A dynamical, non-Euclidean spacetime geometry in general relativity theory implies the possibility of gravitational radiation. Here we explore novel methods of detecting such radiation from astrophysical sources by means of matter-wave interferometers (MIGOs), using atomic beams emanating from supersonic atomic sources that are further cooled and collimated by means of optical molasses. While the sensitivities of such MIGOs compare favorably with LIGO and LISA, the sizes of MIGOs can be orders of magnitude smaller, and their bandwidths wider. Using a pedagogical approach, we place this problem into the broader context of problems at the intersection of quantum mechanics with general relativity.worldline. If spacetime were strictly flat (as shown in figure 2(a)), these two straight and parallel worldlines would never meet. However, like the great circles on a sphere, or geodesics on a pseudosphere, in curved spacetimes these lines can either converge, or diverge from each other, respectively, depending on the sign of the components of the Riemann curvature tensor. An observer fixed on O would see the object P either come towards him, or go away from him, and would interpret this motion as being due to an effective 'tidal' force acting on P (see Chapter 1 of [4] for a description of this). In figure 2(b) we have drawn the specific case where components of the local Riemann curvature tensor are negative, and the worldlines diverge from one another. Since both the observer and the object are simply following their own worldlines-which are geodesics in the absence of all other forces on both the observer and the object-the equivalence principle still holds, and consequently this force is proportional to the mass of the object. The resulting motion of P as observed by O is therefore independent of the mass, composition, or thermodynamic state of the test object at P, and herein lies the geometrical meaning of the equivalence principle. Moreover, when the local Riemann curvature tensor between O and P can be approximated as a constant, this force, like the action of the Moon's gravity on the tides, increases linearly with the distance between O and P (see figure 2(b)), and indeed, the use of the name 'tidal' for this effective force comes from the tidal force on the oceans by the Moon's gravitational field.It is important to note that while the worldline illustrations in figure 2 are a visualization of geometry, and its connection to gravity and gravitational tidal forces, they are only a convenience, and do not represent any frame that can be achieved physically. As drawn, the perspectives are that of an Observer who is removed from the spacetime, and standing outside of it looking in. This is unphysical. Every experimental measurement is made through an experimental apparatus-which may be as simple a device as a pair eyes used to see the motion of an object P a small distance away-that is, by necessity, a physical object. As such, the apparatus must lie along a worldline in the spacetime of the universe, and cannot be...
Guided by the symmetries of the Euler-Lagrange equations of motion, a study of the constrained dynamics of singular Lagrangians is presented. We find that these equations of motion admit a generalized Lie symmetry, and on the Lagrangian phase space the generators of this symmetry lie in the kernel of the Lagrangian two-form. Solutions of the energy equation-called second-order, Euler-Lagrange vector fields (SOELVFs)-with integral flows that have this symmetry are determined. Importantly, while second-order, Lagrangian vector fields are not such a solution, it is always possible to construct from them a SOELVF that is. We find that all SOELVFs are projectable to the Hamiltonian phase space, as are all the dynamical structures in the Lagrangian phase space needed for their evolution. In particular, the primary Hamiltonian constraints can be constructed from vectors that lie in the kernel of the Lagrangian two-form, and with this construction, we show that the Lagrangian constraint algorithm for the SOELVF is equivalent to the stability analysis of the total Hamiltonian. Importantly, the end result of this stability analysis gives a Hamiltonian vector field that is the projection of the SOELVF obtained from the Lagrangian constraint algorithm. The Lagrangian and Hamiltonian formulations of mechanics for singular Lagrangians are in this way equivalent.
With the discovery of Dark Energy, DE , there is now a universal length scale, DE = c/( DE G) 1/2 , associated with the universe that allows for an extension of the geodesic equations of motion. In this paper, we will study a specific class of such extensions, and show that contrary to expectations, they are not automatically ruled out by either theoretical considerations or experimental constraints. In particular, we show that while these extensions affect the motion of massive particles, the motion of massless particles are not changed; such phenomena as gravitational lensing remain unchanged. We also show that these extensions do not violate the equivalence principal, and that because DE = 14010 800 820 Mpc, a specific choice of this extension can be made so that effects of this extension are not be measurable either from terrestrial experiments, or through observations of the motion of solar system bodies. A lower bound for the only parameter used in this extension is set.
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