Can we make objects of zero mass?

Gravity is an attractive force. This fact has surprising consequences, as we will see below.

Positive and negative energies:    Einstein taught us that a mass \( m \) has an intrinsic energy

\[ E=mc^2 \]

Now suppose we place this mass \( m \) near a heavy mass \( M \):

What energy should we now attribute to \( m \)? The gravitational effect of the heavy mass \( M \) leads to a potential energy (PE) for the mass \( m \). Let us use the Newtonian expression for this PE: \[ PE=-{GMm\over r} \] Then the total energy of \( m \), which we call \( E_m \), will be the sum of the intrinsic energy and the potential energy \[ E_m=mc^2-\frac{GMm}{r} \] This is in fact correct for large values of \( r \) , where the gravitational field is weak. At smaller values of \( r \), the gravitational field is strong, and a proper treatment should use the general theory of relativity. But suppose for a moment that we trust the above Newtonian expression even for small \( r \). Then at a critical value of \( r \) given by \[ r_0=\frac {GM}{c^2} \] we see that \( E_m \) vanishes . If we reduce r even further, then \( E_m \) becomes negative .

Implications of a negative \( E_m \) :    The above observation has an interesting implication. The heavy mass \( M \) has an energy of its own, given by \( E=Mc^2 \). If we add in the small mass \( m \) at a point that is sufficiently close, then \( m \) brings in negative energy. So the overall energy of the \( M \) and \( m \) is less than the energy of \( M \) alone. This is strange, because adding something to \( M \) lowered the overall energy.

As we noted above, the correct way to deal with strong gravitational fields is to use the general theory of relativity. But even when we analyze the situation using general relativity, the conclusions remain essentially the same. All that changes is that the critical radius turns out to be larger by a factor of 2 \[ r_0=\frac {2GM}{c^2} \] The critical radius \( r_0 \) where the energy of \( m \) changes sign is called the horizon radius. The region inside this critical radius is called a black hole.

Making massless objects :     We can now imagine putting several small masses \( m \) near \( M \), thus reducing the overall energy more and more. There seems to be nothing stopping us from bringing the overall energy close to zero, or perhaps even all the way to zero.

Note that as the overall energy is decreased, we also reduce the size of the horizon. This happens because energy and mass are related by \( E=mc^2 \). So as the overall energy is lowered, the effective mass becomes lower. From the formula for the horizon radius \( r_0 \), we see that the horizon radius become smaller.

Summary :     Gravity is an attractive force. This means that we can extract energy by bringing one mass near another mass. If we place a small mass \( m \) sufficiently close to a big mass \( M \), then the small mass contributes a net negative energy. Then the mass of the combined system \( M \) and \( m \) is lower than the mass of \( M \) alone. By using many such small masses \( m \), it appears that we can make an object whose total mass is close to zero, or perhaps even zero.

This observation is curious by itself. But as we will see later, more is true: these objects with small mass pose a deep problem for physics. To understand the problem, we turn to the next question: how many such objects are there?