ON THE SUPERSYMMETRIC DIRAC EQUATION AND THE HETEROTIC SUPERSTRING THEORY

2002 ◽  
Vol 11 (09) ◽  
pp. 1409-1418 ◽  
Author(s):  
M. D. POLLOCK
Keyword(s):  
Dirac Equation ◽  
Momentum Tensor ◽  
Approximation Formula ◽  
Internal Space ◽  
Fermion Field ◽  
Linear Quadratic ◽  
Superstring Theory ◽  
Higher Derivative ◽  
Tensor Formula ◽  
Weyl Equation

The supersymmetry transformations under which the four-dimensional massless Dirac equation for a two-component, spin-1/2 fermion field ψ (the Weyl equation) remains invariant were obtained by Volkov and Akulov, who used the result to construct the action S = a-1 ∫ |W| d4x in terms of the energy-momentum tensor [Formula: see text], where Wij = δij + aTij and a is a constant. Here, we show, in the approximation [Formula: see text], that the terms linear, quadratic and quartic in Tij are contained in the bosonic sector of the dimensionally reduced, heterotic superstring action, including higher-derivative gravitational terms up to order ℛ4. By comparison of coefficients, we derive the value B r ≈ 3.5 for the radius squared of the internal space in units of the Regge slope parameter α′, slightly greater than the Hagedorn radius squared [Formula: see text]. The cubic terms are also discussed.

On the internal state of the Schwarzschild black hole

2017 ◽  
Vol 26 (09) ◽  
pp. 1750088
Author(s):  
M. D. Pollock
Keyword(s):  
Black Hole ◽  
Internal State ◽  
Energy Condition ◽  
Momentum Tensor ◽  
Schwarzschild Black Hole ◽  
Superstring Theory ◽  
Higher Derivative ◽  
Central Singularity ◽  
Symmetric Line ◽  
Tensor Formula

If the classical gravitational Lagrangian contains higher-derivative terms [Formula: see text], where [Formula: see text], then vacuum solutions of the Einstein–Hilbert theory [Formula: see text] are subject to modification at sufficiently large spacetime curvatures. Previously, we have calculated the effective energy–momentum tensor [Formula: see text] due to the quartic gravitational terms [Formula: see text] of the heterotic superstring theory in the four-dimensional background spacetime of the Schwarzschild black hole, obtaining an expression which satisfies the strong energy condition, and thereby suggests that the [Formula: see text] might not remove the central singularity. This conjecture was put forward from a different viewpoint by Horowitz and Myers, who argued that a non-singular black-hole interior resulting from the [Formula: see text] would be unstable, necessitating reappraisal of the notion of a singular interior spacetime. Here, we show that the chief features of the solution can be simulated by a Bardeen-type ansatz, assuming the spherically symmetric line element [Formula: see text], where [Formula: see text], which, when [Formula: see text], can explain heuristically why [Formula: see text] in the shell region [Formula: see text] of a macroscopic black hole for which [Formula: see text], while [Formula: see text] remains finite at [Formula: see text].

scholarly journals Higher-derivative f(R,□R,T) theories of gravity

2017 ◽  
Vol 26 (03) ◽  
pp. 1750024 ◽  
Author(s):  
M. J. S. Houndjo ◽  
M. E. Rodrigues ◽  
N. S. Mazhari ◽  
D. Momeni ◽  
R. Myrzakulov
Keyword(s):  
Modified Gravity ◽  
Equations Of Motion ◽  
Momentum Tensor ◽  
De Sitter ◽  
Higher Derivative ◽  
Stress Energy ◽  
Derivative Term ◽  
Theories Of Gravity ◽  
Tensor Formula ◽  
Stress Energy Momentum Tensor

In literature, there is a model of modified gravity in which the matter Lagrangian is coupled to the geometry via trace of the stress–energy–momentum tensor [Formula: see text]. This type of modified gravity is denoted [Formula: see text] in which [Formula: see text] is Ricci scalar [Formula: see text]. We extend manifestly this model to include the higher derivative term [Formula: see text]. We derived equations of motion (EOM) for the model by starting from the basic variational principle. Later we investigate FLRW cosmology for our model. We show that de Sitter (dS) solution is unstable for a generic type of [Formula: see text] model. Furthermore we investigate an inflationary scenario based on this model. A graceful exit from inflation is guaranteed in this type of modified gravity.

ON THE SUPERSTRING BLACK HOLE

2007 ◽  
Vol 16 (01) ◽  
pp. 123-140 ◽  
Author(s):  
M. D. POLLOCK
Keyword(s):  
Black Hole ◽  
Energy Momentum Tensor ◽  
Energy Content ◽  
Momentum Tensor ◽  
Back Reaction ◽  
Effective Energy ◽  
De Sitter ◽  
Superstring Theory ◽  
Higher Derivative ◽  
Effective Lagrangian

The effective Lagrangian for the heterotic superstring theory of Gross et al. contains higher-derivative gravitational terms [Formula: see text], n ≥ 2, which become important at large curvatures. This leads to a natural realization of the limiting-curvature hypothesis of Frolov et al., which was formulated to describe the interior of black holes. Assuming a purely geometrical, four-dimensional Schwarzschild black hole, for which all matter fields are zero, this interior consists of two regions: a shell of effective energy-density ρ immediately beyond the event horizon at r+ = 2M, due to the back reaction of the [Formula: see text] on the Schwarzschild metric, extending inward to a transition radius r0 ≈ M⅓, where the shell signature (- + - -) reverts to the exterior Lorentzian form (+ - - -), and an innermost core tending asymptotically to anti-de Sitter space as r → 0. The total mass-energy content of the hole M can be expressed in terms of the effective energy–momentum tensor Sij as the Nordström mass [Formula: see text], since the space–time is static and free of physical singularities. The conjecture that ρ N (r) is positive in the shell, which is necessary for the contribution to M N to be positive, is shown to be true for the term [Formula: see text], due to the unrenormalized [Formula: see text]. The corresponding "potential" energy–momentum tensor calculated in the Schwarzschild background is isotropic in the region r0 ≪ r ≪ r+, where [Formula: see text], while the dominant "kinetic" contribution is [Formula: see text], so that [Formula: see text].

INFLATION FROM THE SUPERSTRING VACUUM

2009 ◽  
Vol 24 (20n21) ◽  
pp. 4021-4037
Author(s):  
M. D. POLLOCK
Keyword(s):  
Background Radiation ◽  
Physical Space ◽  
Reduction Process ◽  
Tree Level ◽  
Internal Space ◽  
De Sitter ◽  
Superstring Theory ◽  
Higher Derivative ◽  
Effective Lagrangian ◽  
Microwave Background Radiation

Quartic higher-derivative gravitational terms in the effective Lagrangian of the heterotic superstring theory renormalize the bare, four-dimensional gravitational coupling [Formula: see text], due to the reduction process [Formula: see text], according to the formula [Formula: see text], where A r and B r are the moduli for the physical space gij(xk) and internal space [Formula: see text], respectively. The Euler characteristic [Formula: see text] is negative for a three-generation Calabi–Yau manifold, and therefore both the additional terms, of tree-level and one-loop origin, produce a decrease in κ-2, which changes sign when κ-2 = 0. The corresponding tree-level critical point is [Formula: see text], if we set [Formula: see text] and λ = 15π2, for compactification onto a torus. Values [Formula: see text] yield the anti-gravity region κ-2 < 0, which is analytically accessible from the normal gravity region κ-2 > 0. The only non-singular, vacuum minimum of the potential [Formula: see text] is located at the point [Formula: see text], where [Formula: see text], the quadratic trace anomaly [Formula: see text] dominates over [Formula: see text], and a phase of de Sitter expansion may occur, as first envisaged by Starobinsky, in approximate agreement with the constraint due to the effect of gravitational waves upon the anisotropy of the cosmic microwave background radiation. There is no non-singular minimum of the potential [Formula: see text].

ON THE QUARTIC HIGHER-DERIVATIVE GRAVITATIONAL TERMS IN THE HETEROTIC SUPERSTRING THEORY

2006 ◽  
Vol 21 (02) ◽  
pp. 373-404 ◽  
Author(s):  
M. D. POLLOCK
Keyword(s):  
Dimensional Space ◽  
Flat Space ◽  
Momentum Tensor ◽  
Space Time ◽  
Einstein Equations ◽  
Quadratic Term ◽  
Additional Contribution ◽  
Superstring Theory ◽  
Weyl Spinor ◽  
Higher Derivative

The quartic higher-derivative gravitational terms [Formula: see text] in the heterotic-superstring effective Lagrangian [Formula: see text], defined from the Riemann ten-tensor [Formula: see text], are expanded, after reduction to the conformally-flat physical D-space gij, in terms of the Ricci tensor Rij and scalar R. The resulting quadratic term [Formula: see text] is tachyon-free and agrees exactly with the prediction from global supersymmetry in the nonlinear realization of Volkov and Akulov of the flat-space, quadratic fermionic Lagrangian [Formula: see text] for a massless Dirac or Weyl spinor, only when D = 4, assuming the Einstein equation [Formula: see text] for the energy–momentum tensor. This proves that the heterotic superstring has to be reduced from ten to four dimensions if supersymmetry is to be correctly incorporated into the theory, and it rules out the bosonic string and type-II superstring, for which [Formula: see text] has the different a priori forms ±(R2-4RijRij) derived from [Formula: see text], which also contain tachyons (that seem to remain after the inclusion of a further contribution to [Formula: see text] from [Formula: see text]). The curvature of space–time introduces a mass into the Dirac equation, [Formula: see text], while quadratic, higher-derivative terms [Formula: see text] make an additional contribution to the Einstein equations, these two effects causing a difference between [Formula: see text] and [Formula: see text] on the one hand, and the predictions from [Formula: see text] and [Formula: see text] on the other. The quartic terms [Formula: see text] still possess some residual symmetry, however, enabling us to estimate the radius-squared of the internal six-dimensional space [Formula: see text] in units of the Regge slope-parameter α′ as B r ≈ 1.75, indicating that compactification occurs essentially at the Planck era, due to quantum mechanical processes, when the action evaluated within the causal horizon is S h ~ 1. This symmetry is also discussed with regard to the zero-action hypothesis. The dimensionality D = 4 of space–time is rederived from the Wheeler–DeWitt equation (Schrödinger equation) of quantum cosmology in the mini-superspace approximation, by demanding invariance and positive-semi-definiteness of the potential [Formula: see text] under Wick rotation of the time coordinate, which also determines the three-space to be flat, so that K = 0, and again involves the nonlinearity of gravitation.

scholarly journals WEYL EQUATION IN GÖDEL TYPE UNIVERSES

1994 ◽  
Vol 09 (40) ◽  
pp. 3703-3706 ◽  
Author(s):  
LUIS O. PIMENTEL ◽  
ABEL CAMACHO ◽  
ALFREDO MACIAS
Keyword(s):  
Field Equation ◽  
Dirac Equation ◽  
The Family ◽  
Weyl Equation

The Weyl equation (massless Dirac equation) is studied in a family of metrics of the Gödel type. The field equation is solved exactly for one member of the family.

scholarly journals Bounds on higher derivative f(R,□R,T) models from energy conditions

2019 ◽  
Vol 34 (11) ◽  
pp. 1950082 ◽  
Author(s):  
M. Ilyas ◽  
Z. Yousaf ◽  
M. Z. Bhatti
Keyword(s):  
Perfect Fluid ◽  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Ricci Scalar ◽  
Energy Conditions ◽  
Higher Derivative ◽  
Derivative Formula ◽  
Cosmological Solutions ◽  
Walker Metric ◽  
Cosmic Parameters

This paper studies the viable regions of some cosmic models in a higher derivative [Formula: see text] theory with the help of energy conditions (where [Formula: see text], [Formula: see text] and [Formula: see text] are the Ricci scalar, d’Alembert’s operator and trace of energy–momentum tensor, respectively). For this purpose, we assume a flat Friedmann–Lemaître–Robertson–Walker metric which is assumed to be filled with perfect fluid configurations. We take two distinct realistic models that might be helpful to explore stable regimes of cosmological solutions. After taking some numerical values of cosmic parameters, like crackle, snap, jerk (etc.) as well as viable constraints from energy conditions, the viable zones for the under observed [Formula: see text] models are examined.

DOES THE HETEROTIC SUPERSTRING THEORY CONTAIN CLOSED TIME-LIKE LINES?

2009 ◽  
Vol 18 (04) ◽  
pp. 559-586 ◽  
Author(s):  
M. D. POLLOCK
Keyword(s):  
Scalar Field ◽  
Line Element ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Causality Condition ◽  
Superstring Theory ◽  
Lorentzian Space ◽  
Higher Derivative ◽  
Gravitational Theories ◽  
Rotational Space

Causal solutions of the Gödel type, for which the line element is ds2 = dt2 - 2b e mxdtdv - c e 2mxdv2 - dx2 - dz2 with c = 0, are known to exist for gravitational theories containing a cosmological constant Λ and quadratic higher-derivative terms defined by the Lagrangian L = -(1/2)κ-2(R + 2Λ) + A1R2 + A2RijRij. Here, we show that acausal solutions, for which c < 0, containing closed time-like lines, can be constructed only if A2 = 0. Extension of this analysis to the heterotic superstring theory, including a generic massless scalar field ϕ plus quadratic and quartic gravitational terms [Formula: see text] and [Formula: see text], again yields a causal solution with c = 0, and also Λ = 0, as required for anomaly freedom, while solutions with c < 0 are ruled out. More general rotational space–times appear to be intractable analytically, and therefore it remains a matter of conjecture that the heterotic superstring admits only classical Lorentzian solutions which respect causality. For the energy density ρ(ϕ) of the scalar field is positive semi-definite only when g00 ≥ 0, which is equivalent to the causality condition g11 ≤ 0 or c ≥ 0 in a Lorentzian space–time for which det gij < 0; while ρ(ϕ) is unbounded from below in the presence of closed time-like lines, when g11 > 0, implying instability of ϕ, which will react back on the metric until it becomes causal.

The Superstring Propagator and the Signature of Space-Time

1997 ◽  
Vol 12 (14) ◽  
pp. 987-998 ◽  
Author(s):  
M. D. Pollock
Keyword(s):  
Wave Function ◽  
Apparent Horizon ◽  
Scalar Particle ◽  
Space Time ◽  
Functional Integral ◽  
Global Symmetry ◽  
Hamiltonian Constraint ◽  
Superstring Theory ◽  
Acceptable Solution ◽  
Higher Derivative

The Faddeev (Newton–Wigner) propagator K for the heterotic superstring theory is derived from the Wheeler–DeWitt equation for the wave function of the Universe Ψ, obtained in the four-dimensional (mini-superspace) Friedmann space-time ds2=dt2-a2(t)dx2, after reduction from the ten-action. The effect of higher-derivative terms ℛ2 is to break the local invariance under time reparametrization to a global symmetry t→λt, and consequently there are no ghost or gauge-fixing contributions, a functional integral over the (constant) Lagrange multiplier λ being sufficient to enforce the Hamiltonian constraint implicitly. After Wick rotation of the time, [Formula: see text], the only physically acceptable solution for K decreases exponentially on the Planck time-scale ~ t P , explaining from the quantum cosmological viewpoint why the signature of space-time is Lorentzian rather than Euclidean. This is analogous to the case of the (two-dimensional) free relativistic scalar particle, discussed recently by Redmount and Suen, who found that the propagator decreases exponentially outside the light-cone on the scale of the Compton wavelength of the particle (in accordance with the Heisenberg indeterminacy principle). These two seemingly different forms of acausality are thus physically excluded in the same way. The propagator for the Schwarzschild black hole of mass M is also obtained from the Schrödinger equation for the wave function on the apparent horizon, due to Tomimatsu, and the Hawking temperature T H =(8π M)-1 is derived from the Euclidean form of this equation.

HOW DID THE UNIVERSE BEGIN?

1998 ◽  
Vol 07 (05) ◽  
pp. 727-735 ◽  
Author(s):  
M. D. POLLOCK
Keyword(s):  
Time Interval ◽  
Field Equations ◽  
Superstring Theory ◽  
Higher Derivative ◽  
Planck Time ◽  
Contracting Phase ◽  
Three Space ◽  
Previous Phase ◽  
The Universe ◽  
Radius Function

The question of the initial configuration of the Universe — did the expanding Friedmann space-time ds2 = dt2 - a2(t)dx2 tend to a singularity when extrapolated back in time, or was there a turning point, indicating a previous phase of contraction? — is re-examined in the context of the heterotic superstring theory of Gross et al. If the adiabatic index tends to the value γ = 1, then the higher-derivative terms ℛ2 in the Lagrangian L dominate the Einstein–Hilbert term R/16πG in the time interval t p ≲ t ≲ 4t p , during which the action is S ≈ 25ℏ, guaranteeing the approximate validity of the classical field equations (if the compactification process is ignored), where [Formula: see text] is the Newton gravitational constant and t p is the Planck time. Under these conditions, Ruzmaĭkina and Ruzmaĭkina have shown, for a flat three-space with K = 0, that the initial singularity can only be avoided at all if there is a spin-zero tachyon, a conclusion modified by Barrow and Ottewill if K = ±1. We have previously shown, however, that the theory is tachyon-free, and have argued that K has to vanish for the existence of a well-defined, quantum-mechanical ground state. Also, if there is no inflation, the radius function is always much too large for the terms in K to exert any effect, a(t) ≳ 5 × 1029t p . While if γ = 2, then ℛ2 never dominates R/16πG. Accordingly, we conjecture that the Universe did not bounce, irrespective of the value of γ, the absence of a prior contracting phase thus being an aspect of causality.

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