# Equivalence Principles of Gravitation

by Ron Kurtus (revised 15 February 2016)

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. Useful tool: Units Conversion

## Weak Equivalence Principle

The Weak Equivalence Principle (also called the Uniqueness of Free Fall Principle) states that gravitational causes objects to fall or move toward an attracting body at the same rate, independent of their mass.

Objects fall at the same rate

### Proof

The proof of this principle is pretty straightforward. Consider two objects that are the same separation from a larger body. Their equations are:

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 attracting body
• R is the separation from the centers of the objects to the center of the attracting body

Since F = ma, the acceleration is GM/R2 and is the same for both objects. Thus, they will fall at the same rate.

### Restrictions

However, there are some restrictions on this principle.

#### No outside forces

It is assumed that there are no outside forces such as air resistance acting on the falling objects. In other words, they are falling freely.

#### Mass much less than attracting body

A major restriction on the Weak Equivalence Principle is that the mass of each falling object must be much less than that of the attracting body.

The gravitational force causes both the falling object and the attracting body to move toward each other and their center of mass. Thus, the mass of the falling object much be so small with respect to the attracting body that its movement is negligible.

For example, the mass of the Earth is 5.974*1024 kg. An object that had a mass of 6,000,000 kg (6*106 kg) falling from a displacement of 105 km would result in movement of the Earth of:

rM = mR/(M + m) km

where

• rM is the separation between the center of the attracting body and the center of mass between the objects
• m is the mass of the smaller object
• R is the separation between the objects
• M is the mass of the larger, attracting object

Thus:

rM = (6*106)105/(5.974*1024 + 6*106) km

rM = 6*1011/5.974*1024 km

This is approximately:

rM = 10−13 km = 10−5 cm

That is a tiny movement for a mass of that size.

#### Objects must be of similar size

Another restriction is that the objects must be similar in physical size, such that the center of mass for each is at approximately the same displacement from the attracting body. If the separations between the centers of mass are different, the objects 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

and the inertial mass is:

mi = Fi/a

where

• Fi is the force needed to overcome inertia
• mi is the inertial mass
• a is the acceleration on the object

### Gravitational mass

Likewise, the gravitational force is:

Fg = GmgM/R2

and the gravitational mass is:

mg = FgR2/GM

where

• Fg is the gravitational force on the object
• G is the Universal Gravitational Constant
• mg is the gravitational mass of the object
• M is the mass of the attracting object
• R is the separation between the objects, as measured from their centers of mass

### Equivalence

Since the time of Newton, scientists have wondered if the inertial mass (mi) was the same as the gravitational mass (mg). Many experiments verify the equivalence.

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 gravitation.

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 a reference frame that is not accelerated.

## 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 gravitation.

Physics is amazing

## Resources and references

Ron Kurtus' Credentials

### Websites

Weak Equivalence Principle - Smoot Group Astrophysics

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

Gravitation Resources

### Books

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