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Transcript of Cosmological Perspectives of Modified Einstein Field EquationsFinal
2009PH10739-1
Cosmological Perspectives of Modified Einstein
Field Equations
R. Srivatsan 2009PH10739
Supervisor: Prof. Ajit Kumar
Abstract: The Ostrogradsky instability of higher derivative Lagrangians is derived from first principles
using Lyapunov Stability Analysis. We argue that Born-Infeld Lagrangians are viable modifications of the
Einstein-Hilbert action. The Born-Infeld (B-I) version of the FRW equations are derived and the cosmological
dynamics is studied for matter dominated closed and open universes. We compare the results with the usual
cosmology. We then investigate the quantized versions of Born Infeld electrodynamics and Born Infeld
Gravity. We derive Feynman rules for B-I electrodynamics by deriving an effective Lagrangian with the
square root removed using the Faddeev-Popov method. In the case of B-I gravity, the square root in the
Lagrangian is removed by the introduction of the Vierbien fields. This approach has the advantage that
SO(3,1) can be consistently regarded to be the gauge group of gravity. Finally, the quantum fluctuations of
the radii of spatial hypersurfaces in flat space are shown to undergo accelerated increase with time.
Email: [email protected]
INTRODUCTION
The light curves plotted from several hundred Type Ia supernovae [1] indicate that the
universe is expanding at an accelerated rate. In fact, the Friedmann equations derived
from Einsteinβs General Relativity, on the other hand, show that for a fluid that satisfies
the Weak Energy Condition, the universe should decelerate as it expands [2] .This
discrepancy has prompted researchers to look for phenomenological fixes for the
gravitational field equations to make theory agree with experiment.
Two choices are usually discussed at this juncture. One is to assume the existence of
an unseen dark energy [3] that presumably drives the accelerated expansion. But, in
order to make the Einstein field equation with dark matter agree with the observed
anisotropies in the cosmic microwave background, the amount of dark matter that
should be posited is eighteen times the observed ordinary matter. The implausibility of
this assumption has led to the recent interest in modifying the gravitational Lagrangian
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itself. Our aim therefore, is to investigate the possible modifications of general relativity
and their cosmological implications.
However, gravity cannot be modified in an offhand manner. There is a no-go theorem
which actually constrains the form of the Lagrangians that can be used to describe
nature. Basically, it is related to Newtonβs observation, that the equations of physics,
when written in terms of the fundamental quantities, are always second order in time [3].
This result is called Ostrogradskyβs theorem which states that βa system whose
Lagrangian depends non-degenerately on the second and higher order derivatives of
the dynamical quantities is necessarily unstable.β The word βnon-degeneratelyβ means
that the higher derivatives can be solved for in terms of the lower derivatives. When this
condition is satisfied, the system has an instability.
This βtheoremβ has been invoked in the literature to support the claim that modifications
that depend on the traces of the Riemann and Ricci tensors must be excluded from
consideration [3]. But the justification of this important result has been done by bringing
ideas of second quantization into the classical setting. Moreover, the sign indefiniteness
of the energy only shows that the energy function is not a Lyapunov function. It does not
indicate that no Lyapunov function can exist. However, as we have shown, the proof
can be made more precise within classical mechanics, without discussing the particle
picture.
The quantized version of Born Infeld gravity is then investigated. For fixing ideas, the
case of Born-Infeld electrodynamics is treated. The principal difficulty in quantizing this
theory is caused by the appearance of a square root in the Lagrangian. This is removed
by the application of the Faddeev-Popov method and an effective Lagrangian is derived.
It turned out that in the case of gravity, there is a more natural procedure to remove the
square root. This is done by the introduction of Vielbien (or Tetrad) fields. The
gravitational connection then becomes a gauge field. An important point of consistency
is that throughout our work, we have been using the Palatini formalism where the metric
and connection are independent. Hence, in the language of tetrad fields, the Vielbien
(which is determined solely by the metric) and the connection (which is the gauge field)
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become independent dynamical variables. Here, we have chosen to concentrate on the
gauge field and have taken the tetrads to be those of Minkowski space where it is easier
to do quantum field theory.
THEORY AND PRINCIPLES
We briefly discuss the classical theory of general relativity. An excellent reference for
this is [2].
The field equations of general relativity can be derived from an Action principle, as was
first discovered by David Hilbert [3]. The unmodified gravitational action is given by
ππ» = π βπ π4π₯ + ππ (1)
This is called the Einstein-Hilbert Action. The π stands for the Ricci scalar. The quantity
π represents the determinant of the metric tensor. ππ represents the matter part of the
action. Two dynamical quantities enter the action given in (1). They are the metric and
the connection. The Palatini formalism results if these quantities are considered
independent degrees of freedom. We vary the action with respect to both the metric and
the connection. The explicit expression for the Ricci tensor in terms of the connection
Ξπ½πΎπΌ (not necessarily a Levi-Civita connection) is given by
π π½πΏ = βΞ±Ξπ½πΏπΌ β ππΏΞπ½πΌ
πΌ + ΞππΌπΌ Ξπ½πΏ
π β ΞππΏπΌ ΞπΌπ½
π (2)
The metric formalism results if we assume at the outset that the connection by default is
the Levi-Civita connection which is compatible with the metric. We vary (1) only with
respect to the metric in this case. The Levi-Civita connection is given by
Ξπ½πΎπΌ =
1
2 ππΌπΏ (ππΎππ½πΏ + ππ½ππΎπΏ β ππΏππ½πΎ ) (3)
Here and in the previous equation, the repeated indices are summed. In the case of the
Einstein-Hilbert action given in (1), it can be proved [4] that the metric and the Palatini
formalism lead to the same field equations, which can be shown (after a variational
calculation) to be as given below
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π ππ β1
2ππππ =
8ππΊ
π4 πππ (4)
Here, πππ represents the energy momentum tensor of the sources. However, this
circumstance of the metric and Palatini formalisms giving the same equations of motion
is not obtained for more general Lagrangians. For example, for the Born-Infeld Action
ππ΅ = π det πππ + π π ππ 1/2
π4π₯ (5)
the metric and Palatini formalisms, give radically different equations of motion. As we
shall argue, consistency with our first result described below requires that (5) be varied
in the Palatini formalism. Varying (5) in the Palatini approach results in the following
equations of motion [5] (after varying with respect to the metric)
πππ = πππ + π π ππ (6)
π
π πππ = ππππ β π πππ (7)
Here, πππ represents an auxiliary metric while πππ represents its inverse. π is a coupling
constant while π is a Cosmological constant. π and |π| are the determinants of πππ and
πππ respectively. The variation with respect to the connection yields the following
equation
Ξπ½πΎπΌ =
1
2 ππΌπΏ (ππΎππ½πΏ + ππ½ππΎπΏ β ππΏππ½πΎ ) (8)
On the other hand, varying (5) in the metric formalism yields the following mess
βπ πππ =
β1
2πΉπππ + πππ ππΌπ
πΌ β βπβπ πΉπ + πΉπ π ππ + 2βπΌβπ πΉπ π ππΌ β πππβπ½βπΌ πΉπ π
πΌπ½ β
ππΌππΌ πΉπ π ππ β 2πΉπ π π
πΌπ ππΌ + πππ ππΌππΌ πΉπ π β βπβπ πΉπ π + πΉπ π π ππ + β¦
The other quantities need not be defined as they are not used in the subsequent
discussion.
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Our work on the quantum version of B-I electrodynamics and gravity uses some new
tools which are briefly summarized here.
The Faddeev Popov [12] method uses the path integral formalism to remove redundant
infinities while performing the integral. The procedure is as follows:
The generating functional for a Quantum Field theory that is gauge invariant is given by
π = π·π΄π πππ (9)
Here π is the action evaluated along a particular path in the space of the gauge field.
Since the action is gauge invariant, the generating functional above involves sums over
fields that are related to each other by gauge transformations. This over counting results
in infinities. The way to fix this is to insert unity in the form:
ΞπΉβ1 π΄π = π·π πΏ(πΉπ(π΄π
π)) (10)
Here, πΉπ π΄ππ = 0 is the gauge condition one would like to impose. π is a gauge group
element, π·π represents integration over the group elements. πΏ represents the delta
functional and π represents a color index (in case of SU(2)). As argued in [12], the
quantity given above is gauge invariant. Finally, writing 1 as 1 = ΞπΉ(π΄π ) π·π πΏ(πΉπ(π΄ππ))
in the generating functional, we get
π = π·π π·π΄π πππ ΞπΉ(π΄π )πΏ(πΉπ(π΄π )) (11)
Thus the integral over the group space only contributes an overall multiplicative
constant and hence can be ignored. The correct expression is therefore,
π = π·π΄π πππ ΞπΉ(π΄π )πΏ(πΉπ(π΄π )) (12)
This ΞπΉ(π΄π) is the origin of the famous Faddeev-Popov ghosts. However since the
gauge group of Electrodynamics is Abelian, this term is independent of the π΄π in this
case and hence can be taken out of the integral.
Another tool that is subsequently required is the Vielbien (or Tetrad) formalism for the
gravitational field. In the usual formulation of gravity, the dynamical field is taken to be
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the metric πππ and the connection as stated above is the Levi-Civita connection Ξπ½πΎπΌ .
We have used here the Palatini formalism that considers these quantities to be
independent. An equivalent approach to gravity would be the introduction of the Tetrad
fields πππ
[13,14] which have both a spacetime index π and an internal index πΌ. They are
defined to satisfy the following relation
ππΌπ½ = πππ ππΌπππ½π (13)
Here πππ is the Minkowski metric. This equation is then inverted to obtain the following
equation which is also used as a general definition of the Cotetrad fields
πππ = ππΌπ½πππΌππ
π½ (14)
The internal indices of the Tetrad transform in such a way that the tensor ππΌπ½ is left
invariant. As we know, the set of all such transformations is precisely the Lorentz Group
SO(3,1). Hence, in a particular coordinate system, we treat gravity as a gauge theory
with the gauge group SO(3,1). The transformation is given by
πππΌβ² (π₯) = Ξπ½
πΌβ² (π₯)πππ½ (π₯) (15)
Here, Ξπ½πΌβ² is an element of SO(3,1) and as seen above it depends explicitly on the
spacetime point at which it is carried out since it is a local gauge transformation in the
theory. The analogue of the Riemann tensor here is the βfield strengthβ of a gauge field
π΄πΌπΌπ½
and in perfect analogy to the Yang-Mills field is given by [13]
πΉπΌπ½πΌπ½ = ππΌπ΄π½
πΌπ½ β ππ½π΄πΌπΌπ½ + π΄πΌ ,π΄π½
πΌπ½
= ππΌπ΄π½πΌπ½β ππ½π΄πΌ
πΌπ½+ ππππ π
πΌπ½π΄πΌπππ΄π½π π (16)
where, ππππ ππΌπ½
are the structure constants of the SO(3,1) group. Once we have the field
strength πΉπΌπ½πΌπ½
, the Riemann and Ricci tensors can be defined as follows
π πΎπΏπΌπ½
= πΉπΎπΏπΌπ½ ππΌ
πΌππ½π½ (17)
2009PH10739-7
Finally, we review a theorem from control theory. This result is very important for the
subsequent discussion with regard to the proof of the Ostrogradsky instability.
Consider the following system of differential equations
π = π(π) (18)
Here, π represents an n-dimensional vector, while π(π) is a scalar point function. The
dot represents a time derivative. Without losing generality, we can assume that π = 0 is
the equilibrium point of the system [6]. We now state the result. The proof can be looked
up in [6].
Chaetevβs Theorem [6] Let π = 0 be an equilibrium point for (1). Let π βΆ π· β π be a
continuously differentiable function such that π 0 = 0 and π π0 > 0 for some ππ with
arbitrarily small | π0 |. For π > 0, let π΅π denote the set π₯ Ο΅ Rn π± β€ r} contained in π·
and let
π = π π π΅π π π > 0} (19)
With π defined as in (10), suppose that π > 0 throughout π. Then, the point π = 0 is
unstable.
Here, the instability referred to, is instability in the Lyapunov sense. When translated in
terms of the particle picture, this amounts to exactly the same sense as conveyed in [3].
However, this result is more general and can be applied even to a classical field theory,
like General Relativity, which cannot be quantized unambiguously at present. Now, we
apply this result to the case of the system with a Lagrangian that depends non-
degenerately on the second and higher order derivatives of the dynamical quantities.
For simplicity and without loss of generality, we treat below the case of a system with a
finite number of degrees of freedom.
RESULTS AND DISCUSSION
Part 1 of the Thesis:
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Proof of Ostrogradsky instability: In [3], the Ostrogradsky Hamiltonian for one degree of
freedom has been constructed explicitly for a higher derivative Lagrangian πΏ(π, π , π ) to
be of the following form
π» π1,π2,π1,π2 = π1π2 + π π1,π2,π2 β πΏ(π1, π2, π) (20)
Here, the choice of canonical coordinates is as follows [3]:
π1 = π, π1 =π
ππ πΏ β
π
ππ‘
ππΏ
ππ
π2 = π , π2 =π
ππ πΏ (21)
An approach to proving the instability of a system was already carried out in [7], in the
context of charged solitons. The ideas presented below are a combination of the steps
taken in that paper together with the result quoted above.
Now, we come to the crux of the result. Phase space translations are canonical
transformations [8]. Therefore, by a suitable canonical transformation, the Hamiltonian
can be thrown into the following form:
π» π,π = π1π2 + π π,π (22)
Here, π(π,π) is a function that contains no linear terms. Therefore, in a neighborhood
of the origin, the first term dominates. Now, if we choose the Chaetevβs function as
π π,π = π1π2, we get for the derivative of this function along the solution trajectories
π = π1 π2 + π1π2
, so that, using the equations of motion contained in (22),
π = π12 + π’(π,π) (23)
Again, the function π’ π,π is dominated by the first term in a neighborhood of the origin.
Therefore, in a small neighborhood around the origin, the conditions of Chaetevβs
theorem are satisfied so that this theory is unstable at the origin. Finally, by a suitable
translation, we can reach the same conclusion at any point of the trajectory. Hence, we
conclude that such a theory is unstable everywhere.
2009PH10739-9
Now that we have shown that any higher derivative Lagrangian is unstable in the
Lyapunov sense, we can return to the question of modifying gravity to explain
accelerated expansion. In the literature, π(π ) modifications are discussed very
extensively, where π is the Ricci scalar [3], [9]. It is claimed that no other non-trivial
modification can satisfy the constraints imposed by the result proved above [3].
However, Born-Infeld (B-I) Lagrangians are particularly attractive because of their
intriguing properties [10].
The B-I Lagrangian made its appearance with the modification of the electrodynamic
action by Born and Infeld. This was done to remove the infinite self energy of point
charges by introducing an upper bound on the magnitude of the electric field. It was
further pointed out that the B-I electromagnetic field propagates without birefringence
[10]. Analogous work has been carried out for the case of the gravitational field.
However, the cosmology discussed has been restricted to the case where the spatial
hyperspaces are flat [11]. We have extended these results to the case where the spatial
hyperspaces can be curved. Before proceeding further, we must mention here an
important consequence of the Ostrogradsky instability.
The Born Infeld action in (5) can only be varied in the Palatini approach: This
statement can be justified by referring to (2), (3) and (5). Suppose (5) is varied in the
metric formalism. Then as stated in the introduction, the connection is fixed as the Levi
Civita connection given in (3). In that case, the determinant in (5), can be expanded in
terms of the trace as follows (upto third order):
det πππ + π π ππ β 1 +1
2π (π β π πΎ β π 2π) (24)
where
πΎ = π πππ ππ β
1
2π 2 and π = 8π πππ ππΌπ π
πΌ β 6π π πππ ππ + π 3 (25)
Therefore, if the metric is considered the sole dynamical quantity, the Lagrangian would
involve the second and higher derivatives of the metric. But this would result in an
unstable theory in accordance with the result proved above.
2009PH10739-10
However, in the Palatini case, only the first derivatives of the connection coefficients
appear in the expression for the Ricci tensor as can be seen from (2). Therefore, in this
case, there is no instability provided the action in (5) is varied by taking the metric and
connection as separate degrees of freedom and therefore, Born-Infeld Lagrangians are
also viable as alternative theories of gravity.
The FRW equations in Born Infeld Gravity: With the question of viability settled, we
can now set up and solve the FRW equations for B-I gravity. We generalize here the flat
space results of [11] to curved spatial hyper surfaces.
As usual, we assume that the universe is homogenous and isotropic. This results in the
following ansatz for the metric in comoving coordinates [2]
πππ ππ₯πππ₯π = β ππ₯0 2 + π π‘ 2
ππ 2
1βππ2 + π2 ππ 2 + sin2 π ππ 2 (26)
Here, a(t) is the cosmic radius. Other symbols have their usual meanings. For the B-I
equations, we also need an ansatz for the auxiliary metric πππ . This is at once supplied
by the demand for homogeneity and isotropy as [6]
π00 = βπ π‘ 2 and πππ = π π‘ 2πΎππ (27)
Here, πΎππ represents the spatial components of the metric. Equation (27) together with
(8) then determines the non-zero connection coefficients as (here, Uβ represents a time
derivative)
Ξπ‘π‘π‘ =
π β²
π Ξππ
π‘ =π2ππ β²
1βππ2 π2+
ππ β²π2
1βππ2 π2 (28)
Ξπππ‘ =
π2ππ β²
π2+
ππ β²π2
π2 Ξππ
π‘ = Ξπππ‘ sin2 π (29)
Ξπππ =
ππ
1βππ2 Ξπππ = βπ 1β ππ2 Ξππ
π = Ξπππ sin2 π (30)
Ξπππ = β sinπ cos π Ξππ
π= cot π (31)
Ξπππ = Ξππ
π=
1
π (32)
2009PH10739-11
Ξππ‘π = Ξππ‘
π = Ξππ‘π
= π β²
π+
π β²
π (33)
For use in (6), we also need the ππ = 00 and 11 components of the Ricci tensor, which
are given by (as usual an apostrophe represents a time derivative)
π 00 = 3π»1 π» + π»2 β 6π»π»2 β 3 π β²β²
π+
π β²β²
π (34)
π 11 =1
1βππ2 2π +
π2
π2 2π»ππ β² + π β²2 + ππ β²β² + π2 π»2 +
π β²β²
π + ππ β² + π»π2 π» + π»1 β
π»2 (35)
Here π» =π β²
π,π»1 =
π β²
π,π»2 =
π β²
π
For the matter distribution, we assume a perfect fluid with pressure π and density π. It
can further be shown that the fluid satisfies the equation of continuity. Finally, we can
write the following equations of motion using (6) and (7)
1 β π2 = π 3π»1 π» + π»2 β 6π»π»2 β 3 π β²β²
π+
π β²β²
π (36)
π2 β 1 =2ππ
π2 +π
π2 {2π»ππ β² + π β²2 + ππ β²β² + π2 π»2 +π β²β²
π + ππ β² + π»π2 π» + π»1 β π»2 }
(37)
ππ = π + π π (38)
π3
π= π + π π (39)
The fluid also has an equation of state that is given by the following relation
π = π€π (40)
Finally, the continuity equation can be written as
π β²
π= β3π» 1 + π€ (41)
2009PH10739-12
We can immediately integrate (41) to yield the density as a function of radius
π = π0πβ3 1+π€ (42)
Here, π€ is a constant that determines the kind of fluid that is present in the universe. It
is 0 for matter, -1 for vacuum and -1/3 for radiation. The second derivatives can be
eliminated between (36) and (37). Moreover, (38) and (39) can be solved to give π and
π as a function of π and hence as a function of π(π‘) as follows
π = π + π π€π0πβ3 1+π€
1
4 π + π π0πβ3 1+π€
1
4 (43)
π = π+π π€π0π
β3 1+π€ 34
π+π π0πβ3 1+π€ 14
(44)
After doing all the manipulations, we end up with the following first order equation for π
π» =π β²
π=
Β± πΌ2+ 4 πΌ2+ 2π½+2 +πΌπ½ β πΌ π
π
2 π2
π π2β1 β2π +
(π2β1)
3π βπΌ
2 πΌ2+ 2π½+2 +πΌπ½ β πΌ (45)
Here
πΌ = β3π π π 1 + π€
4
π€
π + π π€π π +
1
π + π€π π
π½ = β3π π π 1 + π€
4
3π€
π + π π€π π β
1
π + π€π π
Equation (45) is therefore the final result of our work for the first part of the thesis. It
gives an equation for the evolution of π(π‘). This can be used to discuss any differences
or similarities with the equation for the Hubble rate π β²
π in Einsteinβs relativity. This
equation was simulated using the odeint routine in SciPy and the results are shown
below.
As illustrated in Figure 1 and 2, the main point of difference between the solutions of the
FLRW equations in Einsteinβs gravity and in the present case is that closed universes in
2009PH10739-13
B-I gravity do not experience a βbig-crunchβ even after the elapse of a large amount of
time. Moreover, the radius always accelerates in this theory, which can be contrasted
with the case of general relativity, where it always decelerates. Finally, the Born-Infeld
model gives qualitative agreement with the various features expected of the variation of
π vs time..
Fig. 1 The Hubble rate as a function of time for a matter dominated open universe. Present time corresponds to
abscissa = 0.01.
Fig. 2 The Hubble rate (left) and the radius (right) as functions of time for a matter dominated closed universe.
Present time corresponds to abscissa = 0.01. A comparison with the previous figure indicates that the Hubble rate
increases much faster for a closed universe than an open universe. As discussed in the text, the acceleration of the
universe is clearly apparent
Another important fact to observe is that the Born-Infeld theory predicts an acceleration
that becomes appreciable after a time that agrees well with observations. The actual
2009PH10739-14
variation of the cosmic radius and Hubble rate predicted by the B-I gravity is given
below. Therefore, by tuning the values of the coupling constant π and the cosmological
constant π properly, one can, in principle, make the actual values of the radius agree
with the experiment. On the other hand, an advantage of π(π ) modifications is that any
history of the evolution of the universe can be supported by a proper choice of the
function π(π ) [3]. This flexibility has been lost in the case of the Born-Infeld modification.
Nevertheless, as the preceding plots show, the history predicted by our calculations is
to a tolerable extent, consistent with observations. Figure 2 above illustrates the major
differences between conventional cosmology and B-I cosmology which were mentioned
earlier. In addition to the points indicated above, we also observe that closed universes
always accelerate. This is evident from the fact that the Hubble rate keeps on
increasing. Moreover, we have finally achieved our goal : to show that B-I cosmologies
give believable results as far as cosmology is concerned and give a natural framework
to explain the observed accelerated expansion of the universe. Moreover, as discussed
above, the tuning of the coupling constants and the cosmological constant can make
the results agree with any observed value of the acceleration.
Part 2 of the thesis:
The Quantisation of the Born-Infeld Electrodynamical Field:
The Lagrangian density for the B-I field is given by:
πΏ = 1 +πΉππ πΉ
ππ
π½2 βπΉππ π΅
ππ πΉππ π΅ππ
π½4 1/2
(46)
Here, π½ is a free parameter of the theory. Moreover, πΉππ is the field strength and π΅ππ is
the dual of the field strength.
In order to remove the square root, we introduce an auxiliary field π after [15] as follows:
πΏ = βπ
2 1 +
πΉππ πΉππ
π½2 +πΉππ π΅
ππ πΉππ π΅ππ
π½4 β1
2π (47)
We observe that the π field enters the Lagrangian without time derivatives and hence, it
has no independent dynamics. Its equations of motion are given by
2009PH10739-15
π
πππΏ = 0 = β
1
2 1 +
πΉππ πΉππ
π½2+
πΉππ π΅ππ πΉππ π΅
ππ
π½4 +
1
2π2= 0 (48)
This fixes π in terms of the original dynamical fields which when substituted in the
Lagrangian, gives the same equation as the original one.
The advantage of introducing this field is that on passing to the path integral, π can be
taken to be independent of the other fields, since in the path integral, the action is not
stationary but acquires all values with same probability.
The Faddeev-Popov procedure is then used here, and consists of exploiting the gauge
invariance to factor out the infinite volume of the Gauge space as discussed in the
Theory section. For the Abelian case of electrodynamics, this throws in a factor that is
independent of the π΄π and is not important for our approach. After doing all that, the
normal course is to insert an arbitrary field π and averaging over all such fields. In our
approach, we choose as π the extra auxiliary field which we have introduced in the
Lagrangian above. The argument for justifying this is given below:
Proof: The path integral (or the generating functional) is given by:
π = π·π π·π΄π πππ π ,π΄π (49)
Since π and π΄π are independent degrees of freedom, we can proceed with the standard
F-P integration for π΄π to get
π = π·π π·π΄π ΞπΉ π΄π πΏ π π΄π β π πππ π ,π΄π (50)
Here, ΞπΉ[π΄π ] is the term arising out of neglecting the infinite volume element of the
gauge group space. For Abelian gauge theory, it is independent of π΄π and can be taken
out of the integral. Also, π(π΄π ) is the gauge fixing condition that one would like to
impose.
The important point to observe here is that the arbitrary field π introduced above is
completely independent of π΄π . Therefore, it can be taken to be π itself. The justification
for this is that there is gauge invariance in the equations. This freedom can be used to
set the gauge function π π΄π equal to any arbitrary function. Explicitly,
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π΄β²π = π΄π + πππ
so that, if πππ΄π = 0, then, π π΄π β² = πππ΄
π β² = πππππ = π which can be solved for π. Thus
in the path integral, the gauge fixing function can be set equal to π.
The main point of difference is that instead of the traditional gauge breaking term in the
effective Lagrangian, which is quadratic in the π(π΄π), we get the following effective
Lagrangian after integrating over the π.
πΏπππ = βπ π΄π
2 1 +
πΉππ πΉππ
π½2 β πΉππ π΅ππ
2
π½4 β1
2π π΄π (51)
Finally, since as we said above, the auxiliary field π is not dynamical, we can restrict to
variations of the Lagrangian such that πΏ π π΄π = 0. Therefore, the first and last terms
can be neglected to get
πΏπππ = π π΄π
2 πΉππ πΉ
ππ
π½2 β πΉππ π΅ππ
2
π½4 (52)
Hence, we have achieved our aim: we have removed the square root and brought the
Lagrangian to polynomial form in the π΄πand their derivatives. Now, we come to the
problem of getting the Feynman rules for this effective Lagrangian.
The Feynman Rules for Born-Infeld Electrodynamics
Using the effective Lagrangian derived above, we now give the derivation of the
Feynman rules for B-I Electrodynamics.
πΏπππ =πππ΄
π
2 πΉππ π΅
ππ 2
π½4 + πΉππ πΉ
ππ
π½2 (53)
The effective action is given by
ππππ =1
2 ππΌπ΄
πΌ πΉππ π΅
ππ 2
π½4 + πΉππ πΉ
ππ
π½2 π4π₯ (54)
Assuming the vector fields go to zero at infinity, we can use integration by parts to
rewrite this as follows:
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ππππ = β1
2 π΄πΌππΌ
πΉππ π΅ππ
2
π½4 +
πΉππ πΉππ
π½2 π4π₯
= β1
2 π΄πΌ
πΉππ π΅ππ ππΌ πΉππ π΅
ππ
π½4 + ππΌπΉππ πΉ
ππ
π½2 π4π₯ (55)
Here, we would like to define a few symbols before proceeding further; we define
ππ1π1
ππ to represent the quantity that is +1 for (π1π1) = (ππ) and -1 for the cyclic
permutation. ππΌπ½πΎπΏ is the usual permutation tensor defined for four indices. With these
definitions, the action above becomes
ππππ = β π΄πΌ
π½4 ππππΎπΏ πΉ
πππΉπΎπΏ ππΌ ππ1π1πΎπΏπΉπ1π1πΉπΎπΏ +
π΄πΌ
π½2 πΉππ ππΌπΉππ π4π₯
(56)
Renaming a few indices, and expanding, we get the integrand
π΄πΌ
π½4 ππππΎπΏ ππ1π1 ππ
ππ1π1 πΎπΏ
ππ1π΄π1ππ1π΄π1 ππΌ ππ1π1πΎπΏ ππππ1π1ππΎ1πΏ1
πΎπΏπππ΄πππΎ1π΄πΏ1 π4π₯
+ π΄πΌ
π½2 ππ1π1 ππ
πππ πΌ1π½1 ππ1π΄π1ππΌππΌ1
π΄π½1 π4π₯ (57)
We thus see that the effective action consists of two terms. One which is third order in
the field (the second term) and another which is fifth order in the field (the first term).
Writing the π΄ field in terms of its Fourier components, i.e.,
π΄π½ π₯π = π΄π½ ππ exp(ππππ₯π ) π4π (58)
we get for the second term in the equation above :
π2πππ= βπ
π΄πΌ π1 ππ1π1
ππππππΌ1π½1π2
π1π΄π1 π2 π3πΌπ3πΌ1
π΄π½1 π3
exp(π π1 + π2 + π3 ππ₯π )π4π1π
4π2π4π3π
4π₯ (59)
The integration over the π₯ coordinates gives a delta function. Finally, finding the
functional derivatives [12] with respect to the π΄(π), we get the amplitude in
momentum space for this coupling to be (i.e., the Feynman Rule for this three-
field vertex)
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πΏ3π2πππ
πΏπ΄ π1 πΏπ΄ π2 πΏπ΄ π3 = β
π
π½2 πΏ π1 + π2 + π3 ππ1π1
ππππππΌ1π½1π2
π1π3πΌπ3πΌ1
+ πππππ’π‘ππ‘ππππ
(60)
The permutations arise because the fields are indistinguishable. An important point of
difference from the Yang-Mills propagator is that the coupling amplitude is cubic in the
momentum while it is linear there; this in itself points to the intense self coupling that
occurs in our case.
Similarly the fifth order terms which occur in the first term can be handled using Fourier
Transformations to get:
πΏ5π1πππ
πΏπ΄ π1 πΏπ΄ π2 πΏπ΄ π3 πΏπ΄ π4 πΏπ΄ π5 = β
π
π½4πΏ π1 + π2 + π3 + π4 + π5
ππππΎπΏ ππ1π1
ππππ1π1
πΎπΏππ1π1πΎπΏ πππ
π1π1ππΎ1πΏ1 πΎπΏ
π2π1π3
π1π4ππ5πΎ1 π4πΌ + π5πΌ + πππππ’π‘ππ‘ππππ
(61)
The Quantisation of Born-Infeld Gravity:
Having disposed of the Electrodynamics case, we now come to the treatment of gravity.
Here, there is a natural way in which the square root in the Lagrangian can be removed.
This is achieved as stated in the theory section, by employing Tetrad fields. In terms of
the tetrads, as shown in [16], the Lagrangian for B-I gravity is given by
π = π4 π4π₯ π + π3 π4π₯ π π +π2
2! π4π₯ π π 2 β π πππ
ππ
+π
3! π4π₯ π π 3 β 3π π πππ ππ + 2π ππΌπ πΌπ½π π
π½ + π4π₯ π detπ ππ
(62)
Here, π = βπ = det ππΌπ, π is the Ricci scalar, π ππ is the Ricci tensor. Finally π =
2
Ξ is
related to the reciprocal of the cosmological constant. We now present below the
arguments used in deriving the analogy to the Yang-Mills field.
2009PH10739-19
Analogy with the Yang-Mills field:
First we start with the observation that the cosmological constant is very small in value.
Thus only the leading order terms in π in the Lagrangian above need to be taken into
account.
Moreover, since we are using the Palatini formalism, the Tetrad ππΌπ and the connection
πΉπΌπ½πΌπ½
are independent. Since the signature of gravity is contained in the derivatives of the
connection and not in the metric, we assume here that the ππΌπ are non-dynamical and
have been fixed as the Tetrads of some convenient background metric (Minkowski
spacetime in our case). Thus, the first term in the action above is irrelevant. Since we
are interested in two-point correlation functions, the second term can also be neglected.
Finally, the fourth and fifth terms are ignored because of the smallness of the
cosmological constant argument above.
The interesting term is therefore, given by
π =π2
2! π4π₯ π 2 β π πππ
ππ (63)
where π = det ππΌπ = 1 since the Tetrads are those of flat spacetime. π ππ π ππ can be
written in terms of the connection as
π πππ ππ = πππππ
πΎ πΉππΎπππΉππππ (64)
The product of the Tetrads can be decomposed as
πππππ
πΎ=
1
4πππΎπππ + π΄ππ
ππΎ (65)
Here, π΄ππππΎ
is some tensor that depends on the metric. The only condition that should be
satisfied by this tensor is that πππΎπ΄ππππΎ
= 0. If this holds, then
πππΎπππππ
πΎ= πππ + πππΎπ΄ππ
ππΎ= πππ (66)
With this decomposition, the second term in the Lagrangian can be written as
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π πππ ππ =1
4πππΎππππΉππ
πππΉππππ + π΄ππ
ππΎ πΉππΎπππΉππππ (67)
and hence, the Lagrangian can be thrown into the form
πΏ = β1
4 ππ πΉππ πΉ
ππ + π΄ππππΎ
πΉππΎπππΉππππ (68)
Thus, we see that the first term is exactly equivalent to the kinetic energy part of Yang
Mills field except with two indices instead of one as in conventional Y-M theory. One
main difference is that the gauge group here is non-compact (SO(3,1)) unlike Y-M which
has a compact gauge group (SU(2)). Moreover, as in Y-M, there is no mass term.
Feynman Rules for Quantized B-I gravity:
From the kinetic part, we can now directly get the propagator [12] for this theory to be
π·πΉ π₯ β π¦ = βππΏπππΏππ ππππβππ . π₯βπ¦
π2+ππ π4π (69)
Note that πππ is now the flat space Minkowski metric since as argued above.
To show that this is actually different from the usual Yang-Mills theory, we show below
the result of the derivation of the three vertex Feynman rule (following the method used
for B-I electrodynamics) from the kinetic part of the Lagrangian above
πΊ π, π, π = πΏ3ππππ
πΏπ΄πβ² πβ²πΌ π πΏπ΄
πβ² π β²π½ π πΏπ΄
πβ² πβ²πΎ π
=
βππΏ π + π + π ππ½ππΎπΌ β ππΎππ½πΌ πππ β² πππ
β²πΆπππ β² π β² π β² π β²
+ ππΌππ½πΎ β ππΎππΌπ½ πππβ²πππβ²πΆππ
π β² π β² π β² π β²+
ππΎππ½πΌ β ππ½ππΌπΎ πππβ²πππ
β²πΆπππ β² π β² π β² π β² (70)
Here, πΆππππππ are the structure constants of the SO(3,1) group.
Equations (69) and (70) are the final results of the work of the second part of this thesis.
2009PH10739-21
A heuristic argument for the curvature fluctuations of spatial hypersurfaces:
Finally, we show that the kinetic energy part of the Lagrangian in (68) results in an
accelerated increase in the quantum fluctuations of spatial hypersurfaces. This is shown
by first considering the propagator in (69). The two point correlation function of same
components of the spatial Gauge field at a local point as a function of time are given by
π΄π 0 π΄π π‘ = π·πΉ π‘ = β ππ ππ4π‘
π2β π42+ππ
π4π (71)
Using the analyticity of the path integral, we perform analytic continuation to imaginary
time and replace π‘ = ππ. Also, we use Wick rotation of the contour along which (71) is
evaluated. Under these transformations, the integral in (71) is evaluated to
π·πΉ π = βπΎ
4ππ2 = πΎ
4ππ‘2 (72)
where, again by analytic continuation, we have replaced π by β ππ‘. Here, πΎ is a positive
constant. Then, π·πΉ(π‘) is proportional to the fluctuations in the extrinsic curvature of the
spatial hypersurfaces. Its reciprocal is therefore, proportional to the fluctuation in the
radius of the spatial hypersurfaces. As can clearly be seen, this fluctuation has a
positive second derivative.
FUTURE SCOPE OF THE THESIS
We have carried out an exhaustive study both of Born-Infeld Gravity and Born-Infeld
electromagnetism. Though these are certainly not new topics, interest in this area has
been renewed recently because the Born-Infeld action arises as the low-energy
effective action in open strings and as the world-volume effective action in D-3 branes
[17,18,19].
The Palatini formalism can be observed to play an important role in our work. It was
seen to provide a way to go around the constraint imposed by the Ostrogradsky
instability when the Born-Infeld modification was applied on the Einstein-Hilbert action.
But it is curious that, as has been pointed out above, the metric formalism and the
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Palatini formalism give rise to radically different equations of motion. This discrepancy
deserves to be investigated further.
Another direction in which this work can be extended is to further explore the relation
between Yang-Mills fields and Gravity. As can be seen, there is rich structure available
when tetrads are employed in the description of gravitation. We have here left the
possibility open of following the same procedure in arbitrary curved spacetimes.
Interesting effects are expected to occur because the tetrad fields are no longer trivial.
ACKNOWLEDGEMENTS
I would like to thank my thesis advisor, Prof. Ajit Kumar, for his timely and helpful
guidance without which this work would not have been possible.
REFERENCES
[1] S. Carroll et al., Is Cosmic Speed-Up Due To New Gravitational Physics? Phys. Rev. D vol. 70 Issue 4 Aug. 2004
[2] S. Carroll, Lecture Notes on General Relativity, arXiv : [gr-qc]/ 9712019v1, Dec. 1997
[3] R.P. Woodard, Avoiding Dark Matter with 1/R modifications of Gravity, Lecture Notes in Physics, Springer, vol.
720, 2007, pp 403-433
[4] S. Capozziello et al., A birdβs eye view of f(R)-gravity, Open Astronomy Journal, ISSN 1874-3811, Oct. 2009
[5] M. BaΓ±ados and P.G. Ferreira, Eddingtonβs theory of gravity and its progeny, Phys. Rev. Lett vol. 105, Issue 1,
Jul. 2010
[6] H. Khalil, Non-Linear systems, 2nd
Edition, Prentice Hall, pp. 110-115, 1996
[7] A.K. Kumar et al., Stability of Charged Solitons, International Journal of Theoretical Physics, vol. 18, Issue 6, pp
425-432, Jun. 1979
[8] Goldstein, Safko, Poole, Classical Mechanics, 3rd
Edition, Pearson Education, pp. 400-410, 2002
[9] T. Sotiriou and V. Faraoni, f(R) theories of gravity, Rev. Mod. Phys. vol. 82, Issue 1, Mar. 2010
[10] D. Tennant, Coulomb field Scattering in Born-Infeld electrodynamics Phys. Rev. D vol. 83, Issue 4, Feb. 2011
[11] M. BaΓ±ados, P.G. Ferreira and C.Skordis, Eddington-Born-Infeld gravity and the large scale structure of the
universe, Phys. Rev. D vol. 79, Issue 6, Mar. 2009
[12] L.H.Ryder, Quantum Field Theory, 2nd
Edition, Press Syndicate of the University of Cambridge, pp.240-260,
1996
2009PH10739-23
[13] M.Carmeli, Group Theory and General Relativity, 1st Edition, Imperial College Press, pp.168-190, 1977
[14] J.Baez, J.P.Muniain, Gauge Fields, Knots and Gravity, 1st Edition, World Scientific, pp.405-420, 1994
[15] M. B. Cantcheff, Eur. Phys. J. C46, 247, 2006
[16] J. Nieto, Born-Infeld Gravity in any dimension, arXiv:[hep-th]/0402071v2, June 2004
[17] G.Gibbons et al., arXiv:[hep-th]/020934
[18] S.Deser, Phys. Rev. Lett. 103, 101302 (2009)
[19] S.V. Ketov, Born-Infeld Non-Linear Electrodynamics and String theory, PIERS Proceedings, Moscow, Russia,
2009