Pub Date : 2021-11-02DOI: 10.1093/oso/9780192895646.003.0027
A. Steane
Classical field theory, as it is applied to the most simple scalar, vector and spinor fields in flat spacetime, is described. The Klein-Gordan, Weyl and Dirac equations are obtained, and some features of their solutions are discussed. The Yukawa potential, the plane wave solutions, and the conserved currents are obtained. Spinors are introduced, both through physical pictures (flagpole and flag) and algebraic defintions (complex vectors). The relationship between spinors and four-vectors is given, and related to the Lie groups SU(2) and SO(3). The Dirac spinor is introduced.
{"title":"First steps in classical field theory","authors":"A. Steane","doi":"10.1093/oso/9780192895646.003.0027","DOIUrl":"https://doi.org/10.1093/oso/9780192895646.003.0027","url":null,"abstract":"Classical field theory, as it is applied to the most simple scalar, vector and spinor fields in flat spacetime, is described. The Klein-Gordan, Weyl and Dirac equations are obtained, and some features of their solutions are discussed. The Yukawa potential, the plane wave solutions, and the conserved currents are obtained. Spinors are introduced, both through physical pictures (flagpole and flag) and algebraic defintions (complex vectors). The relationship between spinors and four-vectors is given, and related to the Lie groups SU(2) and SO(3). The Dirac spinor is introduced.","PeriodicalId":365636,"journal":{"name":"Relativity Made Relatively Easy Volume 2","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127871194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-02DOI: 10.1093/oso/9780192895646.003.0022
A. Steane
The main features of the universe and its history, and the application of GR to the universe as a whole are presented. The observed near-isotropy and homogeneity of the universe are described, along with a survey of its history. The Saha equation is applied to the recombination process. Cosmic proper time and comoving coordinates are defined, and the form of the metric (Friedmann-Lemaitre-Robertson-Walker) applicable to such a universe is obtained. The main features of the resulting geometry are discussed at length, with a view to both accurate calculation and sound intuition. Redshift and the cosmic expansion are described from several perspectives. Distance measures (luminosity, angular diameter) are defined and the main elements of the observational cosmic distance ladder are outlined.
{"title":"Cosmology","authors":"A. Steane","doi":"10.1093/oso/9780192895646.003.0022","DOIUrl":"https://doi.org/10.1093/oso/9780192895646.003.0022","url":null,"abstract":"The main features of the universe and its history, and the application of GR to the universe as a whole are presented. The observed near-isotropy and homogeneity of the universe are described, along with a survey of its history. The Saha equation is applied to the recombination process. Cosmic proper time and comoving coordinates are defined, and the form of the metric (Friedmann-Lemaitre-Robertson-Walker) applicable to such a universe is obtained. The main features of the resulting geometry are discussed at length, with a view to both accurate calculation and sound intuition. Redshift and the cosmic expansion are described from several perspectives. Distance measures (luminosity, angular diameter) are defined and the main elements of the observational cosmic distance ladder are outlined.","PeriodicalId":365636,"journal":{"name":"Relativity Made Relatively Easy Volume 2","volume":"312 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134212371","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-02DOI: 10.1093/oso/9780192895646.003.0004
A. Steane
We imagine a group of people living on the inner surface of a huge rotating cylinder in flat spacetime. Their experiences are described and calculated. Thus we introduce gravimagnetic effects and the connection between gravitational time dilation and gravitational acceleration. Gravimagnetic effects such as the force on moving particles and the precession of gyroscopes are derived. The Thomas precession is obtained. These observations illustrate GR ideas that are applicable more generally. Some properties of the general stationary metric are introduced.
{"title":"Life in a rotating world","authors":"A. Steane","doi":"10.1093/oso/9780192895646.003.0004","DOIUrl":"https://doi.org/10.1093/oso/9780192895646.003.0004","url":null,"abstract":"We imagine a group of people living on the inner surface of a huge rotating cylinder in flat spacetime. Their experiences are described and calculated. Thus we introduce gravimagnetic effects and the connection between gravitational time dilation and gravitational acceleration. Gravimagnetic effects such as the force on moving particles and the precession of gyroscopes are derived. The Thomas precession is obtained. These observations illustrate GR ideas that are applicable more generally. Some properties of the general stationary metric are introduced.","PeriodicalId":365636,"journal":{"name":"Relativity Made Relatively Easy Volume 2","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125393283","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-02DOI: 10.1093/oso/9780192895646.003.0016
A. Steane
Various aspects of the Einstein field equation are presented. First the field equation is obtained by arguing that it is the simplest equation that respects the fundamental geometric insight into gravity. Then we consider whether the equation is stable, and introduce the weak energy and dominant energy conditions. The connection between inertial motion and the distant universe (Mach’s principle) is discussed. The equation of motion of matter is obtained from the field equation, and a comparison made with electromagnetic field theory. The energy and momentum of gravitational fields in stationary conditions is discussed, and the Komar energy obtained.
{"title":"The Einstein field equation","authors":"A. Steane","doi":"10.1093/oso/9780192895646.003.0016","DOIUrl":"https://doi.org/10.1093/oso/9780192895646.003.0016","url":null,"abstract":"Various aspects of the Einstein field equation are presented. First the field equation is obtained by arguing that it is the simplest equation that respects the fundamental geometric insight into gravity. Then we consider whether the equation is stable, and introduce the weak energy and dominant energy conditions. The connection between inertial motion and the distant universe (Mach’s principle) is discussed. The equation of motion of matter is obtained from the field equation, and a comparison made with electromagnetic field theory. The energy and momentum of gravitational fields in stationary conditions is discussed, and the Komar energy obtained.","PeriodicalId":365636,"journal":{"name":"Relativity Made Relatively Easy Volume 2","volume":"20 3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122318838","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-02DOI: 10.1093/oso/9780192895646.003.0006
A. Steane
The linearized theory is applied to sources such as ordinary stars whose speed is small compared to the speed of light. This yields the “gravitoelectromagnetic” theory. The gravitoelectromagnetic field equations are obtained, along with their general solution via scalar and vector potentials. It is shown how to calculate the metric perturbation, and hence the field, due to a rotating ring or a ball, and thus how to calculate orbits, timing, and the Lense-Thirring precession.
{"title":"Slow stationary sources","authors":"A. Steane","doi":"10.1093/oso/9780192895646.003.0006","DOIUrl":"https://doi.org/10.1093/oso/9780192895646.003.0006","url":null,"abstract":"The linearized theory is applied to sources such as ordinary stars whose speed is small compared to the speed of light. This yields the “gravitoelectromagnetic” theory. The gravitoelectromagnetic field equations are obtained, along with their general solution via scalar and vector potentials. It is shown how to calculate the metric perturbation, and hence the field, due to a rotating ring or a ball, and thus how to calculate orbits, timing, and the Lense-Thirring precession.","PeriodicalId":365636,"journal":{"name":"Relativity Made Relatively Easy Volume 2","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134526203","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-02DOI: 10.1093/oso/9780192895646.003.0017
A. Steane
The spherically symmetric vacuum solution to the Einstein field equation (Schwarzschild-Droste solution) is derived and associated physical phenomena derived and explained. It is shown how to obtain the Christoffel symbols by the Euler-Lagrange method, and hence the metric for the general spherically symmetric vacuum. Equations for general orbits are presented, and their solution for radial motion and for circular motion. Geodetic (de Sitter) precession is calculated exactly for circular orbits. The null geodesics (photon worldlines) are obtained, and the gravitational redshift. Emission from an accretion disc is calculated.
{"title":"Schwarzschild–Droste solution","authors":"A. Steane","doi":"10.1093/oso/9780192895646.003.0017","DOIUrl":"https://doi.org/10.1093/oso/9780192895646.003.0017","url":null,"abstract":"The spherically symmetric vacuum solution to the Einstein field equation (Schwarzschild-Droste solution) is derived and associated physical phenomena derived and explained. It is shown how to obtain the Christoffel symbols by the Euler-Lagrange method, and hence the metric for the general spherically symmetric vacuum. Equations for general orbits are presented, and their solution for radial motion and for circular motion. Geodetic (de Sitter) precession is calculated exactly for circular orbits. The null geodesics (photon worldlines) are obtained, and the gravitational redshift. Emission from an accretion disc is calculated.","PeriodicalId":365636,"journal":{"name":"Relativity Made Relatively Easy Volume 2","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129759908","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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