Gravity and Strings

146 Action principles for gravity
There is another invariant I4, which is quadratic only in d = 4:
I4 = ɛ µ1···µd−3ν1ν2ν3 ρ1 ρ2 ρd−3
µ1ρ1
µ2ρ2 ···µd−3ρd−3 ν1ν2ν3 , (4.163)
but itisnot invariant under parity transformations (a further requirement) and it is usually
not considered. Also, e by itself is another invariant (a cosmological-constant term) that we
will not consider. Observe that all the Weitzenböck invariants involve the inverse Vielbeins
ea µ and are, therefore, highly non-linear in the Vielbeins.
The general teleparallel Lagrangian of Pellegrini and Plebański [761] is the integral of a
linear combination of the Weitzenböck invariants with arbitrary coefficients:
LT =
3
c i Ii. (4.164)
i=1
Only two of them are really independent since we can choose the overall normalization.
This general Lagrangian, written in differential-forms language to relate it to the Poincaré
gauge theory of gravity which is customarily written in it (see e.g. [524]), is known as the
Rumpf Lagrangian [815] (see also [482, 704]).
There are other ways to parametrize this Lagrangian, for instance, by splitting abc into
several pieces (1) , (2) , and (3) (tentor, trator, and axitor, respectively, [482]). First we
define
va ≡ ab b (2) , abc = 2
1 − d ηa[bvc],
(3) (1)
abc = [abc], abc = abc − (2) abc − (3) (4.165)
abc.
Then, a Lagrangian equivalent to Pellegrini and Plebański’s is [482]
LT = e abc
The relation between these two parametrizations is
a1 = c1 + 1
2 c2, a2 = c1 + 1
2 c2 +
Another parametrization based on va, the tensors
3
ai
i=1
(i) abc. (4.166)
d − 1
c3, a3 = c1 − c2. (4.167)
2
a a1···ad−3 = 1
3! ɛa1···ad−3b1b2b3 b1b2b3 , tabc = a(bc) − (2) a(bc), (4.168)
and the invariants v 2 , t 2 , and a 2 can be found in [522].
4.6.1 The linearized limit
Now, our goal is to try to understand which kind of theories are those defined by the
Lagrangian Eq. (4.164). First, we observe with Møller [702] that, for c1 = 1, c2 = 2,
and c4 =−4, this Lagrangian is identical (up to total derivatives) to the Einstein–Hilbert