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Equivalence Principles of Gravitation
by Ron Kurtus (revised 29 July 2010)
There are several Equivalence Principles that refer to related gravitational concepts.
The Weak Equivalence Principle states that objects fall at the same rate, provided that are freely falling. The equivalence of inertial and gravitational mass states that mass determined by inertia is the same as mass determined by gravitation. The Strong Equivalence Principle extends the equivalence of masses to state that observations of acceleration cannot be distinguished from gravity.
Questions you may have include:
- What is the Weak Equivalence Principle?
- What is the equivalence of inertial and gravitational mass?
- What is the Strong Equivalence Principle?
This lesson will answer those questions. There is a mini-quiz near the end of the lesson.
Weak Equivalence Principle
The Weak Equivalence Principle (also called the Uniqueness of Free Fall Principle) applies to relatively small objects falling to the Earth or other large astronomical body. It says that objects fall at the same rate, independent of their mass.
Objects fall at same rate
This means that two objects dropped together from the same height will strike the ground at the same time. Of course, air resistance is considered negligible, and there are no other forces acting on the objects that can affect their free fall.
Objects close to Earth
For objects relatively close to Earth, the approximately of the force of gravity is used. This is because the acceleration due to gravity, g, is constant.
(See Equivalence Principle of Gravity for more information.)
Same size objects far from Earth
For objects further from Earth that are approximately the same size, application of the Universal Gravitation Equation shows that—although the acceleration or rate the object falls varies with distance—acceleration is the same for both objects falling from the same distance. Consider the equations:
F1 = Gm1M/R2
F2 = Gm2M/R2
where
- F1 and F2 are the forces on objects 1 and 2 respectively
- G is the Universal Gravitational Constant
- m1 and m2 are the masses of objects 1 and 2 respectively
- M is the mass of the Earth
- R is the distance from the center of the objects to the center of the Earth
Since GM/R2 is the same for both objects, they will fall at the same rate.
Exception when objects different size
An exception to the Weak Equivalence Principle is when one object is much larger than the other. In such a case, the distance from the centers of mass would would be different, and they would fall at slightly different rates.
Exception when objects are much different in size
This exception is seldom considered when studying the principle.
Equivalence of inertial and gravitational mass
There is an equivalence of inertial and gravitational mass. You can see this by examining the forces from both inertial mass and gravitational mass.
Inertial mass
If you accelerate an object, the force required to overcome its inertia is:
Fi = mia
where
- Fi is the force needed to overcome inertia
- mi is the inertial mass
- a is the acceleration on the object
Gravity or gravitational mass
Likewise, the acceleration due to gravity is:
Fg = mgg
where
- Fg is the force of gravity on the object (also its measured weight)
- mg is the gravitational mass
- g is the acceleration due to the force of gravity
Note that this equation could also be stated in gravitational terms:
Fg = GmgM/R2
Equivalence
Since the time of Newton, scientists have wondered if the inertial mass was the same as the gravitational mass. Does mi = mg?
Albert Einstein stated that a gravitational force, as experienced locally while on a massive body such as the Earth, is actually the same as the pseudo-force experienced by an observer in an accelerated frame of reference.
Einstein used the equivalence of inertial and gravitational mass as a basic framework for the General Theory of Relativity.
Strong Equivalence Principle
The Strong Equivalence Principle (also known as the Einstein Equivalence Principle) states that the effects of acceleration are indistinguishable from those of gravitation.
(See Artificial Gravity for an example of this.)
Experiments by observer
This means that an observer cannot determine by experiment whether he or she is accelerating or in a gravitational field. In other words, results from experiments in an accelerating spaceship would be the same as those obtained from gravity.
Experiment in accelerating spaceship
Note: One problem with this concept is that acceleration cannot be applied for too long a period, because the spaceship would soon reach the speed of light. On the other hand, gravitation is continuously present.
Einstein's conclusion
Einstein concluded that gravitation and motion through spacetime are related and that the Strong Equivalence Principle suggests that gravitation is geometrical by nature.
Difference between strong and weak
The difference between the Strong Principle of Equivalence and the Weak Principle of Equivalence is that strong equivalence states all the laws of nature are the same in a uniform static gravitational field and the equivalent accelerated reference frame, while weak equivalence states all the laws of motion for freely falling particles are the same as in an unaccelerated reference frame.
Summary
The Weak Equivalence Principle states that objects fall at the same rate, provided that are much smaller than the attracting body and are freely falling. The equivalence of inertial and gravitational mass states that mass determined by inertia is the same as mass determined by gravitation. The Strong Equivalence Principle extends the equivalence of masses to state that observations of acceleration cannot be distinguished from gravity.
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Resources
The following resources provide information on this subject:
Websites
Weak Equivalence Principle - Smoot Group Astrophysics
The Equivalence Principle - Univ.of Washington Laboratory Tests of Gravitational and sub-Gravitational Physics
Equivalence Principle of Gravitation - Living Reviews in Relativity Journal - Max Planck Institute for Gravitational Physics
Equivalence principle - Wikipedia
The Principle of Equivalence - University of Tennessee-Knoxville Astronomy
Gravity and Gravitation Resources
Books
Top-rated
books on Simple Gravity Science
Top-rated
books on Advanced Gravity Physics
Mini-quiz to check your understanding
If you got all three correct, you are on your way to becoming a Champion in Physics. If you had problems, you had better look over the material again.
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