66
T. K. Mathew
To substantiate further the intriguing connection between the flat space gauge
potential and curved space geometry, consider the curved space metric g μν which
can also be defined in terms of the tetrad field at every point in the space as
g μν = e
i
μ e iν = η i j e
i
μ e
j
ν
(5)
where η i j is the local flat space (Lorentzian) metric. As a continuation, it is possible
to define the curved space representatives of the gauge potentials as A
a
μ = e
a
μ A
a
i and
the field tensor as F
a
μν = e
i
μ e
j
ν F
a
i j . Consequently, one can construct a fourth rank
tensor corresponding to Eq. (4), as (for details see the original paper, Ref. [3]),
˜
R μνρλ = e
i
μ e
j
ν e
k
ρ e
l
λ
˜
R i jkl ,
(6)
which has all the symmetry properties of ˜
R i jkl . One may immediately be tempted
to identify ˜
R μνρλ with the usual curvature tensor of the Riemannian manifold. But
such a correspondence will not be exactly correct. On the other hand, as shown by
the authors, it is possible to bifurcate the tensor in to a sum of two as
˜
R μνρλ = R μνρλ + μνρλ
(7)
where R μνρλ is the curvature tensor of the curved manifold and μνρλ is a fourth rank
tensor in the curved space with the same algebraic properties of the curvature tensor.
So, finally the authors have beautifully associated the flat space gauge field to the
metric of the curved space and thus find the curvature of the manifold. This is a truly
remarkable achievement; firstly, because it does not have the noted shortcoming of
the prior theory due to DG; secondly, the theory is not particular about the symmetry
group, in fact the theory is true for a large class of symmetry groups.
Further, they have formulated of the field equation of gravity. One can obtain the
field equation by varying the Einstein–Hilbert action [4],
S E−H =
√
−g R d
4 x
(8)
where g is the metric scalar and R is the curvature scalar. Following Eq. (6) one can
re-write the action as
S E−H =
√ −g
˜
R −
.
(9)
This action is essentially a function of the gauge fields and hence can be viewed
as the action for the U (2) × U (2) gauge theory. But this action is not of the YM
form. While in the linear limit, where the metric takes the form g μν = η μν + h μν ,
the action will exactly reduce to the YM form. Varying this action with respect to
g μν , equivalently treating gauge fields as function of the metric components, one
will get the second rank curvature tensor as [3],
T. K. Mathew
To substantiate further the intriguing connection between the flat space gauge
potential and curved space geometry, consider the curved space metric g μν which
can also be defined in terms of the tetrad field at every point in the space as
g μν = e
i
μ e iν = η i j e
i
μ e
j
ν
(5)
where η i j is the local flat space (Lorentzian) metric. As a continuation, it is possible
to define the curved space representatives of the gauge potentials as A
a
μ = e
a
μ A
a
i and
the field tensor as F
a
μν = e
i
μ e
j
ν F
a
i j . Consequently, one can construct a fourth rank
tensor corresponding to Eq. (4), as (for details see the original paper, Ref. [3]),
˜
R μνρλ = e
i
μ e
j
ν e
k
ρ e
l
λ
˜
R i jkl ,
(6)
which has all the symmetry properties of ˜
R i jkl . One may immediately be tempted
to identify ˜
R μνρλ with the usual curvature tensor of the Riemannian manifold. But
such a correspondence will not be exactly correct. On the other hand, as shown by
the authors, it is possible to bifurcate the tensor in to a sum of two as
˜
R μνρλ = R μνρλ + μνρλ
(7)
where R μνρλ is the curvature tensor of the curved manifold and μνρλ is a fourth rank
tensor in the curved space with the same algebraic properties of the curvature tensor.
So, finally the authors have beautifully associated the flat space gauge field to the
metric of the curved space and thus find the curvature of the manifold. This is a truly
remarkable achievement; firstly, because it does not have the noted shortcoming of
the prior theory due to DG; secondly, the theory is not particular about the symmetry
group, in fact the theory is true for a large class of symmetry groups.
Further, they have formulated of the field equation of gravity. One can obtain the
field equation by varying the Einstein–Hilbert action [4],
S E−H =
√
−g R d
4 x
(8)
where g is the metric scalar and R is the curvature scalar. Following Eq. (6) one can
re-write the action as
S E−H =
√ −g
˜
R −
.
(9)
This action is essentially a function of the gauge fields and hence can be viewed
as the action for the U (2) × U (2) gauge theory. But this action is not of the YM
form. While in the linear limit, where the metric takes the form g μν = η μν + h μν ,
the action will exactly reduce to the YM form. Varying this action with respect to
g μν , equivalently treating gauge fields as function of the metric components, one
will get the second rank curvature tensor as [3],