140
9 Spherically Symmetric Gravitational Fields
In brief summary, solar system experiments and observations indicate that β can
differ from unity by less than about a part in 10
3 and γ can differ from unity by less
than about a part in 10
3 (Will 1993). General relativity is a well-tested theory.
Exercises
9.1 Verify the connections in (9.5) for the Schwarzschild metric in the form (9.4).
9.2 Verify the metric determinant function in (9.6).
9.3 Verify the field equations (9.9) and (9.10).
9.4 Check that the field equations R 22 = 0 and R 33 = 0 are satisfied by the
Schwarzschild solution.
9.5 Check that the off-diagonal terms of the Ricci tensor are zero for the
Schwarzschild form of metric (9.4). Thus the Einstein equations are entirely
satisfied by the Schwarzschild solution.
9.6 What is the Schwarzschild radius for the sun? For the earth? For a proton? For
a typical galaxy?
9.7 Show explicitly that the classical orbit given by (9.31) is an ellipse.
9.8 Calculate the numerical value for the precession of Mercury’s orbit using
(9.43) and data from an astronomy text. Also compare the eccentricity e of
Mercury’s orbit to that of the other planets.
9.9 Calculate the numerical value for the deflection of starlight just grazing the
sun from (9.57).
9.10 Consider the region r < 2m for the Schwarzschild metric. What is the sign
of the 0,0 component and what is the sign of the 1,1 component? Is it thus
clear that t cannot be interpreted as a time marker and r cannot be interpreted as
a radial marker in this region? What happens if we reverse the interpretation of
the two? There has been much written on appropriate coordinates in this region,
the simplest and best known being called the Kruskal-Zekeres coordinates
(Kruskal 1960; Adler 1975).
9.11 Do you foresee any problems in obtaining observational information from the
interior Schwarzschild region r < 2m? See Chap. 11 for further discussion of
this, and also (Adler 2005).
9.12 Use the references (Will 1993, 2014) and see how the three classic tests of
relativity depend on the Eddington parameters, and also how the Shapiro time
delay depends on the Eddington parameters.
9.13 Why is it that the Eddington expansion (9.60) includes quadratic terms in the
time part of the metric but only includes linear terms in the space part of the
metric?
9 Spherically Symmetric Gravitational Fields
In brief summary, solar system experiments and observations indicate that β can
differ from unity by less than about a part in 10
3 and γ can differ from unity by less
than about a part in 10
3 (Will 1993). General relativity is a well-tested theory.
Exercises
9.1 Verify the connections in (9.5) for the Schwarzschild metric in the form (9.4).
9.2 Verify the metric determinant function in (9.6).
9.3 Verify the field equations (9.9) and (9.10).
9.4 Check that the field equations R 22 = 0 and R 33 = 0 are satisfied by the
Schwarzschild solution.
9.5 Check that the off-diagonal terms of the Ricci tensor are zero for the
Schwarzschild form of metric (9.4). Thus the Einstein equations are entirely
satisfied by the Schwarzschild solution.
9.6 What is the Schwarzschild radius for the sun? For the earth? For a proton? For
a typical galaxy?
9.7 Show explicitly that the classical orbit given by (9.31) is an ellipse.
9.8 Calculate the numerical value for the precession of Mercury’s orbit using
(9.43) and data from an astronomy text. Also compare the eccentricity e of
Mercury’s orbit to that of the other planets.
9.9 Calculate the numerical value for the deflection of starlight just grazing the
sun from (9.57).
9.10 Consider the region r < 2m for the Schwarzschild metric. What is the sign
of the 0,0 component and what is the sign of the 1,1 component? Is it thus
clear that t cannot be interpreted as a time marker and r cannot be interpreted as
a radial marker in this region? What happens if we reverse the interpretation of
the two? There has been much written on appropriate coordinates in this region,
the simplest and best known being called the Kruskal-Zekeres coordinates
(Kruskal 1960; Adler 1975).
9.11 Do you foresee any problems in obtaining observational information from the
interior Schwarzschild region r < 2m? See Chap. 11 for further discussion of
this, and also (Adler 2005).
9.12 Use the references (Will 1993, 2014) and see how the three classic tests of
relativity depend on the Eddington parameters, and also how the Shapiro time
delay depends on the Eddington parameters.
9.13 Why is it that the Eddington expansion (9.60) includes quadratic terms in the
time part of the metric but only includes linear terms in the space part of the
metric?