# What is a spacetime continuum?

A very common expression I see in pop science is "the spacetime continuum". This expression isn't commonly used in modern discussions of general relativity, but looking at some older papers on the topic, it does seem to pop up occasionally back then.

Is there a formal definition of what a continuum is? Is it some terminology for a manifold of the era, or a more general space?

• You've got it the wrong way around, the modern notion of manifold was in part motivated by the notion of a spacetime continuum. Mar 17, 2020 at 23:17
• At the end of 19th beginning of 20th century there was a lot of reflection on the nature of the "continuum", with various formal and intuitive notions intensely debated. The original, Aristotelian, continuum was replaced by the arithmetized one of Cantor-Dedekind, but back then some favored alternative versions, intuitionistic (Brouwer, Weyl), non-Archimedean (Veronese), etc. Weyl, whose paper you linked, even wrote a treatise Das Kontinuum. However, the differences do not affect the physics of spacetime, hence the decline in use. Mar 18, 2020 at 6:07
• Riemann had a paper about "manifolds", too, c. 1855. Mar 18, 2020 at 16:27
• In modern usage they just say spacetime, or spacetime manifold. The formal definition of manifold is of later date, it is not due to Riemann but to Weyl (who explained Riemann's intuitive ideas). Mar 19, 2020 at 2:11

William Clifford didn't define a spacetime continuum, but he did write in On the Spacetime theory of Matter:

Riemann has shewn that as there are different kinds of lines and surfaces, so there are different kinds of space of three dimensions; and that we can only find out by experience to which of these kinds the space in which we live belongs. In particular, the axioms of plane geometry are true within the limits of experiment on the surface of a sheet of paper, and yet we know that the sheet is really covered with a number of small ridges and furrows, upon which (the total curvature not being zero) these axioms are not true. Similarly, he says, although the axioms of solid geometry are true within the limits of experiment for finite portions of our space, yet we have no reason to conclude that they are true for very small portions; and if any help can be got thereby for the explanation of physical phenomena, we may have reason to conclude that they are not true for very small portions of space.

I wish here to indicate a manner in which these speculations may be applied to the investigation of physical phenomena. I hold in fact

(1) That small portions of space are in fact of a nature analogous to little hills on a surface which is on the average flat; namely, that the ordinary laws of geometry are not valid in them.

(2) That this property of being curved or distorted is continually being passed on from one portion of space to another after the manner of a wave.

(3) That this variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or etherial.

(4) That in the physical world nothing else takes place but this variation, subject (possibly) to the law of continuity.

This conjecture was formed in 1876, fifty years before Einstein wrote down his equations for GR! Moreover, he went onto write:

I am endeavouring in a general way to explain the laws of double refraction on this hypothesis ...

So although he didn't dream up gravitational lensing, he could see that, given his conjecture, there were implications for how light travelled in space.

Another important precursor is Hertz who, thinking that there were two fundamental concepts - inertia and force - felt that this was one concept too many and theorised that all forces could be rethought as constraint forces and to accomplish this introduced the notion of higher spaces, that is higher dimensional spaces. This is an important conceptual advance for both GR (gravity is a constraint force) and Kaluza-Klien unification which was revived in the much later String Theory.

It's also worth noting that Schrodinger in his essay Science & Humanism cautioned identifying the mathematical or geometrical continuum with the physical continuum:

However painful its loss may be, by losing it we probably lose something that is very well worth losing. It seems simple to us, because the idea of the continuum seems simple to us. We have somehow lost sight of the difficulties it implies ...

And in fact the same had been said by Einstein, who against the popular view of GR as a geometric theory insisted on its physicality by pointing out that the equation of motion of a free particle in GR, geometrically seen as the geodesic equation, was for him, pre-eminently a unification of inertia and gravity - both physical rather than geometric concepts.