PHY423B, Summer '99

A. CONTENTS

1. Historical Introduction to Special Relativity

2. Lorentz Transformations

3. Minkowski Diagrams, Space-Time Intervals

4. Relativistic Kinematics and Einsteinian Optics

5. Four-Tensors and the Lorentz Group

6. Covariance of Electrodynamics

7. Relativistic Mechanics

8. Introduction to General Relativity

9. Introduction to General Relativity

10. Vacuum Field Equations and Schwarzschild's Solution

11. Classical Tests of General Relativity

12. Einstein's Field Equations

13. Weak-Field Approximation and Gravitational Waves

B. THE SKILLS

Historical Introduction to Special Relativity

Knowledge

• Give a synopsis of the historical development of special relativity from Newton to Einstein.
• Describe each of the following and state its significance for the theory of relativity: (a) Michelson-Morley experiment, (b) Stellar aberration, (c) Fizeau drag effect.
• Compare the Newtonian and Einsteinian conceptions of (a) the character of space and time, (b) the relativity principle.
• State the postulates of special relativity.

Lorentz Transformations

Knowledge

• Define simultaneity (as in special relativity). Show that simultaneity depends on the motion of the observer.
• Outline the argument for establishing the form of the Lorentz transformation for a boost along a coordinate axis.
• Discuss in relation to relativity: (a) group property of Lorentz transformations; (b) uniqueness of the invariant speed c; (c) limiting nature of the speed c.
• State the Lorentz transformation for a boost along any coordinate axis in two forms (Lorentz factor γ, rapidity φ) and the relationship between the forms.

Rule Application

• Given the coordinates of an event in one inertial frame, find its coordinates in another. Given the coordinates of an event in two inertial frames, find the relative velocity of the frames.

Problem Solving

• Given information about the relative velocities of a series of inertial frames find the Lorentz transformation from any one of the frames to another.

Minkowski Diagrams, Space-Time Intervals

Knowledge

• State the definition of the squared interval between two events. Define the three kinds of intervals. Identify the kind(s) of interval-connecting cause-and-effect-related events. Show that the squared interval is invariant under standard Lorentz transformations.
• Draw the Minkowski diagram representing the boost connecting two inertial frames. Establish: (a) the relative orientation of the coordinate axes representing the two frames, (b) the relationship between the scales along the coordinate axes representing the two frames. Indicate: (a) an event's coordinates in the two frames, (b) possible world-lines for particles and for light-signals.
• Derive the Lorentz length-contraction. State the length hypothesis. Derive the Einstein time-dilation. State the clock hypothesis.

Problem Solving

• Transform time and space separations of two events from one inertial frame to another. Given the length and orientation of a rod in one frame, find its length and orientation in another.
• Given information about two inertial frames, find appropriate information about events as seen from each frame, e.g. event times, event intervals, and the light travel-time between the two origins.

Relativistic Kinematics and Einsteinian Optics

Knowledge

• Derive the velocity and acceleration transformation equations for a standard Lorentz transformation.
• Derive the correct relativistic expressions for (a) the Fizeau drag effect, (b) the Doppler effect, (c) stellar aberration.

Rule Application

• Given information about the velocity and acceleration (magnitudes and directions) of a body in one inertial frame, find the corresponding quantities in another frame.
• Given the wavelength of light a source emits at rest find its observed wavelength when it is in motion relative to some observer.

Four-Tensors and the Lorentz Group

Knowledge

• Define general Lorentz transformations in terms of the invariance of the squared interval. Express general Lorentz transformations using index notation (including the summation convention) and matrices. Show that the transformation matrix must satisfy: a^T g a = g; a^-1 = g a^T g.
• Concerning four-tensors: (a) Define four-tensors in terms of the transformation properties of their components using contravariant or covariant indices. (b) Demonstrate the use of the metric tensor in raising or lowering indices. (c) Construct new tensors from given tensors by addition, multiplication, contraction and differentiation. (d) Show that the four-gradient or "box" operator behaves as a covariant vector while the D'Alembertian behaves as a scalar.
• Define the velocity and acceleration four-vectors. Use their transformation properties to establish the transformation properties of the corresponding three-vectors.

Covariance of Electrodynamics

Knowledge

• Starting with Maxwell's equations: (a) define the scalar and vector potentials, and derive the wave equations for these potentials, (b) define the 4-vector potential, and justify the 4-vector character of each.
• (a) Define the electromagnetic field tensor and express it in terms of the electric and magnetic fields. (b) Express Maxwell's equations in a manifestly covariant form. (c) Derive and explain the significance of the continuity equation. (d) Express the Lorentz condition in a manifestly covariant form.
• Use the transformation character of the electromagnetic field tensor to determine the transformation rules for the electric and magnetic fields in special cases.
• Express the Lorentz force law in a manifestly covariant form and thereby obtain the transformation character of a 3-force.

Relativistic Mechanics

Knowledge

• Give a plausibility argument (including a definition of 4-momentum) for the relativistic form of Newton's 2nd law. Base the argument on the classical form of the 2nd law, the special relativity postulate, and the 3-force transformation law.
• (a) Derive the form of the Minkowski 4-force using plausibility arguments. (b) Determine the transformation equations for the 3-force. (c) Express the relativistic form of Newton's 2nd law in terms of longitudinal and transverse masses for the appropriate special cases.
• (a) Given an interpretation of the time component of the momentum 4-vector in terms of the particle's energy. (b) Establish conservation of 4-momentum for a closed system.

Problem Solving

• Given information about the particles involved in a collision or decay use conservation of 4-momentum and the center of mass frame to determine quantities (momenta, energies, velocities) related to the process.

Introduction to General Relativity

Knowledge

• Define or explain each of the following: (a) primary motivation which led Einstein to the general theory of relativity, (b) inertial mass and gravitational mass, (c) active and passive gravitational mass, (d) weak, semi-strong and strong equivalence principles, (e) local inertial frame, and the dependence of a frame's extension upon the desired degree of accuracy (f) special relativity as a local theory, (g) bending of light and the gravitational Doppler shift as qualitative consequences of the equivalence principle.
• Define or explain: (a) Riemannian space and metric, (b) indefinite metric and signature, (c) geodesics, (d) geodesic separation (define and derive formula for), (e) curvature (for 2-dim and n-dim surfaces), (f) geodesic deviation, (g) isometric spaces.
• Outline the basic scheme of general relativity with special emphasis on the roles played by the equivalence principle, by geodesics and by masses.
• Derive expressions for (a) the gravitational Doppler shift (and time dilation), (b) the spacetime metric around a spherical mass.

Introduction to General Relativity

Knowledge

• Define or explain each of the following: (a) primary motivation which led Einstein to the general theory of relativity, (b) inertial mass and gravitational mass, (c) active and passive gravitational mass, (d) weak, semi-strong and strong equivalence principles, (e) local inertial frame, and the dependence of a frame's extension upon the desired degree of accuracy (f) special relativity as a local theory, (g) bending of light and the gravitational Doppler shift as qualitative consequences of the equivalence principle.
• Define or explain: (a) Riemannian space and metric, (b) indefinite metric and signature, (c) geodesics, (d) geodesic separation (define and derive formula for), (e) curvature (for 2-dim and n-dim surfaces), (f) geodesic deviation, (g) isometric spaces.
• Outline the basic scheme of general relativity with special emphasis on the roles played by the equivalence principle, by geodesics and by masses.
• Derive expressions for (a) the gravitational Doppler shift (and time dilation), (b) the spacetime metric around a spherical mass.

Vacuum Field Equations and Schwarzschild's Solution

Knowledge

• Give a plausibility argument (as in the Procedures) for the form of the vacuum field equations of general relativity, including a definition of the Ricci tensor and a statement of its symmetry property.
• Establish the form of the Schwarzschild metric by: (a) writing down and justifying the general form of the metric exterior to a spherically symmetric static mass distribution subject to the constraints that (i) θ and φ are the usual spherical coordinate angles; (ii) r² is the square of the radial coordinate, equal to the proper area of a sphere concentric with the mass, divided by 4π; (iii) the metric is stationary, i.e. ∂ g_μν/∂ t; (iv) the coordinates are orthogonal, i.e. cross terms do not occur in the metric; (b) Given the non-zero components of the Ricci tensor for the general spherically symmetric static metric to be able to determine the unknown functions so that the metric is a solution (the Schwarzschild solution) of the vacuum field equations.
• Concerning the Schwarzschild metric: (a) determine the geodesic equations in terms of r, θ, φ, t given the non-zero Christoffel symbols; (b) determine the difference between radar distance and ruler distance along a radial line; (c) explain the significance or lack of significance of the Schwarzschild radius; (d) show that the proper time required for a particle to fall from a finite height to the origin is finite while the coordinate time is infinite; (e) determine the coordinate transformation from the curvature coordinates (or Schwarzschild coordinates) to isotropic coordinates and interpret the isotropic coordinates.

Classical Tests of General Relativity

Knowledge

• Derive the formulae for the three classical tests of general relativity using the Schwarzschild metric and summarize the observational evidence pertinent to each: (a) gravitational Doppler (or red) shift; (b) anomalous planetary perihelion advance; (c) gravitational deflection of light.

Problem Solving

• Solve problems of the type given in the Procedures.

Einstein's Field Equations

Knowledge

• Justify the form of the matter tensor M_μν for dust and verify its properties:
  M_μν = M_νμ;
  ∂_ν M^μν = 0.
• Describe briefly the general manner whereby the laws of physics (other than those dealing with gravitation) are usually generalized from special relativity to general relativity.
• Give a plausibility argument for Einstein's field equations.
• Summarize the fundamental ideas of general relativity in terms of the field equations, the geodesic equations and the equivalence principle.

Weak-Field Approximation and Gravitational Waves

Knowledge

• Identify a certain part of the metric tensor with the Newtonian potential by taking the weak-field approximation of the geodesic equations.
• Obtain in harmonic coordinates the linearized (weak-field) approximation of Einstein's field equations.
• Relate the constant K in Einstein's field equations with Newton's gravitational constant.
• Define or explain each of the following: (a) primary motivation which led Einstein to the general theory of relativity, (b) inertial mass and gravitational mass, (c) active and passive gravitational mass, (d) weak, semi-strong and strong equivalence principles, (e) local inertial frame, and the dependence of a frame's extension upon the desired degree of accuracy (f) special relativity as a local theory, (g) bending of light and the gravitational Doppler shift as qualitative consequences of the equivalence principle.
• Define or explain: (a) Riemannian space and metric, (b) indefinite metric and signature, (c) geodesics, (d) geodesic separation (define and derive formula for), (e) curvature (for 2-dim and n-dim surfaces), (f) geodesic deviation, (g) isometric spaces.
• Outline the basic scheme of general relativity with special emphasis on the roles played by the equivalence principle, by geodesics and by masses.
• Derive expressions for (a) the gravitational Doppler shift (and time dilation), (b) the spacetime metric around a spherical mass.