Download Gravity in Curved Phase-Spaces, Finsler Geometry and Two

Gravity in Curved Phase-Spaces, Finsler
Geometry and Two-Times Physics
Carlos Castro
Center for Theoretical Studies of Physical Systems
Clark Atlanta University, Atlanta, Georgia. 30314, [email protected]
August 2011, revised February 2012
Abstract
The generalized (vacuum) field equations corresponding to gravity on curved
2d-dimensional tangent bundle/phase spaces and associated to the geometry of
the (co)tangent bundle T Md−1,1 (T ∗ Md−1,1 ) of a d-dim spacetime Md−1,1 are
investigated following the strict formalism of Lagrange-Finsler and HamiltonCartan geometry. It is found that there is no mathematical equivalence with
Einstein’s vacuum field equations in spacetimes of 2d-dimensions, with two
times, after a d + d Kaluza-Klein-like decomposition of the 2d-dim scalar curvature R is performed and involving the introduction of a nonlinear connection
Aaµ (xµ , y b ). The physical applications of the 4-dim phase space metric solutions
found in this work, corresponding to the cotangent space of a 2-dim spacetime,
deserve further investigation. The physics of two times may be relevant in the
solution to the problem of time in Quantum Gravity and in the explanation
of Dark Matter. Finding nontrivial solutions of the generalized gravitational
field equations corresponding to the 8-dim cotangent bundle (phase space) of
the 4-dim spacetime remains a challenging task.
1
Introduction : Born’s Reciprocal Relativity in
Phase Space
Born’s reciprocal (”dual”) relativity [1] was proposed long ago based on the
idea that coordinates and momenta should be unified on the same footing, and
consequently, if there is a limiting speed (temporal derivative of the position
coordinates) in Nature there should be a maximal force as well, since force is the
temporal derivative of the momentum. A maximal speed limit (speed of light)
must be accompanied with a maximal proper force (which is also compatible
1
with a maximal and minimal length duality). The generalized velocity and
acceleration boosts (rotations) transformations of the 8D Phase space, where
X i , T, E, P i ; i = 1, 2, 3 are all boosted (rotated) into each-other, were given by
[2] based on the group U (1, 3) and which is the Born version of the Lorentz
group SO(1, 3).
The U (1, 3) = SU (1, 3) ⊗ U (1) group transformations leave invariant the
symplectic 2-form Ω = − dt∧dp0 +δij dxi ∧dpj ; i, j = 1, 2, 3 and also the following
Born-Green line interval in the 8D phase-space (in natural units h̄ = c = 1)
(dσ)2 = (dt)2 − (dx)2 − (dy)2 − (dz)2 +
1
b2
(dE)2 − (dpx )2 − (dpy )2 − (dpz )2
(1.1)
the rotations, velocity and force (acceleration) boosts leaving invariant the symplectic 2-form and the line interval in the 8D phase-space are rather elaborate,
see [2] for details. These transformations can be simplified drastically when
the velocity and force (acceleration) boosts are both parallel to the x-direction
and leave the transverse directions y, z, py , pz intact. There is now a subgroup
U (1, 1) = SU (1, 1) ⊗ U (1) ⊂ U (1, 3) which leaves invariant the following line
interval
(dE)2 − (dP )2
(dω)2 = (dT )2 − (dX)2 +
=
b2
F2
(dE/dτ )2 − (dP/dτ )2
2
2
= (dτ )
1 − 2
(dτ )
1 +
b2
Fmax
(1.2)
where one has factored out the proper time infinitesimal (dτ )2 = dT 2 − dX 2 in
(2.2). The proper force interval (dE/dτ )2 − (dP/dτ )2 = −F 2 < 0 is ”spacelike”
when the proper velocity interval (dT /dτ )2 − (dX/dτ )2 > 0 is timelike. The
analog of the Lorentz relativistic factor in eq-(2.2) involves the ratios of two
proper forces.
If (in natural units h̄ = c = 1) one sets the maximal proper-force to be given
by b ≡ mP Amax , where mP = (1/LP ) is the Planck mass and Amax = (1/Lp ),
then b = (1/LP )2 may also be interpreted as the maximal string tension. The
units of b would be of (mass)2 . In the most general case there are four scales
of time, energy, momentum and length that can be constructed from the three
constants b, c, h̄ as follows
r
r
r
√
h̄
h̄ c
h̄ b
λt =
; λl =
; λp =
; λe = h̄ b c
(1.3)
bc
b
c
The gravitational constant can be written as G = αG c4 /b where αG is a dimensionless parameter to be determined experimentally. If αG = 1, then the
four scales (2.3) coincide with the P lanck time, length, momentum and energy,
respectively.
The U (1, 1) group transformation laws of the phase-space coordinates X, T, P, E
which leave the interval (2.2) invariant are [2]
2
T 0 = T coshξ + (
ξa P sinhξ
ξv X
+
)
2
c
b2
ξ
(1.4a)
sinhξ
ξ
(1.4b)
ξa E sinhξ
)
b2
ξ
(1.4c)
sinhξ
ξv E
+ ξa T )
c2
ξ
(1.4d)
E 0 = E coshξ + (−ξa X + ξv P )
X 0 = X coshξ + (ξv T −
P 0 = P coshξ + (
ξv is the velocity-boost rapidity parameter and the ξa is the force (acceleration)
boost rapidity parameter of the primed-reference frame. These parameters are
defined respectively in terms of the velocity v = dX/dT and force f = dP/dT
(related to acceleration) as
tanh(
ξv
v
)= ;
c
c
tanh(
ξa
f
) =
b
Fmax
(1.5)
It is straightforwad to verify that the transformations (1.4) leave invariant
the phase space interval c2 (dT )2 − (dX)2 + ((dE)2 − c2 (dP )2 )/b2 but do not
leave separately invariant the proper time interval (dτ )2 = dT 2 − dX 2 , nor the
interval in energy-momentum space b12 [(dE)2 − c2 (dP )2 ]. Only the combination
2
(dσ)
2
= (dτ )
F2
1 − 2
Fmax
(1.6)
is truly left invariant under force (acceleration) boosts (1.4).
The physics of a limiting value of the proper acceleration in spacetime [4] has
been studied by Brandt [3] from the perspective of the tangent bundle geometry.
Generalized 8D gravitational equations reduce to ordinary Einstein-Riemannian
gravitational equations in the inf inite acceleration limit. A pedagogical monograph on Finsler geometry can be found in [9] and [10] where, in particular,
Clifford/spinor structures were defined with respect to nonlinear connections
associated with certain nonholonomic modifications of Riemann–Cartan gravity.
We explored in [5] some novel consequences of Born’s reciprocal Relativity
theory in flat phase-space and generalized the theory to the curved spacetime
scenario. We provided, in particular, six specific results resulting from Born’s
reciprocal Relativity and which are not present in Special Relativity. These are
: momentum-dependent time delay in the emission and detection of photons;
energy-dependent notion of locality; superluminal behavior; relative rotation
of photon trajectories due to the aberration of light; invariance of areas-cells
in phase-space and modified dispersion relations. A different approach to the
physics of curved momentum space has been undertaken by [25], [26], [27], [28].
3
The purpose of this work is to analyze the geometry of the curved (co)
tangent bundle within the context of the geometry of Lagrange-Finsler (tangent space) and Hamilton-Cartan spaces (phase spaces) and to compare it with
the Kaluza-Klein approach. We viewed Born’s reciprocal complex gravitational
theory in [6] as an 8D local gauge theory of the Quaplectic group [2] that is
given by the semi-direct product of the pseudo-unitary group U (1, 3) with the
Weyl-Heisenberg group involving four coordinates and momenta. This is not the
approach taken in this work. We shall work with real metrics, instead of complex metrics having symmetric real components and antisymmetric imaginary
ones [6]. We must emphasize that Reciprocity symmetry and U (1, 3) invariance
will not be invoked in this work.
2
Two-Times Physics and Phase Space Metrics
Through the years it has become evident that the (2, 2)-signature is not only
mathematically interesting [16] (see also Refs. [17], [18]) but also physically. In
fact, the (2, 2) signature emerges in several physical contexts, including self-dual
gravity a la Plebanski (see Ref. [19] and references therein), consistent N = 2
superstring theory as discussed by Ooguri and Vafa [20], and the N = (2, 1)
heterotic string [22].
In [21] it was shown how a N = 2 Supersymmetric Wess-Zumino-NovikovWitten model valued in the area-preserving (super) diffeomorphisms group is
Self Dual Supergravity in 2+2 and 3+1 dimensions depending on the signatures
of the base manifold and target space. The interplay among W∞ gravity, N = 2
Strings, self dual membranes, SU (∞) Toda lattices and SU ∗ (∞) Yang-Mills
instantons in 2 + 2 dimensions can be found also [21] .
More recently, using the requirement of SL(2, R) and Lorentz symmetries
it has been proved by [24] that the 2 + 2-target spacetime of a 0-brane is an
exceptional signature, where a special kind of 0-brane called quatl shows that
the 2 + 2-target spacetime can be understood either as the 2 + 2-world-volume
or as a 1 + 1 matrix-brane. Another recent motivation of physical interest
is that the 2 + 2-signature emerges in the discovery of hidden symmetries of
the Nambu-Goto action found by [23]. In fact, Duff was able to rewrite the
Nambu-Goto action in a 2 + 2-target spacetime in terms of Cayley’s hyperdeterminant, revealing apparently new hidden symmetries of such an action.
Duff’s observation has been linked with the matrix-brane idea [24]. Black-holelike solutions ( spacetimes with singularities ) of Einstein field equations in 3 + 1
and 2 + 2-dimensions were studied in [8].
Bars [15] has proposed a gauge symmetry in phase space. One of the consequences of this gauge symmetry is a new formulation of physics in spacetime.
Instead of one time there must be two times, while phenomena described by
one-time physics in 3 + 1 dimensions appear as various shadows of the same
phenomena that occur in 4 + 2 dimensions with one extra space and one extra
4
time dimensions (more generally, d + 2). Problems of ghosts and causality are
resolved automatically by the Sp(2, R) gauge symmetry in phase space.
Instead of working with d+2-dim spaces with two times [15] we shall propose
a different picture and view the two temporal directions as those corresponding
to the time and energy variables of phase space, whereas the remaining phase
space variables correspond to the spatial coordinates and momenta. Following
this premise, firstly, we shall search for candidate 4-dim phase space metrics of
signature (2, 2) which are associated with metrics in 4-dim curved spacetimes
of signature (2, 2) and that are solutions to the ordinary vacuum Einstein field
equations.
In the next section, we shall formulate the rigorous construction of the geometry of the cotangent bundle (phase space) of spacetime within the context of
the geometry of Hamilton-Cartan and Lagrange-Finsler spaces [9], [10], [11]. In
particular, we shall focus on Lagrange-Finsler and/or Hamilton-Cartan theories
constructed on (co) tangent bundles to Lorentz manifolds which are canonically
related to one-time physics on the base manifold (spacetime). Even though the
signature of the 8D metric may be of type (− + ++; − + ++), the geometric
and physical objects on the total space are canonical lifts from the Lorentz base
manifold. For instance, the canonical Cartan nonlinear connection. For such
models, the construction are determined by ”one temporal” dynamics/evolution
and nonholonomic constraints via a generalized relativity and covariance principle. Hence, one must distinguish these constructions from those theories based
on more general (co) tangent bundles with two independent time-like coordinates along the base manifold.
Hamilton-Cartan (dual) constructions are based on maps that are naturally
defined and related to Lagrange mechanics in the case of ”one-time” physics.
A multi-time Hamilton-Jacobi and Euler-Lagrange theory involving multi-time
extensions of symplecto-morphisms and Legendre transforms can be found in
[35], [36] and references therein. A monograph on a multi-temporal theory of
Relativity and its physical implications can be found in [34]. When there are
multiple times, for example two-times and four spatial dimensions in a higherdim 6D space, one can have particles outside the light-cones of other particles,
so that these appear dark with respect to each other and it might explain the
origin of dark matter [37], [38]. The problem is that Gravitational interactions
are mediated by gravitons that also travel at the speed of light so this raises
the question : Why can those particles, outside the light-cones, be affected by
gravity (the gravitons) and not by Electromagnetism ( photons ) ?
More precisely, let us imagine 3 objects A, B, C such that B, C are both
dark with respect to A, but are not dark with respect to each other. Therefore,
B and C can interact gravitationally, meaning that B can bend the light sent
by C. However, the question remains : how is A able to see that (bent) light
which is being received by B and sent from C ? The bending of light ( at
B ) would occur within the light-cone of B and which is not accessible to A.
To answer this question one may argue that the particles are confined to live
along different four-dim sheets (branes) embedded in the 6D space, so gravity
(gravitons) propagates in the 6D bulk of the 6D space, while EM (the photons)
5
only propagates along the four-dim sheets (branes), like it occurs in the braneworld models. Since the multi-time Hamilton-Jacobi description of [35], [36]
involves a multi-sheeted temporal structure, this possibility warrants further
investigation.
In this work we shall not be concerned with these more general extensions
of ordinary mechanics and Relativity involving multi-times. Let us begin with
4-dim spacetimes of signature (2, 2). Given a 4-dim spacetime of signature
+, +, −, − (two times, t1 , t2 ), a parametrization of the t2 , x, y coordinates in
terms of the variables
r ≥ 0;
− ∞ ≤ ξ ≤ +∞; 0 ≤ θ ≤ 2π;
− ∞ ≤ t2 ≤ +∞
(2.1)
y = r cosh ξ sin θ.
(2.2)
given by
t2 = r sinh ξ;
x = r cosh ξ cos θ;
2
where r is the variable throat size of the 2-dim hyperboloid H embedded in
3-dim and that can be defined analytically in terms of t2 , x, y as
−(t2 )2 + x2 + y 2 = r2 .
(2.3)
shows that the flat spacetime metric
(ds)2 = − (dt1 )2 − (dt2 )2 + (dx)2 + (dy)2
(2.4)
can be recast in terms of the new coordinates as
ds2 = −(dt1 )2 + (dr)2 + r2 [ cosh2 ξ (dθ)2 − (dξ)2 ].
(2.5)
Notice that we have a two times (−, +, +, −) signature in eq-(2.5), as one should.
The topology corresponding to metric in eq-(2.5) is R × R∗ × H2 . R∗ is the
half-interval [0, ∞] representing the values of the radial coordinate. One temporal variable t1 is characterized by the real line R and whose values range
from (−∞, +∞), and the other temporal variable t2 is one of the 3 coordinates
(t2 , x, y) which parametrize the two-dim hyperboloid H2 described by eq-(2.3).
A curved spacetime version of the metric of eq-(2.5) is
ds2 = −eµ(r) (dt1 )2 + eν(r) (dr)2 + (R(r))2 [ cosh2 ξ (dθ)2 − (dξ)2 ]. (2.6)
The metric in eq-(2.6) whose signature is (2, 2) = (−, +, +, −) is the hyperbolic
version of the Schwarzschild metric. In general, due to radial reparametrization
invariance, one can replace the radial variable r → R(r) for another variable
called the area-radius, since Einstein’s equations do not determine the form of
the radial function R(r) as explained in the appendix. We still must determine
what are the functional forms of µ(r) and ν(r). In the appendix we find the
solutions to Einstein’s vacuum field equations in D-dimensions for metrics associated with a D − 2-dim homogeneous space of constant positive (negative
) scalar curvature. In particular when D = 4 and the two-dim homogeneous
6
space H2 has a constant positive scalar curvature, like two-dim de Sitter space,
the metric components, in natural units G = h̄ = c = 1, are given by
2M
2M −1
) (dt)2 + (1 −
) (dr)2 + r2 ( cosh2 ξ (dθ)2 − (dξ)2 )
r
r
(2.7)
The 2-dim hyperboloid homogeneous space whose metric is defined by (ds)2 =
r2 (cosh2 ξ (dθ)2 − (dξ)2 ) coincides with the metric of a 2-dim de Sitter space
of constant positive scalar curvature and whose throat-size is r. Anti de Sitter
space has a constant negative scalar curvature. There is a physical singularity
at r = 0, the location of the point mass source, when the hyperboloid H2
degenerates to a cone since the throat size r has been pinched to zero.
By postulating that the momentum coordinates E, P are curled-up along
the 2-dim de Sitter space allows to set the correspondence λEe ↔ ξ; λPp ↔ θ,
while the spacetime coordinates correspondence is X ↔ r; T ↔ t. λp , λe are the
scales of momentum and energy that can be constructed from the three constants
b, c, h̄ as described in eq-(1.3). A nontrivial metric associated to a curved 4D
phase-space and which is a putative solution to the vacuum field (generalized
gravitational) equations corresponding to the cotangent bundle T ∗ M of a twodim spacetime M is
(ds)2 = − (1 −
2M
2M −1
E (dP )2
(dE)2
) (dT )2 + (1−
) (dX)2 + X 2 [ cosh2 ( )
−
]
2
X
X
λe
λp
λ2e
(2.8)
One has the correct (2, 2) signature −, +, +, − and the domain of values of the
X, T, E, P variables are respectively
(ds)2 = −(1−
X ≥ 0;
− ∞ ≤ T ≤ +∞;
−∞≤
E
P
≤ +∞; 0 ≤
≤ 2π
λe
λp
(2.9)
A different identification for the phase space variables can be found from the
following correspondence
−(t2 )2 + x2 + y 2 = r2 ↔ −e2 + x2 + p2 = r2 = X 2
so that
T = t1 = t; e = t2 = r sinhξ = X sinh(
x = r coshξ cosθ = X cosh(
E
P
) cos( )
λe
λp
p = r coshξ sinθ = X cosh(
E
P
) sin( )
λe
λp
E
)
λe
P
p
1
p
= θ = arctan( ) ⇒ dθ =
d( )
λp
x
1 + (p/x)2
x
7
(2.10)
(2.11)
(2.12)
(2.13)
(dX)2 = (dr)2 = [−e2 + x2 + p2 ]−1 (−ede + xdx + pdp)2 ;
r2 (dξ)2 = X 2
x2
(2.14)
(dE)2
=
λ2e
1
e2
[ (−e2 + x2 + p2 )(de)2 +
(−ede + xdx + pdp)2 ] −
2
2
+p
−e + x2 + p2
1
[ 2e de (−ede + xdx + pdp) ]
(2.15)
x2 + p2
etc .... leading to a very complicated expression for the metric (2.8) rewritten
in terms of the t, x, e, p variables. In the f lat phase limit, M = 0, the metric
(2.8) can be rewritten in the standard pseudo-Euclidean form
(ds)2 = −(dt)2 −(de)2 +(dx)2 +(dp)2 ↔ −(dt1 )2 −(dt2 )2 +(dx)2 +(dy)2 (2.16)
as expected.
We have shown that a particular 4D metric of topology R × S 1 × R × S 1
which solves the 4D vacuum Einstein field equations in spaces of signature (2, 2)
is of the form [8]
ds2 = − (λl /ρ) (dt1 )2 + (ρ/λl ) (dρ)2 + ρ2 ( (dθ)2 −
(dt2 )2
) =
λ2t
−(λl /ρ) (dt1 )2 + (ρ/λl ) (dρ)2 + ρ2 ( (dθ)2 − (dφ)2 )
(2.17)
where φ ≡ t2 /λt and λt , λl are constants. By assigning in eq-(2.17) the phase
space variables
t1 = T ; ρ = X; θ =
P
E
; t2 = E; φ =
;
λp
λe
(2.18)
the 4-dim phase space metric of signature (2, 2) becomes
(ds)2 = −
2
X
(dE)2
(dT )2
2 (dP )
+
(dX)2 − X 2
+
X
(X/λl )
λl
λ2e
λ2p
(2.19)
where λt , λe , λp , λl are the four scales of time, energy, momentum and length
that can be constructed from the three constants b, c, h̄ as described in eq-(1.3).
In natural units h̄ = c = G = b = 1 one has that λt = λe = λp = λl = 1.
Since the cotangent space of the circle S 1 is a cylinder R × S 1 , the 4D space
of topology R × S 1 × R × S 1 can be interpreted as the cotangent bundle (phasespace) of the 2-dim torus S 1 × S 1 . The latter torus is also the Shilov boundary
of the 4D phase space. Another approach to gravity in curved phase spaces
from the perspective of the geometry of bounded homogenous complex domains
8
has been undertaken by [30]. Phase space is interpreted as the upper tubular
regions of these complex domains, like the Siegel upper half plane (obtained
from a conformal map of the Poincare disk), and spacetime corresponds to
their Shilov boundary. Applications of curved phase spaces in Quantum Field
Theories can be found in [31].
Physically, eq-(2.18) assigns the energy-momentum E, P variables to be confined along the two-torus S 1 × S 1 , and the space-time T, X variables are the
fibers of the cotangent space of the two-torus. Therefore, the 4D phase-space is
the cotangent bundle of the two-torus whose topology is R × S 1 × R × S 1 . One
may reverse the assignment so that the space-time T, X variables are confined
along the two-torus S 1 × S 1 , while the energy-momentum E, P variables are
the fibers of the cotangent space of the torus, setting aside for the moment the
caveat behind having closed-timelike variables. One can bypass this problem
by going to the covering space of the circle, like it is done with the temporal
variable in AdS spacetime.
As shown in the Appendix, another metric solution in D = 4 when the twodim homogeneous space has a constant negative scalar curvature, like two-dim
de Anti de Sitter space AdS2 , in natural units G = h̄ = c = 1, is given by
2M −1
2M
) (dt)2 − (1 +
) (dr)2 + r2 ( − cosh2 ξ (dθ)2 + (dξ)2 )
r
r
(2.20)
in this case one has a signature f lip from the signature of the metric in eq-(2.7).
By exchanging r ↔ t, like it occurs with the Kantowski-Sachs cosmological
solution, eq-(2.20) becomes
(ds)2 = (1 +
2M
2M −1
) (dr)2 − (1 +
) (dt)2 + t2 ( − cosh2 ξ (dθ)2 + (dξ)2 )
t
t
(2.21)
If the energy-momentum coordinates E, P are curled-up along the 2-dim Anti
de Sitter space, it allows to set the correspondence λEe ↔ θ; λPp ↔ ξ, and
X ↔ r; T ↔ t so that eq-(2.21) becomes
(ds)2 = (1 +
2M
2M −1
P (dE)2
(dP )2
) (dX)2 − (1+
) (dT )2 + T 2 [ − cosh2 ( )
+
]
2
T
T
λp
λe
λ2p
(2.22)
which is another nontrivial metric associated to a curved 4-dim phase-space
and which is a putative solution to the vacuum field (generalized gravitational)
equations corresponding to the cotangent bundle T ∗ M of a two-dim spacetime
M . One has the correct (2, 2) signature +, −, −, + in eq-(2.22) : (dX)2 , (dP )2
appear with a positive sign, whereas (dT )2 , (dE)2 appear with a negative sign.
The domain of values of the X, T, P, E variables in eq-(2.22) are respectively
(ds)2 = (1+
X ≥ 0;
− ∞ ≤ T ≤ +∞;
−∞≤
9
E
P
≤ +∞; 0 ≤
≤ 2π
λp
λe
(2.23)
To sum up, we have displayed three nontrivial metrics in phase space given
by eqs-(2.8, 2.19, 2.22) with split signature (2, 2) and which are obtained from
solutions to the 4-dim Einstein field equations in spacetimes of signature (2, 2)
by identifying one of the temporal variables with the energy and one of the
spatial variables with the momentum. In the next section we shall study the
geometry of the (co) tangent space within the framework of Lagrange-Finsler
geometry and Hamilton-Cartan spaces and explore the field equations in order
to verify the validity of the proposed metric solutions gIJ (xi , pi ) in phase spaces,
where the indices span i, j = 1, 2, 3, ....d and I, J = 1, 2, 3, ...., 2d.
3
A Kaluza-Klein like approach
Some time ago, it was shown by [7] that a Kaluza-Klein-like formalism of Einstein’s theory, based on the (2+2)-fibration of a generic 4-dimensional spacetime,
describes General Relativity as a Yang-Mills gauge theory on the 2-dimensional
base manifold, where the local gauge symmetry is the group of the diffeomorphisms of the 2-dimensional fibre manifold. They found the Schwarzschild solution by solving the field equations after a very laborious procedure. Their formalism was valid for any d+n decomposition of the D-dim spacetime D = d+n.
The Kaluza-Klein-like formalism associated with the line element in d + d dimensions is parametrized as follows
ds2 = φab dy a dy b + (γµν + φab Aaµ Abν ) dxµ dxν + 2φab Abµ dxµ dy a . (3.1)
where all fields depend on the xµ , y a coordinates. In particular, the (nonlinear) gauge connection is given by Aaµ (xρ , y b ) and the span of indices is µ, ν, ρ =
1, 2, ....., d and a, b, c = 1, 2, ....., d. To find the (d + d)-dimensional action principle of general relativity we must compute the scalar curvature of space-times
in the (d + d)-decomposition. For this purpose it is convenient to introduce the
following non-holonomic (non-coordinate basis) ∂ˆA = (∂ˆµ , ∂ˆa ) where
∂ˆµ ≡ ∂µ − Aµa (xρ , y b ) ∂a ,
∂ˆa ≡ ∂a .
(3.2)
From this definition we have
[ ∂ˆA , ∂ˆB ] = fABC (xρ , y b ) ∂ˆC ,
where the structure coefficients (non-holonomic coefficients) fABC are given by
fµνa = − Fµνa , fµab = − faµb = ∂a Aµb ,
fABC = 0, otherwise
(3.3)
The virtue of this non-holonomic basis is that it brings the metric (3.1) into a
block diagonal form
γµν
0
gAB =
,
(3.4)
0
φab
10
which drastically simplifies the computation of the scalar curvature. In this
non-holonomic (non-coordinate) basis the Levi-Civita connection is given by
ΓABC =
1
1 CD ˆ
g (∂A gBD +∂ˆB gAD −∂ˆD gAB )+ g CD ( fABD −fBDA −fADB ) (3.4)
2
2
where fABC = gCD fABD . The field strength Fµνa corresponding to the (nonlinear) gauge connection Aµa is defined as
Fµνa = ∂µ Aaν − ∂ν Aaµ − Acµ ∂c Aaν + Acν ∂c Aaµ
(3.5)
For completeness, the connection coefficients components are [7]
Γµνα =
1 αβ ˆ
∂µ γνβ + ∂ˆν γµβ − ∂ˆβ γµν
γ
2
1
1
Γµνa = − φab ∂b γµν − Fµνa
2
2
Γµaν = Γaµν =
1 να
1
γ ∂a γµα + γ να φab Fµαb
2
2
Γµab =
1
1
1 bc ˆ
φ ∂µ φac + ∂a Aµb − φbc φae ∂c Aµe
2
2
2
Γaµb =
1
1
1 bc ˆ
φ ∂µ φac − ∂a Aµb − φbc φae ∂c Aµe
2
2
2
1
1
1
Γabµ = − γ µν ∂ˆν φab + γ µν φac ∂b Aνc + γ µν φbc ∂a Aνc
2
2
2
1
Γabc = φcd ∂a φbd + ∂b φad − ∂d φab .
2
The Torsion is defined as
A
A
A
TBC
= ΓA
BC − ΓCB − fBC
(3.6)
(3.7a)
giving vanishing torsion components, consistent with the fact that the LeviCivita connection is torsionless by definition. For example, from eqs-(3.3,3.6)
one arrives at the vanishing values
a
a
a
Tµν
= − Fµν
+ Fµν
= 0,
b
b
Tµa
= − Taµ
= ∂a Abµ − ∂a Abµ = 0, ...... (3.7b)
The curvature tensors are defined as
RABCD = ∂ˆA ΓBCD − ∂ˆB ΓACD + ΓAED ΓBCE − ΓBED ΓACE − fABE ΓECD
RAC = RABCB ,
11
R = g AC RAC .
(3.8a)
Explicitly, the scalar curvature R is given by
R = γ µν (Rµανα + Rµaνa ) + φab (Racbc + Raµbµ )
(3.8b)
which becomes, after a very lengthy computation
1
φab γ µν γ αβ Fµαa Fνβb +
4
o
1 µν ab cd n
γ φ φ (Dµ φac )(Dν φbd ) − (Dµ φab )(Dν φcd ) +
4
o
1 ab µν αβ n
(∂a γµα )(∂b γνβ ) − (∂a γµν )(∂b γαβ ) + ∇A j A
(3.9)
φ γ γ
4
where the ”gauged” Ricci tensor Rµν in the base manifold and the internal
space Ricci tensor Rac are defined by
R = γ µν Rµν + φac Rac +
Rµν = ∂ˆµ Γανα − ∂ˆα Γµνα + Γµβα Γανβ − Γβαβ Γµνα
Rac = ∂a Γbcb − ∂b Γacb + Γadb Γbcd − Γdbd Γacb .
(3.10)
The derivative terms ∇A j A in (3.9) are
∇A j A = ∇µ j µ + ∇a j a , ∇µ j µ = ∂ˆµ + Γαµα + Γcµc j µ
∇a j a = ∂a + Γcac + Γαaα j a ,
where j µ and j a are given by
j µ = γ µν φab ∂ˆν φab − 2∂a Aνa ,
j a = φab γ µν ∂b γµν .
(3.11)
(3.12)
As shown by [7], the line element eq-(3.1) in 4 = 2 + 2 dimensions, in the light
cone coordinates
Aau
1
v = √ (t − r).
2
1
Aav = √ (Aat − Aar ).
2
1
u = √ (t + r),
2
1
= √ (Aat + Aar ),
2
(3.13)
(3.14)
after using the Polyakov ansatz
gµν =
−2 h(t, r) −1
−1
0
(3.15)
becomes
ds2 = φab dy a dy b − 2du dv − 2h(u) du2 +
φab (Aau du + Abv dv) (Abu du + Abv dv) + 2φab (Aau du + Abv dv) dy a .
12
(3.16)
Upon setting φab = eσ ρab such that det ρab = 1 and after a very laborious
calculation Yoon [7] arrived finally at the expression for the scalar curvature in
the light-cone coordinates given by
1 2σ
a
b
e ρab F+−
F+−
+ eσ R2 + eσ D+ σ D− σ −
2
1 σ ab cd
1 σ ab cd
e ρ ρ (D+ ρab ) (D− ρcd ) +
e ρ ρ (D+ ρac ) (D− ρbd ) +
2
2
1
1
2
2h++ eσ [ D−
σ + (D− σ)2 + ρab ρcd (D− ρac ) (D− ρbd ) ]
(3.17)
2
4
plus surface terms. The Lie-bracket is
R = −
[ Aµ , φab ] = (∂a Acµ (xµ , y a )) φbc (xµ , y a ) + (∂b Acµ (xµ , y a )) φac (xµ , y a ) +
Acµ (xµ , y a ) ∂c φab (xµ , y a ).
(3.18)
the Yang-Mills-like field strength is
a
Fµν
= ∂µ Aaν − ∂ν Aaµ − [Aµ , Aν ]a =
∂µ Aaν − ∂ν Aaµ − Acµ ∂c Aaν + Acν ∂c Aaµ .
(3.19)
The covariant derivative of a tensor density ρab with weight 1 is
Dµ ρab = ∂µ ρab − [ Aµ , ρ ]ab + (∂c Acµ ) ρab =
∂µ ρab − Acµ ∂c ρab − (∂a Acµ ) ρcb − (∂b Acµ ) ρac + (∂c Acµ ) ρab .
(3.20)
the covariant derivative on the scalar density Ω = eσ of weight −1 is
Dµ Ω = ∂µ Ω − Aaµ ∂a Ω − (∂a Aaµ )Ω ⇒ .
(3.21)
Dµ σ = ∂µ σ − Aaµ ∂a σ − (∂a Aaµ ).
(3.22)
σ
after factoring the e terms.
The authors [7] were able to solve the equations of motion associated with
the Einstein-Hilbert action
Z
S =
du dv d2 y R.
(3.23)
by varying the Einstein-Hilbert action bef ore imposing the light-cone gauge
fixing conditions and det ρab = 1 giving a total of 10 equations for the 10 fields
σ, h++ , h−− , Aa+ , Aa− , ρab with a, b = 1, 2. These 10 fields match the same number of idependent components of the metric gµν in 4D. After a very laborious
procedure the authors found solutions for the ”vacuum” field configurations
Aa+ = 0, Aa− = 0 given by
ds2 = 2 du dv − (1 −
2GM
) dv 2 + u2 dΩ2 .
u
13
(3.24)
2GM
) du2 + v 2 dΩ2 .
(3.25)
v
which have the same f unctional f orm as the Schwarzschild-Hilbert solution
in the retarded and advanced temporal Eddington-Finkelstein coordinates, u =
t − r∗ , v = t + r∗
ds2 = − 2 du dv − (1 −
2GM
) dv 2 + r2 dΩ2 .
(3.26)
r
2GM
ds2 = − 2 du dr − (1 −
) du2 + r2 dΩ2 .
(3.27)
r
with the subtle technicality that r∗ appearing in the definitions u = t − r∗ , v =
t + r∗ in eqs-(2.15, 2.16) is the tortoise radial coordinate r∗ (r) given by
ds2 = 2 dr dv − (1 −
Z
Z
dr∗ =
r
dr
⇒ r∗ = r + 2GM ln |
− 1|.
1 − 2GM/r
2GM
(3.28)
It is well known [29] that one can introduce afterwards the Fronsdal-KruskalSzekeres coordinates in terms of the Eddington-Finkelstein coordinates and
which allow an analytical extension of the Schwarzschild-Hilbert solution into
the interior region of the black hole beyond the horizon.
The solutions in eqs-(3.24-3.27) reflect the spherical symmetry where r2 (dΩ)2
is the standard metric of the two-dim sphere S 2 . In the split signature case (2, 2),
one must replace the r2 (dΩ)2 metric interval by the one corresponding to an
internal two-dim hyperboloid H2 , a two-dim de Sitter space dS2 , and given by
r2 (cosh2 ξ(dθ)2 − (dξ)2 ) as indicated by eq-(2.7).
Secondly, in order to avoid confusion with the notation, let us label t0 , r0 for
the variables of the 1 + 1 dim spacetime M1+1 used by [7], such that the equivalence of eq-(3.25) with eq-(3.27) is established by setting the correspondence
t0 − r0 ↔ r;
t0 + r0 ↔ t − r∗ = t − ( r + 2GM ln |
r
− 1| ) (3.29)
2GM
similar equivalence of eq-(3.24) with eq-(3.26) is established by setting the correspondence
t0 + r0 ↔ r;
t0 − r0 ↔ t + r∗ = t + ( r + 2GM ln |
r
− 1| ) (3.30)
2GM
To sum up, the equivalence among the above metric solutions corroborates once
more that the metric (2.7) is a solution of the 4D Einstein field equations, as
well as being a solution to the field equations after a Kaluza-Klein-like 2 + 2
decomposition of the 4D space is performed, in the split signature (2, 2) case.
Having written the explicit 2 + 2 decomposition and above solutions in eqs(3.26, 3.27), one may assign the internal variables, that parametrize the hyperbolic two-dim de Sitter space dS2 , to correspond to the energy E and momentum
P , and the 1 + 1 base spacetime variables to correspond to the X, T coordinates
of the 4-dim phase space. In order to have consistent units one must have that
14
y a ↔ pa /b; ∂/∂y a ↔ b ∂/∂pa and Aai ∂ya ↔ bAia ∂pa , with i, j = 1, 2; a = 1, 2.
In natural units h̄ = G = c = 1 it yields b = 1, as shown in eq-(1.3), such that
it simplifies matters.
Concluding, the 4-dim curved phase-space metric (2.8) is a viable solution
to the gravitational field equations following the Kaluza-Klein-like 2 + 2 decomposition and after identifying the internal coordinates with the E, P variables.
The other solutions in eqs-(2.19, 2.22) are harder to verify. A more rigorous
method to find and verify solutions to the generalized gravitational field equations in tangent spaces/phase spaces requires to study directly the geometry of
M
the (co)tangent bundle T1+1
(T ∗ M1+1 ) of the two-dim spacetime M1+1 , rather
than following the Kaluza-Klein-like method of [7] described above. We find
that the Kaluza-Klein decomposition is not equivalent to the more rigorous
methods of Finsler-Lagrange and Hamilton-Cartan spaces [9]. We proceed to
show this in the next section. See also the related work by [3].
4
Gravity in Curved Phase Spaces, LagrangeFinsler and Hamilton-Cartan Spaces
In this section we shall present the essentials behind the geometry of the tangent
and cotangent space. We will follow closely the description by authors [9], [10],
[11]. The metric associated with the tangent space T Md can be written in the
in the following block diagonal form
(ds)2 = gij (xk , y a ) dxi d xj + hab (xi , y a ) δy a δy b
(4.1)
(i, j, k = 1, 2, 3, ....d; a, b, c = 1, 2, 3, ....d) if instead of the standard coordinatebasis one introduces the anholonomic frames (non-coordinate basis) defined as
δi = ∂i − Nib (x, y) ∂b = ∂/∂xi − Nib (x, y) ∂b ;
∂a =
∂
dy a
(4.2)
and its dual basis is
δ α ≡ δuα = (δ i = dxi , δ a = dy a + Nka (x, y) dxk )
(4.3)
where the N –coefficients define a nonlinear connection, N–connection structure,
see details in [9], [10], [11]. As a very particular case one recovers the ordinary
linear connections if Nia (x, y) = Γabi (x) y b .
The N–connection structures can be naturally defined on (pseudo) Riemannian spacetimes and one can relate them with some anholonomic frame fields
γ
(vielbeins) satisfying the relations δα δβ − δβ δα = Wαβ
δγ , with nontrivial anholonomy coefficients
k
k
k
c
Wijk = 0; Waj
= 0; Wia
= 0; Wab
= 0; Wab
=0
15
a
b
Wija = − Ωaij ; Wbj
= − ∂b Nja ; Wia
= ∂a Njb
(4.4)
Ωaij = δj Nia − δi Nja
(4.5)
where
is the nonlinear connection curvature (N–curvature). This is the same object as
a
Fµν
described in the previous section when Nja ↔ Aaµ .
A metric of type given by eq-(4.1) with arbitrary coefficients gij (xk , y a ) and
hab (xk , y a ) defined with respect to a N –elongated basis is called a distinguished
metric. A linear connection Dδγ δβ = Γαβγ (x, y)δα associated to an operator of
covariant derivation D is such that is compatible with a metric gαβ and N –
connection structure on a pseudo–Riemannian spacetime. The compatibility of
D with the nonlinear connection N (i.e., preserving the distributions generated
by N ) and the metricity of D are independent conditions; i.e. the relations
Dα (gβγ ) = 0 are only equivalent to metricity, but do not say anything about N
-compatibility. 1
The anholonomic coefficients wγαβ and N–elongated derivatives yield nontrivial coefficients for the torsion tensor, T (δγ , δβ ) = T αβγ δα . The Torsion is
T αβγ = Γαβγ − Γαγβ + wαβγ ,
(4.6)
and the curvature tensor R(δτ , δγ )δβ = Rβαγτ δα
Rβαγτ = δτ Γαβγ − δγ Γαβτ + Γσβγ Γαστ − Γσβτ Γασγ + Γαβσ wσγτ
(4.7)
One should note the key presence of the last terms in eqs-(4.6, 4.7) due to the
non-vanishing anholonomic coefficients wγαβ . One is not accustomed to see
this term in ordinary textbooks. The same occurred in section 3 due to the
evaluation of the curvature and torsion in the non-holonomic basis.
The torsion distinguished
tensor has the following
irreducible, non-vanishing,
h–v–components, T αβγ = T ijk , C ija , S abc , T aij , T abi given by
i
i
i
i
i
Tjk
= Lijk − Likj ; Tja
= Cja
; Taj
= − Cja
i
a
a
a
a
Tja
= 0; Tbc
= Sbc
= Cbc
− Ccb
Tija = − Ωaij ; Tbia = ∂b Nia − Labi ; Tiba = − Tbia
(4.8)
and where Ωaij = δj Nia − δi Nja can be interpreted as the ”field strength” associated with the nonlinear connection Nia .
A canonical distinguished connection is also metric compatible, but the Tori
sion vanishes only along the horizontal and vertical subspaces; i.e. T̂jk
= 0 and
a
i
a
T̂bc = 0, but it is non-vanishing for the other mixed components T̂ja , T̂bia , T̂ji
.
It is denoted by a hat Γ̂ and is parametrized
(hori by the following irreducible
zontal, vertical ) h–v–components, Γ̂αβγ = L̂i jk , L̂abk , Ĉ ijc , Ĉ abc such that [9],
[10], [11].
1 we
thank one of the referees for insisting on this point.
16
1 in
g (δk gnj + δj gnk − δn gjk )
2
1
L̂abk = ∂b Nka + hac δk hbc − hdc ∂b Nkd − hdb ∂c Nkd
2
1
1
Ĉ ijc = g ik ∂c gjk ; Ĉ abc = had (∂c hdb + ∂b hdc − ∂d hbc ) .
(4.9)
2
2
The curvature distinguished tensor associated with the distinguished connection has the following irreducible, nonvanishing h–v–components Rβαγτ =
i
a
i
c
i
a
Rhjk
, Rbjk
, Pjka
, Pbka
, Sjbc
, Sbcd
given by [9], [10], [11]
L̂i jk =
i
i
m i
i
a
Rhjk
= δk Lihj − δj Lihk + Lm
hj Lmk − Lhk Lmj − Cha Ωjk
a
a c
Rbjk
= δk Labj − δj Labk + Lcbj Lack − Lcbk Lacj − Cbc
Ωjk
i
i
b
Pjka
= ∂a Lijk + Cjb
Tka
−
i
l
i
i
δk Cja
+ Lilk Cja
− Lljk Cla
− Lcak Cjc
c
c
d
Pbka
= ∂a Lcbk + Cbd
Tka
−
c
d
c
c
δk Cba
+ Lcdk Cba
− Ldbk Cda
− Ldak Cbd
i
i
i
h i
h i
Sjbc
= ∂c Cjb
− ∂b Cjc
+ Cjb
Chc − Cjc
Chb
a
a
a
e
a
e
a
Sbcd
= ∂d Cbc
− ∂c Cbd
+ Cbc
Ced
− Cbd
Cec
(4.10)
In the canonical distinguished connection case, one replaces all the quantities
in (4.10) by a hat. Those components were given in eq-(4.9).
Having reviewed the geometry of the tangent bundle T M we proceed with
the cotangent bundle case T ∗ M (phase space). In the case of the cotangent
space of a d-dim manifold T ∗ Md the metric can be equivalently rewritten in the
block diagonal form [9] as
(ds)2 = gij (xk , pa ) dxi d xj + hab (xk , pc ) δpa δpb
(4.11)
i, j, k = 1, 2, 3, .....d, a, b, c = 1, 2, 3, .....d, if instead of the standard coordinate
basis one introduces the following anholonomic frames (non-coordinate basis)
δi = δ/δxi = ∂xi + Nia ∂ a = ∂xi + Nia ∂pa ; ∂ a ≡ ∂pa =
∂
(4.12)
∂pa
One should note the key position of the indices that allows us to distinguish
between derivatives with respect to xi and those with respect to pa . The dual
basis of (δi = δ/δxi ; ∂ a = ∂/∂pa ) is
dxi , δpa = dpa − Nja dxj
17
(4.13)
where the N –coefficients define a nonlinear connection, N–connection structure.
An N-linear connection D on T ∗ M can be uniquely represented in the adapted
basis in the following form
k
a
Dδj (δi ) = Hij
δk ; Dδj (∂ a ) = − Hbj
∂b;
(4.14)
D∂ a (δi ) = Cika δk ; D∂ a (∂ b ) = − Ccba ∂ c
(4.15)
a
k
(x, p), Hbj
(x, p), Cika (x, p), Ccba (x, p) are the connection coefficients.
where Hij
For a N-linear connection D with the above coefficients, the torsion 2-forms
are
1 i
T dxj ∧ dxk + Cjia dxj ∧ δpa
2 jk
1
1
b
dxj ∧ δpb + Sabc δpb ∧ δpc
Ωa = Rjka dxj ∧ dxk + Paj
2
2
and the curvature 2-forms are
Ωi =
(4.16)
Ωij =
1
1 i
ia
Rjkm dxk ∧ dxm + Pjk
dxk ∧ δpa + Sjiab δpa ∧ δpb
2
2
(4.17a)
Ωab =
1 a
1
ac
R
dxk ∧ dxm + Pbk
dxk ∧ δpc + Sbacd δpc ∧ δpd
2 bkm
2
(4.17b)
where one must recall that the dual basis of δi = δ/δxi , ∂ a = ∂/∂pa is given by
dxi , δpa = dpa − Nja dxj .
The distinguished torsion tensors are of the form [9]
i
i
i
a
a
Tjk
= Hjk
− Hkj
; Scab = Ccab − Ccba ; Pbj
= Hbj
− ∂ a Njb
δNia
δNja
−
(4.18)
δxi
δxj
a
the last tensor Rija has a one to one correspondence with the field strength Fµν
of section 3. The distinguished tensors of the curvature are of the form
Rija =
i
i
i
l
i
l
i
Rkjh
= δh Hkj
− δj Hkh
+ Hkj
Hlh
− Hkh
Hlj
− Ckia Rjha
ab
b
b
Pcj
= ∂ a Hcj
+ Ccad Pdj
−
b
a
d
δj Ccab + Hdj
Ccda + Hdj
Ccbd − Hcj
Cdab
k
Pijak = ∂ a Hij
+ Cial Tljk −
a
k
l
δj Ciak + Hbj
Cibk + Hlj
Cial − Hij
Clak
18
Sdabc = ∂ c Cdab − ∂ b Cdac + Cdeb Ceac − Cdec Ceab ; etc..........
(4.19)
where we have omitted the other components and once again we have for our
notation ∂ a = ∂/∂pa and δ/δxi = ∂xi + Nia ∂ a . Equipped with these curvature tensors one can perform suitable contractions involving gij , hij , to obtain
curvature scalars of the R, S type
i
R = δij Rkjl
g kl ; S = δbd Sdabc hac
(4.20)
and construct a phase space Lagrangian involving a linear combination of the
curvature scalars L = aR + bS, where a, b are real-valued numerical coefficients.
Similarly, one can perform suitable contractions in the tangent bundle case to
build curvature scalars
i
a
R = δij Rkjl
g kl ; S = δac Sbcd
hbd
(4.21)
and construct similarly a tangent space Lagrangian.
One may add also that one cannot write contractions in eq-(4.20) involving the metric and nonlinear connection g jk Nka to build expressions in the
ab
ab jk
Nbk g kj ; ..... because the latter terms do not
Lagrangian like δbc Pcj
g Nka ; δac Pcj
transform as a scalars under coordinate transformations x0 = x0 (x); p0 = p0 (x, p)
due to the fact that Nak is not a tensor. 2 Similar results apply to the tangent space case. Nevertheless, one should not disregard the possibility that a
suitable linear combination of all the terms involving all possible contractions
might furnish a scalar, as a whole, resulting from a plausible cancellation of all
the inhomogeneous terms that occur under coordinate transformations of the
nonlinear connection.
A Kaluza-Klein-like decomposition of the scalar curvature in higher dimensions given in section 3 involved a very specif ic linear combination of the following nonvanishing curvature tensor contractions given by
µ
α
b
c
R = GM N RM N = γ µν (Rµαν
+ Rµbν
) + φab (Racb
+ Raµb
)
(4.22)
based on the Levi-Civita connection Γνµa (computed in the non-holonomic basis)
given in eqs-(3.4, 3.6).
In the Kaluza-Klein and tangent space case, there is a direct and exact match
c
between the scalar curvatures in their internal spaces S ↔ φab Racb
. However
i
kl
µν α
there is no exact match between R = Rkil g and γ Rµαν because there is no
i
exact match between the components Rhjk
given by eq-(4.10) and the compoα
nents Rµβν given by eq-(3.8a), due to the fact that the distinguished connection
i
coefficients Cha
do not have the same functional form (are not the same) as
the coefficients associated with the Levi-Civita connection Γνµa (computed in
the non-holonomic basis) and given in eq-(3.6). More precisely, the canonically
2 we
thank one of the referees for pointing this
19
i
distinguished connection coefficients Ĉha
given by eq-(4.9) do not have the same
ν
functional form as the coefficients Γµa appearing in eq-(3.6)
i
Ĉha
=
1 il
g (∂a ghl ),
2
Γνµa = Γνaµ =
1 να
1
γ ∂a γµα + γ να φab Fµαb
2
2
(4.23)
To compare both expressions in eq-(4.23) requires the correspondence g il ↔
γ να , ghl ↔ γµα , hab ↔ φab , ....
Furthermore, there is no match in the tangent bundle case with the following
µ
b
nontrivial and nonvanishing scalar contractions γ µν Rµbν
+ φab Raµb
6= 0, stemming from the Kaluza-Klein-like decomposition described in section 3. This
can be verified directly by examining all of the terms in the right hand side of
eq-(3.9). The reason being that in the tangent bundle case, when one evaluates the curvatures associated with the distinguished d-connection, it gives the
a
i
vanishing result Rjbk
= 0 and Rajb
= 0, as can be seen by a simple inspection
a
i
in eqs-(4.10) : there are no Rjbk , Rajb
terms in (4.10) because they are zero.
On the other hand, the curvatures corresponding to the Levi-Civita connection,
evaluated in a non-holonomic basis by eq-(3.8a), lead to the particular non vanµ
a
6= 0 and Raνb
6= 0 and which contribute to the terms
ishing components Rµbν
in the right hand side of eq-(3.9).
In particular, a ”dictionary” between the components of the Levi-Civita
A
ΥA
BC and the canonical distinguished connection Γ̂BC can be found in [14], [10].
One may write the correspondence symbolically between those connections as
A
A
A
ΥA
BC = Γ̂BC + ∆BC , where ∆BC is a distorsion tensor [14], [10], so that
A
A
A
R[gAB ; ΥA
BC ] = R[gAB ; Γ̂BC + ∆BC ] 6= R[gAB ; Γ̂BC ]
(4.24)
implying that the scalar curvature based on the Levi-Civita connection is not
equal to the scalar curvature based on the canonical distinguished connection, as one would expect. Only in very restricted and/or special cases one
can have the equality of the scalar curvatures [10]. Therefore the expression
(R + S)tangent−space obtained from eq-(4.21) is not equal to the expression
RKaluza−Klein in eq-(4.22) and given explicitly by eq-(3.9).
Because in a nonholonomic coordinate basis the metric gAB can be written
in a block diagonal form, let us write the gravitational actions in the absence of
matter as
SLC =
SdC =
1
2κ2
Z
1
2κ2
Z
d d x dd y
d d x dd y
q
q
p
|detgµν | |dethab | R[gAB ; ΥA
BC ]
|detgµν |
(4.25a)
p
A
|dethab | R[gAB ; Γ̂A
BC ] + S[gAB ; Γ̂BC ]
(4.25b)
where κ2 is suitable gravitational coupling in d + d-dim. The first action SLC
is the Einstein-Hilbert action in d + d dimensions whose scalar curvature R is
evaluated with the Levi-Civita connection in a non-holonomic basis as indicated
20
by eq-(3.9). The second action SdC corresponds to the geometry of the 2d-dim
tangent space and involves the sum of the two curvature scalars R + S in eq(4.21) and which are evaluated using the canonical distinguished connection
whose coefficients are given by eq-(4.9). One could have written the action in
the cotangent space instead, but for simplicity let us focus in the tangent space
case. As we have argued above SLC 6= SdC , the functional form of the two
gravitational actions is not the same.
The vacuum field equations obtained by a variation of the actions are given,
respectively, as
δSLC
δSLC
δSLC
= 0
= 0,
= 0,
δgµν
δhab
δNia
(4.26a)
δSdC
δSdC
δSdC
= 0
= 0,
= 0,
δgµν
δhab
δNia
(4.26b)
A
A
It is true that by having ΥA
BC = Γ̂BC + ∆BC one can recast the action SLC in
A
A
terms of Γ̂BC plus the distorsion ∆BC terms in the form
SLC =
1
2κ2
Z
d d x dd y
q
p
|detgµν | |dethab | R[gAB ; Γ̂A
]
+
∆R(distorsion)
BC
(4.27)
Since in the canonical d-connection case one has that R[gAB ; Γ̂A
BC ] = R +S, one
can see that the action in eq-(4.27) becomes the action SdC of eq-(4.25b) but
modif ied by the addition of the correction terms ∆R(distorsion) and which now
play the role of effective ”source” terms induced by the distorsion. Therefore,
if one finds solutions to the vacuum field equations corresponding to the action
SLC one does not have solutions to the vacuum field equations corresponding to
the action SdC , but instead one has solutions to the effective field equations with
”matter” terms corresponding to the induced source terms due to the distorsion.
And vice versa, vacuum solutions based on the action SdC do not correspond
to vacuum solutions based on the action SLC . Therefore, the actions SLC , SdC
correspond to two physically distinct situations : in one case there is no torsion,
in the other there is torsion, and from this point of view they are not physically
equivalent, mainly because the notion of what is the vacuum differs in both
cases. Therefore, despite that there is a ”dictionary” ( loosely speaking an
”equivalence” using the language in [10] ) between the two actions SLC , SdC
this does not mean that they represent the same physics since those two actions
clearly differ and lead to different field equations and solutions.
The field equations (4.26b) are of the form
1
1
δR
δS
( R + S ) gij = 0 ; Sab − ( R + S ) hab = 0,
+
= 0
2
2
δNia
δNia
(4.28)
When i, j = 1, 2, 3, ...., d; a, b = 1, 2, 3, ...., d, one has the correct counting in the
number of field equations eq-(4.28) giving 12 d(d+1)+ 12 d(d+1)+d2 = 21 2d(2d+1)
Rij −
21
and which match the number of degrees of freedom of the metric gAB in 2d =
d + d dimensions. In the next section we will provide solutions to a different
set of equations provided by Vacaru [10] where he replaces the d2 equations
δSdC
2
2
1
δN a = 0 in (4.28) by 2d equations of the form Pia = 0; Pai = 0. The latter
i
k
b
tensors are obtained from the curvature contractions of Pika
, Paib
, respectively.
In doing so, one has an over-determined system of equations unless out of the
latter 2d2 equations only d2 equations are actually independent.
Another difference between the Kaluza-Klein like approach of section 3 based
on the Levi-Civita connection and the (co) tangent space approach of section 4
based on the distinguished connection is that in the former one has covariance
under coordinate transformations of the form x0i = x0i (xj ); y 0i = y 0i (xj , y k )
where we use the same index labels i, j, k, ... for both the x and y coordinates.
In the (co) tangent space case one has covariance instead under a more restricted
set of coordinate transformations of the form [9]
x0i = x0i (xj ), y 0i = y j
∂x0i
∂xj
(4.29a)
∂xj
(4.29b)
∂x0i
To conclude, the 2d-dimensional tangent space (phase space) Lagrangian corresponding to the (co) tangent bundle of the 2-dim spacetime, does not coincide
with the Lagrangian in d+d dimensions after a Kaluza-Klein-like d+d decomposition of the Einstein-Hilbert Lagrangian R described in section 3 is performed.
Consequently, the vacuum field equations obtained from a variational principle
of the tangent space (phase space) Lagrangian are not equivalent to the vacuum gravitational field equations in 2d-dimensions corresponding to the d + d
decomposition of the Einstein-Hilbert Lagrangian R as described in section 3
and based on the Levi-Civita connection.
Hence, the putative phase space metrics in eqs-(2.8,2.19, 2.22), with split
signature (2, 2) are viable solutions to the vacuum field equations obtained from
the Kaluza-Klein-like decomposition of the 4-dim space, but are not solutions
to the vacuum field equations in the 4-dim phase space associated with the
cotangent bundle T ∗ M of the 2-dim spacetime of signature (1, 1).
The generalized (vacuum) field equations corresponding to gravity on the
curved 2d-dimensional tangent bundle/phase spaces associated to the geometry
of the (co)tangent bundle T Md−1,1 (T ∗ Md−1,1 ) of a d-dim spacetime Md−1,1
with one time, and which are obtained from a direct variation of the tangent
space/phase space actions with respect to the respective fields
x0i = x0i (xj ), p0i = pj
gij (xk , y a ), hab (xk , y a ), Nia (xk , y a );
gij (xk , pk ), hab (xk , pa ), Nia (xk , pa )
(4.30)
needs to be investigated further because there is no mathematical equivalence
with the ordinary Einstein vacuum field equations in spacetimes of 2d-dimensions
22
and based on the Levi-Civita connection
RM N (X) −
1
gM N (X) R(X) = 0; M, N = 1, 2, 3, ......., 2d
2
(4.31)
To display explicitly the dif f iculty in solving the field equations in LagrangeFinsler and Hamilton-Cartan spaces, we devote the next section to provide
specific nontrivial solutions in a ”simple” case.
5
Integrable solutions in the Tangent Bundle
T M2 case
To solve the most general solutions for gij (xk , y a ), hab (xk , y a ), Nia (xk , y a ) is a
very difficult task even in the simpler case of actions involving linear terms in
the scalar curvature. Einstein’s equations in dimensions 2 + 2 as a toy model
of Einstein-Finsler gravity in the tangent bundle T M2 over a 2-dimensional
manifold M2 were studied by [10], [11], [12]. An integrable class of solutions
was found which are dif f erent from the solutions in section 3 provided by [7].
Solutions in dimensions 2 + 2 + 2.....; 3 + 2 + 2 + ... have also been found by [11].
The field equations used by Vacaru are
Rij −
2
1
( R + S ) gij = 0 ;
2
Pia = 0;
1
Pai = 0;
2
Sab −
Pia 6=
1
Pai ;
1
( R + S ) hab = 0
2
i, j, k = 1, 2; a, b, c = 1, 2
(5.1)
(5.2)
k
c
k
b
Rij , Sab ,2 Pia ,1 Pai are obtained from the curvature contractions Rikj
, Sacb
, Pika
, Pabi
,
ij
ab
respectively. A further contraction g Rij = R; h Sab = S yields the curvature
scalars.
A special family of solutions was found by [10] for the case
gij = gij (xk ); hab = hab (xk , y 1 ); Nia = Nia (xk , y 1 )
(5.3)
i.e. when the solutions do not depend on the coordinate y 2 . The solutions
are determined by the generating functions f (xi , y 1 ), the integration functions
0
f (xi ),0 h(xk ),0 wj (xk ),0 ni (xk ) and the signature coefficients 1 = ±1, 2 =
±1, 3 = ±1, 4 = ±1. As explained by [10] one should chose and/or fix such
functions following additional assumptions on symmetry of solutions, boundary
conditions etc. For instance, the solutions to the wave equation in two dimensions are given in terms of two generating functions Ψ = f (x + ct) + g(x − ct).
By defining Ni1 = wi , Ni2 = ni , i = 1, 2, the integrable solutions to the
vacuum equations (5.1, 5.2) found by Vacaru [10], [11] are given by
k
k
g11 = 1 eΨ(x ) ; g22 = 2 eΨ(x
)
f or
23
1 ∂x21 Ψ + 2 ∂x22 Ψ = 0
(5.4)
h11 = 3 0 h(xi ) [ ∂y1 f (xi , y 1 ) ]2 ; h22 = 4 [ f (xi , y 1 ) −
0
y1
Z
k
wj = wj (x ) exp
2h11 ∂y1 A
]
dy [
∂y1 h11
1
−
0
y1
! Z
0
f (xi ) ]2
h11 Bj
] exp
dy [
∂y1 h11
1
0
(5.5)
Z
y1
−
0
(5.6)
for j = 1, 2.
ni =
0
ni (xk ) +
Z
dy 1 h11 Ki ; i = 1, 2.
(5.7)
The expressions for A, Bk , Ki are given by
A =
Bk =
K1 = −
1
2
∂y1 h22
2h22
∂y1 h22
∂y1 h11
+
2h11
2h22
∂xk g11
∂ k g22
− x
2g11
2g22
∂x2 g11
∂ 1 g22
+ x
g22 h11
g22 h22
; K2 =
(5.8)
− ∂xk A; k = 1, 2.
1
2
∂x1 g22
∂ 2 g22
− x
g11 h11
g22 h22
(5.9)
(5.10)
Due to the duality between the tangent and cotangent bundles of spacetime,
one expects analogous (not identical) field equations and solutions for the 4-dim
phase space case (associated with the 2-dim space). In this particular case one
could have solutions of the form gij (X, T ); hab (X, T, E); Nia (X, T, E), reminiscent of the so-called rainbow metrics (energy dependent). Solutions of the form
gij (X, T ); hab (X, T, P ); Nia (X, T, P ) are equally valid, or solutions depending
on X, T and the combinations like E + P , or E − P but not both.
One must add that a nonvanishing 2 Pia and/or 1 Pai results, in general,
in nonsymmetric metrics via classical or quantum geometric Ricci flows, see
details in [13]. Solutions with zero 2 Pia ,1 Pai were originally chosen because
this allows to decouple the generalized Einstein equations in general form and
such that there are straightforward limits of the generic off-diagonal solutions.
Via nonholonomic deformations and geometric evolution it is possible to find
a whole class of integrable solutions. Whether or not integrable solutions can
be found also in the case of non-symmetric metrics by having nonvanishing
2
Pia ,1 Pai components, is unknown. Furthermore, there are no known nonsymmetric energy-momentum tensors.
The real challenge is to find solutions to Einstein’s equations associated
with the 8-dim cotangent bundle (phase space) of the 4-dim spacetime; i.e.
one has a 8 = 4 + 4 decomposition with two temporal directions T, E and 6
spatial ones X1 , X2 , X3 , P1 , P2 , P3 . Furthermore, more general actions than the
Einstein-Hilbert type, including torsion-squared and curvature-squared terms
24
2h11 ∂y1 A
]
dy [
∂y1 h11
1
!
in phase space should be studied. To the author, one of the most physically
interesting extensions of gravity in phase spaces (Born reciprocal gravity) is the
study of higher order (co) tangent spaces (Jet bundles) which are related to
higher order accelerations and naturally involve higher dimensions than D = 8.
The principle of maximal speed (special relativity), maximal acceleration/force
(Born reciprocal relativity) would extend to a generalized relativity of maximal
higher order accelerations/forces. Physics in phase space uniting space-timemomentum-energy should reveal important clues to the true origins of mass
that do not rely on the Higgs mechanism.
Finally, it is desirable to find if there are any connections among the latter theories (higher order accelerations/forces) and the higher spin theories of
Vasiliev. The connection between Finsler geometry and the higher conformal
spin theories behind W∞ gravity/strings was suggested by Hull long ago [32].
Yoon also has discussed the emergence of W∞ gravity Lagrangians from the
Lagrangian of eq-(3.17). The physical applications of the phase space metrics
found in section 2 deserve further investigation, as well as how the physics of
two times may resolve the problem of time in Quantum Gravity. For a recent
discussion on the problem of time see [33]. Finding nontrivial solutions of the
generalized field equations corresponding to the 8-dim cotangent bundle (phase
space) of the 4-dim spacetime is a challenging task of paramount importance.
Acknowledgements
We thank M. Bowers for very kind assistance and to the many referees
for providing very useful and critical comments to improve this work. Special
thanks to Matej Pavsic for discussions and sending us the article by Cole on the
prediction of dark matter using six-dimensional relativity. To J.F. Gonzales for
providing the reference to Kalitzin’s book.
APPENDIX
Let us start with the line element
ds2 = −eµ(r) (dt1 )2 + eν(r) (dr)2 + R2 (r)g̃ij dξ i dξ j .
(A.1)
Here, the metric g̃ij corresponds to a homogeneous space and i, j = 3, 4, ..., D−2.
The only nonvanishing Christoffel symbols are
Γ121 = 21 µ0 ,
Γ222 = 12 ν 0 ,
Γ2ij = −e−ν RR0 g̃ij ,
Γi2j =
R0 i
R δj ,
and the only nonvanishing Riemann tensor are
25
Γ211 = 12 µ0 eµ−ν ,
(A.2)
Γijk = Γ̃ijk ,
R1212 = − 12 µ00 − 14 µ02 + 14 ν 0 µ0 ,
R1i1j = − 12 µ0 e−ν RR0 g̃ij ,
R2121 = eµ−ν ( 21 µ00 + 41 µ02 − 14 ν 0 µ0 ),
R2i2j = e−ν ( 21 ν 0 RR0 − RR00 )g̃ij ,
i
Rijkl = R̃jkl
− R02 e−ν (δki g̃jl − δli g̃jk ).
(A.3)
The field equations are
1
1
1
(D − 2) 0 R0
R11 = eµ−ν ( µ00 + µ02 − µ0 ν 0 +
µ
) = 0,
2
4
4
2
R
(A.4)
1
1
1 R0
R00
1
−
) = 0,
R22 = − µ00 − µ02 + µ0 ν 0 + (D − 2)( ν 0
2
4
4
2 R
R
(A.5)
and
Rij =
e−ν 1 0
k
( (ν − µ0 )RR0 − RR00 − (D − 3)R02 )g̃ij + 2 (D − 3)g̃ij = 0, (A.6)
2
R 2
R
where k = ±1, depending if g̃ij refers to positive or negative curvature. From
the combination e−µ+ν R11 + R22 = 0 we get
µ0 + ν 0 =
2R00
.
R0
(A.7)
The solution of this equation is
µ + ν = ln R02 + a,
(A.8)
where a is a constant.
Substituting (A.7) into the equation (A.6) we find
e−ν (ν 0 RR0 − 2RR00 − (D − 3)R02 = −k(D − 3)
(A.9)
γ 0 RR0 + 2γRR00 + (D − 3)γR02 = k(D − 3),
(A.10)
γ = e−ν .
(A.11)
or
where
The solution of (A.10) for an ordinary D-dim spacetime ( one temporal
dimension ) corresponding to a D − 2-dim sphere for the homogeneous space
can be written as
γ = (1 −
dR −2
16πGD M
)(
) ⇒
(D − 2)ΩD−2 RD−3
dr
26
grr = eν = (1 −
16πGD M
dR 2
)−1 (
) .
D−3
(D − 2)ΩD−2 R
dr
(A.12)
where ΩD−2 is the appropriate solid angle in D − 2-dim and GD is the D-dim
gravitational constant whose units are (length)D−2 . Thus GD M has units of
(length)D−3 as it should. When D = 4 as a result that the 2-dim solid angle
is Ω2 = 4π one recovers from eq-(A.12) the 4-dim Schwarzchild solution. The
solution in eq-(A.12) is consistent with Gauss law and Poisson’s equation in
D − 1 spatial dimensions obtained in the Newtonian limit.
Thus, according to (A.8) we get
βD GD M
) + constant.
(A.13)
RD−3
we can set the constant to zero, and this means the line element (A.1) can be
written as
µ = ln(k −
ds2 = −(k −
(dR/dr)2
βD GD M
)(dt1 )2 +
(dr)2 + R2 (r)g̃ij dξ i dξ j . (A.14)
D−3
DM
R
)
(k − βDRG
D−3
One can verify, taking for instance (A.5), that the equations (A.4)-(A.6) do not
determine the form R(r). It is also interesting to observe that the only effect
of the homogeneous metric g̃ij is reflected in the k = ±1 parameter, associated
with a positive ( negative ) constant scalar curvature of the homogeneous D − 2dim space.
References
[1] M. Born, Proc. Royal Society A 165 (1938) 291. Rev. Mod. Physics 21
(1949) 463.
[2] S. Low, Jour. Phys A Math. Gen 35 (2002) 5711; Il Nuovo Cimento B
108 (1993) 841; Found. Phys. 36 (2007) 1036. J. Math. Phys 38 (1997)
2197.
[3] H. Brandt: Contemporary Mathematics 196 (1996) 273. Found Phys.
Letts 4 (1991) 523. Chaos, Solitons and Fractals 10 (2-3) (1999) 267.
[4] E. Caianiello, Lett. Nuovo Cimento 32 (1981) 65.
[5] C. Castro, ” Some consequences of Born’s Reciprocal Relativity in Phase
Spaces” to appear in the IJMPA, (2011); Progress in Physics 1 (2005) 20;
Foundations of Physics 35, no.6 (2005) 971.
[6] C. Castro, Phys. Letts B 668 (2008) 442.
27
[7] Y.Cho, K, Soh, Q. Park and J. Yoon, Phys. Lets B 286 (1992) 251. J.
Yoon, Phys. Letts B 308 (1993) 240. J. Yoon, Phys. Lett A 292 (2001)
166. J. Yoon, Class. Quan. Grav 18 (1999) 1863 .
[8] C. Castro and J.A. Nieto, Int. J. Mod. Phys. A 22, no. 11 (2007) 2021.
[9] R. Miron, D. Hrimiuc, H. Shimada and S. Sabau, The Geometry of
Hamilton and Lagrange Spaces ( Kluwer Academic Publishers, Dordrecht,
Boston, 2001 ).
[10] S. Vacaru, ”Finsler-Lagrange Geometries and Standard Theories in
Physics: New Methods in Einstein and String Gravity” [arXiv : hepth/0707.1524]. S. Vacaru, P. Stavrinos, E. Gaburov, and D. Gonta, ”Clifford and Riemann-Finsler Structures in Geometric Mechanics and Gravity” (Geometry Balkan Press, 693 pages).
[11] S. Vacaru, ”Decoupling of EYMH equations, Off-diagonal Solutions and
Black Ellipsois and Solitons” arxiv : 1108. 2022. S. Vacaru and O.
Tintareanu-Mircea, Nuc. Phys. B 626 (2002) 239.
[12] S. Vacaru, ” Finsler Branes and Quantum Gravity Phenomenology with
Lorentz Symmetry Violations ” arXiv: 1008.4912. Class. Quant. Grav 28
(2011) 215991.
[13] S. Vacaru, Int. J. Theor. Phys. 48 (2009) 579.
[14] R. Miron and M. Anastasiei, The Geometry of Lagrange Spaces : Theory
and Applications (Kluwer, 1994).
[15] I. Bars, Int. J. Mod. Phys. A 25 (2010). 5235. Phys. Rev. D 54 (1996)
5203, Phys. Rev. D 55 (1997) 2373.
[16] M. F. Atiyah and R. S. Ward, Comm. Math. Phys. 55, 117 (1977). R. S.
Ward, Phil. Trans. R. Soc. London A 315 451 (1985).
[17] M. A. De Andrade, O. M. Del Cima and L. P. Colatto, Phys.Lett. B 370;
M. A. De Andrade and O. M. Del Cima, Int. J. Mod. Phys. A 11, 1367
(1996).
[18] M. Carvalho and M. W. de Oliveira Phys. Rev. D 55 ,7574 (1997).
[19] H. Garcia-Compean, ”N = 2 String geometry and the heavenly equations”,
Proceedings of Conference on Topics in Mathematical Physics, General
Relativity, and Cosmology on the Occasion of the 75th Birthday of Jerzy
F. Plebanski, Mexico City, Mexico; 17-20 Sep 2002; hep-th/0405197.
[20] H. Ooguri and C. Vafa, Nucl. Phys. B 367, 83 (1991); Nucl. Phys. B 361,
469 (1991)
28
[21] C. Castro, Journal of Mathematical Physics 35, 920 (1994); Jour. Math.
Physics 35, 3013 (1994) ; Physics Letters B 288, 291 (1992) ; Journal of
Mathematical Physics 34, 681 (1993).
[22] D. Kutasov, E. J. Martinec and M. O’Loughlin, Nucl. Phys. B 477, 675
(1996); hep-th/9603116.
[23] J. Duff, Hidden symmetries of the Nambu-Goto action, hep-th/0602160.
Phys.Lett. B 641, 335 (2006).
[24] J. A. Nieto, Nuovo Cim. B 120, 135 (2005); J.A. Nieto, ”The 2+2 signature and the 1+1 matrix-brane”, hep-th/0606219; J. A. Nieto, M. P.
Ryan, O. Velarde, C. M. Yee, Int. J. Mod. Phys. A 19, 2131.
[25] S. Majid, ”Meaning of Noncommutative Geometry and the Planck Scale
Quantum Group” hep-th/0006166. ”Quantum Groups and Noncommutative Geometry” hep-th/0006167.
[26] J. Moffat, ”Quantum Gravity Momentum Representations and Maximum
Invariant Energy” gr-qc/0401117.
[27] G. Amelino-Camelia, L. Freidel, J. Kowalski-Glikman and L. Smolin, ”
The principle of relative locality” arXiv.org : 1101.0931. L. Freidel and L.
Smolin, ”Gamma ray burst delay times probe the geometry of momentum
space” arXiv.org : 1103.5626
[28] G. Gibbons and S. M. Gielen, Class. Quantum Grav. 26, 135005 (2009).
[29] R. Wald, General Relativity (University of Chicago Press, 1984).
[30] R. Coquereaux and A. Jadczyk, Revs. in Math. Phys 2, no. 1 (1990) 1.
[31] I. Batalin and E. Fradkin, Nuc. Phys. B 326 (1989) 701.
[32] C. Hull, ” W Gravity” hep-th/9211113; ” Classical and Quantum Gravity”
hep-th/9201057; ”Geometry and W Gravity ” hep-th/9301074.
[33] M. Pavsic, ”Klein-Gordon-Wheeler-DeWitt-Schroedinger
arXiv : 1106.2017.
Equation”
[34] N. S. Kalitsin Multitemporal theory of relativity (Publishing House of the
Bulgarian Academy of Sciences, 1975 - Science - 123 pages).
[35] C. Udriste, L. Matei and I. Duca, ” Multitime Hamilton-Jacobi Theory”
Proceedings of the 8th WSEAS International Conference on Applied Computer and Applied Computational Science, Hangzhou 2009.
[36] C. Udriste, O. Dogaru, I. Tevy and D. Bala, Balkan Journal of Geometry
and Its Appications 15, no.2 (2010) 92.
[37] M. Pavsic, ”Extra Time Like Dimensions, Superluminal Motion, and Dark
Matter” arXiv : 1110.4754.
29
[38] E. Cole, Il Nuovo Cimento, 115 B, no. 10 (2000) 1149.
30