I don’t have any mathematical beliefs along those lines. I like to simplify things by abstracting them. Then I can ignore the underlying details.Originally Posted by Fortis
My thinking in developing the new equation for escape velocity went like this: Section 1 shows that eq. 19 is the relativistic equation for free-fall velocity in a uniform gravitational field. Section 2 shows that free-fall velocity from rest at infinity to a given altitude is the escape velocity at that altitude. All that remains for me to do is to create an equation that integrates eq. 19 from infinity to a given altitude r, using some formula that returns the g for each r. Newton already did that for eq. 1 to derive eq. 4, and I find no fault with his inverse square law of gravity applied to r. There is a one-to-one relationship between Newtonian velocity and relativistic velocity, shown by eq. 3. Then I can implicitly integrate, piggybacking on Newton’s integration, by using eq. 3 to convert Newton’s equation for escape velocity into a relativistic equation, eq. 6.
At that point I have an equation for escape velocity that is the desired integration of eq. 19 from infinity to a given altitude r. At that point I’ve abstracted to a level where I don’t need to know more mathematical details about Newtonian gravity.