# Gravity creating other dimensions [duplicate]

It is a well known fact that gravity can bend space, kind of like a rock placed on a sheet of paper, bending the paper. But, doing this will make another third dimension, that the marble actually travels through, since the paper gets stretched down, making the marble go down the z-axis. Does this mean that gravity in 3-d space will create another fourth dimension, in which planets are also going through, even though we are not able to observe it? As you can see, the bend in the plane makes it 3d.

It is crucial to understand that curvature of a manifold is an intrinsic property of that geometry, that does not need us to embed the manifold in a higher dimensional Euclidean space. It is studied by measuring distances between points on the manifold itself. A better analogy would be the following: take a sheet of paper, draw two points on it ($$P=(x_1,y_1), Q=(x_2,y_2)$$). Now, instead of considering the distance between them to be $$PQ=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$ define some other notion of separation between points (define a different metric). This would allow us to define a manifold that is topologically the same as a sheet of paper, but has different local geometrical properties. What is important is that at no point do we need to care whether the paper is floating around in a three dimensional room or a fifteen-dimensional one. All questions about the geomtery can be answered within the two dimensions of the sheet.