A simple action that one may introduce to describe gravity involves the scalar curvatureR.x/, given in (1.1.52), defined by
W D 1 2
Z .dx/p
g.x/R.x/; R.x/ D R.x/g.x/; g.x/ D detŒg.x/ ; (1.2.1) where we have necessarily introduced the invariant volume elementp
g.x/.dx/ in (1.1.26) to ensure invariance under general coordinate transformations. The Ricci tensorRis given in (1.1.50).2is a parameter introduced for dimensional reasons.
The action in (1.2.1) is referred to the Einstein-Hilbert action. The theory based on this action will be compared to that of Newton’s theory of gravitation.
Under an infinitesimal transformation22•g
•.p
g gR/ D p
g.R1
2gR/ •g; (1.2.2)
22See Problem1.9.
leading to the field equation R1
2gRD0; r R1
2gR
D0; r R1
2gR D0;
(1.2.3) whereR 12gR G is the Einstein tensor in (1.1.54). The second equality in the above equation was derived in (1.1.53), and the third one follows from the symmetry ofR,g in their indices.
Partial integration reads slightly in a different manner in curved spacetime. In view of applications of partial integrations we note, for example, that for a scalar field.x/, and a vector fieldS.x/one has
pg.x/
r.x/
S.x/ D p
g.x/ .x/rS.x/; (1.2.4) up to ordinary total derivatives, wherer denotes the covariant derivative intro- duced in the previous section. For a vector fieldS.x/, and a tensor fieldF.x/,
pg.x/F.x/rS.x/D p g.x/
rF.x/
S.x/; (1.2.5) up to ordinary total derivatives. The presence of thep
g.x/factor should be noted in these two equations. For proofs of (1.2.4), (1.2.5), with further generalizations, see Problem1.10.
From (1.2.2), (1.2.5), (1.1.53), in particular, we learn that the action W is invariant under the gauge transformations
•g.x/ D .r.x/C r.x//; (1.2.6) where the minus sign is introduced for convenience, for given vector fields.x/, as a consequence of the fact that
pg.R1
2gR/ •g D p
gh
r.R1
2gR/Cr.R1 2gR/i
D0; (1.2.7) up to ordinary total derivatives in its integrand.
Also the equality g•g D g•g, gives the following equivalent infinitesimal gauge transformations for the metricg,
•g.x/ D
r.x/C r.x/
: (1.2.8)
with an overall plus sign for the variation of the metric relative to its inverse matrix in (1.2.6).
For future reference, the above transformation in (1.2.8) may be rewritten in an equivalent form as follows. To this end, we may write, in detail,
•g.x/ D gr.x/C gr.x/ Dg@.x/Cg@C@gC
gCg@g ; (1.2.9) where in the writing the second line, we have used the definition of the covariant derivative of a vector field, and then added and subtracted the term@g @g. We recognize the last term on the right-hand side of (1.2.9) asrg which is zero. Hence (1.2.8) may be equivalently rewritten simply as
•g.x/ D g@Cg@.x/C@g: (1.2.10) At this stage, it is important to bring the above theory, based on the Einstein- Hilbert action, in contact with that of Newton’s theory of gravitation. The potential energy'.X/, in the latter, due to a mass distribution with mass density.X/is given by
'.X/ D G Z
d3X0 .X0/
jXX0j; (1.2.11)
where G GN is Newton’s gravitational constant. The potential energy'.X/
satisfies the equation23
@j@j'.X/D4 G.X/: (1.2.12)
Now we introduce a sourceS term to the field equation in (1.2.3) and write, more generally,
R1
2gR D S: (1.2.13)
We may rewrite this as
R D S1
2gS
; (1.2.14)
whereSDS. In particular
R00D S001
2g00S
: (1.2.15)
23We have used the familiar relation@j@j
1=jXX0j
D 4 ı3.XX0/.
From (1.1.50),R00may be written in terms of components of the Riemann tensor as R00DR00 0 0CRi0i0 D Ri0 0i; (1.2.16) where, in writing the last equality, we have, from (1.1.42), used the facts that R00 D0,Ri0j0D Ri00j.
We further consider the equation of geodesic deviation equation in (1.1.38) for Di, which may be rewritten as
D2•xi
d2 D Ri0 0j.x/ •xjdx0 d
dx0
d C ; (1.2.17)
while the Newtonian one is given by d2•Xi
dt2 D 1 c2
@i@j'.X/
•XjdX0 d
dX0
d : (1.2.18)
Working in units, in which c D 1, we may, upon comparison of these two equations and making use, in the process, of (1.2.12) and (1.2.16), that we may takeR00 4 GT00, where we have identified the mass density withT00 of the energy-momentum tensor associated with matter. We may identifyS00 in (1.2.15) with4 GT00 by noting that withg00 1,T D T00CTi i T00, withTi i
negligible in comparison toT00, for low speeds of the underlying matter particles, we obtain for the right-hand side of (1.2.15) the expression8 G.T00=2/. Hence as a tensor equation, the field equation (1.2.14) is taken as
R D8 G T1
2gT
; (1.2.19)
or equivalently as
R1
2gR D 8 GT; (1.2.20)
which is the celebrated Einstein’s equation of general relativity.
The above doesn’t yet tell us how to generate the energy-momentum tensor and, in turn, define the parameter in (1.2.1). To this end, we introduce a Lagrangian density for matter, and generalize the expression of the action in (1.2.1) to
W D 1 2
Z .dx/p
g.x/R.x/ C Z
.dx/p
g.x/Lmatter.x/: (1.2.21) In order to obtain (1.2.20), we define the energy-momentum tensor of matter by
T.x/ D p2 g.x/
•
•g.x/ Z
.dx0/p
g.x0/Lmatter.x0/; (1.2.22)
which leads from (1.2.1), (1.2.2), (1.2.21) to a parametergiven by
2 D 16 G: (1.2.23)
in units„ D1;cD1. Re-inserting„and c, the fundamental Planck length is given byp
„G=c3 1:6161033cm, with GGND6:7091039„c5=GeV2. For example, for a scalar field.x/, we may take for the action for matter the integral
WmatterD 1 2
Z .dx/p
g.g@ @Cm22/: (1.2.24) The corresponding energy-momentum tensor is derived in Problem1.11.
Immediate generalizations of the Einstein-Hilbert Action in (1.2.1), are obtained by adding such terms to it as
Z p
g.dx/ ; Z p
g.dx/R2; Z p
g.dx/RR; : : : ; (1.2.25) up to proportionality constants, whereis referred to as a cosmological constant.
The remaining actions are of higher orders in derivatives than the Einstein-Hilbert action and are more problematic in conventional field theory studies.24 We will consider such terms in Sect.1.8.