gravity physics general relativity

1. Short Write Wrap GR1
SW2GR1
Set A
Roa, F. J. P.
A bit of General Relativity (GR) in the form of an exercise concerning Einstein-Hilbert
action (EH) that excludes as yet the cosmological and matter terms. The EH action is
simply given by
(1)
∫ −= RgxdS HE
4
as prescribed in 3 + 1 dimensions of spacetime, where g is the determinant of the metric
tensor νµg and R is the Ricci scalar.
In Einstein’s GR, gravity is thought not as a force that acts instantaneously at a distance
but rather the effects that manifest as gravity are due to the geometry of a given
spacetime. This is the part where the metric tensor comes in and in a small enough region
of spacetime, the said geometry starts out with a fundamental line element
(2)
νµ
νµ dxdxgSd =2
where the components νµg of the metric tensor play dynamical roles.
Gravity is manifested by particle’s motion along geometric paths that satisfy the geodesic
equation which can be derived from the given fundamental line element. We can express
the fundamental line element (2) in terms of an appropriate parametric variable then
extremize the resulting expression to get the geodesic equation
(3)
02
2
=Γ+
τττ
ωµ
ρ
ωµ
ρ
d
xd
d
xd
d
xd
This equation is dynamically governed by the geometry defined by the metric tensor via
the connection
(4)
( )ωµνµνωωνµ
νρρ
ωµ gggg ∂−∂+∂=Γ
2
1

2. This is where gravity affects on the motions of particles through spacetime. In turn, the
metric tensor must satisfy gravity’s own equations of motion (the field equations in GR)
that are derived from EH action, say for instance, from action (1)
(5)
0
2
=− vuvu g
R
R
In this particular exercise (pp. 438 [1] ), we are asked to derive an effective action of the
form
(6)
( )∫ ΓΓ−ΓΓ−= β
βσ
σ
µν
σ
λν
λ
µσ
νµ
ggxdS ffeHE
4
)(
from action (1). Then alternatively, proceeding from this action, we are to derive
Einstein’s field equations (5).
We are to highlight the major details involved in the process of this exercise. Ofcourse,
we should start with action (1) from which we take note of the Ricci scalar R as the inner
product of the inverse metric µν
g and the Ricci tensor νµR
(7)
µν
νµ
RgR =
The Ricci tensor is obtained from the Riemann curvature tensor
(8)
ρ
µλ
λ
σν
ρ
σλ
λ
µν
ρ
µνσ
ρ
σνµ
ρ
µσν ΓΓ−ΓΓ+Γ∂+Γ∂−=R
by one-index contraction
(9)
σ
µσνµν RR =
Basically, the Riemann tensor is associated with parallel transporting a vector ρ
V and in
general curved spacetime, the commutator-covariant differentiation is non vanishing

3. (10)
[ ] 0, ≠∇∇ ρ
νµ V
which simply implies that spacetime is curved as quantified by the Riemann tensor.
Moving forward, let’s write
(11)
σ
µλ
λ
σν
νµ
σ
σλ
λ
µν
νµσ
µνσ
νµσ
σνµ
νµ
ΓΓ−
−ΓΓ−+Γ∂−+Γ∂−−=−
gg
ggggggRg
To specify, the connections involved here are torsion-free, those connections derived
from the metric tensor as given by (4).
Much of our effort would be in re-writing
(12)
σ
µνσ
νµσ
σνµ
νµ
Γ∂−+Γ∂−− gggg
into an appropriate form that would enable us to split up action (1) into two major terms.
To do so, we would be utilizing some of the useful results we have learned from our basic
course involving matrices. One of these is that given a matrix, say with original
components νµg as that of the metric tensor, we can write then each component of the
inverse matrix in terms of the determinant of the original matrix and this determinant’s
partial derivative with respect to a particular component νµg . This, provided that the
original matrix )( νµg is non-singular that is, with a non-zero (non-vanishing)
determinant.
So we can write each inverse component as
(13)
νµ
νµ
g
g
g
g
∂
−∂
−
=
)(
)(
1
From (4) we form the contraction involving this connection,

4. (14)
σων
ωσσ
σν gg ∂=Γ
2
1
Then use (13) in (14) to write this as
(15)
σων
ωσ
σ
σν g
g
g
g
∂
∂
−∂
−
=Γ
)(
)(2
1
and by chain-rule, we can re-write this compactly as
(16)
g−∂=Γ lnν
σ
σν
Following this, we also have
(17)
gggg −∂=Γ− ν
νµσ
σν
νµ
A connection derived from the metric such as (4) is metric compatible and such
connection imlies
(18.1)
0=∇ ωνµ g
and so consequently,
(18.2)
0=∇ ων
µ g
These will enable us to write
(18.3)
σ
ωµσν
σ
νµωσωνµ Γ+Γ=∂ ggg

5. and conversely,
(18.4)
ω
σµ
σνν
σµ
ωσων
µ Γ−Γ−=∂ ggg
Let us take
(18.5)
σ
σν
νµ
µ
σ
σνµ
νµσ
σν
νµ
µ Γ−∂+Γ∂−=Γ−∂ ))(()( gggggg
and find a way to re-write the second major term of the rhs using those results of the
preceding pages.
We can use (17) and (18.4) as well to express
(18.6)
ν
λµ
λµνµ
µ Γ−−=−∂ gggg )(
and substitute this in (18.5) to write for
(18.7)
σ
σν
ν
λµ
λµσ
σν
νµ
µ
σ
σνµ
νµ
ΓΓ−+Γ−∂=Γ∂− gggggg )(
As counterpart to (18.5), we also take
(18.8)
σ
µν
νµ
σ
σ
µνσ
νµσ
µν
νµ
σ Γ−∂+Γ∂−=Γ−∂ ))(()( gggggg
and utilize (18.4) in this to write this consequently as
(18.9)
σ
µν
µ
λσ
νλ
β
βσ
σ
µν
νµσ
µνσ
νµσ
µν
νµ
σ
ΓΓ−
−ΓΓ−+Γ∂−=Γ−∂
gg
gggggg
2
)(

6. We make use of this latest result together with (18.7) to get for the needed expression of
(12). Thus, we arrive at
(18.10)
β
βσ
σ
µν
νµσ
µν
µ
λσ
νλ
σ
σν
νµ
µ
σ
µν
νµ
σ
σ
µνσ
νµσ
σνµ
νµ
ΓΓ−−ΓΓ−
+Γ−∂−Γ−∂=Γ∂−+Γ∂−−
gggg
gggggggg
22
)()(
This is the appropriate expression that we need in (11) and after some suitable re-labeling
of indices, we can thus, split the EH action (1) up into two major separate terms, one of
which is a four integral of a covariant divergence of a vector that we can actually ignore.
(18.11)
( )( ) ( )∫∫ ΓΓ−ΓΓ−+Γ−Γ−∂= β
βσ
σ
µν
σ
λν
λ
µσ
νµµ
µν
νσσ
µν
νµ
σ ggxdgggxdS HE
44
Noting that we can write something involving the covariant divergence of a vector as
(18.12)
( )µ
µ
µ
µ VgVg −∂=∇−
we can then reflect on the first part of the right-hand-side of (18.11) as the said four
integral of a covariant divergence of a vector, and we drop this term off to obtain for the
effective action from the given EH action (1).
(18.13)
( )∫ ΓΓ−ΓΓ−= β
βσ
σ
µν
σ
λν
λ
µσ
νµ
ggxdS ffeHE
4
)(
In the second part of this exercise, we proceed with this effective action to obtain for the
equation of motion of the metric.
(stopped: pp. 16, cited notes)
Ref’s
[1]Ohanian, H. C., GRAVITATION AND SPACETIME, New York: W. W. Norton and Company Inc.,
copyright 1976
[2]Townsend, P. K., Blackholes – Lecture Notes, http://xxx.lanl.gov/abs/gr-qc/9707012
[3]Carroll, S. M., Lecture notes On General Relativity, http://www.arxiv.org/abs/gr-qc/9712019