Star got swallowed by a Warp in Space-Time

No! The title is wrong! The star was not swallowed, she is fine, thanks for asking. The article explain the situation better.

The pulsar is preceding (you can simplify this as "rotating") so the pulsar is no longer "pointing" sometimes to the Earth. We can see directly the pulsar, because it's small and far away. We only can see the emission beams, I like the lighthouse analogy. It's "pointing" elsewhere, so it's still there but we can see it.