Reader's corner. October
1988/10/01 Agirregabiria, Juan M. Iturria: Elhuyar aldizkaria
Question: According to Einstein's theory of General Relativity, Gravity is the spatial curvature of space/time around each mass. Therefore the Universe would be 4 dimensions, but not all are spatial, one is time. But Martin Gardner in his book "Left and Right in the Cosmos" says: According to Einstein's theory, if an astronaut travels as directly as possible, he would arrive at his starting point after a sufficient distance: the three-dimensional world is considered the hypersurface of a hypersphere of 4. The question is: How is this deduced? Is it a consequence of what was said above or another way of saying the same thing? The local curvature (created by masses) is not the same as that of the entire Universe. On the other hand, are these 4 dimensions spatial in this case? (if so, it is not the only one proposed by this Einstein, as I think).
Answer: If I see some aspects in the reader's letter, I will try to clarify them one by one.
1.- According to the reader, in General Relativity each mass (each star, for example) creates a local curvature around it. But in Cosmology we must take into account the mass of the whole Universe. In fact, on the cosmological scale (that is, on the scale of the entire Universe), in all places there is an average mass distribution and, therefore, an average curvature not null. Thus, the local curvature generated by each star is produced microscopically from the global cosmological point of view and only the global mean curvature generated by all the masses is analyzed. The same happens in all areas of physics: despite being a fundamental local theory, global effects occur as a result of the sum of local effects.
2.- Taking into account the fundamental equations of Einstein's theory and some simple hypotheses such as the homogeneity of space and isotropy, the simplest model of the Universe is obtained. As with all models of Physics, this is neither final nor final. But it seems that within its limits is a precise, useful and productive approach. Known as Robertson-Walker, it is the core of the standard model. In theory, the geometry of this model can be of three different types, and in one of them can occur what the reader mentions: ...if an astronaut goes as correctly as possible, he would reach his starting point... Here we want to emphasize that this is a possible theoretical conclusion (but not necessarily).
3.- On the other hand, the reader is not wrong. In General Relativity there are only three spatial dimensions.
4.- But let us make some mathematical considerations. For us to consider the three-dimensional space in the model mentioned as a four-dimensional hyperspace (and as a spherical hypersurface) from the mathematical point of view (we will insist). But there is another equivalent possibility: to think that it is in itself (and not in any other space). To explain it better, take as an example a plane. It is possible to assume it or think it is within space (or in a hyperspace with four, five, ..., unlimited dimensions). In any case it has only two dimensions. The plane has its own character and does not depend on its location within another space. The same goes for a spherical surface.
At school we studied the definition of the spherical surface within space, but as Gauss and Riemann demonstrated in the last century, the definition of the same spherical surface can be done without any other space and by two dimensions. Both definitions are totally equivalent and in both the same surface has only two dimensions. Similarly, the space of the Robertson-Walker Universe (apart from time) has only three dimensions, although it is possible (but in no case necessary) to think that it is within another hyperspace.
5.- From the physical point of view this imaginary hyperspace is possible, but totally useless, since the hypothesis of its existence has no effect or way of demonstrating it. The image of space as a spherical hypersurface of a hyperspace can sometimes be useful to better understand some properties, but it is not at all casual and sometimes can be harmful (it seems that has been so in the case of the reader). It is true that understanding this type of curved spaces without the help of another flat space is not easy, because our common sense is very limited. Perhaps we cannot really understand this kind of thing (surely the study of this possibility would be a profound problem of psychology), but it is possible to acquire the habit of using this type of concepts and, in the end, we think we have understood them.
6.- Finally, a note. Unlike General Relativity, in some current theories the dimensions are more than four. But this is another thing, because the other dimensions are not what we usually call space or time.
Juan Mª Agirregabiria
University of the Basque
Country Basque Summer University