Question

Can you please mathematically derive for me why the weight of a person inside a freely falling elevator is zero?

Answer 1

The weight of an object can be and is defined in two ways;

1) the force due to gravity on a body
2) the force required to hold a body at rest, i.e. to prevent it from falling, in a gravitational field.

Let's use the second definition since most of us think that way. When you're in free fall there are no forces other than gravity acting on your body. The force of gravity is not something you can feel when you're in free fall because it causes every single particle in your body to fall at exactly the same rate. In your frame of reference, i.e. the frame in free fall, there is, for all practical purposes, no gravitational field. In fact in general relativity there really isn't a gravitational field in a free fall frame. It's said to have been transformed away.