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PRINCIPLE OF EQUIVALENCE |
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The Principle of Equivalence lies at the very heart of general relativity. In essence, it equates gravitational fields with accelerated frames of reference. It provides a valuable tool for determining the effects of a gravitational field, since the principle states that the phenomenon can be studied in an equivalent accelerating frame of reference. |
Gravitational and Inertial Mass
In the article on the law of gravitation, we noted that the mass that appears in the expression for the force of gravity,
and the mass that appears in Newton’s second law,
are not strictly the same. In the law of gravitation, the mass m is the “gravitational mass” of the body under consideration, and it describes the response of the body to the pre-existing gravitational field. In Newton’s second law, the mass m is the “inertial mass” of the body, and it provides a measure of the resistance of the body to motion under an applied force.
Therefore, the motion of a body in a gravitational field should strictly be written in the form
Now it would seem that the gravitational mass and inertial mass of a body are very different concepts. We would therefore not be surprised if the two types of mass were to take different values for the same body.
But, experiments have shown something quite remarkable:
The gravitational mass and inertial mass of a body are equal to an accuracy of at least one part in 1012.
The accuracy quoted here results from the inherent precision that is available from the experimental techniques used. (Details of the experimental work on this can be found on Wikipedia or by a Google search). However, the results indicate that, to all intents and purposes, the gravitational mass and inertial mass are numerically identical.
This is an amazing result. It hints that there is a very close relationship between gravitational phenomena, as embodied in the expression F = mg, and dynamic phenomena, described by Newton’s second law F = ma.
Principle of Equivalence
Albert Einstein gave a great deal of thought to this notion, and came to the conclusion that there is an equivalence between motion in an accelerating frame of reference and motion in a gravitational field. Einstein attached such weight to this notion that he formulated a physical principle, called the Principle of Equivalence. In technical language, the principle states that:
All local, freely falling, non-rotating laboratories are fully equivalent for the performance of all physical experiments.
(This is how the principle is expressed in Longair’s book Theoretical Concepts in Physics).
A slightly more comprehensible way of expressing the principle is as follows (this is taken from Cassell’s Laws of Nature):
It is impossible to tell by experiment whether you are in accelerated frame of reference or in a gravitational field.
This formulation of the Principle of Equivalence is more intelligible, bearing in mind our discussion about inertial and gravitational mass. We noted that inertial and gravitational mass are identical, which implies a close relationship between gravitational and dynamic phenomena. The above statement of the Principle of Equivalence formalises this idea by asserting that gravitational and dynamic phenomena are so closely related that in fact you cannot tell the difference between them by test or by experiment.
The major consequence of the Principle of Equivalence is that any phenomenon that is experienced in a gravitational field should also be experienced in an accelerating frame of reference. This is of the greatest importance in the theory of general relativity. Indeed, the problem of determining the gravitational effects of matter in the universe becomes one of finding suitable accelerated frames of reference that take these effects into account. Much of the mathematical complexity of general relativity stems from this procedure.
Examples of the Principle of Equivalence
To illustrate the Principle of Equivalence, consider the following diagram:
This diagram shows two frames of reference (or laboratories, in Einstein's language), both of which have a mass suspended by a spring from the ceiling of the frame of reference. In the diagram on the left, the frame of reference is stationary, but in a gravitational field (for example, the frame of reference could be located on the surface of the earth). The gravitational field exerts a force on the mass, and the spring will stretch slightly in response.
In the diagram on the right, the same laboratory is in a region of zero gravity, but is subject to an upwards acceleration equal to the gravitational acceleration felt by the frame of reference on the left. The same extension of the spring is observed, because the tension in the spring now has to provide an upwards acceleration to the mass on the end of the spring.
In this case, the effects of the gravitational field and the accelerated frame of reference are the same (same extension to spring).
Now consider the following diagram, also showing two frames of reference with springs and masses suspended from the ceilings.
The diagram on the left shows the situation in a region where there is no gravitational field, and no external acceleration of the frame of reference. Since there are no forces acting on the mass at the end of the spring, the spring shows no extension. Compared with the previous diagram, a < d.
In the diagram on the right, the same frame of reference is shown in free fall in a gravitational field. Again there is no extension of the string, as the force to accelerate the mass downwards is provided by the gravitational field itself – no additional force is required from the spring.
In this case, the effects of no gravitational field and a frame of reference in free fall are the same (no extension to spring). The feeling of no gravitational field is familiar to anyone who has ridden in a lift going downwards. As the lift accelerates down, a feeling of lesser weight can be felt, until the lift reaches a constant speed, when the feeling of “normal” weight returns. Similarly, if an aircraft descends rapidly from altitude, a feeling of less weight can be felt. Indeed, zero gravity conditions can be simulated for astronauts by travelling in an aircraft that is put into free fall (for a short period of time!)
All of this discussion can be summarised very concisely:
At any point, we can replace the local effects of gravity by an accelerated frame of reference. However, different frames of reference are required for different points in space, since the magnitude and direction of gravity varies between locations.