After reading Rudolf Carnap on measurement for my dissertation, I happened to stray into the second part of the book, where he starts to talk about non-Euclidean geometry. This was not stuff I needed to read, but it is a topic that has fascinated me ever since, some 15 years ago now, I first tried to get my head round general relativity (GR).

One of the things about this topic that interests me is my suspicion that there are a lot of people who talk about such things as “curved spacetime” and “curved space” without actually knowing much about what they are talking about, or being able to define their terms, etc.

Now, I’m not necessarily saying I don’t believe in curved space or curved spacetime – it’s just that it’s a concept I find it difficult to get my head round, and I’d be surprised if others did not have a lot of difficulty with it too; hence my suspicion that people who claim that it’s all really very straightforward either haven’t really thought about it much, or are bluffing.

There are two main problems. One is the fact that GR assigns to spacetime a *structure* which somehow governs the motion of bodies; this is very different from pre-GR conceptions of spacetime, which view it simply as a coordinate system in which we define the positions of bodies, and I have often felt that textbooks do not really make a big enough issue of this distinction (OK, they may *mention* it, but there is never any attempt to *justify* such a huge conceptual shift). The other concerns the concept of being “curved” when it is applied to three-dimensional space. It is this aspect of GR which my reading of Carnap brought back to me.

In everyday life, curvature can be a property of either a line or a surface. But if we are speaking of a *line*, its curvature only makes sense in relation to points outside the line (and is defined, mathematically, in terms of the distance from the line to the centre of curvature); if we imagined ourselves travelling along the line, I cannot see how we could become aware of any bends in it. Likewise, a *surface* may be curved, but only in relation to points outside the surface. So curvature makes sense for a 1-dimensional object in a 2-dimensional space, or for a 2-dimensional object in a 3-dimensional space. But how can 3-dimensional space itself be curved? Only, presumably, in relation to points outside 3-dimensional space – in other words, points in a fourth dimension.^{1} And even then, we can only argue from analogy with the 2D and 3D cases, which is not necessarily a good way of arguing. Granted, mathematics deals with “spaces” that have more than 4 dimensions, in fact they can have any number of dimensions; but again, this is not a guarantee that those multidimensional “geometries” have counterparts in the real, physical world.

Where does Carnap come into all this? Well, he uses a familiar analogy – he looks at what happens if we try to do geometry on the surface of a sphere. His argument, in common with almost all other arguments on this topic, refers to “triangles” being drawn on a spherical surface; he uses the term without apology and without defining what he means. This caused me some difficulty, since when I learnt geometry at school I was always taught that triangles (and, for that matter, squares, rectangles and other figures) are “plane figures”. On the basis of such definitions, it makes no sense to talk of triangles on the surface of a sphere; and the fact that, as Carnap goes on to show, these “triangles” have “angles” which add up to more than 180 degrees should not raise any eyebrows; nor does the fact that there is no analogue of parallel lines on the surface of a sphere. It is simply a different kind of surface with a different sort of geometry. I doubt, in fact, whether any of my classmates in school geometry would have had a problem with the idea that, if you change the definition of triangles to include figures with curved sides, their properties change too.

Is this simply playing with words? I would argue that it is not – it is more than that. Because all these analogies tend to get piled on top of one another and then relied on as though they were all completely watertight. It’s not uncommon to encounter the argument that, because these spherical “triangles” have angles that add up to more than 180 degrees, and “triangles” drawn on a different sort of curved surface have an angle-sum *less* than 180 degrees, we could somehow draw huge triangles in space, measure the angles and deduce what sort of (four dimensional) universe we live in. For instance, Percy Bridgman (whom I am also reading for my dissertation) mentions the fact that Gauss “checked whether the angles of a large terrestrial triangle add to two right angles and found agreement within experimental error”, but goes on to add that “we now know from the experiments of Michelson that if his measurements had been accurate enough he would not have got a check, but would have had an excess or defect according to the direction in which the beam of light travelled around the triangle with respect to the rotation of the earth”. I’m not sure what all this means, but it seems likely that, given the fact the relativistic corrections to everyday terrestrial quantities are usually tiny, it would be the curved nature of the surface that would produce the larger discrepancy, regardless of the direction of the light beam or the rotation of the earth. Carnap also refers to Gauss’s measurements – which he says concerned a triangle formed by three mountain-tops in Germany, from which Gauss is supposed to have been able to determine whether “space is Euclidean or non-Euclidean”. It’s true that such a triangle would not necessarily be affected by the curvature of the earth, since if Gauss’s instruments allowed him to measure both bearings and angles of elevation, he would indeed be looking at a plane figure, although Carnap does not say whether this is the case.

Of course, we now know that accurate measurements of a triangle such as the one formed by the three mountain tops would *not *give 180 degrees, because the sides of the triangle – which are light beams – would not be straight but curved. Eddington’s experiment in 1919 – which has been corroborated many times since – showed that light rays are bent by gravitational fields. But all this tells us is that light beams are no good for making a straight-sided plane triangle whose angle-sum is 180 degrees – it does not necessarily mean we *can’t* make one.

What makes light beams bend is actually the *metric*. The metric is the physical law that determines what is the shortest distance between two points (in four-dimensional spacetime). We think of the shortest *spatial* distance between two points as the length of the straight line joining them, which equals the square root of the sum of the squares of the differences in the three spatial coordinates (this follows from Pythagoras’s theorem). But Einstein says that in the presence of mass, the metric is altered so that the line is not necessarily straight; light, and any massive particle which is not acted on by a force, follows such a path instead of (as Newton said) a straight line. This is why GR is often summarised as “matter tells space how to curve, and space tells matter how to move”.

Admittedly the idea that the metric is not necessarily the normal “Euclidean” one is a very strange one which suggests that we ought to start thinking about the nature of space in a radically different way.

Even so, I am not convinced that this is the only way of looking at things, or that we could not construct a plane triangle whose angles add up to 180 degrees, regardless of the amount of mass present, though admittedly I do not have a ready-made set of instructions for doing so. Percy Bridgman, who argued that a physical concept only makes sense if we can produce a set of instructions for measuring it, would probably have disagreed with that, however.

^{1} It can be argued that the 4^{th} dimension referred to here *cannot* be time; but that is probably a subject for another blog.

Michael Weiss

said:Hmm, I haven’t read Carnap’s treatment, but I’m guessing, from your description, that explaining GR wasn’t his main priority.

Three points:

(1) You say:

GR assigns to spacetime a structure which somehow governs the motion of bodies; this is very different from pre-GR conceptions of spacetime, which view it simply as a coordinate systemActually this already occurred when Minkowski recast SR in terms of (flat) spacetime.

(2) You say:

Likewise, a surface may be curved, but only in relation to points outside the surface.Nope, not at least in pure math, ever since Gauss laid the foundations of differential geometry. Gauss showed how to define the Gaussian curvature of a surface at a point, purely in terms of data internal to the surface (a so-called intrinsic definition).

I’m reluctant to write more, since I fear a short description will fall prey to the Scylla of inaccuracy, and the Charybdis of unintelligibility — the sort of quick summary that makes sense only to someone who already knows it all from a full treatment. I’ll just add that in the context of a formal development (namely differential geometry), a surface can be “intrinsically” curved — not just curved in relation to points exterior to the surface. This is quite different from a one-dimensional curve (or a line, as you call it) — there indeed there is no such thing as intrinsic curvature.

(3) Are light rays straight? Here we touch on philosophical matters. We can give an abstract, axiomatic, purely mathematical specification of much of GR: spacetime is a 4-dimensional manifold with a semi-definite metric, etc. etc. Pure mathematicians can prove theorems about this. It doesn’t become a physical theory until we say how to hook up the abstract mathematical notions with the “real world”. We need, so to speak, a dictionary correlating pure math concepts with things that can be observed.

Part of the usual dictionary for GR says that geodesics (which can be given a purely mathematical definition) correspond to light rays. (Well, certain geodesics — so-called null geodesics.) In some sense this is a physical hypothesis. I don’t see any way that this hypothesis could be tested

in isolation— in practice, the entire structure of GR, i.e., pure math + dictionary, is tested against various experiments.‘Nuff for now.

Jim Grozier

said:Hmmm … I will have to have another go at GR (have had 2 already, maybe will be third time lucky. Last attempt I fell foul of the Christoffel thingummies). Also must read some differential geometry. However need to get this damned MSc finished first! Do you have any recommendations re textbooks, Michael?

Michael Weiss

said:Hmm, recommendations… Chris Hillman used to have a great website, “Relativity on the World Wide Web”, filled with good links and an excellent off-line bibliography, but I see he’s taken it down.

Take at look at John Baez’s website: http://math.ucr.edu/home/baez/gr/

and also Sean Carroll’s: http://preposterousuniverse.com/grnotes/

“The Phonebook”,

Gravitationby Misner, Thorne, and Wheeler, is probably still the best place to gain real intuition for both GR and basic differential geometry. Problem is, it’s practically black-hole massive (ok, “only” 1279 pages). But for what I think you want, all you need is Chaps 1, 2, and 8.Often in math, older treatments are more intuitive, if messier. But in differential geometry, the opposite seems true. The Christoffel symbols are a technicality, the “machine code” needed to implement parallel transport and covariant differentiation. These can be treated quite conceptually.

With 60 gajillion webpages out there, it seems like the ideal short, intuitive explanation of the basics of curved spacetime

shouldbe out there somewhere. Even if 59.99999 gajillion of the webpages are just cute pictures of cats.Maybe I’ll have to start my own blog…