When constructing general relativity (GR), Einstein required 4D general covariance. In contrast, we derive GR (in the compact, without boundary case) as a theory of evolving 3-dimensional conformal Riemannian geometries obtained by imposing two general principles: 1) time is derived from change; 2) motion and size are relative. We write down an explicit action based on them. We obtain not only GR in the CMC gauge, in its Hamiltonian 3 + 1 reformulation but also all the equations used in York's conformal technique for solving the initial-value problem. This shows that the independent gravitational degrees of freedom obtained by York do not arise from a gauge fixing but from hitherto unrecognized fundamental symmetry principles. They can therefore be identified as the long-sought Hamiltonian physical gravitational degrees of freedom.Since Einstein created GR 4D spacetime covariance has been taken as its axiomatic basis. However, much work has been done in a dynamical approach that uses the 3+1 split into space and time of Arnowitt, Deser, and Misner (ADM) [1]. This work has been stimulated by the needs of astrophysics (especially gravitational-wave research) and by the desire to find a canonical version of GR suitable for quantization.The ADM formalism describes constrained Hamiltonian evolution of 3D spacelike hypersurfaces embedded in 4D spacetime. The intrinsic geometry of the hypersurfaces is represented by a Riemannian 3-metric g ij , which is the ADM canonical coordinate. The corresponding canonical momentum π ij is related to the extrinsic curvature κ ij of the embedding of the hypersurfaces in spacetime by π ij = − √ g(κ ij − g ij κ). The ADM dynamics, which respects full relativity of simultaneity by allowing free choice of the 3+1 split, is driven by two constraints. The linear momentum constraint π ij ;j = 0 reflects the gauge symmetry under 3D diffeomorphisms and is well understood. When it has been quotiented out, the 3 × 3 symmetric matrix g ij has three degrees of freedom. The quadratic Hamiltonian constraint gH = −π ij π ij + π 2 /2 + gR = 0 reflects the relativity of simultaneity -the time coordinate can be freely chosen at each space point. It shows that g ij has only two physical degrees of freedom. The problem is to find them. The solution, if it exists, will break 4D covariance.An important clue was obtained by York [2], who perfected Lichnerowicz's conformal technique [3] for finding initial data that satisfy the initial-value constraints of GR. In the Hamiltonian formalism, these are the ADM Hamiltonian and momentum constraints. Finding such data is far from trivial. York's is the only known effective method. He divides the 6 degrees of freedom in the 3-metric into three groups; 3 are mere coordinate freedoms, 1 is a scale part (a conformal factor), and the two remaining parts represent the conformal geometry, the 'shape of space'. York's method also introduces a distinguished foliation of spacetime -and with it a definition of simultaneity -by hypersurfaces of constant mean (extrinsic) curva...
Excitability is a generic prediction for an optically injected semiconductor laser. However, the details of the phenomenon differ depending on the type of device in question. For quantum-well lasers very complicated multipulse trajectories can be found, while for quantum-dot lasers the situation is much simpler. Experimental observations show the marked differences in the pulse shapes while theoretical considerations reveal the underlying mechanism responsible for the contrast, identifying the increased stability of quantum-dot lasers to perturbations as the root.
The response of an optically injected quantum-dot semiconductor laser (SL) is studied both experimentally and theoretically. In particular, the nature of the locking boundaries is investigated, revealing features more commonly associated with Class A lasers rather than conventional Class B SLs. Experimentally, two features stand out; the first is an absence of instabilities resulting from relaxation oscillations, and the second is the observation of a region of bistability between two locked solutions. Using rate equations appropriate for quantum-dot lasers, we analytically determine the stability diagram in terms of the injection rate and frequency detuning. Of particular interest are the Hopf and saddle-node locking boundaries that explain how the experimentally observed phenomena appear.
An experimental study of the dynamics of a single-mode quantum-dot semiconductor laser undergoing optical injection is described for the first time, to our knowledge. In particular, the first observation of excitable pulses near the locking boundaries for both positive and negative detuning is reported, indicating locking via a saddle-node bifurcation for both signs of the detuning. The phase evolution of the slave electric-field during pulsing was measured and confirmed that the pulses result from 2pi phase slips. The interpulse-time statistics were analyzed, and a Kramers-like distribution was obtained.
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