Not saying I understand GR's spacetime curvature to a great degree, but what I do 'know', it does not require a forth space dimension to 'bend' into. In other words, all curvature is intrinsic to our normal three spatial plus time dimensions. The following is from the book ''The Science of Interstellar'' by Kip Thorne The film made great play of its graphical representation of a black hole. To me, the book seems to mislead with this ''warped in a higher -dimensional bulk.'' Yes, I know it pop-science, but it's the first time I've come across the use of the bulk to explain GR. Please Register or Log in to view the hidden image!
I wouldn't question Dr. Thorne. I think (as I believe you're saying) he's trying to make some every-day sense of what is, inherently, a very difficult concept. Look @ 4.4 [above] now rotate it 360 degrees, one step at a time, clockwise. Now, rotate it another 360 degrees, one step at a time, toward (or away from) you. The 'bulk' is the extra dimension into which that bulge below the Sun (indicative of the gravity well) goes... It is always 90 degrees tangential to the plane, no matter how you rotate the plane. Clearly, though, if you've rotated the plane 90 degrees, the gravity well isn't lying on the plane of some other perspective. The well is always 'down' from whatever frame of reference you choose, so where does that 'well' go? Into the 'bulk.'
I think I can understand where your coming from ''the gravity well isn't lying on the plane of some other perspective.'' Your saying (I think) it's not a real dimension, I can agree with that. My understanding of the Bulk is that, although highly theoretical, it is to be taken as a real dimension for the purpose of use in M-Theory. I don't think Kip Thorn is helping by introducing the Bulk and Branes from M theory into General Relativity. To me, it would be like saying the embedding diagrams or ''rubber sheet pictures'' are real dimensions when explaining GR. https://en.wikipedia.org/wiki/Introduction_to_M-theory https://en.wikipedia.org/wiki/Brane_cosmology
Yes. At least I apparently described it fairly well. Unfortunately for the larger picture, though, these concepts don't translate well into the general Lexicon. I believe Dr. Thorne is trying yet another phraseology, but it suffers from the same obstacle: translating primarily mathematical concepts into understandable lay language. I do have a small nit to pick: by definition, GR & M-theory (or any of the other attempts @ cosmological explanation like 'string' theory) must be integrated, or co-existent. After all, all 'explanations' are in pursuit of the same Grand Unified Theory, & so must be mutually derivative, I think.
