusses solutions to the Klein-Gordon equation in Schwarzschild spacetime near a black hole's event horizon. Very near the horizon, the radial Klein-Gordon equation can be approximated as an oscillatory solution in Regge-Wheeler coordinates. These solutions are then expressed in terms of outgoing and ingoing coordinates, which have distinct analytic properties inside and outside the event horizon.
This document discusses solutions to the Klein-Gordon equation in Schwarzschild spacetime near a black hole's event horizon. Very near the horizon, the radial Klein-Gordon equation can be approximated as an oscillatory solution in Regge-Wheeler coordinates. The time and radial solutions are then expressed in terms of outgoing and ingoing coordinates, which leads to outgoing and ingoing waves with different analytic properties on either side of the event horizon.
Outgoing ingoingkleingordon spvmforminit_proceedfrom12dec18foxtrot jp R
This document discusses outgoing and ingoing Klein-Gordon waves near the event horizons of a black hole. It begins by introducing the Klein-Gordon equation of motion in the background of the Schwarzschild spacetime metric. It then presents the solution to this equation in product form and discusses the radial and time component equations. Finally, it recasts the radial equation using the Regge-Wheeler coordinate and shows that very near the horizon, the equation can be approximated as a simple oscillatory solution. The goal is to obtain outgoing and ingoing wave solutions that have different properties on either side of the black hole's event horizons.
Outgoing ingoingkleingordon spvmforminit_proceedfromfoxtrot jp R
This document discusses outgoing and ingoing Klein-Gordon waves near the event horizons of a black hole. It begins by introducing the Klein-Gordon equation of motion in the background of the Schwarzschild spacetime metric. It then presents the solution to this equation in product form and discusses the radial and time component equations. Finally, it recasts the radial equation using the Regge-Wheeler coordinate and shows that very near the horizon, the equation can be approximated as a simple oscillatory solution. The goal is to obtain outgoing and ingoing wave solutions that have different properties on either side of the black hole's event horizons.
1) The document discusses the exact solution to the Klein-Gordon shutter problem, finding that the wave function does not resemble the optical expression for diffraction but the charge density does show transient oscillations resembling a diffraction pattern.
2) It presents the exact solution for the Klein-Gordon shutter problem using discontinuous initial conditions, finding the wave function solution differs from Moshinsky's approximation.
3) When the exact relativistic charge density is plotted over time, it shows transient oscillations that resemble a diffraction pattern, despite some relativistic differences, demonstrating that diffraction in time exists in relativistic scenarios.
- The Stern-Gerlach experiment in 1922 showed that a beam of silver atoms passed through an inhomogeneous magnetic field split into two beams, providing early evidence that angular momentum is quantized.
- The development of quantum mechanics helped explain phenomena like the fine structure of hydrogen emission spectra, but failed to account for observed splittings until the concept of intrinsic "spin" angular momentum was introduced.
- Angular momentum operators like Jx, Jy, and Jz are defined based on classical angular momentum expressions, with momentum terms replaced by operators involving partial derivatives. These operators obey specific commutation relationships and do not commute with one another.
Paul Dirac developed an equation to describe the dynamics of electrons in a way that was consistent with both special relativity and quantum mechanics. His equation predicted the existence of positrons - particles identical to electrons but with the opposite charge. Dirac's equation allowed for negative energy solutions, which he interpreted as occupied states in a "Dirac sea". Unoccupied positive energy states could be described as holes in the sea, behaving as positively charged particles, which came to be known as positrons. Dirac's prediction was confirmed in 1932 when Carl Anderson discovered the positron through his experiments with cosmic rays.
This document provides an introduction to dynamics and forces acting on particles moving in a straight line. It introduces Newton's second law of motion, F=ma, and defines key concepts like weight, tension, thrust, friction, and normal reaction. It explains how to resolve forces into horizontal and vertical components when multiple forces are acting. Examples show how to set up force diagrams and use Newton's second law to solve for acceleration, distance, and missing forces. Trigonometry is used to resolve forces acting at angles into their x- and y-direction components.
A quick introduction to Mach's principleAdemir Xavier
Mach's principle proposes that inertia is an interaction between a body and the rest of the mass in the universe. The document presents Mach's principle, discusses how it could explain the origin of inertia by deriving kinetic energy as an interaction energy between bodies. It then shows calculations modeling the universe as a spherical shell to derive an expression for a particle's total interaction energy with the universe that is equivalent to its inertial mass.
This document reviews research on the convergence of perturbation series in quantum field theory. It discusses Dyson's argument that perturbation series in quantum electrodynamics (QED) have zero radius of convergence due to vacuum instability when the coupling constant is negative. Large-order estimates show that perturbation series coefficients grow factorially fast in quantum mechanics and field theories. Finally, it describes the method of Borel summation, which may allow extracting the exact physical quantity from a divergent perturbation series through a unique mapping.
Conformal Nonlinear Fluid Dynamics from Gravity in Arbitrary DimensionsMichaelRabinovich
This document summarizes research on constructing solutions to Einstein's equations in arbitrary dimensions d that are dual to fluid dynamics on the boundary. The key points are:
1. Solutions are constructed perturbatively to second order in a boundary derivative expansion and are parameterized by a boundary velocity and temperature field obeying the Navier-Stokes equations.
2. The bulk metric dual to an arbitrary fluid flow on a weakly curved boundary is computed explicitly to second order.
3. The boundary stress tensor dual to these solutions is also computed and expressed in a manifestly Weyl-covariant form involving the velocity, temperature, and their derivatives.
4. Properties of the solutions like the event horizon location and an
The document summarizes key aspects of the Standard Model of particle physics. It describes how the Standard Model accounts for fundamental particles like quarks and leptons that interact via four fundamental forces - gravitation, electromagnetism, weak force, and strong force. These interactions are mediated by exchange of spin-1/2 bosons. The Standard Model has been very successful in explaining experimental observations, but questions remain like incorporating gravity and the origin of particle masses.
SOLUTIONS OF THE SCHRÖDINGER EQUATION WITH INVERSELY QUADRATIC HELLMANN PLUS ...ijrap
The solutions of the Schrödinger equation with inversely quadratic Hellmann plus Mie-type potential for
any angular momentum quantum number have been presented using the Nikiforov-Uvarov method. The
bound state energy eigenvalues and the corresponding un-normalized eigenfunctions are obtained in terms
of the Laguerre polynomials. Several cases of the potential are also considered and their eigen values obtained.
This document discusses the quantization of electromagnetic radiation fields within the framework of quantum electrodynamics (QED). It begins by introducing the classical description of radiation fields in terms of vector potentials that satisfy the transversality condition. Next, it describes quantizing the classical radiation field by treating it as a collection of independent harmonic oscillators, with each oscillator characterized by a wave vector and polarization. Finally, it discusses how the quantization of these radiation oscillators leads to treating their canonical variables as non-commuting operators, in analogy to the quantization of position and momentum in non-relativistic quantum mechanics. This lays the foundation for a quantum description of radiation phenomena using the formalism of QED.
This document provides an introduction to dynamics and forces acting on particles moving in a straight line. It introduces Newton's second law of motion, which states that force is equal to mass times acceleration (F=ma). It defines key concepts like weight, normal reaction force, friction, tension, and thrust. Examples are provided on using F=ma to calculate acceleration given force and mass, or force given mass and acceleration. The document also discusses resolving forces into perpendicular components and using this to solve problems involving multiple forces acting on an object.
The generalization of the Periodic table. The "Periodic table" of "dark matter"Vasil Penchev
The thesis is: the “periodic table” of “dark matter” is equivalent to the standard periodic table of the visible matter being entangled. Thus, it is to consist of all possible entangled states of the atoms of chemical elements as quantum systems. In other words, an atom of any chemical element and as a quantum system, i.e. as a wave function, should be represented as a non-orthogonal in general (i.e. entangled) subspace of the separable complex Hilbert space relevant to the system to which the atom at issue is related as a true part of it. The paper follows previous publications of mine stating that “dark matter” and “dark energy” are projections of arbitrarily entangled states on the cognitive “screen” of Einstein’s “Mach’s principle” in general relativity postulating that gravitational field can be generated only by mass or energy.
The document provides an outline for a course on quantum mechanics. It discusses key topics like the time-dependent Schrodinger equation, eigenvalues and eigenfunctions, boundary conditions for wave functions, and applications like the particle in a box model. Specific solutions to the Schrodinger equation are explored for stationary states with definite energy, including the wave function for a free particle and the quantization of energy for a particle confined to a one-dimensional box.
Conformal Anisotropic Mechanics And Horavas Particleguest9fa195
The document presents a model of classical mechanics that implements anisotropic scaling transformations between space and time consistent with the dispersion relation of Hořava gravity. The resulting action is a generalization of conformal mechanics and reduces to known cases in certain limits. Thermodynamic properties of a gas of free particles under the anisotropic scaling are also derived, matching results from recent literature on anisotropic black branes.
Plutino an accidental_quasi-satellite_of_plutoSérgio Sacani
Plutino 15810 (1994 JR1) is currently in a 1:1 mean motion resonance with Pluto, following a quasi-satellite path where its mean longitude circulates with a superimposed libration. N-body simulations show it will remain in this quasi-satellite state, where it moves around Pluto like a retrograde satellite but with a non-closed trajectory, for nearly 350,000 years. This makes 15810 the first known minor body in a 1:1 resonance with Pluto and the first quasi-satellite discovered in the trans-Neptunian region. The quasi-satellite behavior is triggered by the object's 2:3 mean motion resonance with Neptune.
The stelar mass_growth_of_brightest_cluster_galaxies_in_the_irac_shallow_clus...Sérgio Sacani
This document describes a study of the stellar mass growth of brightest cluster galaxies (BCGs) between z=1.5 and z=0.5 using data from the Spitzer IRAC Shallow Cluster Survey (ISCS). The researchers developed a method to select high-redshift clusters as progenitors of lower-redshift clusters to study the evolution of BCG stellar masses. They find that between z=1.5 and z=0.5, the BCGs grew in stellar mass by a factor of 2.3, matching predictions from a semi-analytic galaxy formation model. Below z=0.5, there are hints of differences between the observed BCG growth and the model predictions.
Topography of northern_hemisphere_of_mercury_from_messenger_altimeterSérgio Sacani
The document describes a study that used laser altimetry from the MESSENGER spacecraft to create a topographic model of Mercury's northern hemisphere. The study found that Mercury has a much smaller range of elevations compared to Mars or the Moon, likely due to Mercury's higher density and gravitational acceleration smoothing out topographic features. The model revealed numerous large impact structures that influence the shape of the hemisphere but do not significantly affect the distribution of elevations.
This document describes a study searching for supermassive black holes (SMBHs) with masses below 107 solar masses by looking for signs of low-level nuclear activity in nearby galaxies that are not known to be active galactic nuclei (AGNs). The study uses Chandra X-ray Observatory data to select for X-ray emission, a signature of AGNs, focusing on late-type spiral and dwarf galaxies most likely to host low-mass SMBHs. Preliminary results applying this technique to six nearby spiral galaxies find nuclear X-ray sources in all six, with NGC 3169 and NGC 4102 likely confirmed as AGNs and NGC 3184 and NGC 5457 possibly hosting AGNs.
The herschel exploitation_of_local_galaxy_andromeda_(helga)_global_farinfrare...Sérgio Sacani
The Herschel satellite was used to observe the Andromeda Galaxy (M31) at far-infrared and submillimeter wavelengths between 100-500 microns. The observations cover an area of about 5.5 by 2.5 degrees centered on M31, the largest area ever observed for this galaxy at these wavelengths. The data reveal dust emission extending significantly beyond M31's optical radius of about 21 kpc. Specifically, annular dust structures are detected on both sides of M31 at a galactocentric distance of about 21 kpc, and two additional annular structures are detected even further out, at about 26 and 31 kpc from the center of M31. In total, the observations estimate the
VIMS images of the Huygens landing site on Titan acquired during Cassini flybys in October and December 2004 provide insight into surface features near the landing site with spatial resolutions of 14.4-19 km/pixel. Ratio images of the brightest and darkest spectra in the 2.03 μm window reveal a particularly contrasted structure north of the landing site, consistent with local enrichment in exposed water ice. The images also show a possible 150 km diameter impact crater with a central peak. While scattering from haze particles dominates Titan's spectrum, spectral ratios of bright and dark areas suggest differences in surface composition and/or topography related to DISR images of the site.
An infrared proper_motion_study_of_bullets_of_orion_nebulaSérgio Sacani
This study measured the proper motions of infrared-emitting bullets in the Orion Nebula using images taken with a four-year time baseline. The [Fe II]-emitting bullets were found to be moving at around 170 km/s, while H2 bullets had smaller motions. Differential motion was not observed between [Fe II] bullets and trailing H2 wakes, suggesting they have reached a steady configuration over at least 100 years. The proper motions provide constraints on the origin and dynamics of the bullet systems, with an impulsive origin appearing likely.
Toward a new_distance_to_the_active_galaxy_ngc4258Sérgio Sacani
This document reports on measurements of centripetal accelerations of maser spectral components in the active galaxy NGC 4258 using data from 51 epochs spanning 1994 to 2004. Accelerations of high-velocity maser emission indicate an origin within 13 degrees of the disk midline. Accelerations do not support a model of trailing shocks from spiral arms but find evidence of spiral structure with a wavelength of 0.75 mas. Accelerations of low-velocity emission are consistent with originating from a concave region where the thin disk is tangent to the line of sight.
This document summarizes a study of the future orbital evolution and merging of the Milky Way, Andromeda Galaxy (M31), and Triangulum Galaxy (M33) systems using N-body simulations and orbit integrations. Key findings include:
1) The M31 velocity vector implies that the Milky Way and M31 will merge in 5.86 billion years, with a 41% chance of a direct collision within 25 kpc.
2) M31 and M33 will have their first pericenter passage in 0.85 billion years at a distance of 80.8 kpc.
3) There is a 9% chance that M33 collides with the Milky Way before M31.
This document summarizes results from simulations of galaxy formation and evolution using hydrodynamical simulations. Higher resolution simulations that include feedback produce galaxies with larger disk scale lengths and smaller bulge-to-disk ratios, in better agreement with observations. Feedback and resolution are necessary to form galaxies with flatter rotation curves and properties matching observed galaxies, like the Tully-Fisher relation. One simulated galaxy has a large disk scale length of 9.2 kpc and small bulge-to-disk ratio of 0.64.
The document summarizes two Chandra X-ray observations of the Type IIn supernova SN 2010jl taken around two months and one year after explosion. The first observation revealed a high temperature (10 keV) and very high absorption column density (1024 cm-2) from circumstellar matter. The second observation, one year later, showed the absorption column had decreased by a factor of three as expected from shock propagation. Both observations detected an unabsorbed luminosity of 7×1041 erg s-1. The Fe line detected in the first observation was not present in the second.
This document summarizes an infrared survey of the massive star forming region RCW 57 (NGC 3576) using L-band (3.5 μm) data from SPIREX and JHKs data from 2MASS. Over 50% of the 209 sources detected showed infrared excess, indicating circumstellar disks. Comparison to other surveys supports a very high initial disk fraction (>80%) around massive stars, though disks may dissipate faster around high-mass stars. 33 sources only detected at L-band indicate heavily embedded, massive Class I protostars. Diffuse PAH emission was also detected throughout RCW 57.
Alignment of th_angular_momentum_vectors_of_planetary_nebulae_in_the_galactic...Sérgio Sacani
This document analyzes the orientations of 130 planetary nebulae (PNe) in the Galactic Bulge to investigate whether there is a preferred alignment. It finds that while the full sample shows a uniform distribution, the bipolar PNe exhibit a non-uniform distribution with a mean orientation along the Galactic plane at a 90 degree position angle, significant at the 0.001 level. This indicates that the orbital planes of binary systems in old stars are oriented perpendicular to the Galactic plane, likely due to strong magnetic fields during star formation that influenced the angular momentum vectors.
Mass and motion_of_globulettes_in_the_rosette_nebulaSérgio Sacani
This document summarizes observations of globulettes (small molecular clumps) in the Rosette Nebula made using radio telescopes and infrared imaging. Radio observations of 16 globulettes detected molecular line emission from 12CO and 13CO, allowing masses to be estimated. Masses ranged from about 50 to 500 Jupiter masses. Infrared imaging found that several globulettes are very opaque and contain dense cores. Analysis of velocities found that globulettes and other nebula structures are expanding at about 22 km/s, with globulettes having only small velocity differences, suggesting they are moving with the general expansion. Some globulettes appear to be detaching from larger structures as they expand outwards.
This document provides a summary of quantum mechanical and solid state physics concepts:
1. It reviews the Schrodinger equation and wavefunction solutions for quantum systems like electrons and atoms. The wavefunction Ψ relates to real physical measurements through the Born interpretation of probability density.
2. When atoms come together to form solids, the discrete energy levels of individual atoms overlap and form continuous energy bands. This allows electrons in solids to take on only certain allowed energy values.
3. Models like the Kronig-Penny model illustrate how the periodic potential of a solid crystal results in an electronic band structure with permitted energy bands separated by forbidden gaps.
This document provides a summary of quantum mechanical concepts and solid state physics. It begins with a review of quantum mechanics and the Schrodinger equation. It then discusses the wave nature of electrons and how the Schrodinger equation describes the wavefunction and probability of finding an electron. It also covers energy band diagrams and how the periodic potential in solids leads to the formation of allowed energy bands. It discusses these concepts for isolated atoms, silicon crystals, and the one-dimensional Kronig-Penny model.
The document provides an overview of quantum mechanics concepts including:
- Erwin Schrodinger developed the Schrodinger wave equation which describes the energy and position of electrons.
- The time-dependent and time-independent Schrodinger equations are presented for 1D systems. Solutions include plane waves and wave packets.
- The postulates of quantum mechanics are outlined including the use of wavefunctions and Hermitian operators to represent physical quantities.
- Differences between the interpretations of quantum mechanics by Heisenberg and Schrodinger are briefly discussed.
General relativity presentation.ragesh,asmitha,m.d.trageshthedon
1. The document discusses Albert Einstein's theory of general relativity and how it improved upon Isaac Newton's theory of gravity.
2. General relativity explains gravity as a distortion of spacetime caused by massive objects, rather than an attractive force.
3. Some key predictions of general relativity include gravitational time dilation, gravitational lensing of light, and gravitational waves. The theory has been confirmed by various observational tests.
1) Albert Einstein developed the theory of relativity, which describes the laws of physics for both non-accelerating and accelerating frames of reference.
2) The theory of special relativity, published in 1905, describes motion at constant velocity while general relativity, published in 1915, describes accelerated frames and gravity.
3) One of the most famous equations in physics, E=mc2, comes from the theory of relativity and shows that mass and energy are equivalent and interconvertible.
Werner Karl Heisenberg was a German theoretical physicist who made foundational contributions to quantum mechanics. He is best known for formulating the uncertainty principle, which states that the more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa. Heisenberg developed the principle at Niels Bohr's institute in Copenhagen in 1927 while working on the mathematical foundations of quantum mechanics.
Einstein's theory of general relativity revolutionized our understanding of gravity by describing it in terms of the warping of space-time by massive objects. It established that large celestial bodies distort the very fabric of space-time around them rather than gravity being an external force. General relativity provided mathematical equations relating the curvature of space-time to the distribution of mass and energy. It made several novel predictions that have been confirmed by experiments, including gravitational lensing and gravitational time dilation. The theory remains crucial to modern astrophysics and cosmology, helping explain phenomena like black holes, gravitational waves, and the expansion of the universe.
Albert Einstein (1879-1955) developed the theories of special and general relativity. Special relativity, published in 1905, established that the laws of physics are the same in all inertial frames of reference and that the speed of light in a vacuum is constant. General relativity, published in 1915, introduced gravitation as a result of the curvature of spacetime caused by the uneven distribution of mass and energy. One of Einstein's most famous equations is E=mc2, which shows that mass and energy are equivalent and interconvertible.
1) The document summarizes the 100-year history of gravitational wave detection, from Einstein's theory of general relativity predicting their existence to the recent direct detection by the LIGO experiment.
2) It describes how LIGO uses laser interferometry to extremely sensitively measure tiny distortions in spacetime caused by passing gravitational waves.
3) The first detected gravitational waves in 2015 matched predictions for the inspiral and merger of two black holes, with the signal analyzed to determine properties of the black holes such as their masses.
Quantum chemistry is the application of quantum mechanics to solve problems in chemistry. It has been widely used in different branches of chemistry including physical chemistry, organic chemistry, analytical chemistry, and inorganic chemistry. The time-independent Schrödinger equation is central to quantum chemistry and can be used to model chemical systems like the particle in a box, harmonic oscillator, and hydrogen atom. Molecular orbital theory is also important in quantum chemistry for describing chemical bonding in molecules.
This document provides an introduction to general relativity. It begins by summarizing the key aspects of special relativity, including that spacetime is four-dimensional and transformations between inertial frames form the Poincare group. It then discusses the equivalence principle and introduces curved coordinates to describe gravity. The document derives the affine connection and Riemann curvature tensor, and introduces the metric tensor. It provides the perturbative expansion leading to Einstein's field equations and discusses solutions like the Schwarzschild metric and gravitational radiation.
This document provides an introduction to quantum mechanics, covering several key topics:
1) It outlines the historical experiments and findings that led to the need for quantum theory, including black-body radiation, atomic spectroscopy, and the photoelectric effect.
2) It discusses the concept of wave-particle duality and experiments demonstrating this duality, such as the Compton effect, electron diffraction, and Young's double slit experiment.
3) It introduces several important figures in the development of quantum mechanics such as Planck, Einstein, de Broglie, Compton, and Davisson and examines their key contributions to establishing quantum theory.
Introduction to quantum mechanics and schrodinger equationGaurav Singh Gusain
Classical mechanics describes macroscopic objects while quantum mechanics describes microscopic objects due to limitations of classical theory. Quantum mechanics was introduced after classical mechanics failed to explain experimental observations involving microscopic particles. Some key aspects of quantum mechanics are the photoelectric effect, blackbody radiation, Compton effect, wave-particle duality, the Heisenberg uncertainty principle, and Schrodinger's wave equation. Schrodinger's equation describes the wave function and probability of finding a particle.
1. Quantum mechanics describes the behavior of matter and light at the atomic scale, which is very different from classical mechanics. Particles have both wave-like and particle-like properties.
2. The de Broglie hypothesis proposed that all particles have an associated wavelength that depends on their momentum. This was confirmed experimentally by observing electron diffraction patterns.
3. Heisenberg's uncertainty principle states that it is impossible to precisely measure both a particle's position and momentum simultaneously. This limits our ability to predict the future behavior of particles.
There is an ongoing debate about the momentum of a photon in a medium, with two main perspectives proposed by Abraham and Minkowski. An experiment is proposed to help resolve this using a Bose-Einstein condensate with a very high refractive index acting as the "medium" and measuring any displacement. Previous experiments involving bouncing light off mirrors attached to torsion balances produced results consistent with both Abraham and Minkowski, not resolving the debate. The topic has been discussed by many prominent physicists over decades without a clear consensus emerging.
Einstein's theories of relativity include the special and general theories. The special theory, published in 1905, deals with inertial frames of reference and the constancy of the speed of light. The general theory, published in 1915, extends these concepts to accelerated frames and explains gravity as a consequence of spacetime curvature. Key findings include spacetime being dynamic, light bending in gravity, and the equivalence of mass and energy. The Michelson-Morley experiment's null results disproved the ether hypothesis and supported Einstein's postulate that the speed of light is independent of motion.
This document provides an introduction to quantum mechanics, covering several key topics:
1) It discusses early discoveries that led to the development of quantum mechanics, including black-body radiation, atomic spectroscopy, the photoelectric effect, and wave-particle duality experiments like the Compton effect and electron diffraction.
2) It introduces Louis de Broglie's hypothesis that particles are also waves, with wavelength related to momentum. This helped establish the wave-particle duality central to quantum mechanics.
3) It describes the development of new mathematical tools in quantum mechanics, including using wave functions and operators to represent physical quantities and their associated measurements. The Schrödinger equation allows determining the wave function and eigenvalues of
Overview of GTR and Introduction to CosmologyPratik Tarafdar
This document provides an overview of general relativity and an introduction to cosmology. It discusses key concepts such as:
- General relativity builds on Einstein's theory that gravity curves spacetime.
- The principle of equivalence states that inertial and gravitational mass are equivalent.
- Einstein's field equations relate the curvature of spacetime to the energy and momentum within it.
- Tests of general relativity include observations of orbiting bodies like Mercury, gravitational lensing, and the detection of gravitational waves.
- The cosmological principle states that the universe is homogeneous and isotropic on large scales.
The document discusses Einstein's field equations and Heisenberg's uncertainty principle. It begins by providing background on Einstein's field equations, which relate the geometry of spacetime to the distribution of mass and energy within it. It then discusses some key mathematical aspects of the field equations, including their nonlinear partial differential form. Finally, it notes that the field equations can be consolidated with Heisenberg's uncertainty principle to provide a unified description of gravity and quantum mechanics.
Relativity theory project & albert einstenSeergio Garcia
Albert Einstein was a German-born theoretical physicist who developed the theory of relativity, one of the pillars of modern physics. He was born in 1879 in Germany and died in 1955 in the United States. He is best known for his mass-energy equivalence formula E=mc2, which has been called the world's most famous equation. He developed the special theory of relativity, which describes the laws of motion at high speeds and led to his famous equation, and the general theory of relativity, which describes gravity as a geometric property of space and time.
Similar to How einstein discovered_dark_energy (20)
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.
Exomoons & Exorings with the Habitable Worlds Observatory I: On the Detection...Sérgio Sacani
The highest priority recommendation of the Astro2020 Decadal Survey for space-based astronomy
was the construction of an observatory capable of characterizing habitable worlds. In this paper series
we explore the detectability of and interference from exomoons and exorings serendipitously observed
with the proposed Habitable Worlds Observatory (HWO) as it seeks to characterize exoplanets, starting
in this manuscript with Earth-Moon analog mutual events. Unlike transits, which only occur in systems
viewed near edge-on, shadow (i.e., solar eclipse) and lunar eclipse mutual events occur in almost every
star-planet-moon system. The cadence of these events can vary widely from ∼yearly to multiple events
per day, as was the case in our younger Earth-Moon system. Leveraging previous space-based (EPOXI)
lightcurves of a Moon transit and performance predictions from the LUVOIR-B concept, we derive
the detectability of Moon analogs with HWO. We determine that Earth-Moon analogs are detectable
with observation of ∼2-20 mutual events for systems within 10 pc, and larger moons should remain
detectable out to 20 pc. We explore the extent to which exomoon mutual events can mimic planet
features and weather. We find that HWO wavelength coverage in the near-IR, specifically in the 1.4 µm
water band where large moons can outshine their host planet, will aid in differentiating exomoon signals
from exoplanet variability. Finally, we predict that exomoons formed through collision processes akin
to our Moon are more likely to be detected in younger systems, where shorter orbital periods and
favorable geometry enhance the probability and frequency of mutual events.
Emergent ribozyme behaviors in oxychlorine brines indicate a unique niche for...Sérgio Sacani
Mars is a particularly attractive candidate among known astronomical objects
to potentially host life. Results from space exploration missions have provided
insights into Martian geochemistry that indicate oxychlorine species, particularly perchlorate, are ubiquitous features of the Martian geochemical landscape. Perchlorate presents potential obstacles for known forms of life due to
its toxicity. However, it can also provide potential benefits, such as producing
brines by deliquescence, like those thought to exist on present-day Mars. Here
we show perchlorate brines support folding and catalysis of functional RNAs,
while inactivating representative protein enzymes. Additionally, we show
perchlorate and other oxychlorine species enable ribozyme functions,
including homeostasis-like regulatory behavior and ribozyme-catalyzed
chlorination of organic molecules. We suggest nucleic acids are uniquely wellsuited to hypersaline Martian environments. Furthermore, Martian near- or
subsurface oxychlorine brines, and brines found in potential lifeforms, could
provide a unique niche for biomolecular evolution.
Continuum emission from within the plunging region of black hole discsSérgio Sacani
The thermal continuum emission observed from accreting black holes across X-ray bands has the potential to be leveraged as a
powerful probe of the mass and spin of the central black hole. The vast majority of existing ‘continuum fitting’ models neglect
emission sourced at and within the innermost stable circular orbit (ISCO) of the black hole. Numerical simulations, however,
find non-zero emission sourced from these regions. In this work, we extend existing techniques by including the emission
sourced from within the plunging region, utilizing new analytical models that reproduce the properties of numerical accretion
simulations. We show that in general the neglected intra-ISCO emission produces a hot-and-small quasi-blackbody component,
but can also produce a weak power-law tail for more extreme parameter regions. A similar hot-and-small blackbody component
has been added in by hand in an ad hoc manner to previous analyses of X-ray binary spectra. We show that the X-ray spectrum
of MAXI J1820+070 in a soft-state outburst is extremely well described by a full Kerr black hole disc, while conventional
models that neglect intra-ISCO emission are unable to reproduce the data. We believe this represents the first robust detection of
intra-ISCO emission in the literature, and allows additional constraints to be placed on the MAXI J1820 + 070 black hole spin
which must be low a• < 0.5 to allow a detectable intra-ISCO region. Emission from within the ISCO is the dominant emission
component in the MAXI J1820 + 070 spectrum between 6 and 10 keV, highlighting the necessity of including this region. Our
continuum fitting model is made publicly available.
Continuum emission from within the plunging region of black hole discs
How einstein discovered_dark_energy
1. How Einstein Discovered Dark Energy
Alex Harveya)
Visiting Scholar
New York University
New York, NY 10003
arXiv:1211.6338v1 [physics.hist-ph] 22 Nov 2012
Abstract
In 1917 Einstein published his Cosmological Considerations Concerning the
General Theory of Relativity. In it was the first use of the cosmological constant.
Shortly thereafter Schr¨dinger presented a note providing a solution to these
o
same equations with the cosmological constant term transposed to the right
hand side thus making it part of the stress-energy tensor. Einstein commented
that if Schr¨dinger had something more than a mere mathematical convenience
o
in mind he should describe its properties. Then Einstein detailed what some
of these properties might be. In so doing, he gave the first description of
Dark Energy. We present a translation of Schr¨dinger’s paper and Einstein’s
o
response.
1 Introduction
In 1917 Einstein published his Cosmological Considerations Concerning the General
Theory of Relativity [1]. The field equations he used included the cosmological con-
stant.
1
Rµν − λgµν = −κ(Tµν − 2 gµν T ) . (1)
Shortly thereafter Schr¨dinger presented a note [2] in which he transposed the cos-
o
mological constant term to the right hand side thus making it part of the source, that
is, part of the stress-energy tensor.
Rµν = λgµν − κ(Tµν − 1 gµν T )
2
(2)
Einstein immediately recognized the implications and responded with an analytic
comment [3]. If the λ term were to remain a constant then the shift was trivial. If,
however, it was truly intended to be a source then its properties would necessarily have
to be specified. There is no record of a response by Schr¨dinger to Einstein’s comment
o
1
2. and there the matter remained. The exchange between Einstein and Schr¨dinger, less
o
technical details, is described by Mehra and Reichenberg [4].
A translation of Schr¨dinger’s paper follows directly. In it all equations, footnotes,
o
and abbreviations are as in the original. Explanatory comments are interpolated in
the text. Footnotes and citations are collected at the end and where obscure, they
are expanded.
2 Schr¨dinger’s Paper
o
Concerning a System of Solutions of the
Generally Covariant Equations for Gravitation
Same time ago Herr Einstein1 presented a system of energy components for matter,
T µν , and gravitational potentials, gµν , which integrate, precisely, the generally covari-
ant field equations. He suggested that this system approximated the real structure
of space and matter in the large. Briefly stated, it describes incoherent matter of
constant density at rest that fills2 a closed spatial continuum of finite volume having
the metric properties of a hypersphere.
For the field equations, a slightly different form than the original one is assumed.3
The “slightly different form”is simply the transposition of λgµν from the l.h.s to the r.h.s.
thereby obtaining Eq. (2) (above).
The concept of a closed total space seems to me to be of extraordinary importance
for general relativity. This is not only, and not mainly, because of the “desolation
argument”from which Herr Einstein started. I see its principle value rather in the
fact that by development of this thought general relativity promises to be what its
name says and what it according to my view only formerly has been, so to say on
paper.4
It is not clear what Schr¨dinger meant by ”desolation argument”. It is possible that he is
o
referring to the problem of constructing a cosmological model within the context of New-
tonian gravitation.
Under these conditions it is also not without interest to remark that the com-
pletely analogous system of solutions exists in its original form without the terms
added by Herr Einstein l.c.. The difference is superficial and slight: The poten-
tials remain unchanged, only the energy tensor of matter takes a different form.
2
3. This refers to the transfer of λgµν to the right hand side.
The system of gµν is also [compare Einstein l.c. equations (7), (8) and (12)]:
g44 = 1, g14 = g24 = g34 = 0 (1)
xµ xν
gµν = − δµν + 2
R − x2 − x2 − x2
1 2 3
0 =
µ, ν = 1, 2, 3, δµν = if ν µ.
1 =
If one now assumes a mixed energy tensor of matter with vanishing off-diagonal
components5 1
T1 0 0 0
2
0 T2 0 0
µ
Tν =
0 0 T3 0 3
(2)
4
0 0 0 T4
the field equations becomet
∂ µ ν µ α ν β
− +
∂xα α β α
2
√ √
∂ log −g µ ν ∂ log −g 1
− − = −κ(Tµν − 2 gµν T ) (3)
∂µ ∂ν α ∂xα
and are satisfied only if
1 2 3 4 1
T1 ≡ T2 ≡ T3 ≡ T4 ≡ ≡ const (4)
κR2
The l.h.s. is just Rµν . The Christoffel symbols were written this way at that time.
The calculation does not present any difficulty; it proceeds exactly as with Einstein
l.c. Most importantly, from a logical point of view, one might remark: k and R
must naturally be specified to be universal constants to start with, i.e., before one
writes potentials (1) and the field equations (3) respectively.
Because the calculation can be carried out easily only for points with the
property x1 = x2 = x3 = 0, condition (4) is satisfied only for those points identical
in x4 ! For the field equations to be satisfied identically it is necessary, permissible
and, sufficient to require, also, the identical validity of (4) for the spatial
3
4. coordinates. In this way these relations become invariant with respect to arbitrary
transformations of the spatial coordinates x1 , x2 , x3 among themselves and the time
x4 into itself. Through simple transformations of this kind one can move any
arbitrary point into one of the two “x4 , axes”.
It is desirable to connect the system of solutions with a reasonable physical
example. This is possible to a certain degree if one accepts Einstein’s hypothesis for
the energy tensor of a continuous compressible fluid. In this case we have6
ν ν dxα dxν
Tµ = −δµ p + gµα ρ
ds ds
ν 0 =
δµ = if ν µ.
1 =
where p are ρ scalars, the “pressure”and density of the fluid. If one sets
dx1 dx2 dx3 dx4
≡ ≡ ≡ 0, ≡1 (5)
ds ds ds ds
which is, because of (1), the basic equation
ds2 = gµν dxµ dxν (6)
and with the equations of motion (that is, a geodesic)
d2 xτ µ ν dxµ dxν
+ =0 (7)
ds2 τ ds ds
we obtain
−p 0 0 0
ν
0 −p 0 0
Tµ =
0
(8)
0 −p 0
0 0 0 p−ρ
In transposing λ to the r.h.s., λ is re-identified as a negative pressure.
This scheme differs (with suitable values for ρ and p) only superficially from our
hypotheses above [(2) and (4)] for the energy tensor.
These hypotheses see matter in the large essentially as a compressible fluid of
constant density at rest under a constant spatially isotropic tension which according
to (4) must be equal to 1/3 of the rest density of energy.7
From the point of view of the old theory, which for Einstein plays the role of a first
approximation,
4
5. What is meant here is not clear. Possibly it refers to the field equations without the λ
term.
It is not at all disturbing that a tension of this sign exists in a matter thought to be
present and uniformly spread over immense spaces and acting on itself under
Newton’s law of force. On the other hand, on closer scrutiny, the following fact is
astonishing: Satisfaction of the 10th field equation (equation [3], µ = ν = 4) is due
to the fact that the expression
4 1 2 3
T4 − T1 − T2 − T3 (9)
vanishes. It now plays, in first approximation, the role of the Newtonian gravitating
mass density, for which as is well known, only the 1Oth field equation needs to be
considered.
To obtain the Newtonian limit of the field equations, only the 4-4 equation is required.
The vanishing of the “mass-term”(10) seemed to me, initially, to be questionable as
to the utility of the given system of solutions as a representation of matter in the
large. But now I find just this result rather satisfactory Firstly, one has to expect, if
not demand, from a theory in which the motion of mass is relative, i.e. determined
only through the interaction of masses, that in any case the total mass of the
universe vanishes.
That the mass-term in our case vanishes not only in toto but at every discrete point
is a consequence of the fiction that matter is exactly at rest and completely,
uniformly spatially distributed. It seems to me completely in the spirit of the
relativity of mass to assume that the interaction function that we call inertial or
gravitational mass appears or comes about only by deviations from the uniform
static distribution. No harm is done when the uniform static distribution in which
the mass can neither act inertial nor gravitating, yields a vanishing mass-term.
Naturally, there is now the task to obtain the real empirical processes for individual
concrete cases, so to say, by variation of this very uniform “inert”integral system.
Then - and only then I feel general relativity would fulfill those postulates that
were for its origin the logical reasons.7 By the way, a reminder, this is also the value
of the light pressure; but the sign is opposite!
5
6. According to equation (4) the quantity 1/κ has the following physical meanings
1 1 2π 2 R3 T4
4
1 E
a) = 2
= 2
κ 6π R 6π R
1 1 2 1 1
b) = · 4πR T1 = p
κ 4π 4π
where E is the total energy of the cosmos, R (as above) the radius, p the one-sided
2
tension in an equatorial plane. For the last results (with κ = 8πk g −1 cm,
c2
k 2 = 6.68 · 10−8 gm−1 cm3 sec−2 ; c = 3 · 1010 cm sec−1;
4π c2
p = = 2 g cm−1 that means g cm light sec−2
κ 2κ
c4
= g cm sec−2 = 6.06 · 1018 dyne .
2κ2
1
A. Einstein, “Kosmologische Betrachttungen zur allgemeinen Relativit¨tstheorie”,
a
Sitzungsberichte der Preussischen Akademie der Wissenschaften 142-150 (1917),
Berlin. In what follows it will be cited by l.c.
2
Perhaps it is correct to say: “constructed”, (instead of fulfilled)
3
A. Einstein, “Die Grundlage der Allgemeine Relativit¨tstheorie”, Annalen der
a
Physik , 49, 769 (1916). An English translation appears in The Collected Papers of
Albert Einstein, vol. 6, A. Engel, translator, E.L.Schucking, consultant, 146-200,
(1999) Princeton University Press, Princeton, NJ.
4
Compare especially to Einstein, l.c., p. 147 [The ds Sitter citation is obscure. It
may refer to Koninklike Akademie van Wettenschappen, 1217-1224 (1917) which
contains comments on Einstein’s Kosmologische Betrachtungen.”
5
If one does not make this assumption, it will appear as a requirement after Eq. (4).
6
See A. Einstein, Die Grundlage ... p. 52. [This reference is unclear. The pagination
of Einstein’s article is 769-822.]
7
This is just the negative of the pressure of light.
8
I thank Herr Flamm for many discussions to clarify these concepts.
9
Compare this with A. Einstein, Die Grundlage ... , Sect. (2).
3 Einstein’s Response
Comment on Schr¨dinger’s Note “Concerning a System of Solutions of
o
6
7. the Generally Covariant Equations for Gravitation”
Einstein begins with
“When I wrote my description of the cosmic gravitational field I
naturally noticed, as the obvious possibility, the variant Herr
Schr¨dinger had discussed. But I must confess that I did not consider
o
this interpretation worthy of mention.”
Later, he finishes with
“The author [Schr¨dinger] is silent about the law according to which p
o
should be determined as a function of the coordinates. We will consider
only two possibilities:
1. p is a universal constant. In this case Herr Schr¨dingers model
o
completely agrees with mine. In order to see this, one merely needs to
exchange the letter p with the letter λ and bring the corresponding term
over to the left-hand side of the field equations. Therefore, this is not
the case the author could have had in mind.
2. p is a variable. Then a differential equation is required which
determines p as a function of x1 . . . x4. This means, one not only has
to start out from the hypothesis of the existence of a nonobservable
negative density in interstellar spaces but also has to postulate a
hypothetical law about the space-time distribution of this mass density.
The course taken by Herr Schr¨dinger does not appear possible to me
o
because it leads too deeply into the thicket of hypotheses.”
Remarkably, in the second possibility, Einstein described not only the central
problem of the search for dark energy but the headaches in formulating its
structure. Of course, this occurred long before the advent of quantum-field theoretic
concerns about the 120 orders of magnitude disparity between the observed vale of
the cosmological constant and the value of the zero-point energy as well as the later
discovery of the Sn(1a) supernovae with their implications, so the discussion
vanished into the archives. Although the exchange disappeared from sight, the
cosmological constant did not. It was used by many cosmologists for various
purposes. (See Harvey [5] for examples, discussion, and an extensive bibliography.)
4 Dark Energy
In the late 1990’s, Einstein’s concern about a “thicket of hypotheses”became real.
Observations on Sn(1a) supernovae by Reiss et al [6] and Perlmutter et al [7]
7
8. indicated that the expansion of the universe was accelerating. This acceleration may
be explained in either of the 2 ways that parallel Einstein’s comments to
Schr¨dinger: the cosmological constant is either a differential-geometric term in the
o
field equations or a manifestation of a “field”driving the acceleration. In the former,
it is simply an observable constant; in the latter it must be extracted from
Einstein’s “... thicket of hypothesis”. This field is called “quintessence”.
Quintessence is a dynamic scalar field with potential energy tuned to provided
appropriate dark energy at different epochs of cosmic evolution. 1 The choice
between regarding the cosmological constant as either a term in the field equations
or a manifestation of dark energy is exacerbated by the circumstance that evidence
for the existence of the latter has yet to be found.
5 Epilogue
Although Schr¨dinger never again discussed the cosmological constant Einstein did.
o
Going from the cosmological to the infinitesimal he utilized it in an attempt to cope
with what was at the time a very important problem. Recall that in 1918 the only
elementary particles known were the electron and the proton. Physicists were
attempting to understand why these were stable despite their internal
electromagnetic repulsion. Most attempts were based solely on electromagnetic
theory. For a review of these efforts see Pauli [12]. Einstein’s effort was to construct
a model in which stability was achieved through the use of gravitational forces. In
particular, he used modified gravitational field equations which included the
cosmological constant [13]. The attempt was not successful and this was the last
time he mentioned the cosmological constant other than to denounce it.
6 Acknowledgment
My deep thanks to Prof. Engelbert Schucking for suggesting this project as well as
providing a translation of Schr¨dinger’s paper.
o
a)
Prof. Emeritus, Queens College, City University of New York; ah30@nyu.edu.
1
See Zlatev [8] or Linder [9] for discussions of specific models. For a broad discussion see Carroll
[10]. For a history of the cosmological constant see See Straumann [11].
8
9. References
[1] A Einstein, “Kosmologische Betrachtungen zur allgemeinen
Relativit¨tstheorie”, Sitzungsberichte der Preussischen Akademie der
a
Wissenschaften 170 (1917), Berlin. An English translation appears in collection
of papers The Principle of Relativity, Dover Publications, New York (1923).
o ¨
[2] E Schr¨dinger, “Uber ein L¨sungssystem der allgemein kovarianten
o
Gravitationsgleichungen”, Physikalische Zeitschrift, 19, 20-22 (1918). The
original article in German is available at the Hathi Trust website.
o ¨
[3] A. Einstein, “Bemerkung zu Herrn Schr¨dingers Notiz Uber ein L¨sungssystem
o
der allgemein kovarianten Gravitationsgleichungen”, Physikalische Zeitschrift,
ad XIX, (1918), 165-166. The English translation appears in The Collected
Papers of Albert Einstein, Volume 7: The Berlin Years: Writings, 1918-1921 ,
ed. M. Jansen et al , Princeton University Press, Princeton, New Jersey (2002),
document 3.
[4] J. Mehra and H. Reichenberg, “The Historical Development of Quantum
Theory - Volume 5 - Erwin Schr¨dinger and the rise of Quantum Mechanics -
o
Part 1 - Schr¨dinger in Vienna and Zurich 1887-1925”, Springer-Verlag, New
o
York (1987).
[5] A. Harvey, “Cosmological Models”, American Journal of Physics, 51, 901-906
(1993).
[6] A. G. Riess et al, ‘Observational Evidence from Supernovae for an Accelerating
Universe and a Cosmological Constant”, arXiv: astro-ph/9805201v1.
[7] S. Perlmutter et al, “Measurements of Lambda and Omega from 42
High-Redshift Supernovae”, arXiv: astro-ph/9812133v1.
[8] I. Zlatev, L. Wang, and P. J. Steinhardt, “Quintessence, Cosmic Coincidence
and the Cosmological Constant, arXiv:astro-ph/9807002v2.
[9] E. V. Linder and D. Huterer, “How Many Dark Energy Parameters“,
arXiv:astro-ph/0505330v1.
[10] S. M. Carroll, “The Cosmological Constant”Living Reviews in Relativity, The
Max Planck Institute for Gravitational Physics, Potsdam, Germany, February
7, 2001 provides a comprehensive review of the physics and cosmology of the
cosmological constant. It is obtainable at http://relativity.livingreviews.org/.
This website is a rich source of comprehensive reviews of many topics in
relativity theory.
9
10. [11] N. Straumann, “The History of the Cosmological Constant Problem”,
arXiv:gr-qc/020027v1.
[12] W. Pauli, Theory of Relativity, Pergamon Press, London (1958). See Part V,
p.184 ff.
[13] A. Einstein, “S-peilen Gravitationfelder in Aufbau der Elementarteilchen eine
Wesentliche Rolle”(Do gravitational fields play an essential role in the structure
of elementary particles), Sitzungsberichte der Preussischen Akademie der
Wissenschaften, (Math. Phys.), 349-356 (1919) Berlin.
10