Why Does the Earth Move Towards the Apple According to the Theory of General Relativity?

It seems intuitive to think that when an apple falls from a tree, it is the apple that moves towards the ground, while the Earth remains stationary. However, the principles of physics, specifically the Theory of General Relativity (GR), suggest that both the apple and the Earth are in motion relative to each other. This article delves into the details of this phenomenon and why, although the movement of the Earth towards the apple is negligible, it is still valid according to the laws of physics.

Introduction to Newton's Third Law and Inertia

John Newton's famous statement, "For every action, there is an equal and opposite reaction," illustrates the principle of conservation of momentum. When the apple falls towards the Earth, the Earth exerts an equal gravitational force on the apple, causing it to accelerate towards the Earth. However, due to the vast difference in mass between the Earth and the apple, the effects are significantly different.

Comparing the Mass of Earth and Apple

The mass of the Earth is approximately (5.972 times 10^{24}) kilograms, while the mass of the apple is only about 200 grams. The Earth's immense mass leads to a fundamental difference in the way it and the apple respond to gravity.

When an apple falls 3 meters from a tree to the ground, it moves 3 meters towards the Earth. But the Earth, which rises by an incredibly tiny distance, experiences a much smaller change in position due to its massive mass. This minuscule displacement, approximately 13.2 yoctometers (0.000000000000000000000000000000132 meters), is far too small to be measured or observed with any available technology.

Differences in Mass and Inertia

Both the Earth and the apple are attracted to each other due to Newton's law of gravitation. However, the acceleration of each object is influenced by their respective masses and inertia. Inertia, which is the resistance of an object to changes in its state of motion, reduces the acceleration of anything affected by an external force.

Because of its high mass, the apple experiences a significant downward acceleration of approximately (9.81 , m/s^2), causing it to fall rapidly towards the Earth. Conversely, the Earth has a much lower acceleration due to its immense mass. Even if the Earth were to move towards the apple, the displacement would be so minute that it is unnoticeable, less than a billionth of an inch (

Illustrating the Effect with Newton's Laws

According to Newton's second law, (F ma), where (F) is the force, (m) is the mass, and (a) is the acceleration. When the apple and Earth are attracted by gravity, they both experience the same force, but their accelerations differ due to their different masses. The apple, with a much smaller mass, experiences a significant acceleration, making its fall perceivable. The Earth, with its enormous mass, experiences an acceleration so small that it is practically unmeasurable.

To further illustrate this, consider the mass ratio of the apple to the Earth: (100 , text{g} : 5.972 times 10^{24} , text{kg}). This extreme difference in mass means that the apple's acceleration is much greater than the Earth's. Thus, the movement of the Earth towards the apple is negligible and is often neglected in practical scenarios.

Conclusion

While it is true that both the Earth and the apple are influenced by gravity, the Earth's massive size and corresponding inertia mean that its movement towards the apple is barely perceptible. This phenomenon is a fascinating aspect of the fundamental laws of physics, reminding us of the delicate balance involved in the interaction between objects of vastly different masses.

Understanding these principles not only deepens our appreciation for the natural world but also highlights the importance of considering all factors in the equations of motion. Whether you are a physicist or a curious individual, the concept that the Earth moves towards the apple, albeit imperceptibly, is a testament to the intricate and beautiful nature of our universe.