[stcredzero]

You could only modulate gravity as fast as you could move energy close to you or far away from you. An implausibly powerful gamma ray flashlight flicking on and off would let you do this. Get the beam powerful enough, and it will have a gravitational field.

[gliese1337]

Get the beam powerful enough and it will have meaurable gravitational field. All beams of any intensity technically do contribute some, albeit infinitesimal, component to the gravitational field.

It's a fun exercise to calculate just how strong a laser beam you'd need to have to make a black hole entirely out of photons.

[stcredzero]

A black hole can actually cause an additional singularity due to the blueshifting of infalling light. IIRC, one researcher studied this phenomenon and called it a "blue sheet."