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1 1 R r r r + R r r r 2 2 1 1 R r r r R r r r 2 2 1 1 R r r R r r r = R r r + Rr r r 2 2
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= R r r r Comparison with (5.34) leads us to conclude that R r r =
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Cartan s Structure Equations
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Similar calculations show that (check) R t t = R t t = R t r t r = R r r = R =
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a2 + k a2 a2 + k , a2 R r r =
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Quiz
Consider spherical polar coordinates, where r 2 sin2 d 2 . Calculate the Ricci rotation coef cients. 1.
r
ds 2 = dr 2 + r 2 d 2 +
(a) (b) (c) (d) 2.
1 r r sin2 r sin cos r sin2
(a) (b) (c) (d) 3.
is given by tan sin sin cot r r 2 sin cos
Applying the appropriate transformation matrix to the Ricci rotation coef cients, one nds that r is (a) r sin2 (b) r sin2 (c) cot r (d) r12 Consider the Rindler metric, ds 2 = u 2 dv 2 du 2 . One nds that nonzero Ricci rotation coef cients are 1 (a) v v v = u u v = u (b) The space is at, so all the Ricci rotation coef cients vanish. (c) u v v = v u v = u12 1 v u (d) uv = u vv =
Cartan s Structure Equations
In spherical polar coordinates, the commutation coef cients Cr , Cr , and C are 1 (a) Cr = Cr = r , C = 0 1 (b) Cr = Cr = r , C = tan 1 (c) Cr = Cr = r , C = cot r (d) Cr = Cr = r12 , C = cot r Consider the Tolman metric studied in Example 5-1. The Ricci rotation coef cient r is given by (a) R e (t,r ) R (b) R e (t,r ) R (c) e (t,r ) (t,r ) (d) e R
The G tt component of the Einstein tensor for the Tolman metric is given by (a) G tt = 0 (b) G tt = 1 R e2 2R + 2R + R 1 R 2 2 R R + 1 + R 2 (c) G tt = (d) G tt =
R 1 R2 1 R2
R e2 R e2
+ 2R + R 1 R 2 2 R R + 1 + R 2 + 2R + R 1 R 2 2 R R + 1 + R 2 2R
For the Robertson-Walker metric in Example 5-2, using a b = diag( 1, 1, 1, 1) to raise and lower indices, one nds that (a) = (b) (c) (d)
= = =
CHAPTER
The Einstein Field Equations
The physical principles that form the basis of Einstein s theory of gravitation have their roots in the famed Tower of Pisa experiments conducted by Galileo in the seventeenth century. Galileo did not actually drop balls from the famed leaning tower, but instead rolled them down the inclined planes. How the experiments were actually conducted is not of importance here; our concern is only with one fundamental fact they reveal that all bodies in a gravitational eld experience the same acceleration regardless of their mass or internal composition. It is this fundamental result that allows us to arrive at our rst principle of equivalence. In basic newtonian physics, a quantity called mass shows up in three basic equations those that describe inertial forces, the force on a body due to a gravitational potential, and the force that a body produces when it is the source of a gravitational eld. There is not really any a priori reason to assume that the mass that shows up in the equations describing each of these situations
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The Einstein Field Equations
is one and the same. However, we will show that Galileo s results prove that this is the case.
Equivalence of Mass in Newtonian Theory
In newtonian theory, there are three types of mass. The rst two types describe the response of a body to inertial and gravitational forces, while the third type is used to describe the gravitational eld that results when a given body acts as a source. More speci cally,
Inertial mass: The rst appearance of mass in an elementary physics course is in the famous equation F = ma. Inertial mass is a measure of the ability of a body to resist changes in motion. In the following, we denote inertial mass by m I . Passive gravitational mass: In newtonian theory, the force that a body feels due to a gravitational eld described by a potential is given by F = m . The mass m in this equation, which describes the reaction of a body to a given gravitational eld, is called passive gravitational mass. We denote it by m p . Active gravitational mass: This type of mass acts as the source of a gravitational eld.
It is not obvious a priori that these types of masses should be equivalent. We now proceed to demonstrate that they are. We begin by considering the motion of two bodies in a gravitational eld. Galileo showed that if we neglect air resistance, two bodies released simultaneously from a height h will reach the ground at the same time. In other words, all bodies in a given gravitational eld have the same acceleration. This is true regardless of their mass or internal compositions. We consider the motion of two bodies in a gravitational eld. The gravitational eld exerts a force on a body and so we can use Newton s second law to write F1 = m I a1 1 F2 = m I a2 2 Now the force on a body due to the gravitational eld can be written in terms of the potential using F = m , where in this case m is the passive gravitational
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