A THING ABOUT FALLING - THE WEAK PRINCIPLE OF EQUIVALENCE

In our last two articles, we have seen how velocities and displacements of objects change when they are in free fall, taking note of whether they are falling in a vacuum or air. However, I have not yet highlighted the central question as to why all objects, irrespective of their mass, accelerate equally in a constant gravitational field, bearing in mind that there is no air resistance. The reason is quite fundamental. At the heart of the Newtonian framework lies a magnificent principle, so indispensable, as we will see in the ensuing paragraphs of this article, that Einstein developed an entirely new theory of gravity, viz., the General Theory of Relativity.  

The principle in question is the Principle of Weak Equivalence or the Weak Principle of Equivalence. But before going further down the road, we have to return to the famous feather and coin experiment, only this time, it has not been performed by Galileo or Newton but by our beloved Brian Cox. In the following video, uploaded by the BBC, we see Brian entering a giant vacuum chamber at NASA's Space Power Facility in Ohio to see what happens if we drop a bunch of feathers and a bowling ball when we suck out all the air. Will they hit the ground as predicted by our equations? 

It turns out that they do follow the equations without deviating a  bit. When there is air, i.e., in ordinary atmospheres, the ball, being heavier, hits the ground before the bunch of feathers because the latter quickly acquires the terminal velocity and falls slower. We see this quite often, so there is nothing unnatural in it. But in the presence of air, we do not get to see the true nature of gravity. When they remove the air and arrive at a near-perfect vacuum, we can see that the ball and the feathers accelerate equally. Neither of them leads the other as they fall together so as to say hand-in-hand, like a young couple. Thus we readily see, right before our eyes, that gravity accelerates all masses equally, just as our equations had predicted. If we could build a large-enough vacuum chamber to accommodate a T. Rex or an excavator, we would only see that they fall at the same rate and at the same time. 

But wait. There is more to it than meets the eye. Newton conceived gravity as a force. An action-at-a-distance long-range attractive type force, pulling every speck of matter towards the other. From the Newtonian point of view, an apple and the Earth simultaneously attract each other in proportion to their masses. But the Earth being more massive, exerts a stronger pull on the apple. This explains why we always see an apple fall towards the Earth but not the reverse, even though the latter is equally true. All in all, siding with the Newtonian view, gravity is an invisible force inherent in matter, by virtue of which they mutually pull each other. But in Einstein's vision, the same scenario turned out to be quite different. He discovered that gravity is not a force at all. What he later said to be ''the happiest thought in my life'' reads as follows:

The young physicist was sitting at his desk in the patent office at Bern when this thought occurred to him: If a person fell freely from the roof of his house, he would not feel his own weight. While he was falling, it would be - at least in his immediate surroundings - as if there is no gravity.  

Einstein was thinking about weightlessness. We all know weight is the measure of the relative strength of gravity. The stronger the pull, the more massive you weigh. This is why astronauts weigh less on the moon, for it has only one-sixth of Earth's gravitational pull. When a body is in free fall, it experiences no reaction force and hence no weight. If we try imagining a scenario where we have a person falling with his eyes closed and he can not see the surrounding, then he will not be able to tell whether he is falling at all. At the end of the (above) video, Brian tells us that if we remove the surroundings and fall simultaneously with the feather and the ball, we will not register them falling. According to GR, from our perspective, i.e., in our local frame, they would appear to be perfectly still. In a similar fashion, we can think of a situation where some brave individual decides to leap from a tall building with an apple in his hand. Halfway through his fall, he suddenly feels the urge to let go of the apple and see what happens. Since they are both attracted by the same force of gravity and hence the same acceleration, the apple would not even leave his hand when he let go. To an outside observer, both the apple and the man fall simultaneously. But with respect to his own frame, if the man could not see his surroundings except the apple, he would unmistakably conclude that there is no gravity because that apple did not fall from his hand. Objects can not fall in the absence of gravity. And motion does not arise unless we compare it to some reference frame/object. So, if one of us were suddenly teleported to deep space where there are no stars in his field of view, he would have no sense of position and hence no conception of up or down. Similarly, if a person suddenly finds himself in free fall, without any reference point to compare his falling motion, he would not even know if he is falling. He would feel being in perfectly empty space. 

Einstein's trouble began with a simple thought experiment in which he imagined himself in an elevator. Einstein thought that if the elevator suddenly snapped, it would immediately plummet downwards by the action of Earth's gravity, and the whole system would be perfectly weightless. He will not feel his weight as if there is no gravity. Now, instead of falling freely, if the elevator were accelerating in deep space, exactly at the rate of 9.8 m/s², i.e., the same as Earth's gravitational acceleration, he would be glued to the floor of the elevator. It would be as if he never left the Earth. The acceleration would mimic the effects of gravity as if acceleration is gravity. Therefore Einstein asked himself, if gravity is not a force, then what is it? And this became the cornerstone of GR.  

Apples fall because they are not falling at all. 
Image Credits: There goes my genuine sketch of Mr Einstein

Unfortunately, in this article, this is as far as I can go with general relativity. All will be revealed in good time. But until then, let us stick to Sir Newton. 

In his Principia: The Mathematical Principles of Natural Philosophy, Newton's three laws of motion or axioms as he writes are as follows: 

• Law I. Every body preserves in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon.  
• Law II.  The alteration of motion is ever proportional to the motive forces impressed; and is in the direction of the right line in which that force is impressed.
• Law III. To every action there is always opposed an equal reaction: or the mutual actions of the two bodies upon each other are always equal, and directed to contrary parts. 

In the standard textbook language, they are usually written as follows: 

• The First Law: Every material body (or a particle) continues to maintain its state of rest or uniform motion in a straight line unless it is compelled to change its state (of rest or uniform motion) by some external force(s) acting on it. 
• The Second Law: The time rate of change of momentum of a body (particle), i.e., the acceleration produced in the said body under an applied external force is directly proportional to the force and is in the direction of the force applied.     
• The Third Law: To every action, there is an equal and opposite reaction or if two bodies exert equal forces on each other, then the mutual action, i.e., the force between the said bodies will have the same magnitude but opposite directions.

The advanced reader must already be aware that Newton never goes into explicitly defining the fundamental concepts such as space, time, motion and force. They can be intuitively grasped. The reader must also be aware of the dozen definitions of force that are in the literature. Anyhow, we need to start somewhere. And the best practice is to start with the F = ma law. This expression is a modern version of Newton's original idea. When a force acts on a body having a mass m, the former imparts momentum to the latter, which causes the body to move. This momentum, as we know, is a product of the body's mass times instantaneous velocity, i.e.,

Equation 1.0

Taking the time derivative of momentum we have, 

Or for a constant mass system, i.e., when m does not vary with time, we have,  

So that we may finally have, 

Equation 1.1

The F = ma law (dropping off the vectors), as you can well see, almost interchangeably relates force, mass and acceleration. It tells us how objects move in the Newtonian universe. If we put the acceleration term to the left-hand side, we get a definition of mass in terms of force and acceleration, as m = F/a. Similarly, if we put m to the left-hand side, we get acceleration in terms of force and mass as a = F/m. The keen reader might ask whether this is a good practice, for it does not explicitly define mass or force, per se. And that would be a perfectly valid question. However, since our sole intention is to determine why all objects fall and experience the same gravitational acceleration, we can step aside from giving too much attention to the meaning of force and look to a particular definition of mass. 

Mass, when defined in terms of the force acting on a body and the resulting acceleration it experiences, is known as inertial mass. Newton's first law tells us of an important property inherent in all material bodies - inertia. In his own words, as Definition III, Newton writes - The vis insita, or innate force of matter, is a power of resisting, by which every body, as much as in it lies, endeavours to persevere in its present state, whether it be of rest, or of moving uniformly in a right line. He further continues,  ''... this vis insita, may, by a most significant name, be called vis inertiæ, or the force of inactivity''. However, the force of inertia does not manifest without reference to an external force. When an external force impresses upon a stationary body, the tendency of the latter to remain at rest or, if it is in motion, then the tendency to maintain that state under the influence of the said external force is solely due to its inertia. In the first case, this inertia is referred to as the body's resistance to motion, while in the latter case, the same is its impulse of motion. Anyhow, Newton remarks, ''... but motion and rest, as commonly conceived, are only relatively distinguished...''.  

 Mass and weight are fundamentally different. When we speak of mass, we mean the net quantity of matter in a body, whereas weight, on the other hand, is the amount of downward gravitational force acting on the same. The mass of an object is a constant quantity unless we think of relativistic or quantum effects. Weight, on the contrary, varies in proportion to the strength of the gravitational field. In spite of this fact, in our daily lives, we recklessly interchange mass and weight. When we say a person weighs 75 kg, we actually mean that his mass is 75 kg, i.e., he has ''75 kg'' of ''material'' in him. If we are to calculate his weight, i.e., the net downward force of gravity acting on him, then we have to multiply his mass (75 kg) with the gravitational acceleration (9.8 m/s²) to arrive at 735 Newton. This quantity, ''735 Newton'' is his actual weight. Then why do we say that the man weighs 75 kg? Well, it is an interesting question, which I will discuss in another article. His mass does not change no matter where he is in the entire universe. But his weight differs depending on what planet he is on. The stronger the gravity, the heavier he becomes and likewise, the lesser the pull, the lighter he weighs. 

Mass is a measure of inertia. The more the mass, the greater its inertia and consequently, an equally large force must be applied if we are to set it into motion. Suppose the same force F acts on two bodies having inertial masses m₁ and m₂ to produce accelerations a₁ and a₂, respectively. This situation occurs when the masses m₁ and m₂ collide elastically, whereby the force exerted on m₁ by m₂ is F₁₂, and that on m₂ by m₁ is given by F₂₁ (indeed they are vector forces). According to the third law of equal action and opposite reaction, we can equate F₁₂ and F₂₁, and thus, from the F = ma law, we can write 

Equation 1.2

Hence,  

If m₂ is known beforehand or considering it as unit mass, we can effortlessly calculate m₁ from the accelerations a₁ and a₂, provided they are non-zero, including the forces themselves. The F = ma law allows us to set up a useful convention, where if a body of mass m, say one kilogram, accelerates at the rate of one metre per second per second, i.e., roughly about one-tenth of Earth's gravitational acceleration, then the body experiences a force of one Newton. Therefore, in the general sense, if a force F acts on a body having inertial mass mᵢ, the acceleration produced in the said body can be expressed as

Equation 1.3 

Let us return to Newton's law of gravitation. The gravitational force acting between two (point) masses m₁ and m₂ at a separation of some distance r is given by 

Equation 2.0

In the above expression, G is the universal gravitational constant, and the negative sign indicates the attractive nature of the gravitational force. Rewriting the above equation in terms of the mass of a planet, Mg, and some object, mg, we have

Equation 2.1

Here Re is the Earth's radius. Note that we have used a different subscript for the masses, whose purpose will be evident in a tick. Since the acceleration due to gravity for any typical planet or the like is given by 

Therefore, from equation 2.1 we have 

Equation 2.2 

Or we can write  

Equation 2.2*

where we have dropped off the negative sign for obvious reasons. 

Thus we arrive at a new definition of mass, the so-called gravitational mass, given in terms of the force of gravity experienced by an object in a uniform gravitational field. Now, if we equate equations 1.3 and 2.2*, we get

Equation 2.3

 or 

Equation 2.3*

When Galileo investigated the nature of free fall, using inclined planes, water clocks and some necessary thought experiments, he discovered that all objects, irrespective of their constituent masses, fall with the same acceleration, i.e., 9.8m/s². If we look at equation 2.3*, we understand that the acceleration of a freely falling body will be equal to the gravitational acceleration only and only if the mg/mi term turns out to be unity, i.e., when 

The above equality between inertial and gravitational masses explains why all objects, irrespective of their mass, shape, size and internal constitution, fall at the same rate (obviously when there is no air). As long as the equality holds, it does not matter whether we throw a T. Rex or an excavator; they will fall at the same rate. The true essence of the Principle of Equivalence, which is also referred to as the Universality of Free Fall or the Galilean Equivalence Principle, is that there is no distinction between inertial and gravitational masses, there is no difference between accelerated frames and static gravitational fields. So much so if a typical observer is accelerating at 9.8 m/s² in an isolated spacecraft, he will not be able to tell if he is on Earth, for acceleration gives rise to the same gravitational effects on Earth.  

 In GR, we find a more fundamental version of the Equivalence Principle known as the Strong Equivalence Principle. The equivalence principle is deep. It is one of the most remarkable discoveries in theoretical physics, for it reveals the true nature of gravity. So much so, since gravity is all about space, time, matter and energy, we can begin to understand that it is one of the core principles of our universe. Since the date of its discovery, the validity of the equivalence principle has been subjected to rigorous tests of which the Eötvös experiment deserves special mention. In all these years, no significant deviation ''of concern'' has been observed. Had the equivalence principle did not hold true, different objects would fall at different rates. 

This article has run its course. But do not presume that it is time to draw the curtains yet. The equivalence principle is not about falling dinosaurs and Einsteins. Before signing off, I must say, if you have read this far, believe me, we have only scratched the surface.