Gravitational redshift of photons traversing a collapsing dust cloud and observable consequences

Abstract

We analyze the frequency shift of photons propagating on an asymptotically flat spacetime describing a collapsing, spherical dust cloud. We focus on the case where the interaction of the photons with the matter can be neglected. Under fairly general assumptions on the initial data characterizing the collapse, we show that photons with zero angular momentum which travel from past to future null infinity, crossing the collapsing cloud through its center, are always redshifted with respect to stationary observers. We compute this redshift as a function of proper time of a distant stationary observer and discuss its dependency on the mass distribution of the cloud. Possible implications of this redshift effect for weak cosmic censorship and light propagation in cosmological spacetimes are also briefly discussed.

Observational distinction between black holes and naked singularities: the role of the redshift function

“…The numerical integration along the radial light rays required for this computation was performed using a fourth-order Runge-Kutta algorithm. Our numerical solutions were checked to be self-convergent to fourth order accuracy and consistent with the analytic results for the case of homogeneous density distributions [29].…”

“…Important information regarding the nature of a collapsing object is revealed from the study of the spectrum of electromagnetic radiation emitted from the surface or the interior of the object, see for example [22][23][24][25][26][27][28]. In this work, following [29], we consider a different physical scenario in which the radiation, instead of being generated from the interior of the collapsing object is originated from a distant source and crosses the collapsing object without interacting with the matter. For instance, this radiation could be emitted from a distant supernova explosion or be part of the cosmic microwave background.…”

“…[22][23][24][25][26][27][28]. In this work, following [29], we consider a different physical scenario in which the radiation, instead of being generated from the interior of the collapsing object is originated from a distant source and crosses the collapsing object without interacting with the matter. For instance, this radiation could be emitted from a distant supernova explosion or be part of the cosmic microwave background.…”

“…The explicit formula for computing this frequency shift function has been derived in Ref. [29]. The numerical integration along the radial light rays required for this computation was performed using a fourth-order Runge-Kutta algorithm.…”

Shadow of a naked singularity

“…Evidence supporting this proposal came from the analysis of radial photons from an external source passing through a collapsing spherical dust cloud. As shown in [24], the total frequency shift of such photons relative to asymptotic static emitters and observers is always toward the red. Moreover, the analysis in I revealed that the photon's energy measured by the asymptotic observer is gradually redshifted toward zero in the black hole case, whereas whenever a globally naked singularity forms the redshift of these radial photons remains finite and their frequency shift exhibits an abrupt cutoff once they have been detected by the observer who has crossed the Cauchy horizon (see Figs.…”