Riemannian Geometry is free of Torsion. Why use it for General Relativity?

[oldman]

As I understand it, Riemannian geometry doesn't allow Torsion (a property of geometry involving certain permutations among the indices of Christoffel Symbols). Does this restrict the geometry of General Relativity (GR) to describing only a curved spacetime with the Riemann curvature tensor? Is curvature distinct from a distortion that involves say shear, as in an elastically or plastically deformed solid? If so, why is GR so restricted?

 

[oldman]

I guess that my original post above is too obscure to have warranted a reply. I'll conclude by trying to explain my motivation a bit more fully.

In the early 20th century, when Einstein was Generalising his (special) theory of Relativity (GR) to describe gravity, physics was what we might now call a mesoscopic enterprise. It quantitatively and predictively treated phenomena on a scale between the microscopically very small (where we now know that discrete quantum phenomena rule the behaviour of atoms) and the macroscopically very large (which we now know is not confined to the Island Universe of the Milky Way).

The mathematical tools which Einstein used, like calculus, tensor analysis and differential geometry, are all based on the concept of a smooth, differentiable continuum tied to the set of Real numbers. Einstein cleverly used ready-made Riemannian geometry to describe gravity with a previously unsuspected peculiarity (curvature) of the assumed continuous space (what we now call a differentiable manifold) of mesoscopic physics. Perhaps this approach, in the light of 20th Century and Post-Millennial knowledge, might be a tad restrictive.

Riemannian geometry treats only the geometry of smooth, continuous manifolds. I’ve been told that without the assumption of smoothness and differentiability, geometry would develop unacceptable and horrendous pathologies, like singularities, cusps and suchlike. But I happen to know that there are aspects of geometry where smoothness doesn’t always rule; as in descriptions of the geometry of dislocated crystals, where topological peculiarities (defects in translational symmetry) interrupt smoothness. This stuff can be found in Nabarro’s Theory of Crystal Dislocations (Oxford University Press 1967), an authoritative monograph.

Nabarro’s Chapter 7 discusses the geometry of generalised spaces and its bearing on defect physics. He mentions that Riemannian spaces have the distinction of being a special species of generalised space; one that is free of something called torsion. As I understand it: if one maps a closed loop from a comparison torsion-free Riemannian space (curved or flat) into a space with torsion (or vice versa) the mapped loop acquires a closure failure. Such a flawed loop then characterises with torsion a localised peculiarity or topological defect. A little like the way the orientation change of a vector parallel-transported around closed loop(s) characterises the local curvature in Riemannian space.

It is well established that gravity on a mesoscopic scale is accurately described by GR, in terms of a Riemannian space curved by mass/energy (think gravitational lensing and Pound-Rebka effect). Is it at all rational to suppose that if GR were only an approximation, on the microscopic scale, to a generalised geometry that incorporated features (like torsion) which could describe other non-continuum geometrical peculiarities like discrete triangulations (think Buckminster-Fuller or Renate Loll) it might alleviate the difficulty of reconciling general relativity with modern microscopic physics?

 

[Mentz114]

I think you've answered your own question. There is a theory of gravity which gives the same predictions as GR but uses the torsion defect instead of curvature. It is a neater looking theory in some ways because it starts from a gauge principle. If you haven't come across it there is an excellent summary in arXiv:gr-qc/0011087v1.

 

[bcrowell]

Is it at all rational to suppose that if GR were only an approximation, on the microscopic scale, to a generalised geometry that incorporated features (like torsion) which could describe other non-continuum geometrical peculiarities like discrete triangulations (think Buckminster-Fuller or Renate Loll) it might alleviate the difficulty of reconciling general relativity with modern microscopic physics?

A lot of the motivation for looking at alternative formulations of GR is that they might be easier to quantize. I think this is the reason for a lot of the interest in the Ashtekar formulation.

Note that there are also theories of gravity with torsion that are *not* equivalent to GR. In these theories, spin is usually taken to be the source of torsion. The EotWash group at UW does high-precision experiments to look for things like this.

 

[oldman]

...If you haven't come across it there is an excellent summary in arXiv:gr-qc/0011087v1.

Thanks very much for pointing me to this article, Mentz, but it's "quite above my fireplace" as folk put it here, where winter rather than summer looms. Thanks also for your helpful reply, bcrowell. The antipodean EotWash group look to me like gallant experimental crusaders. But I've still not found an answer here or elsewhere to what I thought was a simple question. I'll rephrase more specifically, to try and clarify how ignorant I am:

Does general relativity as conventionally formulated treat shear, which seems to me a common sort of continuum distortion of a kind quite distinct from curvature? If not, is there a more general formulation that does? (Shear is quite important in the formation of astronomical structures formed by gravitational condensation). I'm afraid that I'm unlikely to be able to follow such a theory in detail, judging from Andrade et al.'s paper -- I'm much too mathematically disadvantaged (as it can euphemistically be put) and old, to boot --- but I'd like to know if such a treatment has been done.

 

[Daverz]

non-continuum geometrical peculiarities like discrete triangulations (think Buckminster-Fuller or Renate Loll) it might alleviate the difficulty of reconciling general relativity with modern microscopic physics?

This sounds like Regge calculus:

http://en.wikipedia.org/wiki/Regge_calculus

It's also described in Misner, Thorne & Wheeler.

 

[George Jones]

Does general relativity as conventionally formulated treat shear,

If I understand your question correctly, then the answer is "Yes." See, for example, section 15.2 Motion of a Continuous Medium in Relativistic Mechanics in the book An Introduction to General Relativity and Cosmology by Plebanski and Krasinski.

which seems to me a common sort of continuum distortion of a kind quite distinct from curvature?

Einstein's equation does put constraints on shear.

 

[Phrak]

Riemann geomety admits torsion. This means that the connection coefficents can decompose with antisymmetric as well as symmetric parts. But general relativity premises that the connection is symmetric under interchange of it's lower indices. This is just one of the half dozen to ten restictions imposed on the connection. In best recollection, there are about eight or nine constrains imposed on the Christoffel connection to develop general relativity. A theory that loosens this restriction to allow torsion is called einstein-Cartan gravity.

But, in my mind, there is more than this torsion business to consider. General relativity was developed at a time when the best tool at hand was Riemann geometry--a topology impressed with a metric. Physicists have been obsessed, ever since, with a metric imposed on a point set topology and consider little else other than variations within this theme.

I no longer think a metic over a point set is sufficient in describing spacetime and its elements, but that the metric should be supplanted.

 

[oldman]

...Does general relativity as conventionally formulated treat shear, which seems to me a common sort of continuum distortion of a kind quite distinct from curvature? ...

Your welcome comments, Phrak, show thinking somewhat parallel to mine. I'll try and reply to comments by George Jones and Daverz a bit later --- I'm about 80 Km or so from a library that I know has a copy of Misner et al. Not so sure about Plebanski and Krasinski. In the meantime, thanks to both for these replies.

I have one further primitive thought. General relativity does accommodate uniform change of scale, as in the expansion of the universe. And uniform shear is essentially uniform (but anisotropic) scale-change; one can always (in three spatial dimensions at least) rotate axes to change shear to superimposed expansion and contraction. But what about non-uniform shear? Too complicated? Uniform scale-change always makes my simple mind boggle; where's the reference ruler? Made of electromagnetism, I guess.

 

[George Jones]

For the specialized case of geodesics, see sections 2.2 and 2.3 from Poisson's notes,

http://www.physics.uoguelph.ca/poisson/research/agr.pdf,

which evolved into the excellent book, A Relativist's Toolkit: The Mathematics of black hole Mechanics.

 

[oldman]

Uniform scale-change always makes my simple mind boggle; where's the reference ruler? Made of electromagnetism, I guess.[/QUOTE]

Thanks for the reference to Poisson's notes, George. They are simply and lucidly written --- just right for me, with a bit of effort. Leafs to the rescue. The only thing that is missing is an index --- but I'll fix that by ordering his book, which no doubt is where I'll find this need filled.

Taken together with my last post, it has set my mind boggling once again at how the scale of any distortion (especially a simple uniform one-dimensional expansion, but also a four-dimensional mixed time- and space-like uniform shear that must also of necessity involve scale changes) is set in a purely relativistic context -- or is that an oxymoron?

I know that in crystals, scale is provided for strains and defects by an imagined "perfect reference crystal" which is unstrained and free of defects (see Nabarro's book mentioned in post 2). This is a bit off topic, but have you any comment on what could constitute a reference in the case of curvature or torsion? The universe at large? As in lensing?

 

[Phrak]

Your welcome comments, Phrak, show thinking somewhat parallel to mine. I'll try and reply to comments by George Jones and Daverz a bit later --- I'm about 80 Km or so from a library that I know has a copy of Misner et al. Not so sure about Plebanski and Krasinski. In the meantime, thanks to both for these replies.

I have one further primitive thought. General relativity does accommodate uniform change of scale, as in the expansion of the universe. And uniform shear is essentially uniform (but anisotropic) scale-change; one can always (in three spatial dimensions at least) rotate axes to change shear to superimposed expansion and contraction. But what about non-uniform shear? Too complicated? Uniform scale-change always makes my simple mind boggle; where's the reference ruler? Made of electromagnetism, I guess.

I haven't really been keeping up with your thread, but there are a few thing to know about.

Torsion, as the term is applied, means that the connection coefficients can have antisymmetric parts. For each upper index of the connection there is a 4x4 matrix. The 4x4 matrix can be decomposed into symmetric and antisymmetric parts. The antisymmetric part is what is referred to as torsion.

General relativity, you should know, assumes that the connection is torsion free. It premises that the antisymmetric part is zero amplitue. Of all possible connections, gereral relativity relies upon the torsion-free Christoffell connection.

$$\Gamma_{\mu\nu}^{\sigma} = \Gamma_{\nu\mu}^{\sigma}$$

Riemann geometry, on the other hand, is more inclusive than general relativity. I'm not completely clear on the following. If all that we have to work with is a metric with a point-set topology, this defined Riemann geometry.

There are two good questions about the differences between the more general Riemann geometry and the specific instanciation that is general relativity.

1) Does a point-set topology impressed with a metric preclude a torsion component to the connection?

2) If the torsion component is not precluded, does the torsion constitute an additional field on the point-set topology other than the metric?

 

[oldman]

Torsion, as the term is applied, means that the connection coefficients can have antisymmetric parts...

... gereral relativity relies upon the torsion-free Christoffell connection.

$$\Gamma_{\mu\nu}^{\sigma} = \Gamma_{\nu\mu}^{\sigma}$$

...If all that we have to work with is a metric with a point-set topology, this defined Riemann geometry.

There are two good questions about the differences between the more general Riemann geometry and the specific instanciation that is general relativity.

1) Does a point-set topology impressed with a metric preclude a torsion component to the connection?

2) If the torsion component is not precluded, does the torsion constitute an additional field on the point-set topology other than the metric?

Here you're asking questions that are beyond my limited capacity to answer, Phrak. Perhaps someone else can step in.

To me torsion means something less esoteric than connection coefficients, perhaps something similar to but distinct from curvature; involving circuits that display closure failure rather than vector orientation-change (as in curvature). I don't know whether closure failure in a circuit qualifies the material or space inside the circuit as having a strange topological feature, or not --- in fact I'm more familiar with the red poinsettia flowers in my garden than point-set topology! Perhaps you could educate me a bit here?

 

[Phrak]

Here you're asking questions that are beyond my limited capacity to answer, Phrak. Perhaps someone else can step in.

To me torsion means something less esoteric than connection coefficients, perhaps something similar to but distinct from curvature; involving circuits that display closure failure rather than vector orientation-change (as in curvature). I don't know whether closure failure in a circuit qualifies the material or space inside the circuit as having a strange topological feature, or not --- in fact I'm more familiar with the red poinsettia flowers in my garden than point-set topology! Perhaps you could educate me a bit here?

Nevermind point-set topology. I should probably not have brought it up. A manifold is just a way of taking a bunch of points and giving them a notion of closeness by establishing a measure of distance between any two of them in a smooth way, so that you can take derivatives.

The word torsion, as normally used in classical physics and the usage in relativity, differ.

As I recall there is a uniqueness of the Chistoffell connection. All it's attributes, including the requirement that it is torsion free, simplify the equation of a feely falling test particle. We're still free to define as many connections on a manifold as we wish. A connection is just a convention. However, I'm at a loss, right now to identify which wonderfully simplified properties a torsion-free connection endows upon the equations governing the trajectory of a particle.

 

[IsometricPion]

However, I'm at a loss, right now to identify which wonderfully simplified properties a torsion-free connection endows upon the equations governing the trajectory of a particle.

Geodesics (defined as paths of extremal distance) are not necessarily locally straight in geometries with torsion (at least they do not have the same defining equation ref. http://arxiv.org/pdf/0803.3276v3" section 8.1).

 

[Mentz114]

However, I'm at a loss, right now to identify which wonderfully simplified properties a torsion-free connection endows upon the equations governing the trajectory of a particle.

This is a slippery subject. There is a good explanation of the difference in the EOMs with torsion vs with curvature in section 5 of this ref arXiv:gr-qc/0011087v1. The 'geodesic' equations in the torsion <> 0, curvature = 0 case do not represent force free motion except when the torsion tensor is zero ( ie flat spacetime) . So there are no geodesics in the GR sense.

It's obviously a big loss not to have freely falling frames or null geodesics. This also means no Raychaudhuri equation - oh dear, it gets worse.

Having spent a little time doing calculations in TP gravity, I would say that GR is simpler.

It seems that the differences between the two approaches come from the fact that GR incorporates the equivalence of inertial mass and gravitational mass and TP gravity does not.

 

[TrickyDicky]

The 'geodesic' equations in the torsion <> 0, curvature = 0 case do not represent force free motion except when the torsion tensor is zero ( ie flat spacetime) . So there are no geodesics in the GR sense.

It's obviously a big loss not to have freely falling frames or null geodesics. This also means no Raychaudhuri equation - oh dear, it gets worse.

Having spent a little time doing calculations in TP gravity, I would say that GR is simpler.

It seems that the differences between the two approaches come from the fact that GR incorporates the equivalence of inertial mass and gravitational mass and TP gravity does not.

Ths is obviously right. The way this equivalence is built in the EFE and the geodesic motion equation is precisely the torsion free connection. So I gues it is clearly an axiom of GR as there is only one connection with metric compatibility that is torsion free if we want to make the covariant derivative operate in a way independent of coordinates. The essence of GR resides in deriving geodesic motion from curvature thru the appropriate connection. Therefore it has to be torsion free.