*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/9780691174631.003.0014
- Subject:
- Physics, History of Physics

This chapter begins with an epistemological discussion of space and time and of geometry in relation to Einstein's formulation of relativity theory. It introduces the concept of invariants. These are ...
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This chapter begins with an epistemological discussion of space and time and of geometry in relation to Einstein's formulation of relativity theory. It introduces the concept of invariants. These are the only quantities to have an objective significance with respect to a specific geometry. The chapter then introduces the basics of tensor calculus. Tensor formulation of the laws of physics guarantees that they are covariant with respect to transformations, which represent the symmetry of space. The symmetry of space, namely, its isotropy and homogeneity, determines the form of physical laws. Einstein refers to this as “the principle of relativity with respect to direction.” The chapter then demonstrates how this principle is reflected in some of the equations of physics.Less

This chapter begins with an epistemological discussion of space and time and of geometry in relation to Einstein's formulation of relativity theory. It introduces the concept of invariants. These are the only quantities to have an objective significance with respect to a specific geometry. The chapter then introduces the basics of tensor calculus. Tensor formulation of the laws of physics guarantees that they are covariant with respect to transformations, which represent the symmetry of space. The symmetry of space, namely, its isotropy and homogeneity, determines the form of physical laws. Einstein refers to this as “the principle of relativity with respect to direction.” The chapter then demonstrates how this principle is reflected in some of the equations of physics.

*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/9780691174631.003.0015
- Subject:
- Physics, History of Physics

This chapter shows how the principle of special relativity and the principle of the constancy of the velocity of light uniquely determine the Lorentz transformation. Unlike in pre-relativity physics, ...
More

This chapter shows how the principle of special relativity and the principle of the constancy of the velocity of light uniquely determine the Lorentz transformation. Unlike in pre-relativity physics, space and time are not separate entities. They are combined into a four-dimensional spacetime continuum, which is most clearly demonstrated in the formulation of the theory of special relativity due to Hermann Minkowski. The chapter then defines vectors and tensors with respect to the Lorentz transformation, leading to a tensor formulation of Maxwell's equations, of the electromagnetic force acting on charges and currents, and of the energy-momentum of the electromagnetic field and its conservation law. It also introduces the energy-momentum tensor of matter and discusses the basic equations of the hydrodynamics of perfect fluids (the Euler equations).Less

This chapter shows how the principle of special relativity and the principle of the constancy of the velocity of light uniquely determine the Lorentz transformation. Unlike in pre-relativity physics, space and time are not separate entities. They are combined into a four-dimensional spacetime continuum, which is most clearly demonstrated in the formulation of the theory of special relativity due to Hermann Minkowski. The chapter then defines vectors and tensors with respect to the Lorentz transformation, leading to a tensor formulation of Maxwell's equations, of the electromagnetic force acting on charges and currents, and of the energy-momentum of the electromagnetic field and its conservation law. It also introduces the energy-momentum tensor of matter and discusses the basic equations of the hydrodynamics of perfect fluids (the Euler equations).

*Robert E. Newnham*

- Published in print:
- 2004
- Published Online:
- November 2020
- ISBN:
- 9780198520757
- eISBN:
- 9780191916601
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198520757.003.0031
- Subject:
- Earth Sciences and Geography, Geochemistry

In most dielectrics, the linear relation between electric polarization and applied electric field is accurately obeyed even for fairly large fields of 107 V/m. The reason is that the atomic ...
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In most dielectrics, the linear relation between electric polarization and applied electric field is accurately obeyed even for fairly large fields of 107 V/m. The reason is that the atomic displacements are extremely small, in the range of nuclear sizes—millions of times smaller than the size of atoms. Though nonlinear effects such as electrostriction have been known for some time, it was not until the invention of the laser that sufficiently large optical fields became available to produce sizeable nonlinear optical effects. The induced polarization P can be written as a power series in an electric field, … P = χE + dE2 +· · ·, … where χ is the linear electric susceptibility, and the higher-order terms lead to nonlinear effects such as second harmonic generation. The electric field associated with the incident light is sinusoidal, E = E0 sin ωt, and when E is substituted in the expression for P, a power series in sin ωt results. The second term is dE20 sin2 ωt = 1/2dE20 (1 − cos 2ωt), which includes a component of polarization with twice the frequency of the impressed field E. This rapidly oscillating induced dipole moment is the source of second harmonic light. The intensity of the light depends on the size of d, the second order coefficient. Crystal symmetry is a major factor in the second-order effect. The one-dimensional polar chain in Fig. 29.2 illustrates the origin of the quadratic term. When the applied field is directed to the left, the ions and bonding electrons are in very close contact and the displacements will be small because of short range repulsive forces. These forces do not oppose motion in the opposite direction, so that fields directed to the right give larger motions and larger polarizations. A centric chain does not show this effect. Such a chain can give rise to odd-order terms producing saturation but not to even power terms in the P(E) relation. This means that centric crystals are useless as second harmonic generators.
Less

In most dielectrics, the linear relation between electric polarization and applied electric field is accurately obeyed even for fairly large fields of 107 V/m. The reason is that the atomic displacements are extremely small, in the range of nuclear sizes—millions of times smaller than the size of atoms. Though nonlinear effects such as electrostriction have been known for some time, it was not until the invention of the laser that sufficiently large optical fields became available to produce sizeable nonlinear optical effects. The induced polarization P can be written as a power series in an electric field, … P = χE + dE2 +· · ·, … where χ is the linear electric susceptibility, and the higher-order terms lead to nonlinear effects such as second harmonic generation. The electric field associated with the incident light is sinusoidal, E = E0 sin ωt, and when E is substituted in the expression for P, a power series in sin ωt results. The second term is dE20 sin2 ωt = 1/2dE20 (1 − cos 2ωt), which includes a component of polarization with twice the frequency of the impressed field E. This rapidly oscillating induced dipole moment is the source of second harmonic light. The intensity of the light depends on the size of d, the second order coefficient. Crystal symmetry is a major factor in the second-order effect. The one-dimensional polar chain in Fig. 29.2 illustrates the origin of the quadratic term. When the applied field is directed to the left, the ions and bonding electrons are in very close contact and the displacements will be small because of short range repulsive forces. These forces do not oppose motion in the opposite direction, so that fields directed to the right give larger motions and larger polarizations. A centric chain does not show this effect. Such a chain can give rise to odd-order terms producing saturation but not to even power terms in the P(E) relation. This means that centric crystals are useless as second harmonic generators.