- #1

FallenApple

- 566

- 61

Assume that both are flexible and can stretch like balloons at the same time, the particles that comprise them are solid like billiard balls. So stretching/contracting simply means that the spacing of these particles get smaller/bigger with time.

Well, initially, they are both M(in) and M(out) at rest.

But then they attract each other via the inverse square law of gravity. The inner shell doesn't stretch outward at all since we know that the gravitational field due to the outer shell is 0 everywhere inside of it. But from the outer spherical shell's perspective, the inner spherical shell can be considered a point mass. So every piece of the outer shell is being pulled on by M(in). This makes M(out) contract towards M(in) with increasing velocity while M(in) doesn't move.

Here is what happens:

**M(out) contracts inward towards M(in). M(in) will stay fixed since there is no influence from M(out).**Intially, as the outer sphere contracts, due to the symmetry, the vector momentum of each piece of M(out) will cancel out with another piece of M(out), specifically the opposite side. So momentum is conserved if the system is M(out) only.

However, if we consider M(out) as a whole, it would have a radial momentum inward. But M(in) has no radial momentum outward. So momentum is not conserved.

Can this be fixed if we consider momentum of gravitational field?

If we create a mathematical spherical volume

*V*ecapsulating both shells at time t=0, then we can analyze how the momentum changes with time.As the outer sphere contracts, the change in volume delta V becomes filled with the field lines from the outer shell. Since the field due to the M(in) is fixed for any region outside R(in), the displacement in volume corresponds to an increase field density. We see that the field density would increase with time within

*V.*

So there are two contributions to momentum. The momentum of p=m*v and the momentum of the gravitational field. Both are from the outer shell, M(out) only.

So the flux momentum of this field is increasing inwards through time and the radial momentum of M(out) mechanical motion is increasing inwards as well and are both in the same direction.

So does this mean momentum is not conserved for this situation? Perhaps its a time delayed effect? That is the situation changes after the collision. But I thought momentum of a isolated system doesn't depend on time.