Relativistic Frictional Force

[camacazio]

I had some free time between classes today and I started thinking about the force of friction. The macroscopic approximation for it is simply a constant related to the type of materials in contact and proportional to the normal force of the object. Say we had--and I'll make this an idealized situation--a rocket ship with a perfectly flat surface on it's underside. It is travelling near the speed of light (for completeness let's say .8c) when it starts to travel along a frictional surface on Earth. Once it begins sliding, the force of friction will cause an accelleration slowing down the rocket ship. At first the simplified equation for friction between the rocket ship and the surface is all good and fine aside from some minor relativistic changes to force. However, this rocket ship as it slows down will lose it's length contraction gained from it's speed. As it slows more, it will expand more. Our simplistic equation for force of friction does not rely on surface area, so that is not a problem. The equation is, though, an approximation to simplify an assortment of small jagged edges along any real surface. As the space ship slows and expands in its direction of motion, the jagged edges on its underside will spread out--effectively reducing the force of friction. In the reference frame of the rocket ship, the road was lorentz contracted and is now what is expanding. Both the road and rocket ship reference frames should agree in a reduction of force of friction due to an observed change in the nature of the surfaces in contact. To further complicate things, the expansion of the rocket ship will do this: as it's edges lengthen along the ground, it's lengthening is pushed against by the road. As it expands, the friction is opposing it's expansion. The time dialation that will occur between the front and back of the space ship with this force will effectively cause the front and back to expand at diffierent rates, though this part I'm not too sure about. If it's expanding with it's decrease in velocity, then every discreet change in velocity will give a discreet change in length. This discreet change in length across an integral for the change in rocket ship velocity across the surface could give a useful term for the accelleration with which each end expands, and use that in discreet chunks of mass for finding what discreet chunk of opposition to expansion each small expansion has.