While that wouldn't work for heaps of reasons, I like your idea. I guess the issue would be discerning 'when' such a thing happened, and finding the right item to look back at, which would be a difficult challenge...
And this in turn is one of the big reasons why FTL is probably impossible: It is in essence time travel.
It has been suggested that black holes could provide this mirror, but it requires resolving not just something the size of Earth, but something much smaller, at a distance of 30 million light years +. Unfortunately you actually start hitting up against the limits of spacetime's resolution itself fairly quickly. But at least in principle, it is possible that some lucky photon has bounced off a dinosaur, and has traveled all the way out there, and all the way back to Earth, just now, that if you knew where to look in the sky, perhaps such a photon might even enter your eye now. Or perhaps there is some human out there who really has "seen" the dinosaurs. But you'd never really know.
Beyond speculation, I'm not aware of anything showing spacetime to be discrete.
> Beyond speculation, I'm not aware of anything showing spacetime to be discrete.
It is a prediction of loop quantum gravity [1]:
In 1988, Carlo Rovelli, Lee Smolin, and Abhay Ashtekar introduced a theory of quantum gravity called loop quantum gravity. In 1995, Rovelli and Smolin obtained a basis of states of quantum gravity, labelled by Penrose's spin networks, and using this basis they were able to show that the theory predicts that area and volume are quantized. This result indicates the existence of a discrete structure of space at very small scale.
[1] https://en.wikipedia.org/wiki/Carlo_Rovelli#Loop_quantum_gra...
> The relatively rapid spatial and temporal variability of the X-ray radiation from some molecular clouds near the Galactic center shows that this emission component is due to the reflection of X-rays generated by a source that was luminous in the past, most likely the central supermassive black hole, Sagittarius A[star]. Studying the evolution of the molecular cloud reflection features is therefore a key element to reconstruct Sgr A[star]'s past activity. The aim of the present work is to study this emission on small angular scales in order to characterize the source outburst on short time scales. We use Chandra high-resolution data collected from 1999 to 2011 to study the most rapid variations detected so far, those of clouds between 5' and 20' from Sgr A[star] towards positive longitudes. Our systematic spectral-imaging analysis of the reflection emission, notably of the Fe Kalpha line at 6.4 keV and its associated 4-8 keV continuum, allows us to characterize the variations down to 15" angular scale and 1-year time scale. We reveal for the first time abrupt variations of few years only and in particular a short peaked emission, with a factor of 10 increase followed by a comparable decrease, that propagates along the dense filaments of the Bridge cloud. This 2-year peaked feature contrasts with the slower 10-year linear variations we reveal in all the other molecular structures of the region. Based on column density constraints, we argue that these two different behaviors are unlikely to be due to the same illuminating event. The variations are likely due to a highly variable active phase of Sgr A[star] sometime within the past few hundred years, characterized by at least two luminous outbursts of a few-year time scale and during which the Sgr A[star] luminosity went up to at least 10^39 erg/s.
0) https://arxiv.org/abs/1307.3954
Edit: Older image: https://www.chandra.harvard.edu/photo/2007/gcle/
But, darn it, now we're going to have to map the cosmic background light pollution. :)