Dirac style Relaxation of the
Israel Junction Conditions Aharon Davidson and Ilya Gurwich
Physics Department Ben Gurion University Beer Sheva 84105, Israel
Einstein’s Gravity in Higher Dimensions - Jerusalem ‘07 1
* It seems that our potential contribution to this workshop was done long ago (Davidson-Gedalin ‘94)... when deriving the axially symmetric Kaluza-Klein metric where the Euclidean Kerr horizon played the role of a finite length magnetic flux string. By adjusting the dilaton charge, as dictated by string theory, the magnetic flux tube world-sheet was then shown to exhibit a 2-dim black and white dihole structure. An alternative (extremal) interpretation was given later (Emparan ‘99). This talk, however, while fitting the general title of the workshop, is not directly related to Black Holes...
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Abstract Following Dirac's brane variation prescription, the brane must not be deformed during the variation process, or else the linearity of the variation may be lost. Alternatively, the variation of the brane is done, in a special Dirac frame, by varying the bulk coordinate system itself. Imposing covariant Dirac style boundary conditions on the constrained 'sandwiched' gravitational action, the Israel junction conditions get relaxed, but remarkably, in such a way that the original solutions are respected. The Israel junction conditions are traded for a pair of coupled Regge-Teitelboim equations (plus local conservation laws). As far as the classical field equations are concerned, Randall-Sundrum and ReggeTeitelboim brane theories appear to be two faces of the one and the same unified brane theory. Within the framework of such a unified brane cosmology, we examine the dark matter interpretation of the effective energy/momentum deviations from General Relativity.
Phys. Rev. D74, 044023 (2006)
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Extensible model of the Electron [Dirac ‘62] A classical spinless electron is viewed as a breathing bubble in the electromagnetic field ‘with no constrains fixing its size and shape’.
Consider a flat 4-dim Minkowski background
and let r(t) denote the bubble’s radius. r(t) cannot serve as a canonical variable, as otherwise the linearity of the variation may be lost. Alternatively, perform an implicit transformation R = r - f(t), such that in the new frame
• The brane is located at R=0. • The brane is not deformed during the variation. • f(t) serves as a canonical variable.
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Constrained (String-like) Gravity [Regge-Teitelboim ‘75] With Quantum Gravity as the original motivation, the Universe is viewed as a 4-dim extended object (= test brane) floating in a higher N-dim flat background. In turn, the (now induced) metric
ceases to serve as the canonical gravitational field. This job is taken by the embedding vector
.
The Universe dynamics is governed by the Einstein-Hilbert action
The action may look deceptively conventional, but the Einstein field equations get modified
exhibiting both General Relativistic as well as geodesic ingredients.
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The flat background RT-equations form a set of conservation laws
reflecting the maximal symmetry of the embedding space. In a more geometrically oriented language, using the extrinsic curvatures (i=4,...,N), the RT-equations take the form
The advantage of the Regge-Teitelboim approach is threefold:
• • •
All matter field equations remain intact. Energy/momentum is automatically conserved. Every Einstein solution is still an RT-solution.
Note: Only one RT-equation survives when embedding a 4-dim Universe in a 5-dim background.
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A Dark Interpretation ? Alternatively, one may invoke a constrained gravity formalism
In this language,
are kept independent,
with the latter denoting Lagrange multipliers.
It turn,
revealing an effective (conserved) energy/momentum tensor to be regarded a geometrical artifact of Geodesic Brane Gravity. It remains to be seen, however, if Effective = Dark ?
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Here is a brief reminder of RT cosmology (Λ4 brane) γT=PT/ρT
0
Total equation of state
ρd ρd0
1000
Dark ‘matter’ density
Λ-brane
n
io iat
100
rad
n
-1/3
-2/3
now
-1 0
1/3 Λτ 3
2
2/3
re
10
ΛCDM
1
Ωd,m
1
atu v r u c
1+z 1
2
3
4
5
The Hubble Plot
dL(z)
The age of the Universe
n=4 n=11/3 n=10/3 n=3 n=8/3 n=7/3 n=2
30
u
1.5
ΛCDM
25
M D C
20
1
Λ-brane
Λ
15
e n a
r
b Λ
10
0.5
5
now
0 0
0.2
Ω 0.4
0.6
0.8
1
d,m
0 0
2
4
6
8
z
The dark companion of Λ4 can be approximated by dark dust.
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Consider now a test brane (or a stealth brane, Cordero-Vilenkin ‘02) floating in a curved non-dynamical 5-dim background, and allow for different metrics on the two sides (i=L,R ) of the brane
Following Dirac, perform the implicit general coordinate transformation
such that in the z-frame, with bulk metrics
the brane is kept in rest during the variation process. The corresponding variations
do not deform the brane.
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Once bulk gravity is switched on, the gravitational fields themselves may vary where On the bulk, the arbitrariness of whereas the arbitrariness of
leads to Einstein equations, , the essence of re-parametrization
invariance, is ‘swallowed’, i.e. does not lead to any field equation. But can
be fully arbitrary on the brane?
• Arbitrary normal-normal and normal-tangent variations may violate Dirac’s linearity of the variation principle.
• Arbitrary (conventional) tangent-tangent variations would make y(x) redundant, thus driving the Dirac frame practically meaningless.
• Geroch-Traschen inconsistency for co-dimension >1 defects. Adopt restrictive Dirac boundary conditions (leading to relaxed junction conditions)
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Brane variation Dirac style Let the underlying action be
• The constrained gravity brane Lagrangian is sandwiched between the L,R bulk Lagrangians.
• The presence of the Gibbons-Hawking boundary terms is mandatory (to integrate out all
terms on the brane).
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Notations:
At the technical level, to make the various constraints manifest, in particular introduce a set of Lagrange multipliers, and add the following piece to the action
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• The variation with respect to
:
Project this equation on the brane, and orthogonal to the brane
• The variation with respect to
contains contributions from
both i=L,R and after taking all constraints into account we find
• The variation with respect to
lead, as expected, to Einstein
equations on the bulk.
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• The variation with respect to
(for L,R separately)
is where the restrictive Dirac boundary conditions are expected to play a non-trivial role.
Had we equated the r.h.s. of this equation to zero... 1. The NN-component would have reproduced
2. The NT-component would have vanished identically, 3. The TT-component would have resulted in
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Substituting the latter into the
equation would have led
us to the Israel junction conditions, and consequently to the Randall-Sundrum field equations
But in our case, we have
The only contribution comes from the TT-components, namely
Expressing
in terms of
and its derivatives, we find
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At this stage, no trace is left from Dirac’s frame (that is to show that it is no more than a mathematical tool), and the variation acquires the familiar Regge-Teitelboim form
Reflecting the fundamental embedding identity, the velocity and the covariant acceleration
are orthogonal
to each other. In turn, the above equation splits into: 1. Local conservation law
2. Geodesic field equation
The extrinsic curvature is the normal component of the acceleration, hence
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Relaxing the Israel junction conditions Define the combined Einstein-Israel tensor
such that our field equation reads
Given
the Regge-Teitelboim equation splits into
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t t
In two cases, the calculation of the Lagrange multipliers can be practically bypassed
t
• Smooth background such that Recover the original (reduced) RT-equation with
• Z2-symmetry such that Arrive at the generalized (full) RT-equation with the Collins-Holdom relation where
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On pedagogical grounds, define the asterisked tensors
The full RT-equation resembles then its reduced (albeit asterisked) version which, for the Z2-symmetric case, takes the form
Stemming from
conservation, the integrability condition of
the resulting RT-equation requires
(not necessarily
)
to be locally conserved on the brane
This is guaranteed in fact by the Coddazi relation
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Unified Brane Cosmology Let the cosmological FRW brane metric
be embedded within a Z2-symmetric AdS background of cosmological constant The extrinsic curvatures are
where
Given the pair
, and recall a previous discussion, it seems
tenable to define the asterisked pair
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The basic energy/momentum conservation law is given by
The Coddazi relation and the fact that
further imply
The cosmological evolution is then governed by the RT-equation
More explicitly, the second order differential equation to solve is
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Energy/momentum conservation and the fact that allow first integration of the form
The analytic solution can be borrowed from the original RT-theory
In analogy with the dark energy density defined via
which is nothing but the naive interpretation of a physicist equipped with Einstein equations, but totally ignorant of brane gravity, it is convenient to re-arrange the latter equation in the form
The asterisked dark energy density obeys the cubic equation
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Finally, we extract the cubic equation
which determines
as a function of the bare energy density.
The corresponding FRW equation takes the form
On the practical side, the above equations allow us to conveniently navigate within the
parameter space, with all special
cases easily accessible. As far as the classical equations of motion are concerned, we are particularly interested in deviations from the following limits:
• Regge-Teitelboim limit • Randall-Sundrum limit • Dvali-Gabadadze-Porrati and Collins-Holdom limit
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• The Regge-Teitelboim limit Letting
, the dark component is subject to
If the conserved RT-charge does not vanish, even an ‘empty’ brane would intriguingly admit a non-trivial dark component
A natural scale emerges
• At early times
, the evolution is governed by
negative pressure dark cosmic background.
• The inclusion of
is achieved via
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• Maximally symmetric brane limit What are the ingredients needed in order to end up with
We need
which can be translated into
In other words, one should have started from
Note: It has been demonstrated that such a radiation term can in fact be of a dynamical origin. This calls, however, At least at the framework of the original RT, for a minimally coupled scalar field accompanied by a special uniquely quartic potential.
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In particular, a flat brane calls (apart from k=0 and) for a double fine-tuning
To lift the radiation fine-tuning, one may start from
In turn, to the first order of
, we obtain
For a small enough scale factor, one serendipitously encounters a novel ‘inflation from radiation’ scenario, characterized by
The inflation terminates at
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• The Randall-Sundrum limit Taking
, one faces
Choosing the tenable + sign (such ambiguity is known to accompany the RS model), RS cosmology gets RT generalized
For small
(or large enough a), a novel dark radiation term makes
its appearance
Note that such a dark radiation term has nothing to do with the familiar RS radiation term which arises once Z2 is broken.
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• The Dvali-Gabadadze-Porrati limit Once
turns finite, it takes some algebra to single out the
particular solution of the master cubic equation which is the analytic continuation of the RS-favorite solution. Using the notation and assuming that the bare energy density is above critical (as otherwise another solution takes over), we find
and the small
expansion starts now with
Note that for large enough
, not only is Einstein gravity met
again, but the dark companion becomes quite different from a simple radiation term.
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