# Why gravitational mass and inertial mass being same was a problem before Einstein solved it?

Why is it so that Einstein is credited with telling us the reason for the equivalence of inertial and gravitational mass of an object? Did Newton ever make a distinction between the two masses? What was the problem in accepting the fact of a single mass being an inherent property?

• Maybe useful - physics.stackexchange.com/a/427246/133418 – Avantgarde Jun 8 at 7:59
• Just curious, why did you withdraw your acceptance of my answer – Bob D Jun 10 at 11:01
• Sorry, it was a mistake, realised after your comment. – Gariman Singh Jun 10 at 11:06
• Ok thanks I appreciate it – Bob D Jun 10 at 11:14

The equivalency of gravitational and inertial mass was known by Newton and others before Einstein, as Einstein himself acknowledged. In Einstein's words "It is true that this important law had hitherto been recorded in mechanics, but it had not been interpreted" (Einstein book: Relativity. The Special and the General Theory).

It was Einstein who extended the principle using his famous elevator thought experiments. A person in a windowless elevator in free fall in the earth’s gravitational field experiences weightless. The person would have no way of knowing if the person and elevator were in outer space in the absence of a gravitational field or in free fall in a gravitational field.

If the same person and elevator were in outer space and an external force is applied at the top of the elevator ( top meaning the surface of the elevator adjacent to the head of the person) giving the elevator an “upward” acceleration of $$g$$, the person would have no way of knowing that the person is standing on the floor of the elevator on earth or being accelerated in outer space.

These thought experiments helped lead him to the general theory of relativity.

So, exactly why did two different terms for mass arise?

The two different terms for mass arose because the term “gravitational mass” described the unique property of mass that was determined from Newton’s universal law of gravity, whereas the term “inertial mass” described the property of mass in which mass resists a change in motion in response to a net external force. Experiments by Galileo and Newton, and experiments many times since, demonstrated that gravitational mass equals inertial mass. But, as @garyp stated, there was no reason why they should be the same.

Bottom line: Gravitational mass is defined by the force of gravitation. Inertial mass is defined by Newton’s second law.

Why there is a need for a statement on equivalence when you can easily define only one term?

Since “gravitational mass” equals “inertial mass”, in my opinion I see no reason to use two terms. One can simply use the term “mass”. But the statement on equivalency is crucial to the general theory of relativity. The elevator thought experiments discussed above are the basis of Einstein’s idea that the force of gravity as felt locally while standing on a massive body is the same as the pseudo-force experienced by a person in a non-inertial frame of reference, such as the forces experienced in the accelerating elevator in space, or the forces experienced in an accelerating car.

Hope this helps.

• So, exactly why did two different terms for mass arise? Why there is a need for a statement on equivalence when you can easily define only one term? – Gariman Singh Jun 9 at 2:48
• @GarimanSingh See my revised answer in response to your follow up questions. – Bob D Jun 9 at 22:28

The problem was that there was no reason why they should be the same.

The two masses appear in phenomena that seemingly had nothing to do with one another. I apply a force to an object, measure the acceleration, and divide. A principle of dynamics. Why should it have any relationship at all to the nature-provided force that holds the Moon to the Earth?

In Newton, the application of a force is required to overcome the masses inertia. In Einstein, it is curved space that overcomes the inertia with no force directly applied to the object of mass. An object in orbit experiences both kinds simultaneously. An object in direct free-fall has its inertial mass force set to zero. Once an object has been accelerated and is now moving with constant velocity and direction, it is itself in a state of free-fall. If we now bend Newtons first law around a gravitating body, the object will remain in free-fall but will follow a curved path called a geodesic.

It's worth noticing that it's perfectly consistent to imagine a physical theory in which the inertial and gravitational masses of an object are distinct quantities. Indeed, such a strange phenomenon actually exists in our universe: all particles carry an "electromagnetic mass" (aka charge) that dictates the "electromagnetic gravitational force" (aka Coulomb force) acting on the particle, according to the Coulomb force law $$F = \frac{k q_1 q_2}{r^2},$$ the analogue of Newton's universal gravitation law. And in this example the electromagnetic mass is decidedly not equal to the inertial mass, since different particles can carry positive, negative, or zero charge.

This example shows that the fact that gravitational and inertial masses are equal is not self-evident. In both Newtonian and Einsteinian physics it plays the role of a postulate. However, in Einsteinian physics it is a much more fundamental postulate, since it is essentially equivalent to the principle that all objects fall with the same acceleration in the presence of a gravitating body, which is a version of the equivalence principle of general relativity, a cornerstone of the entire theory (this Wikipedia page has a good explanation of how these principles relate to each other). In Newtonian physics it's more a case of "yeah, this is an assumption that is simple and natural and fits our observations of how the world behaves, so let's adopt it as a postulate." Einstein's approach simply makes it seem that much more natural.