Retarded Green’s functions in perturbed spacetimes for cosmology and gravitational physics

Electromagnetic and gravitational radiation do not propagate solely on the null cone in a generic curved spacetime. They develop “tails,” traveling at all speeds equal to and less than unity. If sizeable, this off-the-null-cone effect could mean objects at cosmological distances, such as supernovae, appear dimmer than they really are. Their light curves may be distorted relative to their flat spacetime counterparts. These in turn could affect how we infer the properties and evolution of the Universe or the objects it contains. Within the gravitational context, the tail effect induces a self-force that causes a compact object orbiting a massive black hole to deviate from an otherwise geodesic path. This needs to be taken into account when modeling the gravitational waves expected from such sources. Motivated by these considerations, we develop perturbation theory for solving the massless scalar, photon, and graviton retarded Green’s functions in perturbed spacetimes $g_{\mu \nu }={\overline{g}}_{\mu \nu }+h_{\mu \nu }$, assuming these Green’s functions are known in the background spacetime ${\overline{g}}_{\mu \nu }$. In particular, we elaborate on the theory in perturbed Minkowski spacetime in significant detail; and we apply our techniques to compute the retarded Green’s functions in the weak field limit of the Kerr spacetime to first order in the black hole’s mass $M$ and angular momentum $S$. Our methods build on and generalize work appearing in the literature on this topic to date, and lay the foundation for a thorough, first-principles-based, investigation of how light propagates over cosmological distances, within a spatially flat inhomogeneous Friedmann-Lemaître-Robertson-Walker universe. This perturbative scheme applied to the graviton Green’s function, when pushed to higher orders, may provide approximate analytic (or semianalytic) results for the self-force problem in the weak field limits of the Schwarzschild and Kerr black hole geometries.

Figure 1

The spacetime region $V$ is the volume contained within the two constant time surfaces at times $t$ and $t^{\prime }$, where $t>t^{\prime }$. The observer is located at $x\equiv (t,\overrightarrow{x})$. The dark oval is the region defined by the intersection between the past light cone of $x$ and its interior with that of the constant time surface at $t^{\prime }$. On it, we allow some nontrivial field configuration to be present, and through the Kirchhoff representations in Eqs. (1, 2, 3), the Green’s functions evolve it forward in time. We emphasize that the causal structure of the Green’s function, as exhibited by Eqs. (4, 5, 6) means, in a generic curved spacetime, the observer receives fields not only from her past light cone (edge of the dark oval), but also its interior (dark oval itself). In addition, there is some (scalar, photon, or graviton)-producing source $J$ which sweeps out a world tube, and our observer receives radiation from the portion of this world tube that lies on and within the interior of her past light cone. The picture here is to be contrasted against the Minkowski one, where observers only detect fields from their past null cone.

Figure 2

Top panel: the intersection of the interiors of the future null cone of $x^{\prime }$ and that of the past null cone of $x$ always defines a finite (as opposed to infinite) region of spacetime. Moreover, if (and only if) $x^{\prime }$ lies on or within the interior of the backward light cone of $x$ (or, equivalently, if and only if $x$ lies on or within the interior of the forward light cone of $x^{\prime }$), then there is a nontrivial intersection (indicated by the dark dashed oval) between the forward light cone of $x^{\prime }$ and backward light cone of $x$, which in 3-space we shall show is a prolate ellipsoid, when the background is Minkowski. Bottom panel: if $x^{\prime }$ lies outside the backward light cone of $x$ (or, equivalently, if $x$ lies outside the forward light cone of $x^{\prime }$), then there is no intersection between the forward light cone of $x^{\prime }$ and backward light cone of $x$.

Figure 3

Time elapsed ($t-t^{\prime }$) vs radial distance ($r$) view of the causal structure of the Green’s functions in the weakly curved limit of Kerr spacetime. The spacetime point source is located at radial coordinate distance $r^{\prime }$ from the black hole. The dark gray area represents the early time tail $|r-r^{\prime }|\le |\overrightarrow{x}-{\overrightarrow{x}}^{\prime }|\le t-t^{\prime }\le r+r^{\prime }$; and the light gray region is the late time tail $t-t^{\prime }\ge r+r^{\prime }$. The dashed line is $t-t^{\prime }=r+r^{\prime }$; the two solid black lines are $t-t^{\prime }=|r-r^{\prime }|$. As already noted by DeWitt and DeWitt [1] and Pfenning and Poisson [3], the Green’s functions undergo an abrupt change in behavior when time elapsed transitions from $t-t^{\prime }<r+r^{\prime }$ to $t-t^{\prime }>r+r^{\prime }$. The region $t-t^{\prime }\ge r+r^{\prime }$ only receives contribution from ${\widehat{\mathcal{I}}}^{(S)}$ in (129), which contains angular momentum $S$ terms only when $t-t^{\prime }$ is exactly equal to $r+r^{\prime }$; while the dark gray region $|r-r^{\prime }|\le t-t^{\prime }\le r+r^{\prime }$ only receives contribution from ${\widehat{\mathcal{I}}}_{\mu \nu }^{(A)}$ in (130) and the geometric curvature terms in (133) through (137). The tail part of the massless scalar Green’s functions is only nonzero for $t-t^{\prime }\ge r+r^{\prime }$, because it is entirely governed by ${\widehat{\mathcal{I}}}^{(S)}$. Because the photon and graviton Green’s functions depend on ${\widehat{\mathcal{I}}}_{\mu \nu }^{(A)}$ and the geometric terms, they are nonzero within $|r-r^{\prime }|\le t-t^{\prime }\le r+r^{\prime }$. However, their behaviors are altered once $t-t^{\prime }\ge r+r^{\prime }$, since they too become governed solely by ${\widehat{\mathcal{I}}}^{(S)}$.

Figure 4

At X, let there be a burst of photons from a source of finite duration. If there were no tails, these photons would sweep out a light cone of nonzero thickness proportional to the duration of the event itself. Because of the metric perturbations $h_{\mu \nu }$, we have shown via our photon Green’s function calculation that light develops a tail in a spatially flat inhomogeneous FLRW universe. The dark oval represents the tail of the photon field $A_{\mu }^{(\mathrm{tail})}$ at the present time $t$. Deep within the light cone, we argue that the size of the tail effect in our Universe is primarily governed by the power spectrum of the metric perturbations, which is currently being probed by large scale structure observations.