Equivalence Principle

What is equivalence principle?

In the general theory of relativity developed by Alber einstein, the equivalence principle explains about “equivalence of inertial mass and gravitational mass”. He observed that, the force due to gravity(gravitational force) experienced by a person standing on the massive object (for example earth) is equivalent to the pseudo-force experienced by the observer in the non-inertial frame of reference(accelerated).

History of Equivalence Principle

Albert Einstein realised that there is some complexity between Newton’s theory of gravity and his own theory of special relativity. He looks into the universe in a completely different way!

As we know, during the year 1905 Albert Einstein set a milestone in physics with his new framework on laws of physics, that is his theory of special relativity. Nonetheless one facet of physics seemed to be incomplete; The gravitational force theory developed by Newton. Special theory of relativity enfolded space and time excluding gravity. Years later, Alber Einstein managed to unify gravity with his relativistic ideas on space and time. As a result another revolutionary theory emerged. “The general theory of relativity”.

History of Equivalence Principle

The very first step towards a general theory of relativity was the realisation that, Even with in a gravitational field, there exists a reference frame in which gravity is almost zero/absent. Thus, at least to a certain approximation, one can say gravity free laws of special relativity governs the laws of physics, if and only if the observations are confined within a small region of space and time. This derived from what he formulated as equivalence principle which, in turn, is inspired from the consequence of a freefall.

What is Effective mass?

Mass is a property of any physical object, It is a measure of resistance to the acceleration in the presence of external force. Effective mass is the terminology used in general theory of relativity which talks about gravitational mass and inertial mass.

  • Gravity: Tendency of massive object to attract each other. Gravitational force between two bodies is given by-

\(F=G\frac{Mm_{gravity}}{r^{2}}\) —- (1)

Where, G is the universal gravitational constant.

M is the mass of one body

\(m_{gravity}\) is the mass of second body

r is the distance between them

  • Inertia: Resistance offered by the body against the change in its current state of motion. Force involved here is given by-

\(F=m_{inertia}\times a\) —–(2)

\(m_{inertia}\) is the inertial mass of the object

a is the acceleration of the object

Principle of equivalence comes up with an idea, saying equation (1) and (2) are the same. That is:

\(F= G\frac{Mm_{gravity}}{r^{2}}=m_{inertia}\times a\)

\(\Rightarrow G\frac{Mm_{gravity}}{r^{2}}=m_{inertia}\times a\)

On rearranging the above equation we get-

\(\Rightarrow a=\left [ \frac{m_{gravity}}{m_{inertia}} \right ]\times G\frac{M}{r^{2}}\)

Hence, proves their equivalence mathematically.

Equivalence Principle Example

Imagine you are standing in an elevator or, precisely, in something inside of which looks very similar to an elevator cabin. You are totally isolated from the outer world. Under these circumstances, if you take an object and drop it, it falls down and reach the floor/base. This is the most expected way you expect, considering your experience here on earth. The situation is depicted in the following diagram:

Will the effect be same if the elevator is subjected under the gravitational field, similar to that of earth?

Not necessarily! Theoretically you may be situated in deep space, far away from gravitational influence of any significant mass concentration. The cabin you are standing in can be inside a rocket which is accelerating at 9.8 meter per square second. The situation is illustrated clearly with the following diagram:

In such a scenario, you drop off an object, The object would experience an acceleration of 9.8 meter per square second from cabin floor- exactly the same acceleration as that an object experience during free fall here, right on earth. Now as you, as an observer standing inside a cabin floor can hardly differentiate two situations. From your frame of reference situations will be indistinguishable: are they falling under the gravity of a massive object like earth? Or is it the cabin floor accelerating them?

Thus, Einstein formulates his Principle of equivalence stating that “No experiment can be performed that could distinguish between a uniform gravitational field and an equivalent uniform acceleration.”

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Practise This Question

In the figure shown, for an angle of incidence 45^{\circ}at the top surface, what is the minimum refractive index needed for total internal reflection at vertical face