Stress Energy Paradox

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T.M.Tlas@gmail.com - 13 Aug 2005 18:22 GMT

Hi everyone,

I came accross this puzzle while reading Goldstein's "Mechanics".

As we all know, the i,j th component of the stress tensor gives the
flux of the ith component of momentum, accross
a surface of constant x^j. In other words it is the ith component of
force accross this surface.

Now, the lagrangian density of an elastic rod is given by : 1/2 (d
phi/dt)^2 - 1/2 (d phi/dx)^2 (obtained by taking a
string of springs connected in sequence and then letting their lengths
shrink to zero, i'm setting the Young's modulus and
the mass density to one)

>From it we see that the equation of motion is: d^2 phi/dt^2 - d^2

phi/dx^2 = 0

A solution is : Phi (x,t) = a * x
This corresponds to uniform stretching/compressing when "a" is
positive/negative

Now the xx component of the stress-energy tensor is: -1/2 [ (d
phi/dt)^2 + (d phi/dx)^2 ]. The thing that struck me is that
this expression is always negative!
That didn't make any sense: If you compress the rod linearly, the force
should point in the positive direction, and when you
stretch it, it should point to the negative. Clearly, the sign of the
force should depend on the sign of "a" in the
solution above.

The more i dug into this, the more perplexing this became. For example,
the expression for the force above is equal to (the
negative of) energy density. The expected expression should have been
proportional to d phi/dx (this is the limit of :
N(i+1) - N(i), which gives the extension of the ith spring).

The only explanation i could come up with, was that when we change the
sign of the extension/force we get a negative flux of
a negative quantity, and so the number comes out positive.
Unfortunately, it seems obvious that we can get positive as well
as negative stresses in an elastic rod. So where is the catch? Are the
definitons of the stress tensor in terms of forces
and in terms of fluxes NOT equivalent?

I have a terrible feeling that i'm making some awfully stupid mistake
somewhere, if anybody could find a solution, i would be
more than grateful.

Thanks for reading,
Tim

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Igor Khavkine - 20 Aug 2005 06:04 GMT

> Hi everyone,
>
[quoted text clipped - 25 lines]
> force should depend on the sign of "a" in the
> solution above.

That's an interesting question. I was also stumped by it for a bit. But
here's what I think.

You are right in both cases. The xx component of the stress-energy
tensor is always negative, while the real stresses felt in the rod
do depend on whether it is stretched or compressed. The only remaining
conclusion is that the stress-energy tensor associated to the scalar
field phi is not the same as the stress-energy tensor associated to the
rod.

In retrespect that is somewhat obvious. After all, there are many
possible systems whose linearized equations of motion reduce to that of
a scalar field. The real stress tensor is unlikely to be the same in
each case, while the stress tensor associated to the abstract scalar
field is always the same.

The actual stress tensor S is actually a linear function (Hook's law,
holds as a first approximation) of the gradient of the displacement
field. In this simplified case, the displacement field is always
longitudinal and its magnitude is given by the scalar field phi. So the
real stress tensor will be of the for S ~ dphi/dx ~ a, fully in
accordance with expectation.

But your question still remains puzzling if asked in a different way.
What relation does the stress tensor T of the scalar field phi,
T ~ (dphi/dx)^2, have to the real stress tensor S, S ~ dphi/dx?

> I have a terrible feeling that i'm making some awfully stupid mistake
> somewhere, if anybody could find a solution, i would be
> more than grateful.

Actually, I think you've stumbled onto something quite subtle here.

Igor

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T.M.Tlas@gmail.com - 21 Aug 2005 20:53 GMT

Hi Igor,

I came to the same conclusion, i.e. that the SE tensors are not the
same.
The thing that's still puzlling me is, why?

This is especially strange if we keep in mind the motivation for the
name SE in the case of the scalar field.
It's justified by looking at the T_0i components and noticing that they
are the momentum densities. After that the continuity equation is
invoked which means that the space-space components represent the
fluxes of the densities, i.e. fluxes of momentum, i.e. forces.

After getting puzzled by this i tried to see what happens in other
cases, so i took the simplest: dust. Ironically, the xx,yy,zz
components are now always positive! (they are proportional to v^2)

I know they are pressures, and should be positive, but if you, e.g. sit
in a frame in which the particles of dust have negative x velocity, it
seems that the force SHOULD be negative...

The only solution that i found is to note that the flux of momentum is
never negative. what i mean is that you always have either something
positive flowing in the positive direction, or something negative
flowing in the negative direction, so the signs cancel and we get
positive flux all the time.

This doesn't apply to the elastic rod, because the xx component is not
the absolute value of what's expected but something totally different.
As you said the question is to know the relation between the two
tensors, i'm still totally stumped.

Tim

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Natural Science Forum / Physics / Research / August 2005

Stress Energy Paradox