Lecture note 10/22
Dipsikha Debnath
October 30, 2014
Alternative theories of gravity
U µ ∇µ U µ = lRµν U µ + l2 U µ ∇µ RU ν
Now we do dimensional analysis of the above equation U = dimension less
[l] = L
[∇µ ] = L−1
[R] = L−2
In the second term in right hand side of the above equation we introduce the term l2 to match
dimension.
Action can be written as
R
√
S = d4 x −g((1 + αφ2 )R + Lφ )
L = Lagrangian density for scalar field.
Lagrangian density = f (λ)R + h(λ)g µν ∇µ λ∇ν λ − U (λ)
We define conformal metric as
g̃µν = 16π G̃f (λ)gµν
We define a new scalar field
φ=
where K =
R
K 1/2 dλ
f h + 3h2
16π G̃f 2
so, in terms of φ the action becomes S=
R
√
R̃
1
d4 x −g̃(
− g̃ ρσ ∇ρ φ∇σ φ − V (φ))
16π G̃ 2
U (λ)
V (φ) =
[16π G̃f ]2
f (λ) is well behaved and (g̃µν , φ) can be used instead of (gµν , λ)
Now we include matter in action, in scalar-tensor theories of gravity the action can be written
as S=
R
√
R̃
1
d4 x −g̃(
− g̃ ρσ ∇ρ φ∇σ φ − V (φ) + H(φ)LˆM )
16π G̃ 2
LˆM is for matter dependence.
Extra Dimension
In (4 + d) dimensional spacetime with coordinates X a and metric Gab , where a,b run from
0 to d + 3
ds2 = Gab X a X b = gµν dxµ dxν + b2 (x)γij (y)dy i dy j
where xµ are coordinates in 4 dimensional space time.
y i are coordinates on extra dimensional manifold.
The action is the (4 + d) dimensional action plus a matter term
S=
where
√
R
√
R[Gab ]
+ LˆM )
d4+d x −G(
16πG4+d
−G = square root of minus the determinant of Gab .
and R[Gab ] = R[gµν ] + b−2 R[γij ] − 2db−1 g µν ∇µ ∇ν b − d(d − 1)b−2 g µν ∇µ b∇ν b .
∇µ = covariant derivative in 4 dimensional metric gµν
.
R
√
The volume element V is defined as V = dd y γ.
The 4 dimensional Newton’s constant G4 is related to (4 + d) dimensional analogue by
1
V
=
16πG4
16πG4+d
Now we introduce κ =
R[γij ]
, then the action becomes d(d − 1)
√
d4 x −g(
1
(bd R[gµν ] + d(d − 1)bd−2 g µν ∇µ b∇ν b + d(d − 1)κbd−2 ) + Vbd LˆM )
16πG4
r
d(d + 2)
here scalar field is coupled to gravity , φ =
ln b
16πG
S=
R
Higher order in curvature action
Here Lagrangians contain more than second order in derivatives of the metric.The action
p
R
S = dn x ( − g)(R + α1 R2 + α2 Rµν Rµν + α3 g µν ∇µ R∇ν R + ....
Here α ’s are coupling factors.
So including higher order derivatives in the theory leads to infinite answer but using higher
order terms in Lagrangian we can renormalize the thory but as a consequence we encounter
ghosts (negative energy field excitation).