Diagram showing space and time in space-time. Here space is depicted as a two-dimensional entity in three-dimensional spacetime. Time from the observer's viewpoint is represented as a vertical line. Image by K. Aainsqatsi (Own work) [Public domain], via Wikimedia Commons.
A friend of mine on Twitter, Aaron (a pseudonym), is an overseas, 16-year old boy, who seriously admires Albert Einstein and wants to become a physicist. He continually writes me (Ted, also a pseudonym) questions about the theory of relativity and related topics, and I am sending answers. In this series of blog posts, those questions and answers are reproduced with modifications. I am not an expert in the fields of physics related to relativity. So, my answers might contain errors. If you find any error, please do not hesitate to write a comment for the benefit, not only of the boy and me, but also of other readers.
Aaron: Will there be time, if there is no space?
Ted: Your question does not seem to be a valid one in physics. Physics is the branch of natural science to study matter and its motion through spacetime; and these, i.e., matter, motion and spacetime, are inseparable from space.* From this viewpoint, physicists are not expected to think about the circumstance in which there is no space. In treating a complex problem, physicists often assume a simplified model of the situation, but elimination of space would make the problem non-physical.
However, the following should be noted in relation to your question: Approaches to quantum gravity, being studied for uniting quantum mechanics with general relativity, suggest the possibility that space and time are not fundamental entities but emergent phenomena (see, for example, Ref. 1). If such is the case, the phrase in your question, "there is no space," would have physical meaning in the sense that equations of quantum gravity would dispense with space and time variables.
* Especially in the theory of relativity, space and time (to say precisely, imaginary time) are treated symmetrically, and it can happen that part of time duration of one observer is part of spatial length of the other observer.
- Luboš Motl, Emergent space and emergent time, The Reference Frame (2004).