Narrow Results

Accurate to the first order in the uniform angular velocity, the general relativistic frame dragging effect of the moments of inertia and the radii of gyration of two kinds of neutron stars are calculated in a relativistic σ – ω model. The calculation shows that the dragging effect will diminish the moments of inertia and radii of gyration.

We investigate velocity frame dragging with the boosted Schwarzschild black string solution and the boosted Kaluza–Klein bubble solution, in which a translational symmetry along the boosted z-coordinate is implemented. The velocity frame dragging effect can be nullified by the motion of an observer using the boost symmetry along the z-coordinate if it is not compact. However, in spacetime with the compact z-coordinate, we show that the effect cannot be removed since the compactification breaks the global Lorentz boost symmetry. As a result, the comoving velocity depends on r and the momentum parameter along the z-coordinate becomes an observer independent characteristic quantity of the black string and bubble solutions. The dragging induces a spherical ergo-region around the black string.

We investigate equatorial geodesics in the gravitational field of a rotating and deformed source described by the approximate Hartle-Thorne metric. In the case of massive particles, we derive within the same approximation analytic expressions for the orbital angular velocity, the specific angular momentum and energy, and the radii of marginally stable and marginally bound circular orbits. Moreover, we calculate the orbital angular velocity and the radius of lightlike circular geodesics. We study numerically the frame dragging effect and the influence of the quadrupolar deformation of the source on the motion of test particles. We show that the effects originating from the rotation can be balanced by the effects due to the oblateness of the source.

The Thirring–Lense effect is the phenomenon that an observer near a rotating mass, being in a state which is non-rotating with respect to the rest of the universe, experiences extra inertial forces, i.e. becomes dizzy. The first anticipation of the effect goes back to Ernst Mach; its first quantitative prediction on the basis of general relativity was given by Hans Thirring and Joseph Lense. Almost ninety years later, the effect seems to be experimentally verified.

We will mainly discuss the measurable angle (local angle) of the light ray ${\psi }_P$ at the position of the observer $P$ instead of the total deflection angle (global angle) $\alpha$ in Kerr spacetime. We will investigate not only the effect of the gravito-magnetic field or frame dragging due to the spin of the central object but also the contribution of the motion of the observer with a coordinate radial velocity $v^r=dr/dt$ and a coordinate transverse velocity $b{v}^{\varphi }=bd\varphi /dt,$ where $b\equiv L/E$ is the impact parameter ($L$ and $E$ are the angular momentum and the energy of the light ray, respectively) and $v^{\varphi }=d\varphi /dt$ is a coordinate angular velocity. $v^r$ and $b{v}^{\varphi }$ are computed from the components of the four-velocity of the observer $u^r$ and $u^{\varphi }$, respectively. Because the motion of observer causes an aberration, we will employ the general relativistic aberration equation to obtain the measurable angle ${\psi }_P$ which is determined by the four-momentum of the light ray $k^{\mu }$ and the four-momentum of the radial null geodesic $w^{\mu }$ as well as the four-velocity of the observer $u^{\mu }$. The measurable angle ${\psi }_P$ given in this paper can be applied not only to the case of the observer located in an asymptotically flat region but also to the case of the observer placed within the curved and finite-distance region. Moreover, when the observer is in radial motion, the total deflection angle ${\alpha }_{\mathrm{\text{radial}}}$ can be expressed by ${\alpha }_{\mathrm{\text{radial}}}=(1+v^r){\alpha }_{\mathrm{\text{static}}}$; this is consistent with the overall scaling factor $1-v$ instead of $1-2v$ with respect to the total deflection angle ${\alpha }_{\mathrm{\text{static}}}$ in the static case ($v$ is the velocity of the lens object). On the other hand, when the observer is in transverse motion, the total deflection angle is given by the form ${\alpha }_{\mathrm{\text{transverse}}}=(1+b{v}^{\varphi }/2){\alpha }_{\mathrm{\text{static}}}$ if we define the transverse velocity as having the form $b{v}^{\varphi }$.

In this paper we show that the Thomas precession of the spinning bodies, which is in general case constrained in all rigid bodies, induces magnetic field of the spinning bodies. This is one of the main reasons for the magnetic field of the spinning bodies. The general formula for this magnetic field is deduced and if it is applied to the Earth, its magnetic field changes between 0.295 G at the equator and 0.59 G at the poles, assuming that the density inside the Earth is uniform.