Hints for the Goldstein problems:

 
7.20, 7.21: Use 4-vector relations. State your results for energies in terms of the total energy rather than the kinetic energy.

7.25: Integrate the equation of motion for in rectangular coordinates using the ordinary time. Use the results to find (known in terms of p and m) and . Do the general problem with a nonzero initial momentum perpendicular to the electric field.

7.26: Use the relativistic kinetic energy. The potential V = -k/r is the gravitational analog of an electromagnetic potential , and L = T - V. Use polar coordinates in the plane perpendicular to the angular momentum L (why is it conserved?), and integrate the orbit equation to relate the angle to the radius. The result from Prob. 3.14 will be useful: just identify the constants that appear here with those in 3.14 when the orbit equations are written in the same form. A rederivation of the final analytic result for the rate of precession is not needed, but work out the new number.

LD16: This problem demonstrates the difference between interactions involving scalar fields and those, such as the electromagnetic interaction, which involve 4-vector fields. The "scalar potential" of electromagnetic theory is actually the timelike component of the 4 vector A = ( A, ). As a result, the Hamiltonians for the two cases depend quite differently on or , as you will see.

		LD17(a) is a straightforward relativity problem, but is important in providing the connection between Newtonian gravity and general relativity. The mass m is included to make that connection in a familiar form. However, m is an overall factor in S, so drops out of the equations of motion. This disappearance of m is a consequence, at a deeper level, of Einstein's principle of equivalence. The result for the line element does not distinguish between massless and massive particles, so the equations for geodesic motion can be used for either kind of particle.
		LD17(b) combines ideas from throughout the course - geodesic motion, the orbit equation, turning points, and scattering - in the context of relativity. The methods needed are not new, but are simply used in a new setting. Recall the treatment of the scattering angle in Goldstein, Sec. 3.10, use the constants of the motion to obtain an expression for , and integrate to obtain the scattering angle. It is better to leave the orbit integral in terms of r, with the physical situation clear, than to make Goldstein's substitution u = 1/r. Your result for the deflection should justify the first-order treatment in GM/c2.
		The deflection of light passing the sun is one of the classic predictions and tests of general relativity. The use of starlight, as in the historic observations by Eddington during the solar eclipse of 1919, has been replaced by the deflection of signals from astronomical radio sources and the related measurement of the gravitational time delay in radar signals passing near the sun. The deflection and time delay are now very accurately measured, and agree with the theory. The gravitational deflection of light also leads to imaging of distant galaxies by foreground galaxies as predicted by Einstein. This "gravitational lensing" is a hot topic now.

Send comments or questions to: ldurand@hep.wisc.edu

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© 1997, Loyal Durand