OK. Suppose you have a surface with a closed curve as a boundary. Then suppose that surface grows like a soap bubble but the boundary is stationary like the orifice through which air passes to make the bubble grow. It would seem that the 2D surface grows in both dimensions in the middle of the bubble, but the buble is not growing in at least one dimension along the boundary. What would be the equation for the metric both in the middle and on the boundary and as it approaches the boundary? I wonder all this because in a different thread I explore the possibility that matter may be the boundary of an expanding universe. If so, I wonder what distinction there is in the metic of space as it approaches the boundary (particles). I'm kind of thinking that matter may be like a stationary boundary where the growing space must somehow bend and stretch to accommodate a fixed boundary. But the photon particles may be where the boundary grows right along with the surrounding space. I suppose you could have a boundary that has portions that expand and protions that are fixed.