Existence Of Conjugate Points Research Articles

Complexity from spinning primaries

We define circuits given by unitary representations of Lorentzian conformal field theory in 3 and 4 dimensions. Our circuits start from a spinning primary state, allowing us to generalize formulas for the circuit complexity obtained from circuits starting from scalar primary states. These results are nicely reproduced in terms of the geometry of coadjoint orbits of the conformal group. In contrast to the complexity geometry obtained from scalar primary states, the geometry is more complicated and the existence of conjugate points, signaling the saturation of complexity, remains open.

Analytical Solution for Optimal Low-Thrust Limited-Power Transfers Between Non-Coplanar Coaxial Orbits

In this paper, an analytical solution for time-fixed optimal low-thrust limited-power transfers (no rendezvous) between elliptic coaxial non-coplanar orbits in an inverse-square force field is presented. Two particular classes of maneuvers are related to such transfers: maneuvers with change in the inclination of the orbital plane and maneuvers with change in the longitude of the ascending node. The optimization problem is formulated as a Mayer problem of optimal control with the state defined by semi-major axis, eccentricity, inclination or longitude of the ascending node, according to the class of maneuver considered, and a variable measuring the fuel consumption. After applying Pontryagin’s maximum principle and determining the maximum Hamiltonian, short periodic terms are eliminated through an infinitesimal canonical transformation. The new maximum Hamiltonian resulting from this canonical transformation describes the extremal trajectories for long duration transfers. Closed-form analytical solution is then obtained through Hamilton-Jacobi theory. For long duration maneuvers, the existence of conjugate points is investigated through the Jacobi condition. Simplified solution is determined for transfers between close orbits. The analytical solution is compared to the numerical solution obtained by integration of the canonical system of differential equations describing the extremal trajectories for some sets of initial conditions. Results show a great agreement between these solutions for the class of maneuvers considered in the analysis. The solution of the two-point boundary value problem of going from an initial orbit to a final orbit, based on the analytical solution, is also discussed.

Optimal low-thrust transfers between coplanar orbits with small eccentricities

In this paper, a first-order analytical solution, which includes the short periodic terms, for the problem of optimal time-fixed low-thrust limited-power transfers (no rendezvous), in an inverse-square force field, between coplanar orbits with small eccentricities is obtained through canonical transformation theory. Short periodic terms are eliminated from the maximum Hamiltonian, expressed in non-singular orbital elements, through an infinitesimal canonical transformation built through the Hori method. Closed-form analytical solutions are obtained for the average canonical system by solving the Hamilton–Jacobi equation through the separation of variables technique. For long duration maneuvers, the existence of conjugate points is investigated through the Jacobi condition.

Comparison theory in Lorentzian and Riemannian geometry

A survey is given of selected aspects of comparison theory for Lorentzian and Riemannian manifolds, in which both Jacobi equation and Riccati equation techniques have been employed. Specifically, the existence of conjugate points on a complete geodesic in the presence of positive Ricci curvature and the topic of volume comparison are treated.

The refractive index of curved spacetime: The fate of causality in QED

It has been known for a long time that vacuum polarization in QED leads to a superluminal low-frequency phase velocity for light propagating in curved spacetime. Assuming the validity of the Kramers–Kronig dispersion relation, this would imply a superluminal wavefront velocity and the violation of causality. Here, we calculate for the first time the full frequency dependence of the refractive index using world-line sigma model techniques together with the Penrose plane wave limit of spacetime in the neighbourhood of a null geodesic. We find that the high-frequency limit of the phase velocity (i.e. the wavefront velocity) is always equal to c and causality is assured. However, the Kramers–Kronig dispersion relation is violated due to a non-analyticity of the refractive index in the upper-half complex plane, whose origin may be traced to the generic focusing property of null geodesic congruences and the existence of conjugate points. This makes the issue of micro-causality, i.e. the vanishing of commutators of field operators at spacelike separated points, a subtle one in local quantum field theory in curved spacetime.

Causality and micro-causality in curved spacetime

We consider how causality and micro-causality are realised in QED in curved spacetime. The photon propagator is found to exhibit novel non-analytic behaviour due to vacuum polarization, which invalidates the Kramers–Kronig dispersion relation and calls into question the validity of micro-causality in curved spacetime. This non-analyticity is ultimately related to the generic focusing nature of congruences of geodesics in curved spacetime, as implied by the null energy condition, and the existence of conjugate points. These results arise from a calculation of the complete non-perturbative frequency dependence of the vacuum polarization tensor in QED, using novel world-line path integral methods together with the Penrose plane-wave limit of spacetime in the neighbourhood of a null geodesic. The refractive index of curved spacetime is shown to exhibit superluminal phase velocities, dispersion, absorption (due to γ→e+e−) and bi-refringence, but we demonstrate that the wavefront velocity (the high-frequency limit of the phase velocity) is indeed c, thereby guaranteeing that causality itself is respected.

The timelike cut locus and conjugate points in Lorentz 2-step nilpotent Lie groups

We investigate the timelike cut locus and the locus of conjugate points in Lorentz 2-step nilpotent Lie groups. For these groups with a timelike center, we give some criteria for the existence of conjugate points along timelike geodesics. We show for instance that a nonsingular timelike geodesic which is translated by an element of the group has a conjugate point. For those that we will call of GH-type, and for those which are globally hyperbolic with a timelike center and a one-dimensional derived subgroup, we show that if in addition the derived subgroup is spacelike, then nonsingular timelike geodesics maximize distance up to and including the first conjugate point. Other related results and corollaries are also obtained.

Causality and conjugate points in general plane waves

Let ℳ = ℳ0 × ℝ2 be a pp-wave-type spacetime endowed with the metric ⟨·, ·⟩z = ⟨·, ·⟩x + 2du dv + H(x, u) du2, where (ℳ0, ⟨·, ·⟩x) is any Riemannian manifold and H(x, u) is an arbitrary function. We show that the behaviour of H(x, u) at spatial infinity determines the causality of ℳ, say: (a) if −H(x, u) behaves subquadratically (i.e, essentially −H(x, u) ⩽ R1(u)|x|2−ϵ for some ϵ > 0 and large distance |x| to a fixed point) and the spatial part (ℳ0, ⟨·, ·⟩x) is complete, then the spacetime ℳ is globally hyperbolic, (b) if −H(x, u) grows at most quadratically (i.e, −H(x, u) ⩽ R1(u)|x|2 for large |x|) then it is strongly causal and (c) ℳ is always causal, but there are non-distinguishing examples (and thus, not strongly causal), even when −H(x, u) ⩽ R1(u)|x|2+ϵ, for small ϵ > 0.Therefore, the classical model ℳ0 = ℝ2, H(x, u) = ∑i, j hij(u)xixj(≢ 0), which is known to be strongly causal but not globally hyperbolic, lies in the critical quadratic situation with complete ℳ0. This must be taken into account for realistic applications. In fact, we argue that −H will be subquadratic (and the spacetime globally hyperbolic) if ℳ is asymptotically flat. The relation of these results with the notion of astigmatic conjugacy and the existence of conjugate points is also discussed.

Isoperimetric inequalities and conjugate points on Lorentzian surfaces

Recently, an isoperimetric inequality for a sector on the Minkowski 2-spacetime has been derived by the method of parallels and the relativistic Gauss-Bonnet formula. In the present paper, we derive an isoperimetric inequality for a sector on a Lorentzian surface with curvatureK ≤ C. As a sector can be modeled by a geodesic variation of a timelike geodesic, our isoperimetric inequality gives an upper bound for the spacelike boundary of a sector. As an application of our results, we give an elementary proof of the existence of conjugate points on a Lorentzian surface with curvatureK ≤ C < 0 and we obtain an upper bound for the (timelike) diameter of a globally hyperbolic Lorentzian surface withK ≤ C < 0 by comparison of sectors.

Conjugate points along null geodesics of asymptotically flat spacetimes

We locate pairs of conjugate points on null geodesics along which there is a `barrier' of Weyl curvature. The existence of conjugate points in this case is predicted by general theorems. We also find that the same conjugate points can be obtained perturbatively off flat space, assuming the barrier to be weak. The conjugate points appear at second order in the perturbative approach. These results are relevant to the existence of singularities in the perturbative treatment of lightcone cuts of null infinity in asymptotically flat spacetimes that are close to Minkowski.

Existence of conjugate points for self-adjoint linear differential equations

SynopsisThe conjecture of Muller-Pfeiffer [4] concerning the oscillation behaviour of the differential equation (–l)n(p(x)y(n))(n) + q(x)y = 0 is proved, and a similar conjecture concerning the more general differential equation ∑nk=0(−l)k(Pk(x)y(k)(k + q(x)y= 0 is formulated.

On the Existence of Conjugate Points for a Second Order Ordinary Differential Equation

In this paper we show that a result of S. W. Hawking and R. Penrose [Proc. Roy. Soc. London Ser. A, 314 (1970), pp. 529–548] on the existence of conjugate points for a real second order linear differential equation is a consequence of a much earlier result of M. Yelchin [5]. Yelchin’s original proof is clarified and corrected and his result is extended. As a result, we obtain extensions of the Hawking–Penrose theorem and Tipler’s [J. Differential Equations, 30 (1978), pp. 165–174] a complementary result to said theorem.

Bounds for Eigenvalues and Conditions for Existence of Conjugate Points

Recently, results concerning existence of the first conjugate point of the equation $y^{(iv)} - py = 0,p(x) > 0$ for $x \in [ a,b ]$ when p is linear, concave and convex have been published. Leighton and Nehari have shown that a first conjugate point exists if and only if the lowest eigenvalue $\lambda _1 $ of $u^{(iv)} - \lambda pu = 0$, $u(a) = u'(a) = u(b) = u'(b)$, satisfies the inequality $\lambda _1 \leqq 1$. By using this result and known properties of $\lambda _1 $. as a functional of p, the known results for existence of conjugate points are strengthened and new results are obtained. This method may also be used to obtain existence theorems for conjugate points of other problems.

Conjugate Loci of Constant Order

One class of problems in riemannian geometry is to determine in what way the conjugate locus structure restricts the topological, differentiable, or metric structure of the manifold. Other than applications of the Morse theory, only a few results of this type are known. For example, if there is one point in a simply connected riemannian manifold possessing no conjugate points, then the manifold is diffeomorphic with euclidean space [12]. E. Hopf [8] has shown that a riemannian structure on the torus T2, with no conjugate pairs along any geodesic, must be flat. The only known results where one allows the existence of conjugate points are in the cases where the first conjugate locus is suitably spherical. For a 2-dimensional compact simply connected riemannian manifold M, Green [7] has shown that, if the first conjugate locus in the tangent space at each point is a sphere of the same radius, then M is isometric to the standard sphere S2. The analogous problem in higher dimensions would be for a compact simply connected riemannian manifold M, for which the first conjugate locus in each Mm is a sphere of the same radius consisting of conjugate points of maximal order. Differentiably such a manifold is a sphere, but the isometry conjecture is unsolved. In this higher dimensional case, assume the first conjugate locus is a sphere of the same radius at each point, but consisting of conjugate points of some constant order less than maximal. In this case one might conjecture the manifolds are isometric to one of the compact irreducible riemannian symmetric spaces of rank 1. However, here the topological question is still open. Allamigeon [3] has shown that, in this case, the manifold has the correct integral cohomology ring. In a somewhat different vein, Klingenberg [11] has shown that, if the conjugate locus structure in the tangent space at one point of a simply connected riemannian manifold is similar to that for one of the compact irreducible rank 1 riemannian symmetric spaces, then the manifold has the integral cohomology ring of the space after which it is modeled. In this paper, we draw some differentiable and topological conclusions from the structure of the conjugate locus at a single point. At first we do not assume the conjugate locus is spherical. If there is one point m in a compact simply connected riemannian manifold M for which each point of the