If we know nothing about the Shrike, we know even less about the Time Tombs. The Time Tombs include the Jade Tomb, the Obelisk, and the Sphinx, which (of course) looks nothing like the Sphinx in Egypt but still possesses its own riddle: the Merlin's sickness that infects Rachel.
At least Meina Gladstone has a theory about the Time Tombs: "One hypothesis is that the Tombs relate to some future war. A settling of future scores by rearranging the past, perhaps" (5.1017). She's right about that, because the Consul opened the Time Tombs to avenge Siri's death.
We have a theory of our own about the anti-entropic fields that surround the Time Tombs like waves, messing with time. One of the definitions of "entropic" is "Inevitable and steady deterioration of a system or society." Anti-entropic would therefore be anti-destruction, and that is a very good thing.
And one last idea: At the end of the book as the pilgrims skip toward the Time Tombs singing songs from The Wizard of Oz, the Time Tombs glow, an eerie Emerald City. Perhaps, like the Wizard of Oz, the Shrike is just a hoax. Pay no attention to the giant spiked four-armed killing machine behind the curtain.