The expansion of the Universe refers to the phenomenon of recession (removal) of distant objects and is interpreted as a dilation of space between these objects over large distances (galaxies and clusters of galaxies). Solution to the equations of general relativity, highlighted by the physicist Alexandre Friedmann in 1922, the expansion is observed in 1929 by Edwin Hubble. The phenomenon still raises many questions, one of the most common of which is:in what is the Universe expanding?
The dynamics of the Universe is described, in cosmology, by the Friedmann-Lemaître-Robertson-Walker (FLRW) metric. The FLRW metric is an exact solution to Einstein's equations which, within the framework of general relativity, describes a homogeneous, isotropic and expanding universe. Based on Friedmann's equations — equations describing a homogeneous and isotropic expanding universe — the expansion of the universe is interpreted as a dilation of space over time.
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During this dilation, space "swells" between astrophysical objects which then move away from each other (recession). It is therefore not the objects that move on their own, but the space between them that extends and therefore carries them along in its movement. Expansion only has measurable effects on large scales; locally, gravity is stronger and therefore maintains gravitational cohesion.
In 1916, when Albert Einstein published his work on general relativity, he demonstrated the intimate relationship between space and time. Unlike Newtonian physics where space and time are absolute, rigid and immutable, general relativity describes a relative and dynamic space-time on which energy and matter can influence. Einstein's equation thus describes the direct relationship that exists between the geometry of space-time and its matter-energy content.
The spacetime of general relativity has three dimensions of space and one dimension of time (3+1). It is a four-dimensional curved spacetime, mathematically described by a pseudo-Riemannian Lorentz manifold, a particular class of differential manifold. In geometry, a variety of dimension n (with n a natural integer) is a generalizing topological object.
In this framework, the dimension of a manifold designates the number of parameters necessary to position a point locally on the manifold. In the context of general relativity, an event, to be described in space-time, requires three position parameters (x, y, z) and a temporal parameter (t). This is why the spacetime of general relativity is a 4-dimensional manifold, because the description of an event requires these 4 parameters.
On this Lorentzian pseudo-Riemannian manifold, a pseudo-Riemannian metric is applied. This means that a curvature parameter is needed to describe spacetime. The spacetime of general relativity is therefore a curved spacetime.
It is important to note that curvature is a local property of spacetime; it requires no exterior space to be described and measured. Curvature is therefore an intrinsic property of the surface of the space-time manifold.
The properties of space-time are entirely described by Einstein's equations, in particular by the curvature tensor which expresses the dynamic properties of space-time.
This description of the dynamics of the Universe is completely self-sufficient; general relativity is entirely contained within this curved 4-dimensional space-time. Calculations and measurements are strictly done internally and do not need to be integrated into a larger space.
The mathematical and physical description of the Universe is thus perfectly autonomous, it does not require any external reference to be carried out.
Expansion is a phenomenon taking place in our universe, supported by observations and measurements carried out in the latter. It is therefore a purely internal phenomenon, independent of any notion of container or exterior. Therefore, the answer to the initial question is as follows:the Universe does not expand into anything, it expands in itself, and of itself.
However, this does not mean that there is no outer or greater dimensional space in which our universe operates. It simply means that such an assumption is not necessary to describe or apprehend the expansion.
Unless future observations demonstrate the existence of an external space with more than 4 dimensions in which our universe would be contained, and that this space does indeed influence the expansion, the latter remains a strictly internal phenomenon.
