The Equivalence of Mass

All mass is affected by the curvature of space in the same way because all mass is equivalent. There are three kinds of mass, inertial mass, passive gravitational mass, and active gravitational mass. These three types of masses are equivalent. Inertial mass is the ability of a body to resist a change in motion. Passive gravitational mass is the reaction a body has to a gravitational field. Active gravitational mass acts as the source of gravity. It can be shown mathmatically that the three types of mass are the same, but Einstein showed through thought experiments that the three types of mass are the same. For example, imagin there are 2 elevators, one elevator is here on Earth under the influence of the Earth's gravity, and the other elevator is in deep space far from any gravitational field. If a ball with a certain mass is placed inside of both elevators, then the equivalence of inertial, passive gravitational, and active gravitational mass can be shown. An elevator in deep space with no acceleration will allow the ball to float above the floor where it was released. This type of mass for the floating ball is inertial mass, since it is not under the influence of gravity. An elevator on earth in free fall creates a situation that is identical to an elevator in deep space with no acceleration, allowing the ball to float above the floor where it was released. This type of mass for the floating ball is passive gravitational mass, since it is under the influence of the Earth's gravity. An observer in either elevator can not distinguish by observing the ball if the elevator is floating in deep space or falling down the elevator shaft due to the influence of Earth's gravity. In both cases, the inertial mass of the floating ball in deep space and the passive gravitational mass of the floating ball under the influence of the Earth's gravity, causes the mass of the ball to behave in an identical manner. Therefore, unaccelerated inertial mass and passive gravitational mass are the same. This equivalence of unaccelerated inertial mass and passive gravitational mass is known as the "Weak Equivalence Principle".
An elevator in deep space under an acceleration equal to Earth's gravity will cause the ball to fall to the floor after being released. This type of mass for the accelerated ball is inertial mass, since it is resting on the floor due to its resistance to a change in motion. An elevator sitting on earth creates a situation that is identical to an elevator in deep space accelerating with a value equal to Earth's gravity, causing the ball to rest upon the floor after being released. This type of mass for the resting ball is passive gravitational mass, since it is under the influence of the Earth's gravity. An observer in either elevator can not distinguish by observing the ball if the elevator is accelerating in deep space or sitting on the surface of the Earth. In both cases, the inertial mass of the accelerating ball in deep space and the passive gravitational mass of the resting ball under the influence of the Earth's gravity, causes the mass of the ball to behave in an identical manner. Therefore, accelerated inertial mass and passive gravitational mass are the same. This equivalence of accelerated inertial mass and passive gravitational mass is known as the "Strong Equivalence Principle".
Einstein did not have an explicit thought experiment relating passive gravitational mass and active gravitational mass, but by relating inertial mass to passive gravitational mass, then the equivalence of active gravitational mass can be implied. This relationship can be shown using Newton's third law: For every action there is an opposite and equal reaction. If two balls are placed near each other in deep space, then the mass of both balls will create an attractive force. If the mass of both balls are the same, then the mass of the first ball will attract the mass of the second ball by the same amount that the mass of the second ball attracts the mass of the first ball. This means that the attractive force of ball one is equal to the attractive force of ball two. Ball one's active gravitational mass creates an attractive force and ball two's active gravitational mass creates an attractive force. Ball one's passive gravitational mass is a reaction to the force of ball two's active gravitational mass, and ball two's passive gravitational mass is a reaction to the force of ball one's active gravitational mass. Since, by Newton's third law, the reaction must be equal and opposite to the action, then passive gravitational mass must be equal and opposite to the active gravitational mass. Meaning, bodies of the same mass attract and are attracted by the same amount, and therefore, passive gravitational mass is equivalent to active gravitational mass.

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