For other readers: the components with respect to the coordinate basis in general have no physical meaning. The components wrt a frame, OTH, are the components the observer with said frame would actually measure.
You can get the frame components from the coordinate components computed via index gymnastics by using the expression for the frame vectors (or dual coframe covectors) in terms of the coordinate vectors (or coordinate covectors).
As for the interpretation: we know from Newtonian analysis that our wire should be under tension, i.e. there should be a nonzero diagonal component in the stress tensor computed in the frame. Since we aer using a frame comoving with the matter, the momentum components should vanish. And they do. The sole surviving component of the stress tensor is negative because this is a tension.
A tricky point in comparing with Newtonian analysis: sometimes a frame is actually "spinning". To find out, compute the Fermi derivatives of the spatial frame vector fields along the timelike frame vector field. If these vanish (after projection orthogonal to the timelike frame vector), the frame is nonspinning.
Since we are accelerating our wire (as I recall), we don't expect our frame to be inertial. We can compute the acceleration vector as the covariant derivative of the timelike frame vector along itself.
BTW, pervect, I am a bit confused about what you are doing here. Since you used the Langevin frame for the Born chart (right?) your wire is, I guess, a circular wire which we have set rotating about the axis of symmetry, so that we have a stationary axisymmetric scenario. You never wrote out your line element or your frame, but I guess you found a tension along the length of the wire (orthogonal to
. That would make sense in terms of centrifugal "force" assuming your third frame vector is something like
(Additional: sorry, pervect; I see now that you did say exactly this, so all is well.)
Originally Posted by pervect
What's puzzling me is how this satisfies the Newtonian limit for a slowly rotating disk.
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Well, first of all, if you expand to first order in the angular velocity
, then you are ignoring the tension. To make a comparision with Newton you will probably need to use a weak-field analysis but you'll need to carry out the computations to higher order in the velocity.
Also, to obtain a reasonable model incorporating elasticity you will need to use something more elaborate so that you can try to match the RHS of the EFE to a matter tensor suitable for "hyperelastic" matter.
Here is the short version of how you might begin a more careful analysis for matter which is rotating rigidly:
Start with a chart for a stationary axisymmetric spacetime; that is, a Lorentzian manifold with two commuting Killing vector fields, one timelike and the other spacelike. We don't want a static spacetime so the timelike Killing vector should
not be hypersurface orthogonal, i.e. should have nonzero vorticity.
To be safe, you can take the Weyl canonical chart, although that is awkward to interpret geometrically. So you can play with making some simplifying assumptions.
For example, you could start with
where w,v are functions of z,r only. Here the timelike Killing vector field is
which we naively assume has
unit length, that is we assume that
. The spacelike Killing vector field is
and it has length
, where the function r is defined by
. So the radial coordinate has a known geometric interpretation, as do the time and angular coordinates. Finally we choose z so that
. Turning this around, I have somewhat described how one could try to derive this chart starting from purely coordinate-free considerations, which of course guarantees that all the metric functions appearing in our line element have coordinate-free interpretations.
Read off the coframe
Take the dual frame
and boost it by an undetermined amount (depending only on z,r) in the
direction. It is convenient to write
Then the new frame field is
Now
already has vanishing expansion scalar, so require that the shear tensor also vanish. In GRtensor speak:
Code:
casesplit([seq(seq(grarray(expv(bdn,bdn))[j,k], j=1..4),k=1..4)] );
where expv(bdn,bdn) is the expansion tensor of
Code:
grdef(`expv({((a) (b))} := p{(a) ^(m)}*p{(b) ^(n)}*(v{(m) ;(n)}+v{(n) ;(m)})/2`):
where p(dn,dn) is the projection orthogonal to the timelike unit vector field whose expansion tensor is to be computed, v(dn). We can cause GRTensor to ask this vector field to be input (as a linear combination of the frame vector fields) by
Code:
grdef(`v{ ^(a) }`):
Just make sure the coefficients in your linear combination,
obey the unit contraint
. For example
above, and
.
(Additional: my post seems to have been accidently truncated at this point. I'll try to reconstruct it.)
The condition that the expansion tensor vanish (i.e. that the shear tensor vanish) turns out to give w in terms of p via quadrature.
Next, we want our frame field to be comoving with the matter, so require that the momentum components vanish. Two already do so, and the condition
gives an equation in p only. We should check that these conditions are not mutually inconsistent, although this is pretty obvious in this case. The easiest way to do that with GRTensor is to casesplit the three equations and let maple do its differential ring magic.
The acceleration vector of our matter is orthogonal to
(and automatically orthogonal to
since
is a unit vector). The matter tensor has the form
![LaTeX Code: <BR>T^{\\hat{m} \\hat{n}} = \\left[ \\begin{array}{cccc}<BR>e & 0 & 0 & 0 \\\\<BR>0 & a & f & 0 \\\\<BR>0 & f & b & 0 \\\\<BR>0 & 0 & 0 & c<BR>\\end{array} \\right]<BR>](latex_images/13/1319561-32.png)
where the hats are often used to signal that an expression refers to components with respect to an ONB or frame, rather than a coordinate basis.
At this point, you can start to try use some elasticity theory to further determine the precise form of the matter tensor; so far we only know that some components vanish as above.
Note that pervect's computation,
, which is not sufficiently general for a fluid or for most elastic solids.
The assumption above that
simplifies the analysis, but is rather artificial and inconsistent with dust, for example. So if the above doesn't work out, go back and start over using a more general stationary axisymmetric line element (at worst, you can use the Weyl canonical chart).
(Oh darn, gotta go. I said a lot more but I won't be able to reconstruct it after all.)
(Additional: I'll try to add a bit more reconstruction):
The idea here is to use elasticity to put constraints on the form of the matter tensor, perhaps eventually coming up with sufficiently simply equations to yield a solution with a reasonable interpretation (including a good understanding of what assumptions enter into the derivation). So for example for uniaxial tension in an homogeneous isotropic elastic body you'd expect the stress tensor to be diagonal with form
where f is some function and k is a constant (simply related to standard characterizations of isotropic homogeneous elastic materials such as the two Lame constants), and you'd expect to relate f to the energy density. Note one tricky point: many textbooks on elasticity theory provide tensor equations which are only valid in a Cartesian chart and thus are not "tensorial" in the sense students of gtr might expect. There is nothing wrong with this practice, of course, one just has to be aware of it and to make adjustments. This will be relevant if you consider uniaxial
torsion (as in spinning up a disk by applying a torque) rather than unaxial
tension (as in pulling on a rod).
By the way, one of the many advantages of using frame fields is that this is by far the conceptually and computationally easiest way to make connections with other theories.
Don't forget that the particular metric I chose above involved an assumption,
which certainly doesn't hold for the Schwarzschild vacuum or Schwarzschild fluid, or indeed for very many stationary axisymmetric solutions! It was chosen for mathematical convenience. If you play around you'll soon appreciate that using fewer metric functions of fewer variables helps alot, but there is actually another reason why my choice is convenient: the integrability condition for the quadrature (which gives
as a function of p and its partials) is precisely the equation in p alone which results from the demand that the momentum vanish, i.e. that our new frame be comoving with the matter. This is in fact very typical and similar statements will hold for good analyses starting from more general line elements, so when playing around be sure to look out for this! See the Maple help for casesplit for using this command to find the integrability condition for two quadrature equations of the type we found above.
Those of you who are interested in solitons will want to learn about the Lax pair formulation, which involves basically the same phenomenon, in which we obtain a second order PDE of interest as the integrability condition for two first order equations giving some function of two variables in terms of another by quadrature. This is in turn related to defining a certain connection, whose curvature--- well, see the review paper I've cited elsewhere!
Emphasize a good web reference
![LaTeX Code: <BR>\\left[ \\begin {array}{cccc} \\rho&0&0&0\\\\\\noalign{\\medskip}0&0&0&0\\\\\\noalign{\\m edskip}0&0&-{\\omega}^{2}\\rho\\,{r}^{2}&0<BR>\\\\\\noalign{\\medskip}0&0&0&0\\end {array} \\right] <BR>](latex_images/13/1319515-0.png)
![LaTeX Code: <BR>\\left[ \\begin {array}{cccc} \\left( {\\omega}^{2}{r}^{2}+1 \\right) \\rho&0&\\rho\\,\\omega&0\\\\\\noalign{\\medskip}0&0&0&0\\\\ \\noalign{\\medskip}<BR>\\rho\\,\\omega&0&0&0\\\\\\noalign{\\medskip}0&0&0&0\\end {array} \\right] <BR>](latex_images/13/1319515-2.png)
Frame components or coordinate basis components?
