# Long rod in an expanding universe

Suppose you have a metal rod, of vast length, floating in space. From it's center-of-mass reference frame it is assumed to be at rest with respect to the CMB.

The rod will have some degree of internal forces (eg tension) holding it in shape. Suppose then that the rod is so long that if one were to observe one end of the rod from the other, they would find the remote region of space getting further away from them.

What forces would the rod experience due to expansion, if any?

Would it experience a 'pull' from either end?

If there were dust particles stationary relative to the CMB near each end of the rod, would they see the rod shrink in length / would the rod observe the dust particles moving away from the center of mass?

If the rod is a conductor (thermal or electric) I'm not entirely sure what to expect to see.

(I suspect this may have been answered on this site somewhere already, if so I could not find a similar enough wording of this problem)

• I have solved the problem numerically. Assuming the rod does not break, the entire thing stretches to an equilibrium length. The strain (and thus forces) appear to be an inverted quadratic along the length of the rod, going from zero at the two endpoints to some maximum value in the centre. I won't post it as an answer though since this should be solvable analytically... – lemon Feb 24 '15 at 8:15

Yes the rod would experience a force that stretches it.

The simplest way to understand this is to consider the rod as two point masses at distances $+r$ and $-r$ from you. The four-acceleration of the two masses is then given by the geodesic equation:

$${d^2 x^\mu \over d\tau^2} = - \Gamma^\mu_{\alpha\beta} {dx^\alpha \over d\tau} {dx^\beta \over d\tau}$$

Take the norm of the four-acceleration, multiply by the mass and that gives you the force on the masses. Then integrate along the bar to get the total force on the bar.

But actually doing the calculation seems hard to me. The $x$ coordinates used are comoving coordinates and the ends of the rod are not stationary in comoving coordinates so you'd need to work out their four velocity and plug it back into your equation. No doubt there will be site members who can do the calculation in their heads, but at this point I'm afraid I have to call a halt!

• This is helpful thanks. For parallel, if there is a stationary, static electric field really far away (far enough for expansion, near enough to be 'felt'), to a local stationary observer would the resulting potential be decreasing over time (hence causing a magnetic field due to "changing" electric field)? – Xeren Narcy Feb 25 '15 at 2:44
• Yes, I think so. – John Rennie Feb 25 '15 at 6:36
• Then this becomes interesting... Does the static electric field feel the same stress due to expansion as a result of this? Intuitively it seems like the resulting magnetic effects would relate to the stress though I'm unsure how to express that. – Xeren Narcy Feb 26 '15 at 0:24