# Where does the work come from if tidal forces are stretching elastic objects?

An elastic object e.g. a rubber band will be stretched

• in accelerated expanding FRW-spacetime
• during radial free fall in Schwarzschild spacetime by tidal forces due to Ricci- and Weyl-curvature resp., until equilibrium with its internal cohesive forces is achieved.

The work done to stretch the rubberband is represented by the elastic potential energy stored in it. Where does this work done come from in these two cases?

• I'm having a hard time understanding what FRW et all brings to the table here. How is this different than a Newtonian example? Where is the work in two shotputs held together by an elastic in LEO? – Maury Markowitz May 3 '18 at 19:07
• Maury, in the FRW-universe I mentioned, things (galaxies) are moving away accelerated from each other in free fall. So, a rubber band consisting of matter (things along the band) would be stretched. Likewise if two marbles are falling towards a mass one after the other their distance increases accelerated because the lower marble feels stronger gravity. If instead a rubber band falls it will be stretched. I hope that's the point you are interested in. – timm May 3 '18 at 20:34
• My concern is why introduce FRW at all? Does that not simply add a layer of complexity that has no actual effect on the answer? – Maury Markowitz May 3 '18 at 20:38
• Well one could just refer to tidal gravity due to curved spacetime. But the answer might depend on the particular spacetime model, expanding (FRW) or static (Schwarzschild). The energy is not conserved in the former. I do hope that someone knowledgable will give some input. – timm May 4 '18 at 7:16
• Maybe I'm completely off-topic (tell me if I do) : at a distance $r_{1}$ from your massive object, the rubber band has energy $E_{1}=E_{grav,1}+E_{elastic,1}$. Falling from $r_{1}$ to $r_{2}$, you have an energy transfer from the gravitational potential form to the stored elastic form : $E_{2}=E_{1}=E_{grav,2}+E_{elastic,2}$ for a rubber band whose stretch is conservative. The works thus come from the gravitational attraction of the heavy body on the rubber band. Following Maury, no need for FRW or Schwarzchild to see where does the work come from. – Naptzer May 4 '18 at 8:27

In the case of a rubberband falling into a Schwarzschild blackhole, the work comes from the increasing differential in binding energy between the two ends of the string. This is completely analaguous to the Newtonian case. Binding energy is well defined due to the existence of a time translation symmetry (i.e. timelike Killing field.)

The situation in FRW is slightly different. The rubberband in an FRW metric will only expand if

1. The expansion of the of the FRW spacetime is accelerating

or

1. The intitial conditions of the rubberband are such that there is relative kinetic energy between the two ends of the rubberband.

If neither is the case the size of the rubberband will just stay the same, and no work is done.

In case 1, the work comes from the vacuum energy and effectively slows down the acceleration. Instead of a rubber band it is easier to think about an FRW metric filled with an elastic medium. The effect is that you get an FRW solution with a slightly different equation of state (the pressure is a bit larger).

In case 2, the work simply comes from the relative kinetic energy of the opposite ends. This slows down the expansion of the rubber band.

• Thanks for answering. „In the case of a rubberband falling into a Schwarzschild blackhole, the work comes from the increasing differential in binding energy between the two ends of the string.“ Isn‘t it vice versa,the increasing differential in binding energy comes from the work done? But then where does the work come from? Clearly both are equivalent, perhaps there is no definite causality. „Instead of a rubber band it is easier to think about an FRW metric filled with an elastic medium.“ This is a very good hint. – timm May 4 '18 at 15:53
• "In case 1, the work comes from the vacuum energy and effectively slows down the acceleration. " This means that $\rho_\Lambda$ has to decrease. Hmm quite weird. – timm May 4 '18 at 18:20
• @timm No it would not mean the vacuum energy density that has decreased. It means, that slightly less volume has been created by the expansion. – mmeent May 4 '18 at 18:55
• Sorry @mmeent I can't follow here. If the energy densities are not affected by the stretching of the rubberband the time evolution of the scale factor isn't affected too. How then can less volume be created? – timm May 4 '18 at 19:26
• Recall: $$\ddot{a}/a = -\frac{4\pi G}{3}(\rho+3p)$$ Increasing $p$, which is what adding elasticity to our medium does, thus decreases the acceleration of the growth of the scale factor $a$. Put differently, less volume is created. – mmeent May 4 '18 at 19:59