Spectral line broadening and angular blurring due to spacetime geometry fluctuations

We treat two possible phenomenological effects of quantum fluctuations of spacetime geometry: spectral line broadening and angular blurring of the image of a distance source. A geometrical construction will be used to express both effects in terms of the Riemann tensor correlation function. We apply the resulting expressions to study some explicit examples in which the fluctuations arise from a bath of gravitons in either a squeezed state or a thermal state. In the case of a squeezed state, one has two limits of interest: a coherent state which exhibits classical time variation but no fluctuations, and a squeezed vacuum state, in which the fluctuations are maximized.

Figure 1

A source moves along a world line with tangent $t^{\mu }$ while a detector a proper distance $s$ away moves along a world line with tangent $v^{\mu }$. The source emits a ray at point D which has tangent $k^{\mu }(\lambda =0)$ at point D and tangent $k^{\mu }({\lambda }_0)$ at A. Parallel propagation of $k^{\mu }$ around ABCD results in a slightly rotated vector $k^{\mu }+\Delta k^{\mu }$. The closed path ABCD encloses the spacetime region of interest.

Figure 2

The spacetime region between the source and detector in null coordinates. For each fixed $u=u^{*}$ between $u=0$ and $u=t_0$, one integrates along the line $u=u^{*}$ from $v=u^{*}$ to $v=u^{*}+2s$.