Can Forces Be Interpreted as Christoffel Symbols in General Covariance Contexts?

[pervect]

Does anyone have any references in the literature that talk about interpreting forces as Christoffel symbols?

Particularly interesting would be references which discuss the role of forces (or the lack of a role of forces) in generally covariant re-formulations of classical mechanics.

I would like to do some more reading to compare to my own thoughts on the subject.

 

[atyy]

Is your idea: force = accelerated worldline = even in Fermi normal coordinates Christoffel symbols don't disappear?

 

[pervect]

My idea is that the fictitious forces you need to add in non-inertial coordinate systems prevents forces from transforming like tensors. For instance, a tensor quantity that's zero in one coordinate system is zero in all, but our fictitious forces are zero in inertial frames and are non-zero in accelerated frames.

When we look around at how forces do transform, lo and behold, they actually transform as Christoffel symbols, not tensors.

We can (and apparently do) treat forces as tensors (for instance, the four-force in special relativity), by considering transformations only between inertial frames rather than general transforms, in those special circumstances the Christoffel symbols do transform like tensors. But if we allow truly general transformations, they transform like Christoffel symbols, not like tensors.

It seems pretty simple, to the point of even being obvious, but I'd like to have a reference in hand and read about it more to see if there's any subtle points I might be missing.

 

[atyy]

Why isn't the force a tensor? Isn't it f_u=mDU^u/dτ like http://arxiv.org/abs/0806.0464 Eq 2 or http://arxiv.org/abs/1102.0529 Eq 1.16?

Are you thinking of something like remark 2 on p35 of http://www.blau.itp.unibe.ch/lecturesGR.pdf ?

 

[pervect]

If we want to put accelerated frames on an equal footing with non-accelerated frames, as we do in GR, we need (or at least it's my goal) to treat the fictitious forces just as if they were real forces. Singling out some forces as fictitious happens because we restrict the valid descriptions of phyics to "inertial frames", and we want to treat inertial frames and non-inertial frames equally in our generally covariant reformulation.

But clearly, fictitious forces can't transform as tensors, as per my previous argument. To repeat said argument, a tensor quantity that's zero in one frame is zero in all. And clearly fictitious forces are zero in inertial frames, and non-zero in non-inertial ones,.

So to make fictitious forces just as real as non-fictitious ones, we need to think of all forces as arising from Christoffel symbols. So we replace F=ma with the geodesic equations. The old notion of force gets subsumed as a subset of the Christoffel symbols. So we wind up with force-free particles still following geodesics in the reformulation, and with forces being Christoffel symbols and not tensors, though the geodesics are still Newtonian geodesics, because the theory is just a repackaging of Newtonian theory.

 

[haushofer]

Maybe it's interesting to look in the Newton-Cartan literature, where classical mechanics is reformulated in terms of the geodesic equation. As such, $\Gamma^i_{00}$ plays the role of a central force, whereas $\Gamma^i_{0j}$ is often interpreted as Coriolis force (because $\Gamma^i_{0j} = -\Gamma^j_{0i}$ and it multiplies $\dot{x}^i$ in the geodesic equation).

 

[PhilDSP]

Another alternative is to consider how tensors may decompose into quaternions which may further decompose into spinors. In a situation where a tensor doesn't provide the right transformation characteristics for a force, combinations of quaternions and or spinors might. There are some references I have in mind but need to check the exact name and title. One author had specifically done the analysis of how the Christoffel symbols need to be accounted for where I believe the covariant derivative of a quaternion term was being evaluated.

 

[PhilDSP]

The most interesting and relevant material I've come across on this in the past week is from the "Formal Structure of Electromagnetics - General Covariance and Electromagnetics" by E. J. Post. On page 20 he describes how the Christoffel symbol arises in a geometric transformation of objects which is not homogeneous where a derivative of a transformation element is needed. He gives such a rule to determine the Christoffel symbol and goes on to say

One may therefore expect $\Gamma^\lambda_{VK}$ to occur in association with inertia forces, because an inertia force can never be a vector in the general relativistic sense.

Starting on page 190 Post derives the Eikonal equation, relates that to the Hamiltonian canonical equations and derives first and second order equations from those. He finally relates those to the Geodesic equation with the Christoffel symbol. In general the book contains more revelations than any other physics book I think I've read pound for pound, so it's obviously to be recommended.

 

[pervect]

Thanks PhilDSP- while I don't have this reference (Post) yet, this is exactly the sort of thing I was looking for.

 

[dextercioby]

Post's book seems interesting. I'll search for it too. Thanks for the reccomendation.

EDIT: E.J. Post wrote a great book.

 

[atyy]

This is about GR, not Newton, but maybe it's relevant. http://relativity.livingreviews.org/Articles/lrr-2009-4/ (section 3.3.1): "For example, the Christoffel symbols are not tensorial, but they do have geometric, and hence physical content, namely the linear connection. Indeed, the connection is a non-local geometric object, connecting the fibres of the vector bundle over different points of the base manifold. Hence any expression of the connection coefficients, in particular the gravitational energy-momentum or angular momentum, must also be non-local. In fact, although the connection coefficients at a given point can be taken zero by an appropriate coordinate/gauge transformation, they cannot be transformed to zero on an open domain unless the connection is flat."

Actually, in GR aren't the inertial frames still privileged because of local flatness and the EP? I think it's universally accepted that the Christoffel symbols in GR represent inertial forces or gravity, eg. in Fermi normal coordinates. But you are thinking about Newton, not GR?