The Significance of a Non-Constant Scalar Field in 4-Dimensional Spacetimes

[MathematicalPhysicist]

this is from his website:
"In what class of 4-dimensional spacetimes does there exist a real, non-constant scalar field φ with the following properties:

It obeys the wave equation: ◻φ=0
Its gradient is everywhere null: ∇φ.∇φ=0"
somehow i think it is concerned with his work, but what is its siginificance?

 

[Wave's_Hand_Particle]

I believe this is the link:http://xxx.lanl.gov/ftp/quant-ph/papers/0401/0401024.pdf

As to the significance :) I'll leave that for now, while Deustch is a remarkable thinker, I wonder if you could place a link with relevance to the context of your post, others may then want to comment on any significance?

 

[MathematicalPhysicist]

i got the question from deustch's website, in the question section.