What is the mass of Stephen Hawking? What difference does it make?
Einstein explained The Equivalence Principle of his general theory of relativity as follows:
A little reflection will show that the law of the equality of the inertial and gravitational mass is equivalent to the assertion that the acceleration imparted to a body by a gravitational field is independent of the nature of the body. For Newton’s equation of motion in a gravitational field, written out in full, it is:
- (Inertial mass) x (Acceleration) = (Intensity of the gravitational field) x (Gravitational mass).
It is only when there is numerical equality between the inertial and gravitational mass that the acceleration is independent of the nature of the body.
Let us analyze this statement and see if it makes sense. Let us ask if there are two different forms of mass, inertial mass and gravitational mass, and if so why they are equal?
Let us then start with Galileo’s Free Fall Principle stating that all bodies accelerate equally fast in a gravitational field of strength , where is a gravitational potential:
where is velocity and acceleration. In this setting of free fall there is no reason to introduce something called gravitational mass , since it would only enter as a multiplicative factor:
where we would view as the gravitational force acting on the body. Since all bodies fall in the same way irrespective of size and composition, there would be no reason to make a distinction measured as gravitational mass. We note that Galileo’s Principle connects motion ( to gravitational potential gradient with a balance of kinetic (motion) energy and gravitational energy: what is lost in kinetic energy is gained in gravitational energy and vice versa.
However, if we expand the scope and include motion subject to general not necessarily gravitational forces, we find reason to measure the magnitude or modulus of force by comparison with gravitational force using e.g. a spring scale:
Once we are able to measure force magnitude, we can define the inertial mass of a body moving with velocity under a force by the relation
But by Galileo’s Principle we also have and thus , that is, inertial mass is equal to gravitational mass. Galileo’s law guarantee’s that the definition of does not depend on the magnitude of : doubling must give rise to a doubling of , because a doubling of has the same effect, and is compared to .
We conclude that there is only one mass, because by definition inertial mass = gravitational mass. Further, Newton’s 2nd Law is valid just because we have agreed to define inertial mass = gravitational mass.
The essential physics is introduced by Galileo’s Principle of Free Fall, which makes it possible to scale force by a spring scale, and then define inertial mass (or simply mass) as force divided by acceleration.
We see here a clear distinction between definition (inertial mass equal to gravitational mass), which is a tautology empty of physics which cannot be false, and Galileo’s Principle of Free Fall which is a statement about physics, which may be true or false.
Einstein did not tell whether his law of equality of inertial and gravitational mass is a definition or physical fact, which distorts his science into pseudo-science.