When Einstein was thinking about the theory of general relativity based on the elimination of especial relativity constraints (especially the geometric relationship of space and time), he understood the first limitation of especial relativity is ignoring changes over time. Because in especial relativity, only the curvature of the space was considered. Therefore, tensor calculations should be to...

We construct exact solutions of the Einstein-Dirac equation, which couples the gravitational field with an eigenspinor of the Dirac operator via the energymomentum tensor. For this purpose we introduce a new field equation generalizing the notion of Killing spinors. The solutions of this spinor field equation are called weak Killing spinors (WK-spinors). They are special solutions of the Einste...

Interpretations of Einstein’s equation differ primarily concerning whether E = mc entails that mass and energy are the same property of physical systems, and hence whether there is any sense in which mass is ever “converted” into energy (or vice versa). In this paper, I examine six interpretations of Einstein’s equation and argue that all but one fail to satisfy a minimal set of conditions that...

The conformal space M was introduced by Dirac in 1936. It is an algebraic manifold with a spin structure and possesses naturally an invariant Lorentz metric. By carefully studying the birational transformations of M, we obtain explicitly the transition functions of the spin bundle over M. Since the transition functions are closely related to the propagation in physics, we get a kind of solution...

Beginning with a special form of the Einstein-Rosen metric, we find new cosmological solutions of the Einstein equations, having two hypersurface-orthogonal Killing vectors , with ideal fluid. The equation of state is in the most cases of the form p = γρ.

DARK-BRIGHT SOLITONS AND VORTICES IN BOSE-EINSTEIN CONDENSATES MAY 2014 DONG YAN B.S., ZHEJIANG UNIVERSITY, CHINA M.S., UNIVERSITY OF MASSACHUSETTS AMHERST Ph.D., UNIVERSITY OF MASSACHUSETTS AMHERST Directed by: Professor Panayotis G. Kevrekidis This dissertation focuses on the properties of nonlinear waves in Bose-Einstein condensates (BECs). The fundamental model here is the nonlinear Schrödi...

I discuss the problem of inflation in the context of Friedmann-Robertson-Walker Cosmology and show how, after a simple change of variables, to quantize the problem in a way which parallels the classical discussion. The result is that two of the Einstein equations arise as exact equations of motion and one of the usual Einstein equations (suitably quantized) survives as a constraint equation to ...

Assuming SO(3)-spherical symmetry, the 4{dimensional Einstein equation reduces to an equation conformally related to the eld equation for 2{dimensional gravity following from the Lagrangian L = jRj 1=3. Solutions for 2{dimensional gravity always possess a local isometry because the traceless part of its Ricci tensor identically vanishes. Combining both facts, we get a new proof of Birkhoo's the...

We present two constructions of new solutions to the dispersionless KP (dKP) equation arising from the first two Painlevé transcendents. The first construction is a hodograph transformation based on Einstein–Weyl geometry, the generalised Nahm's equation and the isomonodromy problem. The second construction, motivated by the first, is a direct characterisation of solutions to dKP which are cons...

We present two constructions of new solutions to the dispersionless KP (dKP) equation arising from the first two Painlevé transcendents. The first construction is a hodograph transformation based on Einstein–Weyl geometry, the generalised Nahm's equation and the isomonodromy problem. The second construction, motivated by the first, is a direct characterisation of solutions to dKP which are cons...