Is it possible to explain general relativity without tensors?

Question

I do not know much about tensors. So I wonder:

Is it possible to explain general relativity without tensors?

I have some understanding of special relativity.

I also have some understanding about differential equations and matrices.

Answer 1

One could take your question two ways. I'll deal with the first way simply by grabbing Jinawee's most excellent comment:

If you are asking this to avoid studying tensors, the answer is no. But it would be interesting to know if there is a reformulation without tensors (maybe geometric algebra or something like that). Of course, it wouldn't be simpler that tensors, like if you try to solve 2x+3=5 using an obscure theorem of algebraic geometry to avoid using normal algebra.

and leave it at that: you are not going to spare yourself any intellectual work by finding a way of talking about general relativity in the detail that physicists and geometers can do with tensors.

However, let's explore your idea a little further from a theoretical standpoint. Even from the theoretical stanpoint, it would be hard to imagine GR done without tensor-like ideas: GR needfully involves geometry, and tensor methods are the intellectual legacy of the mathematicians who thought about the geometry. So, if we want a new way to describe these geometrical ideas not involving tensors, then that might be like asking for an alternative to the Weierstrass/Bolzano limit conception of the foundations of calculus when we solve problems described by differential equations. It may happen eventually, but if it does it will be an advance in mathematics, not physics. Nonstandard analysis as an alternative formulation of calculus was presented to the world in the 1960s by Abraham Robinson: new ways of looking at differential geometry may likewise in the future be born. There are two essential ideas in Einstein's theory:

1. Highly colloquially: "Geometry = Stuff";

2. Spacetime is a smooth manifold which is locally Minkowskian; that is, the tangent space everywhere can be construed as a Minkowsky spacetime by taking, at a given spacetime point, the spatial basis vectors to be defined by (x,y,z) axes stationary (no acceleration, no rotation) with respect to a freefalling observer passing through the spacetime point in question and the time basis vector to be defined by that observer's clock: in short "special relativity holds locally for a freefalling observer";

The second point, essentially the principle of equivalence, tells us how to interpret the spacetime manifold defined by the first point (which is the Einstein field equations). Manifolds simply have tangent bundles - full stop - the second point simply tells us which World Lines through the manifold are those of observers who feel no forces. Put together, the two points are kind of "application note" for Newton's first law: they define when Newton's first law holds and tell us that if we want a body to have motion relative to these manifold and equivalence principle defined freefalling observers, then we must impart a force as behested by Newton's second law. Freefalling frames that are momentarily co-moving with you as you sit on your seat are heading down through your feet accelerating relative to you at $g{\rm ms^{-2}}$; accordingly, your seat must push you upwards with a force $m\,g$ ("your weight") to accelerate you relative to these freefalling frames.

To even have a theory at all, we must find a way to describe the first point "geometry = stuff", written as $G = T$. More specifically:

1. The left hand side of this equation, "geometry", means the deviation of geometry from Euclid's parallel postulate: it quantifies how badly the parallel postulate fails. So in any description of general relativity, we must find a way to encode this idea.

2. The right-hand side, "stuff", is fairly self-explanatory; it must encode the distribution of mass / energy in the manifold.

3. Both sides of the equation must separately tell their tales in a way such that, when we look at the manifold very "close-up" or "in-the-small" (i.e. we look at essentially the tangent space), we see all the symmetries - invariance with respect to translations, rotations and boosts - as well as the conservation laws - those of momentum, angular momentum and energy - they imply. The equations must be such that, locally, energy and momentum are conserved.

Okay: now how shall we do this without tensors? I'll concentrate on the geometry bit; this was the bit that befuddled Einstein for so long, and, moreover, it was the bit that many people before Einstein had wrestled with in probing the physical world. William Kingdon Clifford, for example, had the vague idea that the deviation of "real", physical geometry from the Euclidean parallel postulate was what the matter - the "material things" - in the World was. Riemann thought along very like lines. Gauss before him took the idea of the potential, real world failure of the parallel postulate so seriously that he had big triangles between mountain peaks measured precisely by surveyors to see whether their insight angles all summed to 180°.

One unconventional way might be to define an "absolute system of coordinates" - a high dimensional Euclidean or Minkowskian space that we then embed the physical spacetime in. There are three theorems - the strong Whitney, weak Whitney and the Nash embedding theorems that might justify such an approach. For a 3+1 dimensional spacetime, we would have to use something like an eight dimensional Euclidean space (or probably a 6+2 dimensional Minkowskian space) to do this. (I'm not sure whether you would need the full eight dimensions for relativity: I thought of asking a question on this forum about it, but have not yet gotten my ideas clear about it). Wulf Rossmann uses this approach to introduce the notion of "curved space" (i.e. one wherein the parallel postulate does not hold) at the beginning of chapter 3 in his Lecture notes on differential geometry. It would be a fair bet that someone has tried this kind of approach general relativity somewhere, although I don't know of any approach along these lines nor indeed whether there is some fundamental flaw in this idea. I'm fairly sure of a few things, though, amongst which that this would not be a simpler way to study general relativity:

1. One would be dealing with both the spacetime manifold's topology and that of its complement; presumably only the former would be "physically real" and so we have a great deal of added complexity;

2. The manifold would be defined by complicated nonlinear constraints in the higher dimensional space;

3. These constraints would encode both intrinsic and extrinsic curvature. Again, only the former would be "physically real"; the latter would show itself as maybe a rich but also as maybe an impossibly tangled and intractable family of gauge invariances. To give an example: suppose "physically real" space were a 2D cylinder, or a 2D cone, or even a 2D Möbius strip (the kind made out of a flat strip of paper) embedded in a 3D abstract Euclidean space; this is a geometrical object with only extrinsic but no intrinsic curvature. In this theory, the extrinsic curvature is given being only by the relationship between the cylinder/cone/strip and its complement in 3D space. If all that were "physically real" were the cylinder/cone/strip itself, then this curvature would have no physical reality. A local theory like GR should and would show that the cylinder's surface were wholly Euclidean, for one can develop a cylinder/cone/strip isometrically into a flat plane: we just cut the cylinder parallel to its axis and "unroll it", or cut the Möbius strip and return it to its flatness or let the cone relax to a flat circle with a sector removed. The point is that any geometry done on these surfaces is wholly and exactly Euclidean: angles in triangles sum to 180° and the parallel postulate holds. The only thing we would need to describe beyond Euclid to fully describe reality would be global topology: the fact that if you wander off in a certain constant direction you'll get back to your beginning point in the end.

4. There may need to be a different embedding for each particular manifold fulfilling the theory, thus further complicating it;

5. Most importantly for this question, one would invariably want to exploit symmetries present in special cases to find solutions to the relevant equations in e.g. spherically / cylindrically symmetric problems. So one would have to deal with the problem of finding transformation laws between the geometric objects such as vectors or whatever in different co-ordinate systems to make sure that the geometric objects stayed the same relative to the "absolute" co-ordinates". In other words, one would be drawing very near to the tensor idea. If the theory were local, and we've no reason to believe physics is not, then we are forced into the idea of a tangent space, and therefore fundamentally we must confront the idea of pullback and push forward. It's truly hard to see how one wouldn't ultimately come to this idea unless a wholly new way of doing differential geometry were found. Maybe, (and this is a real shot in the dark) far in the future, a geometry formulated on mathematical axioms grounded on mereology might look different enough to get around the tensor idea.

So now, let's forget about embeddings and stick with a 3+1 dimensional manifold. How to describe a deviation from Euclid now? It can be shown that a general method to detect deviation from the parallel postulate in any manifold is to define two matrix valued functions of tangent vectors to the manifold called the curvature $R(X,Y)$ and the torsion $T(X,Y)$. Just as the electric field is a vector valued function of space, the curvature and torsion are matrix valued functions of $T\,M \times T\,M$, where $T\,M$ is the tangent bundle of the manifold $M$ (essentially the collection of the tangent spaces at all points of $M$). If your're at a point in the manifold, you measure its local deviation from the Euclid parallel postulate by (1) checking how a vector (i.e. a member of the tangent space) changes as it is parallel transported around a little loop in the manifold and (2) checking how a frame twists as it is parallel transported along a geodesic in the manifold (see also my answer here and here). Once one has chosen the notion of "distance" (metric) in a manifold, one can always arrange one's notion of parallel transport with that metric such that the torsion is absorbed, i.e. we only have to worry about the curvature $R(X,Y)$. The curvature works like this: you define your little loop in the manifold as the parallelogram defined by the tangent vectors $X$ and $Y$ to the point in the manifold. You put them into the curvature function and then $R(X,Y)$ gives you a matrix, i.e. a linear transformation, that defines how any vector in the tangent space is changed as it is parallel transported around the loop. It can be shown that there is an open set, i.e. a finite patch of the manifold, wherein the Euclidean axioms precisely hold if and only if $R(X,Y)$ vanishes at all points within that patch (as long as we have absorbed the torsion, which we do in GR). More generally, the pair $R$ and $T$ quantify the deviation from Euclid (or Minkowsky in GR): they both vanish in a patch if and only if that patch can be given a precisely Euclidean (or Minkowskian) geometry holding for all its points. In contrast with my imagined "embedded" theory above, these two quantities work with only the manifold, so there is no "nonphysical" extrinsic deviations from Euclid as with our cylinder/cone/Möbius strip example.

So this is how we define deviation from the parallel postulate. Actually GR doesn't use the whole curvature $R(X,Y)$ it uses a "contracted form" with some other terms thrown in to make sure that our local energy / momentum conservation can happen when we put the geometry together with "Stuff". Again, although we can talk about it abstractly, we shall inevitably in practical problems want to shift co-ordinates and so shall have to talk about in detail how $R(X,Y)$ changes with co-ordinates so that it gives the same answer for the transformation wrought by the loop whatever co-ordinates we use. We shall inevitably be forced to confront the tensor nature of this geometric description.

So in summary, I don't think it is likely that one can fully forsake tensors and describe GR fully even in principle without at least some tensor ideas. I think you would be led to the same ideas, maybe under a different name.

If tensors are bothering you, there are really good overviews given in the chapter "Calculus on Manifolds" and nearby chapters in Roger Penrose's "Road to Reality", the first few chapters of Misner, Thorne and Wheeler "Gravitation" (or indeed the first chapter of Kip Thorne's Classical Physics course) - both these references have a Feynman Lectures on Physics feel to them or the early chapters of Bernard Schutz's "First Course in General Relativity". I recall getting some joy from chapter 8 of Murray Spiegel's "Vector Analysis" many years ago, but browsing it now it seems a bit clunky and dated - but then I'm not reading it the first time. Try giving it a go.

Answer 2

Of course it can be. But then, instead of a single object which incorporates certain parameters (for example, a metric tensor), which has a very useful properties, you are going to work with 16 (!) parameters (which are tensor elements), each having relations with other tensor elements. It is also more convenient to express energy, momentum in tensor formalism. It's naive to think that those who work with general relativity use the hard way. So it happens, that tensor formalism is most likely the most easy one.

Answer 3

John Baez does just that on his site, The Meaning of Einstein's Equation. Here, he gives an easily readable explanation of Einstein's equation and derives some of it's consequences, even Newton's inverse square law. The abstract:

This is a brief introduction to general relativity, designed for both students and teachers of the subject. While there are many excellent expositions of general relativity, few adequately explain the geometrical meaning of the basic equation of the theory: Einstein's equation. Here we give a simple formulation of this equation in terms of the motion of freely falling test particles. We also sketch some of the consequences of this formulation and explain how it is equivalent to the usual one in terms of tensors. Finally, we include an annotated bibliography of books, articles and websites suitable for the student of relativity.

Answer 4

I suggest this book: A Most Incomprehensible Thing. It assumes very little math background and does a good job introducing you to tensors and GR.

Answer 5

Einstein (The meaning of Relativity) and Feynman (available in audio from a CalTech lecture from his Lectures in Physics) went a long ways but I think if you want to understand it which means working some with it the answer is: No.

Answer 6

This has been mentioned before in some of the other answers but let me just expand on it. Einstein's Field equation is a tensor equation relating the curvature of space-time to the distribution of energy and momentum. Feynman states in his lectures on physics (vol 2 chapter 42) that the field equation is equivalent to the statement that the scalar curvature of space (not space-time) at any point is proportional to the energy density at that point, and this must be true for any local inertial observer. (As you know from special relativity, two observers with different velocities disagree on measurements of space/time and momentum/energy, so Feynman's condition means different things to different observers.)

It can be shown that Feynman's statement is equivalent to the field equation (mathematically they both imply each other), so in that sense you can state General Relativity in a way which avoids the use of tensors, but the problem comes when you want to calculate anything. For example, working out what the field equation implies for the space-time curvature around a spherical star is relatively straightforward, but working out what Feynman's condition implies in the same situation is very complicated (I know because I tried... and failed).

Like most mathematical tools, tensors take some getting used to but eventually make your life easier. The question you need to answer is how deep do you want to go into General Relativity? If it's more than a passing acquaintence then learning tensors will save you time in the long run.