Undergrad Adding Extra Dimension for Calculations in Curved Spaces

[Kairos]

as calculations are technically difficult in curved spaces, I wonder if we would obtain the same results by adding one additional (virtual) dimension in order to embed the space in a higher order Euclidean volume, just to facilitate the treatments? (for example embed a 3D hypersphere in a 4D euclidean space)
thank you in advance

 

[fresh_42]

Does this become easier, because it is on a flat plane?

You can calculate with spacetime as if it was flat, at least either locally or on a global scale. Since the embedding doesn't change the geometric shape, there will be no gain in doing so.

 

[Orodruin]

The entire point of differential geometry is not to have to rely on an embedding space because it generally complicates things.

 

[A.T.]

as calculations are technically difficult in curved spaces, I wonder if we would obtain the same results by adding one additional (virtual) dimension in order to embed the space in a higher order Euclidean volume, just to facilitate the treatments? (for example embed a 3D hypersphere in a 4D euclidean space)
thank you in advance

Embedding curved spaces in flat higher dimensionless spaces is usually done to help with visualization. But for calculations, you usually want as few variables and as little redundancy as possible.

 

[Kairos]

OK thanks
I supposed that the simplicity of euclidean rules would compensate the additional spatial variable. bad idea !

 

[pervect]

Kip Thorne has written some popularizations in "Interstellar" about the embedding approach. But I haven't seen any non-popularized treatment of General Relativity using embeddings.

So, if you're attracted to the approach and you don't mind reading popularizations (which are usually limited, even when well written), you could try "Interstellar", but for a serious, textbook study you'd want to learn the differential geometry approach.

There's at least one other approach to GR, that uses funky fields that warp rulers and clocks in a flat space-time. This is akin to Einstein's discussion of rulers on a heated marble slab as a model for non_euclidean spatial geometries. This approach has been outlined by Straumann in "Reflections on Gravity" <<link>>. I rather suspect that Straumann's approach has some limits in regards to modelling some of the topological features that the full theory handles, and that this matters in such topics as understanding black holes, but the author doesn't discusss these limitations, unfortunately.