Harvey R. Brown
- Published in print:
- 2005
- Published Online:
- September 2006
- ISBN:
- 9780199275830
- eISBN:
- 9780191603914
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199275831.003.0009
- Subject:
- Philosophy, Philosophy of Science
In his 1923 book The Mathematical Theory of Relativity, Arthur Eddington distinguished between two chains of reasoning in general relativity. The first familiar one starts with the existence of the ...
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In his 1923 book The Mathematical Theory of Relativity, Arthur Eddington distinguished between two chains of reasoning in general relativity. The first familiar one starts with the existence of the four-dimensional space-time interval ds, whose meaning is the usual one associated with the readings of physical rods and clocks and possibly light rays. The other less familiar chain of reasoning ‘binds the physical manifestations of the energy tensor and the interval; it passes from matter as now defined by the energy-tensor to the interval regarded as the result of measurements made with this matter.’ This chapter takes up the challenge of outlining this second chain of reasoning. In developing this reasoning, it argues that the dynamical underpinning of relativistic kinematics that has been defended in this book is consistent with the structure and logic of GR.Less
In his 1923 book The Mathematical Theory of Relativity, Arthur Eddington distinguished between two chains of reasoning in general relativity. The first familiar one starts with the existence of the four-dimensional space-time interval ds, whose meaning is the usual one associated with the readings of physical rods and clocks and possibly light rays. The other less familiar chain of reasoning ‘binds the physical manifestations of the energy tensor and the interval; it passes from matter as now defined by the energy-tensor to the interval regarded as the result of measurements made with this matter.’ This chapter takes up the challenge of outlining this second chain of reasoning. In developing this reasoning, it argues that the dynamical underpinning of relativistic kinematics that has been defended in this book is consistent with the structure and logic of GR.
Ta-Pei Cheng
- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199573639
- eISBN:
- 9780191722448
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199573639.003.0004
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
After a review of the Newtonian theory of gravitation in terms of its potential function, we start the study of general relativity (GR) with the introduction of the equivalence principle (EP). The ...
More
After a review of the Newtonian theory of gravitation in terms of its potential function, we start the study of general relativity (GR) with the introduction of the equivalence principle (EP). The Weak EP (the equality of the gravitational and inertial masses) is extended by Einstein to the Strong EP, the equivalence between inertia and gravitation for all interactions. This implies the existence of “local inertial frames” at every spacetime point. In a sufficiently small region, the “local inertial observer” will not sense any gravity effect. The equivalence of acceleration and gravity means that GR (physics laws valid in all coordinate systems, including accelerating frames) must necessarily be a theory of gravitation. The strong EP is used to deduce the results of gravitational redshift and time dilation, as well as gravitational bending of a light ray.Less
After a review of the Newtonian theory of gravitation in terms of its potential function, we start the study of general relativity (GR) with the introduction of the equivalence principle (EP). The Weak EP (the equality of the gravitational and inertial masses) is extended by Einstein to the Strong EP, the equivalence between inertia and gravitation for all interactions. This implies the existence of “local inertial frames” at every spacetime point. In a sufficiently small region, the “local inertial observer” will not sense any gravity effect. The equivalence of acceleration and gravity means that GR (physics laws valid in all coordinate systems, including accelerating frames) must necessarily be a theory of gravitation. The strong EP is used to deduce the results of gravitational redshift and time dilation, as well as gravitational bending of a light ray.
Alexis G. Burgess and John P. Burgess
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691144016
- eISBN:
- 9781400838691
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691144016.003.0004
- Subject:
- Philosophy, General
This chapter examines how the notion of indeterminacy challenges the equivalence principle (“Saying something is true is equivalent to just saying it”). Indeterminacy involves cases where we have a ...
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This chapter examines how the notion of indeterminacy challenges the equivalence principle (“Saying something is true is equivalent to just saying it”). Indeterminacy involves cases where we have a question “Are things thus?” to which neither the affirmative nor the negative answer seems appropriate. If the question “Are things thus?” cannot be answered “Yes” or “No,” then it seems the corresponding declarative “Things are thus” cannot be called “true” or “false,” and so it seems we have a counterexample to the bivalence principle, according to which every proposition is either true or false. The chapter first considers two classes of examples, involving phenomena of presupposition and vagueness, before discussing various conceivable lines of response to the problems they raise. In particular, it looks at denial strategy, disqualification strategy, deviance strategy, doublespeak strategy, dependency strategy, and defeatism. Finally, it analyzes a third type of case, involving a purported special kind of relativity.Less
This chapter examines how the notion of indeterminacy challenges the equivalence principle (“Saying something is true is equivalent to just saying it”). Indeterminacy involves cases where we have a question “Are things thus?” to which neither the affirmative nor the negative answer seems appropriate. If the question “Are things thus?” cannot be answered “Yes” or “No,” then it seems the corresponding declarative “Things are thus” cannot be called “true” or “false,” and so it seems we have a counterexample to the bivalence principle, according to which every proposition is either true or false. The chapter first considers two classes of examples, involving phenomena of presupposition and vagueness, before discussing various conceivable lines of response to the problems they raise. In particular, it looks at denial strategy, disqualification strategy, deviance strategy, doublespeak strategy, dependency strategy, and defeatism. Finally, it analyzes a third type of case, involving a purported special kind of relativity.
Javier Díaz‐Giménez
- Published in print:
- 2001
- Published Online:
- November 2003
- ISBN:
- 9780199248278
- eISBN:
- 9780191596605
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199248273.003.0002
- Subject:
- Economics and Finance, Macro- and Monetary Economics
This is a brief introduction to dynamic programming and the method of using linear quadratic (LQ) approximations to the return function; the method is an approximation because it computes the ...
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This is a brief introduction to dynamic programming and the method of using linear quadratic (LQ) approximations to the return function; the method is an approximation because it computes the solution to a quadratic expansion of the utility function about the steady state or the stable growth path of model economies. The main purpose of the chapter is to review the theoretical basis for the LQ approximation and to illustrate its use with a detailed example (social planning). The author demonstrates that, using the LQ approximation approach and the certainty equivalence principle, solving for the value function is a relatively easy task. The different sections of the chapter describe the standard neoclassical growth model, present a social planner problem that can be used to solve the model, give a recursive formulation of the social planner's problem, and describe an LQ approximation to this problem. Exercises are included throughout, and an appendix presents a MATLAB program to illustrate the LQ method.Less
This is a brief introduction to dynamic programming and the method of using linear quadratic (LQ) approximations to the return function; the method is an approximation because it computes the solution to a quadratic expansion of the utility function about the steady state or the stable growth path of model economies. The main purpose of the chapter is to review the theoretical basis for the LQ approximation and to illustrate its use with a detailed example (social planning). The author demonstrates that, using the LQ approximation approach and the certainty equivalence principle, solving for the value function is a relatively easy task. The different sections of the chapter describe the standard neoclassical growth model, present a social planner problem that can be used to solve the model, give a recursive formulation of the social planner's problem, and describe an LQ approximation to this problem. Exercises are included throughout, and an appendix presents a MATLAB program to illustrate the LQ method.
Ta-Pei Cheng
- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199573639
- eISBN:
- 9780191722448
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199573639.001.0001
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
Einstein's general theory of relativity is introduced in this advanced undergraduate and beginning graduate level textbook. Topics include special relativity, the principle of equivalence, Riemannian ...
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Einstein's general theory of relativity is introduced in this advanced undergraduate and beginning graduate level textbook. Topics include special relativity, the principle of equivalence, Riemannian geometry and tensor analysis, Einstein field equation, as well as many modern cosmological subjects: from primordial inflation, cosmic microwave anisotropy to the dark energy that propels an accelerating universe. The subjects are presented with an emphasis on physical examples and simple applications. One first learns how to describe curved spacetime. At this mathematically more accessible level, the reader can already study the many interesting phenomena such as gravitational lensing, black holes, and cosmology. The full tensor formulation is presented later, when the Einstein equation is solved for a few symmetric cases. Mathematical accessibility, together with the various pedagogical devices (e.g., worked-out solutions of chapter-end problems), make it practical for interested readers to use the book to study general relativity and cosmology on their own. In this new edition of the book, presentations on special relativity and black holes are augmented by new chapters. Other parts of the book are updated to include new observation tests of general relativity (e.g., the double pular system) and more recent evidence for dark matter and dark energy.Less
Einstein's general theory of relativity is introduced in this advanced undergraduate and beginning graduate level textbook. Topics include special relativity, the principle of equivalence, Riemannian geometry and tensor analysis, Einstein field equation, as well as many modern cosmological subjects: from primordial inflation, cosmic microwave anisotropy to the dark energy that propels an accelerating universe. The subjects are presented with an emphasis on physical examples and simple applications. One first learns how to describe curved spacetime. At this mathematically more accessible level, the reader can already study the many interesting phenomena such as gravitational lensing, black holes, and cosmology. The full tensor formulation is presented later, when the Einstein equation is solved for a few symmetric cases. Mathematical accessibility, together with the various pedagogical devices (e.g., worked-out solutions of chapter-end problems), make it practical for interested readers to use the book to study general relativity and cosmology on their own. In this new edition of the book, presentations on special relativity and black holes are augmented by new chapters. Other parts of the book are updated to include new observation tests of general relativity (e.g., the double pular system) and more recent evidence for dark matter and dark energy.
C. Julian Chen
- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780199211500
- eISBN:
- 9780191705991
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199211500.003.0005
- Subject:
- Physics, Condensed Matter Physics / Materials
This chapter presents a unified theory of tunneling phenomenon and covalent bond force, as a result of the similarity between the Bardeen theory of tunneling and the Herring-Landau theory of the ...
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This chapter presents a unified theory of tunneling phenomenon and covalent bond force, as a result of the similarity between the Bardeen theory of tunneling and the Herring-Landau theory of the covalent bond. Three general theoretical treatments are presented, which show that tunneling conductance is proportional to the square of the covalent bond interaction energy, or equivalently, the square of covalent bond force. The constant of proportionality is related to the electronic properties of the materials. For the case of a metal tip and a metal sample, an explicit equation contains only measurable physical quantities is derived. Several experimental verifications are presented. The equivalence of covalent bond energy and tunneling conductance provides a theoretical explanation of the threshold resistance observed in atom-manipulation experiments, and points to a method of predicting the threshold resistance for atom manipulation.Less
This chapter presents a unified theory of tunneling phenomenon and covalent bond force, as a result of the similarity between the Bardeen theory of tunneling and the Herring-Landau theory of the covalent bond. Three general theoretical treatments are presented, which show that tunneling conductance is proportional to the square of the covalent bond interaction energy, or equivalently, the square of covalent bond force. The constant of proportionality is related to the electronic properties of the materials. For the case of a metal tip and a metal sample, an explicit equation contains only measurable physical quantities is derived. Several experimental verifications are presented. The equivalence of covalent bond energy and tunneling conductance provides a theoretical explanation of the threshold resistance observed in atom-manipulation experiments, and points to a method of predicting the threshold resistance for atom manipulation.
A. J. Leggett
- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780199211241
- eISBN:
- 9780191706837
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199211241.003.0003
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter starts by emphasizing that we can construct a picture of the universe as a whole only by extrapolating the laws of physics as we know them on earth to conditions almost unimaginably ...
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This chapter starts by emphasizing that we can construct a picture of the universe as a whole only by extrapolating the laws of physics as we know them on earth to conditions almost unimaginably different from those prevailing here. It then reviews the information we can get from optical and other observations, and the picture of the current universe and its contents which emerges from it, with a discussion inter alia of the cosmic distance scale and the cosmological red shift. The ideas of special and general relativity are sketched, and some consequences such as gravitational radiation and black holes are discussed; possible futures of the universe are considered. Finally, it is indicated how extrapolation of the equations of general relativity into the past leads inexorably to the idea of a “hot big bang”.Less
This chapter starts by emphasizing that we can construct a picture of the universe as a whole only by extrapolating the laws of physics as we know them on earth to conditions almost unimaginably different from those prevailing here. It then reviews the information we can get from optical and other observations, and the picture of the current universe and its contents which emerges from it, with a discussion inter alia of the cosmic distance scale and the cosmological red shift. The ideas of special and general relativity are sketched, and some consequences such as gravitational radiation and black holes are discussed; possible futures of the universe are considered. Finally, it is indicated how extrapolation of the equations of general relativity into the past leads inexorably to the idea of a “hot big bang”.
Alexis G. Burgess and John P. Burgess
- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691144016
- eISBN:
- 9781400838691
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691144016.003.0008
- Subject:
- Philosophy, General
This chapter examines the more purely philosophical aspect of the question of the paradoxes of truth. It first considers a particular paradoxical derivation that uses the equivalence principle not in ...
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This chapter examines the more purely philosophical aspect of the question of the paradoxes of truth. It first considers a particular paradoxical derivation that uses the equivalence principle not in the form of the T-biconditionals (which immediately raise questions about what kind of conditional is involved), but of rules of T-introduction and T-elimination. It then explains the concept of revenge as well as logical and contextualist “solutions” and the so-called “paraconsistency.” It concludes with a discussion of inconsistency theories such as defeatism and deflationism. The background assumptions about meaning behind an integrated deflationist/defeatist theory might run as follows. Meaning can be given by rules, but rules can be inconsistent. There is even a result in mathematical logic (Church's theorem) to the effect that there is no mechanical test for inconsistency of rules, making it unlikely we have any filter preventing us from ever internalizing inconsistencies.Less
This chapter examines the more purely philosophical aspect of the question of the paradoxes of truth. It first considers a particular paradoxical derivation that uses the equivalence principle not in the form of the T-biconditionals (which immediately raise questions about what kind of conditional is involved), but of rules of T-introduction and T-elimination. It then explains the concept of revenge as well as logical and contextualist “solutions” and the so-called “paraconsistency.” It concludes with a discussion of inconsistency theories such as defeatism and deflationism. The background assumptions about meaning behind an integrated deflationist/defeatist theory might run as follows. Meaning can be given by rules, but rules can be inconsistent. There is even a result in mathematical logic (Church's theorem) to the effect that there is no mechanical test for inconsistency of rules, making it unlikely we have any filter preventing us from ever internalizing inconsistencies.
Ta-Pei Cheng
- Published in print:
- 2015
- Published Online:
- August 2015
- ISBN:
- 9780199693405
- eISBN:
- 9780191803130
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199693405.001.0001
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This book introduces Einstein’s general theory of relativity. Topics include the geometric formulation of special relativity, the principle of equivalence, the Einstein field equation and its ...
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This book introduces Einstein’s general theory of relativity. Topics include the geometric formulation of special relativity, the principle of equivalence, the Einstein field equation and its spherical solution, and black holes, as well as cosmology. The subject is presented with an emphasis on physical examples and simple applications without the full tensor apparatus (although, for those wishing to have a glimpse at the proper tensor formulation of Einstein’s field equation, this is presented in a final chapter). The reader first learns the physics of the equivalence principle and how it inspired Einstein’s idea of curved spacetime as the gravitational field. At the mathematically more accessible level of a metric description of a warped space, the reader can already study many interesting phenomena, such as gravitational time dilation, GPS operation, light deflection, precession of Mercury’s perihelion, and black holes. Many modern topics in cosmology are discussed: from primordial inflation and the cosmic microwave background, to the dark energy that propels an accelerating universe.Less
This book introduces Einstein’s general theory of relativity. Topics include the geometric formulation of special relativity, the principle of equivalence, the Einstein field equation and its spherical solution, and black holes, as well as cosmology. The subject is presented with an emphasis on physical examples and simple applications without the full tensor apparatus (although, for those wishing to have a glimpse at the proper tensor formulation of Einstein’s field equation, this is presented in a final chapter). The reader first learns the physics of the equivalence principle and how it inspired Einstein’s idea of curved spacetime as the gravitational field. At the mathematically more accessible level of a metric description of a warped space, the reader can already study many interesting phenomena, such as gravitational time dilation, GPS operation, light deflection, precession of Mercury’s perihelion, and black holes. Many modern topics in cosmology are discussed: from primordial inflation and the cosmic microwave background, to the dark energy that propels an accelerating universe.
Hanoch Gutfreund and Jürgen Renn
- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691175812
- eISBN:
- 9781400865765
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175812.003.0002
- Subject:
- Physics, History of Physics
This section discusses the development of Albert Einstein's ideas and attitudes as he struggled for eight years to come up with a general theory of relativity that would meet the physical and ...
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This section discusses the development of Albert Einstein's ideas and attitudes as he struggled for eight years to come up with a general theory of relativity that would meet the physical and mathematical requirements laid down at the outset. It first considers Einstein's work on gravitation in Prague before analyzing three documents that played a significant role in his search for a theory of general relativity: the Zurich Notebook, the Einstein–Grossmann Entwurf paper, and the Einstein–Besso manuscript. It then looks at Einstein's completion of his general theory of relativity in Berlin in November 1915, along with his development of a new theory of gravitation within the framework of the special theory of relativity. It also examines the formulation of the basic idea that Einstein termed the “equivalence principle,” his Entwurf theory vs. David Hilbert's theory, and the 1916 manuscript of Einstein's work on the general theory of relativity.Less
This section discusses the development of Albert Einstein's ideas and attitudes as he struggled for eight years to come up with a general theory of relativity that would meet the physical and mathematical requirements laid down at the outset. It first considers Einstein's work on gravitation in Prague before analyzing three documents that played a significant role in his search for a theory of general relativity: the Zurich Notebook, the Einstein–Grossmann Entwurf paper, and the Einstein–Besso manuscript. It then looks at Einstein's completion of his general theory of relativity in Berlin in November 1915, along with his development of a new theory of gravitation within the framework of the special theory of relativity. It also examines the formulation of the basic idea that Einstein termed the “equivalence principle,” his Entwurf theory vs. David Hilbert's theory, and the 1916 manuscript of Einstein's work on the general theory of relativity.
Steven Carlip
- Published in print:
- 2019
- Published Online:
- March 2019
- ISBN:
- 9780198822158
- eISBN:
- 9780191861215
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198822158.003.0001
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics
General relativity describes gravity as a byproduct of the geometry of spacetime. This chapter explains why this notion makes sense, introducing the principle of equivalence and the universality of ...
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General relativity describes gravity as a byproduct of the geometry of spacetime. This chapter explains why this notion makes sense, introducing the principle of equivalence and the universality of gravitational interactions.Less
General relativity describes gravity as a byproduct of the geometry of spacetime. This chapter explains why this notion makes sense, introducing the principle of equivalence and the universality of gravitational interactions.
Ta-Pei Cheng
- Published in print:
- 2015
- Published Online:
- August 2015
- ISBN:
- 9780199693405
- eISBN:
- 9780191803130
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199693405.003.0004
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
After a review of the Newtonian theory of gravitation in terms of its potential function, this chapter starts the study of general relativity (GR) with the introduction of the equivalence principle ...
More
After a review of the Newtonian theory of gravitation in terms of its potential function, this chapter starts the study of general relativity (GR) with the introduction of the equivalence principle (EP). The weak EP (equality of gravitational and inertial masses) was extended by Einstein to the strong EP (equivalence between inertia and gravitation for all interactions). This implies the existence of “local inertial frames” at every spacetime point. In a sufficiently small region, a “local inertial observer” will not sense any gravitational effect. The equivalence of acceleration and gravity means that GR (with physics laws valid in all coordinate systems, including accelerating frames) must necessarily be a theory of gravitation. The strong EP is used to deduce results for gravitational redshift and time dilation, as well as gravitational bending of a light ray. The operation of GPS is shown to depend crucially on relativistic time dilation effects.Less
After a review of the Newtonian theory of gravitation in terms of its potential function, this chapter starts the study of general relativity (GR) with the introduction of the equivalence principle (EP). The weak EP (equality of gravitational and inertial masses) was extended by Einstein to the strong EP (equivalence between inertia and gravitation for all interactions). This implies the existence of “local inertial frames” at every spacetime point. In a sufficiently small region, a “local inertial observer” will not sense any gravitational effect. The equivalence of acceleration and gravity means that GR (with physics laws valid in all coordinate systems, including accelerating frames) must necessarily be a theory of gravitation. The strong EP is used to deduce results for gravitational redshift and time dilation, as well as gravitational bending of a light ray. The operation of GPS is shown to depend crucially on relativistic time dilation effects.
Ta-Pei Cheng
- Published in print:
- 2013
- Published Online:
- May 2013
- ISBN:
- 9780199669912
- eISBN:
- 9780191744488
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199669912.003.0012
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
In general relativity (GR) physics laws are not changed under general coordinate transformations. Among Einstein’s motivation for GR was his wish to have a deeper understanding of the empirically ...
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In general relativity (GR) physics laws are not changed under general coordinate transformations. Among Einstein’s motivation for GR was his wish to have a deeper understanding of the empirically observed equality between gravitational and inertial masses. Gravity naturally enters into the theory when accelerated frames are considered. The generalization of this equivalence principle to electromagnetism implies gravitational redshift, gravitational time dilation, and gravitational bending of a light ray. Most importantly, such considerations led Einstein to the idea that the gravitational effect on a body can be attributed directly to some underlying spacetime feature and the gravitational field is simply curved spacetime. We present some elements of Riemannian geometry (the mathematics of curved space): Gaussian coordinates metric tensor, the geodesic equation, and curvature.Less
In general relativity (GR) physics laws are not changed under general coordinate transformations. Among Einstein’s motivation for GR was his wish to have a deeper understanding of the empirically observed equality between gravitational and inertial masses. Gravity naturally enters into the theory when accelerated frames are considered. The generalization of this equivalence principle to electromagnetism implies gravitational redshift, gravitational time dilation, and gravitational bending of a light ray. Most importantly, such considerations led Einstein to the idea that the gravitational effect on a body can be attributed directly to some underlying spacetime feature and the gravitational field is simply curved spacetime. We present some elements of Riemannian geometry (the mathematics of curved space): Gaussian coordinates metric tensor, the geodesic equation, and curvature.
Massimiliano Di Ventra
- Published in print:
- 2022
- Published Online:
- March 2022
- ISBN:
- 9780192845320
- eISBN:
- 9780191937521
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780192845320.003.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This Chapter shows that any physical system performs some type of computation, with the observers providing meaning (semantics) to such a computation. It then introduces an equivalence principle ...
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This Chapter shows that any physical system performs some type of computation, with the observers providing meaning (semantics) to such a computation. It then introduces an equivalence principle between physical (semantic) information and computation. Finally, it discusses the general criteria shared by all MemComputing machines.Less
This Chapter shows that any physical system performs some type of computation, with the observers providing meaning (semantics) to such a computation. It then introduces an equivalence principle between physical (semantic) information and computation. Finally, it discusses the general criteria shared by all MemComputing machines.
David M. Wittman
- Published in print:
- 2018
- Published Online:
- August 2018
- ISBN:
- 9780199658633
- eISBN:
- 9780191863653
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780199658633.003.0013
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics
The equivalence principle is an important thinking tool to bootstrap our thinking from the inertial coordinate systems of special relativity to the more complex coordinate systems that must be used ...
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The equivalence principle is an important thinking tool to bootstrap our thinking from the inertial coordinate systems of special relativity to the more complex coordinate systems that must be used in the presence of gravity (general relativity). The equivalence principle posits that at a given event gravity accelerates everything equally, so gravity is equivalent to an accelerating coordinate system.This conjecture is well supported by precise experiments, so we explore the consequences in depth: gravity curves the trajectory of light as it does other projectiles; the effects of gravity disappear in a freely falling laboratory; and gravitymakes time runmore slowly in the basement than in the attic—a gravitational form of time dilation. We show how this is observable via gravitational redshift. Subsequent chapters will build on this to show how the spacetime metric varies with location.Less
The equivalence principle is an important thinking tool to bootstrap our thinking from the inertial coordinate systems of special relativity to the more complex coordinate systems that must be used in the presence of gravity (general relativity). The equivalence principle posits that at a given event gravity accelerates everything equally, so gravity is equivalent to an accelerating coordinate system.This conjecture is well supported by precise experiments, so we explore the consequences in depth: gravity curves the trajectory of light as it does other projectiles; the effects of gravity disappear in a freely falling laboratory; and gravitymakes time runmore slowly in the basement than in the attic—a gravitational form of time dilation. We show how this is observable via gravitational redshift. Subsequent chapters will build on this to show how the spacetime metric varies with location.
Andrew M. Steane
- Published in print:
- 2021
- Published Online:
- December 2021
- ISBN:
- 9780192895646
- eISBN:
- 9780191943911
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780192895646.003.0002
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics
This chapter is a survey of central ideas and equations in general relativity. The basic equations are written down with a view to seeing where we are heading in the book, and in order to present ...
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This chapter is a survey of central ideas and equations in general relativity. The basic equations are written down with a view to seeing where we are heading in the book, and in order to present both the field theory and the geometric interpretation of gravity. The central role of the metric is introduced, and the equivalence principle is discussed. It is emphasized that spacetime interval is both a mathematical and a physical idea. It is explained how gravity works “behind the scenes” by modifying equations which otherwise look like familiar equations of electromagnetism. The sense in which acceleration is in some respects a relative and in some respects an absolute concept is explained. It is shown why the motion of matter, not just its mass, must influence gravitation. The stress-energy tensor is introduced and defined.Less
This chapter is a survey of central ideas and equations in general relativity. The basic equations are written down with a view to seeing where we are heading in the book, and in order to present both the field theory and the geometric interpretation of gravity. The central role of the metric is introduced, and the equivalence principle is discussed. It is emphasized that spacetime interval is both a mathematical and a physical idea. It is explained how gravity works “behind the scenes” by modifying equations which otherwise look like familiar equations of electromagnetism. The sense in which acceleration is in some respects a relative and in some respects an absolute concept is explained. It is shown why the motion of matter, not just its mass, must influence gravitation. The stress-energy tensor is introduced and defined.
Hanoch Gutfreund and Jürgen Renn
- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691174631
- eISBN:
- 9781400888689
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691174631.003.0022
- Subject:
- Physics, History of Physics
This chapter attempts to formulate a consistent extension of the theory of general relativity. The starting point of the general theory of relativity is the recognition of the unity of gravitation ...
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This chapter attempts to formulate a consistent extension of the theory of general relativity. The starting point of the general theory of relativity is the recognition of the unity of gravitation and inertia (principle of equivalence). From this principle, it follows that the properties of “empty space” were to be represented by a symmetrical tensor expressed in the theory. The principle of equivalence, however, does not give any clue as to what may be the more comprehensive mathematical structure on which to base the treatment of the total field comprising the entire physical reality. As such, this chapter considers the problem of how to find a field structure which is a natural generalization of the symmetrical tensor as well as a system of field equations for this structure which represent a natural generalization of certain equations of pure gravitation.Less
This chapter attempts to formulate a consistent extension of the theory of general relativity. The starting point of the general theory of relativity is the recognition of the unity of gravitation and inertia (principle of equivalence). From this principle, it follows that the properties of “empty space” were to be represented by a symmetrical tensor expressed in the theory. The principle of equivalence, however, does not give any clue as to what may be the more comprehensive mathematical structure on which to base the treatment of the total field comprising the entire physical reality. As such, this chapter considers the problem of how to find a field structure which is a natural generalization of the symmetrical tensor as well as a system of field equations for this structure which represent a natural generalization of certain equations of pure gravitation.
David D. Nolte
- Published in print:
- 2018
- Published Online:
- August 2018
- ISBN:
- 9780198805847
- eISBN:
- 9780191843808
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198805847.003.0007
- Subject:
- Physics, History of Physics
This chapter describes how gravity provided the backdrop for one of the most important paradigm shifts in the history of physics. Prior to Albert Einstein’s general theory of relativity, trajectories ...
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This chapter describes how gravity provided the backdrop for one of the most important paradigm shifts in the history of physics. Prior to Albert Einstein’s general theory of relativity, trajectories were paths described by geometry. After the theory of general relativity, trajectories are paths caused by geometry. This chapter explains how Einstein arrived at his theory of gravity, relying on the space-time geometry of Hermann Minkowski, whose work he had originally harshly criticized. The confirmation of Einstein’s theory was one of the dramatic high points in twentieth-century history of physics when Arthur Eddington journeyed to an island off the coast of Africa to observe stellar deflections during a solar eclipse. If Galileo was the first rock star of physics, then Einstein was the first worldwide rock star of science.Less
This chapter describes how gravity provided the backdrop for one of the most important paradigm shifts in the history of physics. Prior to Albert Einstein’s general theory of relativity, trajectories were paths described by geometry. After the theory of general relativity, trajectories are paths caused by geometry. This chapter explains how Einstein arrived at his theory of gravity, relying on the space-time geometry of Hermann Minkowski, whose work he had originally harshly criticized. The confirmation of Einstein’s theory was one of the dramatic high points in twentieth-century history of physics when Arthur Eddington journeyed to an island off the coast of Africa to observe stellar deflections during a solar eclipse. If Galileo was the first rock star of physics, then Einstein was the first worldwide rock star of science.
Marcus Klamert and Bernhard Schima
- Published in print:
- 2019
- Published Online:
- March 2021
- ISBN:
- 9780198794561
- eISBN:
- 9780191927874
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198759393.003.24
- Subject:
- Law, EU Law
Member States shall provide remedies sufficient to ensure effective legal protection in the fields covered by Union law.
Member States shall provide remedies sufficient to ensure effective legal protection in the fields covered by Union law.
Kartik B. Athreya
- Published in print:
- 2013
- Published Online:
- May 2014
- ISBN:
- 9780262019736
- eISBN:
- 9780262314404
- Item type:
- chapter
- Publisher:
- The MIT Press
- DOI:
- 10.7551/mitpress/9780262019736.003.0003
- Subject:
- Economics and Finance, Macro- and Monetary Economics
This chapter describes two of the three most influential findings macroeconomics has available to it. One is known as the First Fundamental Theorem of Welfare Economics, or the “invisible-hand” ...
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This chapter describes two of the three most influential findings macroeconomics has available to it. One is known as the First Fundamental Theorem of Welfare Economics, or the “invisible-hand” theorem. It tells us that under some conditions, competitive market outcomes deliver stable non-wasteful outcomes. This result holds even if consumers care only about the goods and services they consume, and know nothing about the others, or the world, beyond the prices they face. Next, the chapter describes work showing that in general, there will be prices that, all by themselves, can guide self-interested buyers and sellers to such orderly outcomes. But who cares that this can happen? In practice, will it? To answer this, we discuss some reasons to believe in the relevance of Walrasian Equilibrium for the real-world.Less
This chapter describes two of the three most influential findings macroeconomics has available to it. One is known as the First Fundamental Theorem of Welfare Economics, or the “invisible-hand” theorem. It tells us that under some conditions, competitive market outcomes deliver stable non-wasteful outcomes. This result holds even if consumers care only about the goods and services they consume, and know nothing about the others, or the world, beyond the prices they face. Next, the chapter describes work showing that in general, there will be prices that, all by themselves, can guide self-interested buyers and sellers to such orderly outcomes. But who cares that this can happen? In practice, will it? To answer this, we discuss some reasons to believe in the relevance of Walrasian Equilibrium for the real-world.