## Why is General Relativity so difficult?

Just a brief post following yesterday’s centenary of General Relativity, after which somebody asked me what is so difficult about the theory. I had two answers to that, one mathematical and one conceptual.

The Field Equations of General Relativity are written above. In the notation used they don’t look all that scary, but they are more complicated than they look. For a start it looks like there is only one equation, but the subscripts μ and ν can each take four values (usually 0, 1, 2 or 3), each value standing for one of the dimensions of four-dimensional space time. It therefore looks likes there are actually 16 equations. However, the equations are the same if you swap μ and ν around. This means that there are “only” ten independent equations. The terms on the left hand side are the components of the Einstein Tensor which expresses the effect of gravity through the curvature of space time and the right hand side describes the energy and momentum of “stuff”, prefaced by some familiar constants.

The Einstein Tensor is made up of lots of partial derivatives of another tensor called the metric tensor (which describes the geometry of space time), which relates, through the Field Equations, to how matter and energy are distributed and how these components move and interact. The ten equations that need to be solved simultaneously are second-order non-linear partial different equations. This is to be compared with the case of Newtonian gravity in which only ordinary different equations are involved.

Problems in Newtonian mechanics can be difficult enough to solve but the much greater mathematical complexity in General Relativity means that problems in GR can only be solved in cases of very special symmetry, in which the number of independent equations can be reduced dramatically.

So that’s why it’s difficult mathematically. As for the conceptual problem it’s that most people (I think) consider “space” to be “what’s in between the matter” which seems like it must be “nothing”. But how can “nothing” possess an attribute like curvature? This leads you to conclude that space is much more than nothing. But it’s not a form of matter. So what is it? This chain of thought often leads people to think of space as being like the Ether, but that’s not right either. Hmm.

I tend to avoid this problem by not trying to think about space or space-time at all, and instead think only in terms of particle trajectories or ligh rays and how matter and energy affect them. But that’s because I’m lazy and only have a small brain…

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November 26, 2015 at 3:44 pm

Can space be a bit like Ether in the sense that gravity waves can propagate through it?

November 26, 2015 at 4:39 pm

It depends on what you mean by Ether. There is a technical understanding of what Ether is, and by that definition, no space is not even close to being like Ether. This is the whole point of special relativity, and in particular, of the Michelson-Morley experiments.

November 26, 2015 at 4:53 pm

Thank you I think I mean that space is something, that can be distorted and carry a wave/change in density?

Or is this the wrong way to look at it?

November 27, 2015 at 2:38 pm

Yes, of course space is something. Basically, “there is no Ether” means that this ‘something’ is very different from any medium like water or air.

November 26, 2015 at 9:28 pm

If the gravitational constant G is replaced by opposite of the Coulomb’s constant -k (using charges instead of masses), then it seem that the Einstein field equations can be a compact way to write the Maxwell’s equations at low energy (gravitoelectromagnetism).

November 27, 2015 at 2:33 pm

Well at least on the surface they are different as the main unknown in the Einstein equations is a symmetric (metric) tensor, and the fields in the Maxwell equations are differential forms (or vector fields). But I think there are reformulations of these equations that make them look similar.

November 27, 2015 at 6:51 pm

The unknown values are the metric tensors, that it is symmetric, but Bahram Mashhoon (for me the most complete explanation until now) use a linear perturbation of the Minkowski spacetime to obtain a low energy approximation of the Einstein field equation that are equations for scalar and vector potentials, from which he obtain equations for the electromagnetic fields, that are differential forms; so that it is not a direct correspondence between metric tensors and electromagnetic fields (if I understand precisely the analysis).

November 27, 2015 at 8:21 pm

Aha! I have heard terms such as gravitomagnetic and gravitoelectric, which may have something to do with the correspondence you are talking about.

November 27, 2015 at 8:44 am

I tend to avoid this problem by not trying to think about space or space-time at all, and instead think only in terms of particle trajectories or ligh rays and how matter and energy affect them. But that’s because I’m lazy and only have a small brainIt’s because you weren’t taught GR as a gauge theory! In the Cambridge formulation which uses the mathematical language of Clifford algebra, the gauge theory is expressed in a flat background spacetime, allowing many results of conventional analysis to be applied without need to generalise them first to curved manifolds. The gauging then ensures that all relations between physical quantities are independent of the position and orientation of the matter fields, removing all dependence on the fixed background spacetime. Those who prefer to view GR as a dynamical theory of geometry will need to re-adjust, but what counts are testable predictions and the theory is anchored to physical quantities. And even if you prefer old-fashioned spacetime curvature, that is more transparent in Clifford algebra too.

November 29, 2015 at 12:07 am

I confess I haven’t found Clifford algebra to be all that transparent. However, I’ve never taken a course in it and just relied on self-study from textbooks picked by what looked attractive on the university library shelf. Have you got a recommendation of one that makes Clifford algebras clear to the mathematically-trained-but-still-ignorant?

November 29, 2015 at 5:48 pm

Certainly! The point is to extend the superiority of complex analysis over 2D vector algebra into higher dimensions. The cost is loss of commutativity – but it is well worth it. (Quaternions are the 3D version.) You get a unified language that encompasses and improves on vector and tensor analysis, spinors, etc. Try the Cambridge trio’s paper “Imaginary Numbers Are not Real – the Geometric Algebra of Spacetime” in

Foundations of Physicsvol. 23, p1175-1201 (1993). This is one of several papers in which they recast the ideas of David Hestenes (the key man in this field) and set out what physics looks like in them. The one on emag is a stunner too. Or for fullest details the trio’s 2003 book “Geometric Algebra for Physicists” (Cambridge UP).November 27, 2015 at 9:13 pm

Anton, is this approach same as Feynman’s approach to GR

November 29, 2015 at 5:18 pm

I’m not familiar with Feynman’s approach to GR so I can’t give a Yes/No answer, but I do know that people formulated GR as a gauge theory before the Cambridge trio (Anthony Lasenby, Chris Doran, Steve Gull) did. The earlier results were messy, however, and the trio found a much more natural gauge theory by working in Clifford Algebra. Their theory could of course be translated into conventional mathematical language, but they think – and I agree – that it is worth learning Clifford Algebra for much else in theoretical physics as well as this application. I’ll suggest why in my reply to Joseph’s question posed above at 12.07am on Nov 29, 2015. The Cambridge theory is set out in “Gravity, gauge theories and geometric algebra”,

Philosophical Transactions of the Royal Society of London Avol. 356, pp. 487-582, 1998. The Cambridge theory involves two gauge fields obeying first order differential equations (not the metric which obeys 2nd order ones). I’d bet that more progress can be made quantising the Cambridge fields than the metric…December 1, 2015 at 5:48 pm

Anton I am referring to this for example

http://arxiv.org/abs/gr-qc/0411023 (which refers to 1970 paper

by Stanley Deser)

Feynman’s own work is outlined in his lectures on GR book (which came out in 1995)

December 23, 2015 at 2:40 pm

To this I would add:

1) Differential geometry is fairly hard — or at least fiddly with all the tensors and indices.

2) Even when you grasp it, it seems like you just know how to turn the mathematical crank. Without much geometrical or physical intutition.

I did not understand what the Ricci tensor was untill I read “The Meaning of Einstein’s Equation” here http://math.ucr.edu/home/baez/einstein/

March 15, 2021 at 2:42 am

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