Except matter IS fixed relative to other matter its how geasutres vaguely at all solids/liquids/most gasses works. The space the line is in would stretch, which would stretch the line, and then the line would contract again due to the bonds between its atoms producing tension. Of course since expansion is continuous eventually the line would snap, or if expansion suddenly became linear then the tension would reach an equilibrium and thus you couldn’t extract energy from it.