Starting with a different action and following a different procedure than the construction of strings with dynamical tensions described by Guendelman 1, a variational procedure of our action leads to ...a coupled nonlinear system of D+4 partial differential equations for the D string coordinates Xμ and the quartet of scalar fields φ1,φ2,ϕ,T, including the dilaton ϕ(σ) and the tension T(σ) field. Trivial solutions to this system of complicated equations lead to a constant tension and to the standard string equations of motion. One of the most relevant features of our findings is that the Weyl invariance of the traditional Polyakov string is traded for the invariance under area-preserving diffeomorphisms. The final section is devoted to the physics of maximal proper forces (acceleration), minimal length within the context of Born's Reciprocal Relativity theory 6 and to the Rindler world sheet description of accelerated open and closed strings from a very different approach and perspective than the one undertaken by 7.
The study of the
4
-tachyon off-shell string scattering amplitude
A
4
(
s
,
t
,
u
)
, based on Witten’s open string field theory, reveals the existence of poles in the
s
-channel and associated to a ...continuum of complex “spins”
J
. The latter
J
belong to the Regge trajectories in the
t
,
u
channels which are defined by
-
J
(
t
)
=
-
1
-
1
2
t
=
β
(
t
)
=
1
2
+
i
λ
;
-
J
(
u
)
=
-
1
-
1
2
u
=
γ
(
u
)
=
1
2
-
i
λ
, with
λ
=
r
e
a
l
. These values of
β
(
t
)
,
γ
(
u
)
given by
1
2
±
i
λ
, respectively, coincide precisely with the location of the critical line of nontrivial Riemann zeta zeros
ζ
(
z
n
=
1
2
±
i
λ
n
)
=
0
. It is argued that despite assigning angular momentum (spin) values
J
to the off-shell mass values of the external off-shell tachyons along their Regge trajectories is not physically meaningful, their net zero-spin value
J
(
k
1
)
+
J
(
k
2
)
=
J
(
k
3
)
+
J
(
k
4
)
=
0
is physically meaningful because the on-shell tachyon exchanged in the
s
-channel has a physically well defined zero-spin. We proceed to prove that if there were nontrivial zeta zeros (violating the Riemann Hypothesis) outside the critical line
R
e
a
l
z
=
1
/
2
(but inside the critical strip) these putative zeros
d
o
n
′
t
correspond to any
poles
of the
4
-tachyon off-shell string scattering amplitude
A
4
(
s
,
t
,
u
)
. We finalize with some concluding remarks on the zeros of
sinh
(
z
) given by
z
=
0
+
i
2
π
n
, continuous spins, non-commutative geometry and other relevant topics.
We propose the notion of a classical/quantum duality in the gravitational case (it can be extended to other interactions). By this one means exchanging Bohm's quantum potential for the classical ...potential VQ↔V in the stationary quantum Hamilton–Jacobi equation (QHJE) so that VQ+V=−V0 (ground state energy). Despite that the corresponding Schrödinger equations, and their solutions differ, their associated quantum Hamilton–Jacobi equation, and ground state energy remains the same. This is how the classical/quantum duality is implemented. In this scenario Bohm's quantum potential (which coincides with the attractive Newtonian potential) is now correlated to a classical repulsive gravitational potential (plus a constant). These results suggest that there might be a quantum origin to the classical repulsive gravitational behavior (of the accelerated expansion) of the universe which is based on this notion of classical/quantum duality. We hope that the notion of classical/quantum duality raised in this work in connection to the QHJE may cast further light into the deep interplay between gravity and quantum mechanics.
A very brief introduction of the history of Born’s Reciprocal Relativity Theory, Hopf algebraic deformations of the Poincare algebra, de Sitter algebra, and noncommutative spacetimes paves the road ...for the exploration of gravity in curved phase spaces within the context of the Finsler geometry of the cotangent bundle T∗M of spacetime. A scalar-gravity model is duly studied, and exact nontrivial analytical solutions for the metric and nonlinear connection are found that obey the generalized gravitational field equations, in addition to satisfying the zero torsion conditions for all of the torsion components. The curved base spacetime manifold and internal momentum space both turn out to be (Anti) de Sitter type. The most salient feature is that the solutions capture the very early inflationary and very-late-time de Sitter phases of the Universe. A regularization of the 8-dim phase space action leads naturally to an extremely small effective cosmological constant Λeff, and which in turn, furnishes an extremely small value for the underlying four-dim spacetime cosmological constant Λ, as a direct result of a correlation between Λeff and Λ resulting from the field equations. The rich structure of Finsler geometry deserves to be explored further since it can shine some light into Quantum Gravity, and lead to interesting cosmological phenomenology.
We present the deep connections among (Anti) de Sitter geometry, and complex conformal gravity-Maxwell theory, stemming directly from a gauge theory of gravity based on the complex Clifford algebra
...Cl
(4,
C
). This is attained by simply
promoting
the de (Anti) Sitter algebras
so
(4, 1),
so
(3, 2) to the real Clifford algebras
Cl
(4, 1,
R
),
Cl
(3, 2,
R
), respectively. This interplay between gauge theories of gravity based on
Cl
(4, 1,
R
),
Cl
(3, 2,
R
) , whose bivector-generators encode the de (Anti) Sitter algebras
so
(4, 1),
so
(3, 2), respectively, and 4
D
conformal gravity based on
Cl
(3, 1,
R
) is reminiscent of the
A
d
S
D
+
1
/
C
F
T
D
correspondence between
D
+
1
-dim gravity in the bulk and conformal field theory in the
D
-dim boundary. Although a plausible cancellation mechanism of the cosmological constant terms appearing in the real-valued curvature components associated with complex conformal gravity is possible, it does
not
occur simultaneously in the imaginary curvature components. Nevertheless, by including a Lagrange multiplier term in the action, it is still plausible that one might be able to find a restricted set of on-shell field configurations leading to a cancellation of the cosmological constant in curvature-squared actions due to the coupling among the real and imaginary components of the vierbein. We finalize with a brief discussion related to
U
(
4
)
×
U
(
4
)
grand-unification models with gravity based on
C
l
(
5
,
C
)
=
C
l
(
4
,
C
)
⊕
C
l
(
4
,
C
)
. It is plausible that these grand-unification models could also be traded for models based on
G
L
(
4
,
C
)
×
G
L
(
4
,
C
)
.
A brief review of the essentials of asymptotic safety and the renormalization group (RG) improvement of the Schwarzschild black hole that removes the r = 0 singularity is presented. It is followed ...with a RG improvement of the Kantowski-Sachs metric associated with a Schwarzschild black hole interior such that there is no singularity at t = 0 due to the running Newtonian coupling G(t) vanishing at t = 0. Two temporal horizons at t.sub.- equivalent t.sub.P and t.sub.+ equivalent t.sub.H are found. For times below the Planck scale t < t.sub.P, and above the Hubble time t > t.sub.H, the components of the Kantowski-Sachs metric exhibit a key sign change, so the roles of the spatial z and temporal t coordinates are exchanged, and one recovers a repulsive inflationary de Sitter-like core around z = 0, and a Schwarzschild-like metric in the exterior region z > R.sub.H = 2G.sub.oM. The inclusion of a running cosmological constant LAMBDA(t) follows. We proceed with the study of a dilaton-gravity (scalar-tensor theory) system within the context of Weyl's geometry that permits singling out the expression for the classical potential V(phi) = kappaphi.sup.4, instead of being introduced by hand, and find a family of metric solutions that are conformally equivalent to the (anti) de Sitter metric. To conclude, an ansatz for the truncated effective average action of ordinary dilaton gravity in Riemannian geometry is introduced, and a RG-improved cosmology based on the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric is explored where instead of recurring to the cutoff identification k = k(t) = xiH(t), based on the Hubble function H(t), with xi a positive constant, one now has k = k(t) = xiphi(t), when phi is a positive-definite dilaton scalar field that is monotonically decreasing with time.
We continue to explore the consequences of Thermal Relativity Theory to the physics of black holes. The thermal analog of Lorentz transformations in the
tangent
space of the thermodynamic manifold ...are studied in connection to the Hawking evaporation of Schwarzschild black holes and one finds that there is
no
bound to the thermal analog of proper accelerations despite the maximal bound on the thermal analog of velocity given by the Planck temperature. The proper entropic infinitesimal interval corresponding to the Kerr–Newman black hole involves a
3
×
3
non-Hessian metric with diagonal and off-diagonal terms of the form
(
d
s
)
2
=
g
ab
(
M
,
Q
,
J
)
d
Z
a
d
Z
b
, where
Z
a
=
M
,
Q
,
J
are the mass, charge and angular momentum, respectively. Since the computation of the scalar curvature associated to this metric is very elaborate, to simplify matters, we focused on the singularities of the metric and found that they correspond to the extremal Kerr–Newman black hole case
r
+
=
r
-
=
G
M
with vanishing temperature. Black holes in asymptotically Anti de Sitter spacetimes are more subtle to study since the mass turns out to be related to the
enthalpy
rather that the internal energy. We finalize with some remarks about the thermal-relativistic analog of proper force, the need to extend our analysis of Gibbs-Boltzmann entropy to the case of Reny and Tsallis entropies, and to complexify spacetime.
Starting with a brief description of Born's reciprocal relativity theory (BRRT), based on a maximal proper force, maximal speed of light, and inertial and non-inertial observers, we derive the exact ...thermal relativistic corrections to the Schwarzschild, Reissner-Nordstrom, and Kerr-Newman black hole entropies and provide a detailed analysis of the many novel applications and consequences to the physics of black holes, quantum gravity, minimal area, minimal mass, Yang-Mills mass gap, information paradox, arrow of time, dark matter, and dark energy. We finish by outlining our proposal towards a space-time-matter unification program where matter can be converted into spacetime quanta and vice versa.
Born’s reciprocal relativity theory (BRRT), based on a maximal proper-force, maximal speed of light, and inertial and non-inertial observers, is re-examined in full detail. Relativity of locality and ...chronology are natural consequences of this theory, even in flat phase space. The advantage of BRRT is that Lorentz invariance is preserved and there is no need to introduce Hopf algebraic deformations of the Poincaré algebra, de Sitter algebra, nor non-commutative space–times. After a detailed study of the notion of generalized force, momentum, and mass in phase space, we explain that what one may interpret as “dark matter” in galaxies, for example, is just an effect of observing ordinary galactic matter in different accelerating frames of reference than ours. Explicit calculations are provided that explain these novel relativistic effects due to the accelerated expansion of the Universe, and which may generate the present-day density parameter value Ω
DM
∼ 0.25 of dark matter. The physical origins behind the numerical coincidences in black hole cosmology are also explored. We finalize with a rigorous study of the curved geometry of (co)tangent bundles (phase space) within the formalism of Finsler geometry, and provide a short discussion on Hamilton spaces.
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