Lorentz-invariant Solution Research Articles

ON RELATIVISTIC HARMONIC OSCILLATOR

A relativistic quantum harmonic oscillator in 3+1 dimensions is derived from a quaternionic non-relativistic quantum harmonic oscillator. This quaternionic equation also yields the Klein-Gordon wave equation with a covariant (space-time dependent) mass. This mass is quantized and is given by $m_{*n}^2=m_\omega^2\left(n_r^2-1-\beta\,\left(n+1\right)\right)\,,$ where $m_\omega=\frac{\hbar\omega}{c^2}\,,$ $\beta=\frac{2mc^2}{\hbar\,\omega}\, $, $n$, is the oscillator index, and $n_r$ is the refractive index in which the oscillator travels. The harmonic oscillator in 3+1 dimensions is found to have a total energy of $E_{*n}=(n+1)\,\hbar\,\omega$, where $\omega$ is the oscillator frequency. A Lorentz invariant solution for the oscillator is also obtained. The time coordinate is found to contribute a term $-\frac{1}{2}\,\hbar\,\omega$ to the total energy. The squared interval of a massive oscillator (wave) depends on the medium in which it travels. Massless oscillators have null light cone. The interval of a quantum oscillator is found to be determined by the equation, $c^2t^2-r^2=\lambda^2_c(1-n_r^2)$, where $\lambda_c$ is the Compton wavelength. The space-time inside a medium appears to be curved for a massive wave (field) propagating in it.

Lorentz invariant vacuum solutions in general relativity

All Lorentz invariant solutions of vacuum Einstein's equations are found. It is proved that these solutions describe space-times of constant curvature.

Tree-level S-matrix of Pohlmeyer reduced form of AdS 5 × S 5 superstring theory

With a motivation to find a 2-d Lorentz-invariant solution of the AdS 5 × S 5 superstring we continue the study of the Pohlmeyer-reduced form of this theory. The reduced theory is constructed from currents of the superstring sigma model and is classically equivalent to it. Its action is that of G/H = Sp(2, 2) × Sp(4)/[SU(2)]4 gauged WZW model deformed by an integrable potential and coupled to fermions. This theory is UV finite and is conjectured to be related to the superstring theory also at the quantum level. Expanded near the trivial vacuum it has the same elementary excitations (8+8 massive bosonic and fermionic 2-d degrees of freedom) as the AdS 5 × S 5 superstring in the S 5 light-cone gauge or near plane-wave expansion. In contrast to the superstring case, the interaction terms in the reduced action are manifestly 2-d Lorentz invariant. Since the theory is integrable, its S-matrix should be effectively determined by the two-particle scattering. Here we explicitly compute the tree-level two-particle S-matrix for the elementary excitations of the reduced theory. We find that this S-matrix has the same index structure and group factorization properties as the superstring S-matrix computed in hep-th/0611169 but has simpler coefficients, depending only on the difference of two rapidities. While the gauge-fixed form of the reduced action has only the bosonic [SU(2)]4 part of the PSU(2|2) × PSU(2|2) symmetry of the light-cone superstring spectrum as its manifest symmetry we conjecture that it should also have a hidden fermionic symmetry that effectively interchanges bosons and fermions and which should guide us towards understanding the relation between the two S-matrices.

Post-Newtonian limitations on measurement of the PPN parameters caused by motion of gravitating bodies

We derive explicit Lorentz-invariant solution of the Einstein and null geodesic equations for data processing of the time delay and ranging experiments in gravitational field of moving gravitating bodies of the solar system - the Sun and major planets. We discuss general-relativistic interpretation of these experiments and the limitations imposed by motion of the massive bodies on measurement of the parameters gamma_{PPN}, beta_{PPN} and delta_{PPN} of the parameterized post-Newtonian formalism.

Aberration and the Fundamental Speed of Gravity in the Jovian Deflection Experiment

We describe our explicit Lorentz-invariant solution of the Einstein and null geodesic equations for the deflection experiment of 2002 September 8 when a massive moving body, Jupiter, passed within 3.7' of a line-of-sight to a distant quasar. We develop a general relativistic framework which shows that our measurement of the retarded position of a moving light-ray deflecting body (Jupiter) by making use of the gravitational time delay of quasar's radio wave is equivalent to comparison of the relativistic laws of the Lorentz transformation for gravity and light. Because, according to Einstein, the Lorentz transformation of gravity field variables must depend on a fundamental speed $c$, its measurement through the retarded position of Jupiter in the gravitational time delay allows us to study the causal nature of gravity and to set an upper limit on the speed of propagation of gravity in the near zone of the solar system as contrasted to the speed of the radio waves. We discuss the misconceptions which have inhibited the acceptance of this interpretation of the experiment. We also comment on other interpretations of this experiment by Asada, Will, Samuel, Pascual-Sanchez, and Carlip and show that their `speed of light' interpretations confuse the Lorentz transformation for gravity with that for light, and the fundamental speed of gravity with the physical speed of light from the quasar.

Lorentz-invariant solutions of the classical Yang-Mills equations

A ‘gauge’ representation of the noncompact group SL(2,C) is defined. All the corresponding invariant (singular) solutions of the classical Yang-Mills equations in Minkowski space are found. It is shown that they are related to a family of real SU(2) x SU(2)-invariant Euclidean solutions containing the self-dual (one-instanton) configurations.

Construction of N = 8 supergravity theories by dimensional reduction

In this paper I ask which N = 8 supergravity theories in four dimensions can be obtained by dimensional reduction of the N = 1 supergravity theory in eleven dimensions. Several years ago Scherk and Schwarz produced a particular class of N = 8 theories by giving a dimensional reduction scheme on the restricted class of coset spaces, G/H, with dim H = 0 (and therefore dim G = 7). I generalize their considerations by looking at arbitrary (seven-dimensional) coset spaces. Also, instead of giving a particular ansatz which happens to work, I set about the distinctly more difficult task of determining all ansatzes which produce N = 8 theories. The basic ingredient of my dimensional reduction scheme is the demand that certain symmetries, including supersymmetry, be truncated consistently. I find the surprising result that the only N = 8 theories obtainable within the contexts of my scheme are those theories already written down by Scherk and Schwarz. In particular dim H = 0 and dim G = 7. Independently of these considerations, I prove that any dimensional reduction scheme which consistently truncates supersymmetry must also be consistent with the equations of motion. I discuss Lorentz-invariant solutions of the theories of Scherk and Schwarz, pointing out that since the ansatz of Scherk and Schwarz consistently truncates supersymmetry, any solution of these theories is also a solution of the N = 1 supergravity theory in eleven dimensions and, hence, in particular that there is a Freund-Rubin-type ansatz for these theories. However I demonstrate that for most gauge groups the ansatz must be trivial which implies that for these theories the cosmological constant of any Lorentz-invariant solution must be zero (classically). Finally, I make some comparisons with work by Manton on dimensional reduction.

The asymptotic behavior of the Klein-Gordon equation with external potential

Let U 0( t) and U( t) denote the temporal propagator of the Klein-Gordon equation □ ϑ = m 2 ϑ and the perturbed equation □ ϑ = ( m 2 + V( x)) ϑ, respectively. If (i) V ϵ L p ( E 3) for any 2 ⩽ p </ 3, (ii) V is real-valued and (iii) satisfies a restriction on the size of its negative part, then the (free-to-physical) wave operators, W ±: = s − lim U(− t) U 0( t), as t → ±∞ exist as bounded operators on the finite-energy and Lorentz-invariant solution space of the Klein-Gordon equation.

The lorentz-invariant solutions of the klein–gordon equation. I

The lorentz-invariant solutions of the klein–gordon equation. I