# Why isn't general relativity the obvious thing to try after special relativity?

To preface my question, I ask this as a mathematics student, so I don't have a very good sense of how physicists think.

Here is the historical context I'm imagining (in particular taking into account the development of differential geometry in the 19th century):

Classical mechanics is about Lagrangians of matter fields on $\mathbb{R}^3$ (with the flat metric)

Sometime in the 1820s Gauss speculated about replacing the flat metric on $\mathbb{R}^3$ by one with non-vanishing curvature

Special relativity is about Lagrangians of matter fields on $\mathbb{R}^{3,1}$ (with the flat metric). I'm taking this jump for granted since as I understand it, it was arrived at from experimental results on electromagnetism at the end of the 19th century.

Now in line with Gauss, it seems like it would be extremely natural to speculate about replacing the flat metric on $\mathbb{R}^{3,1}$ by one with non-vanishing curvature (and in the same spirit also considering more exotic topologies for the underlying manifold).

Given that then we would have to ask exactly which metric we are looking for, it seems natural to say that there should be a Lagrangian term corresponding to the metric. The Einstein-Hilbert functional is probably the simplest one to try. And so we get the Einstein equations.

Alternatively (as I heard from someone is the actual history) you could observe that the energy-momentum tensor is of course a 2-tensor, and so for an Euler-Lagrange equation the most natural metric-dependent expression would be $\operatorname{Ric}=T$. Since $T$ is always divergence free it would be natural to replace $\operatorname{Ric}$ by $\operatorname{Ric}-\frac{1}{2}Rg$ just from taking the contracted second Bianchi identity into account. And so again we get the Einstein equations.

I've often heard it said that if Einstein had not come up with special relativity, someone else probably would have in the next five or ten years. However if he had not come up with general relativity, it would have taken much longer to discover. Why is this? I feel that I must be missing something here.

• " However if he had not come up with general relativity, it would have taken much longer to discover." Hilbert came up with it at basically the exact same time. There was also Nordstrom Gravity, which worked off of the trace rather than the full tensor. It really wouldn't have taken nearly as long as people think; the ideas were ripe for the taking at the time they were found – Robert Mastragostino Mar 20 '14 at 1:24
• The curved spacetime approach only works because of the equivalence principle. The deep insight required to be able to come up with general relativity was seeing that the equivalence principle was the main content of Newtonian gravity, and that this led naturally to a metric theory. Only once you have that can you start thinking about field equations and manifold theory. – Jerry Schirmer Mar 20 '14 at 2:00
• @RobertMastragostino: they certainly knew about each others' work, but they were definitely working independently. A bit of a trivia point is that Hilbert published his paper first, but admitted that Einstein deserved the credit for general relativity. – Jerry Schirmer Mar 20 '14 at 2:00
• @youler: it was considered to be a coincidence of the form of the Newtonian force law for a long time. The insight was figuring out that it was the essential feature of gravity from which other things flow. – Jerry Schirmer Mar 20 '14 at 2:27
• I should also add that a modern development of special relativity is very much informed by the subsequent development of general relativity. While the concept of hyperbolic rotation was certainly present in 1905 with Minkowski, it certainly wasn't as developed as it is now, nor was the real significance of hte negative signature metric known. – Jerry Schirmer Mar 20 '14 at 2:59

I do think Jerry Schirmer answered the question in the comments, but I'll try to expand just to make clear how he explained everything.

Let us consider given that special relativity is correctly described by physics in Minkowski spacetime. Then we can ask ourselves how to include gravity without violating causality, which is mandatory by the finite velocity of light.

The idea is to consider Einstein's elevator. Namely that there is no local experiment which can be done that can differentiate between bodies in free fall in a constant gravitational field and the same bodies uniformly accelerated. That's because gravity affects everything the same way. A somewhat formalization of this is called Einstein's equivalence principle (in contrast with Galileo's, that say about coordinate transformation by constant velocities).

Note first that this is not the case for eletromagnetism. One can always use test charges to determine the electromagnetic fields, and it is impossible to do away with them using accelerated frames. Also, the equivalence principles is strictly local. If you look at extend regions gravity will appear through tidal forces.

So, if you think that special relativity is a particular case of general relativity (because it's just the same without gravity) the question is: what looks locally like special relativity but not globally? The answer is curved lorentzian manifolds, that locally are Minkowski.

But, as Jerry stressed, if you think in curved manifolds as generalization of flat ones, that does not, in principle, say anything about gravity. Only by noticing it is a force unlike any other, and formalizing it through the equivalence principle, one can justify the physics behind it, that is the use of curved manifolds. For instance, you suggest it is natural to generalize the situation by allowing curved spaces, but from the mathematical point of view one could just as well argue that there are other forms of generalization, e.g. we could instead try to projectify Minkowski. This is indeed usefull in other contexts, but it has nothing to do with gravity. So for a physicist is important we have "conceptual insights" to guide the process of "generalization for comprehension", or in other words we need principles with physical content.

I'm really unsure about what Gauss could be thinking regarding the metric. He did try to formulate classical mechanics in a differential geometrical way (Lanczos "Vartiational principles of classical mechanics" discusses it), but if that's what you're referring to, then it had nothing to do specifically with gravity.

EDIT: Oh boy, that last sentence is very misleading, I'm sorry. I had a look at Lanczos' book and realized that while Gauss pushed for a different formulation of classical mechanics, it's called Principle of Least Constraint, page 106 in Lanczos, it was only after some time that Hertz gave the principle the geometrical interpretation. So really not relevant to you question. I won't erase the paragraph though, in case anyone is interested.

Also, the equivalence principle argument says nothing about the field equations, and would be true even if the correct equations were different. As a matter of fact, a lot of general relativity independs of Einstein Field Equations, like the causal structure and (to some extend) the singularity theorems. This is why the equivalence principle was formulated as early as 1907 but the field equations came only in 1915.

I'm not a big fan of "what if" questions in history, majorly because they don't seem to have answers, but while Poincaré had the Lorentz trasnformations and a lot of understanding of special relativity, I never heard of anyone who anticipated the equivalence principle. So I hope this makes plausible that while others could have done SR, it did not seem likely that GR was coming, because first it was needed to understand what gravity is. Nordstrom's theory is an extension of ideas of eletromagnetism and was bound to failure. Hilbert indeed got the field equations right on his own, but would not get there without the motivation of curved spacetimes

I'm not sure if your main interest lies on the question on the title of this thread or in the question you pose near the end of your text. I'll try to answer both, in spite of not being an expert neither on GR nor on the education on physics.

Why is it, if it is true, that GR would have taken many more years to discover, had Einstein not discovered it?

I agree with you and the comments to your question in saying that GR would most likely have emerged, in the following years (though "how many years?" is a question which I don't think anybody can answer) as a consequence of the equations given by Hilbert in the paper he published at almost the same time. A detailed account on this subject can be found on the following Physics SE question: Did Hilbert publish GR before Einstein?, and I can quote Pais' biography of Einstein on this subject (for a beautiful account of this particular episode of the development of GR, check chapters 11-14 of "Subtle is the Lord":

Let us come back to Einstein's paper of November 18. It was written at a time in which (by his own admission) he was beside himself about his perihelion discovery (formally announced that same day), very tired, unwell, and still at work on the November 25 paper [the paper called "The Field Equations of Gravitation"]. It seems most implausible to me that he would have been in a frame of mind to absorb the content of the technically difficult paper Hilbert had sent him on November 18. More than a year later, Felix Klein wrote that he found the equations in that paper so complicated that he had not checked them. [...]

I rather subscribe to Klein's opinion that the two men 'talked past each other, which is not rare among simultaneously productive mathematicians'[...] I again agree with Klein 'that there can be no question of priority, since both authors pursued entirely different trains of thought to such an extent that the compatibility of the results did not at once seem assured'. I do believe that Einstein was the sole creator of the physical theory of general relativity and that both he and Hilbert should be credited for the discovery of the fundamental equation.

I am not sure that the two protagonists would have agreed [included as a funny note].

Subtle is the Lord, Chapter 14, page 260. I think this supports my previous statement quite well.

Why isn't general relativity the obvious thing to try after special relativity?

I think this question can only be answered if we are talking about the teaching of the subject to undergrads in college. If we were talking about one teaching him-/herself GR after learning SR, I believe that the answer would be "well, mostly because one doesn't want to/couldn't do it; in fact, one could have done it before", or something along those lines, which doesn't quite help.

When talking about teaching SR to undergrads, you've got to understand this: there isn't a standard for teaching the theories of Relativity to undergrads (written as an as-of-yet undergrad). We mostly get taught versions of the theory with not so much emphasis on the mathematical structures lying below the subject, and get mostly taught to do calculations and think of the dynamics. So we, in these cases, don't always get to know the concepts of metric, of flat spacetime (which we actually use, but almost never think about it for the course), of curved spacetime, among others.

Add to that the fact that, in many universities (such as mine and the one's of many of my acquaintances), we don't get to learn much math (besides Calculus, some Linear Algebra and Complex Variables) before plunging into the vast world of physics subjects, so we are not in a position to say "oh, of course, this concept that I saw when learning SR can be, by analogy with these things which I haven't learned, extended to a more general (and complicated) one, which would link to some variational principle in order to get some new and complicated tensor equations, which we can't solve".

So well, if you see it from a mathematician's point of view (yours), the next step from SR is quite obvious because of the mathematics you know, but if you are a physics undergrad, who has (most likely) not had an intense training in mathematics, you are certainly not going to be able to take that step. Though I may be generalizing horribly, but all that I've said is true, in my experience.