Physics Asked on December 23, 2020
This question is a result of me trying to understand how this universe can be possibly infinite if it isn’t infinitely old. So to compare with an area that is flat it can be constructed both in 2D and in 3D,but a curved area can’t be an element of 2D structure but just at least of a 3D structure. Then the question is why it is so simple to imagine a curved surface of let say a baloon that is curved and finite but it becomes so difficult to imagine a space that is curved and finite as a part of a 4D structure?Is it that other ways of thinking fetch a infinite universe as a result?
You mean embedding into 4-dimensional? 4 won't be enough for all cases. See Klein bottle (its 2D).
Answered by RobertSzili on December 23, 2020
A topological space or manifold can have an intrinsic curvature independent of any embedding or immersion in a higher space. For example the projective plane has intrinsic positive curvature, while the hyperbolic plane has intrinsic negative curvature. It is perfectly possible to map each of these onto the other.
Similarly, a curved 3D space can exist without any containing 4-space. It can then be immersed in your favourite flat 4-space, though possibly not embedded without self-intersection or singularities; some 3-manifolds will require either a compatibly-curved 4-space or a flat container of 5 or more dimensions.
For example you can immerse the projective plane in a projective 3-space but not in a flat (Euclidean) 3-space.
A gentle introduction to the topic is given by Jeffrey R. Weeks in The Shape of Space, 2nd Edition, CRC Press, 2002.
Answered by Guy Inchbald on December 23, 2020
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