Two masses, m and M, are a fixed distance R apart. One of the masses is much larger then the other. At time t the masses start to fall towards each other. Using Newton's Law of Gravitation we can determine the acceleration of the small mass. Can one use the Schwarzschild metric in the following way to find the acceleration of the small mass? Assume the small mass initially has some acceleration a. Treat acceleration a as a variable. After falling for some small time dt, using the acceleration a we can determine dr. We can now plug both dt and dr into the Schwarzschild metric and get a value for dτ, the proper time interval? If we now plot dτ as a function of initial acceleration a we should get a peak for dτ when the acceleration is that value given by Newton's Law? Thanks.