Introduction to supergravity 2015: Exercise 2.
Institute for Theoretical Physics, Masaryk University,
611 37 Brno, Czech Republic
Consider the following action of a scalar field coupled to gravitation
S = d4
x
√
−g −
M2
P
2
R −
1
2
∂mφ∂m
φ . (1)
1. Find the equations of motion for φ by extremizing the action
δS
δφ
= 0. (2)
Verify that they are
m
∂mφ = 0. (3)
2. Find the Einstein equations
Gmn =
1
M2
P
Tmn, (4)
by extremizing the action (1) with respect to the metric
δS
δgmn
= 0. (5)
Find Tmn and Gmn.
3. Using the properties of Rklmn calculate
m
Gmn. (6)
What does this result imply for m
Tmn on-shell?
4. Show that Minkowski space is a vacuum solution to the Einstein equations. In other words
that
gmn = ηmn, (7)
is a consistent background for the action (1).
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