Mass-Energy Equivalence

Introduction

Alright so today's lesson will be on Mass-Energy Equivalence, which is arguably one of the most outstanding ideas that have been made throughout human history. Let's begin with two questions that will warm up our minds a little bit.

The Genius and his Famous Equation

In 1905, Albert Einstein came up with the first derivation of the formula for mass-energy equivalence in his paper Does the inertia of a body depend upon its energy content?

Nice, however this wasn't the first equation that Einstein proposed (at least, it wasn't quite in this form). Let's talk a bit about the history of this idea, and how we got to the definition that we are all so familiar with today.

The History behind the Definition

Issac Newton

While our man Einstein was the one who gave the formal, and accepted, definition of mass-energy equivalence the idea goes almost 200 years before Einstein made his big move. In the year of 1717, Issac Newton pondered upon the idea that light particles and matter particles were inter-convertible. He made in inquiry in the Opticks where he asked: "Are not the gross bodies and light convertible into one another, and may not bodies receive much of their activity from the particles of light which enter their composition?"

Emmanuel Swedenborg

Next up is the swedish scientist and theologian Emmanuel Swedenborg who, in his work entitled Principia, theorized that all matter is composed of dimensionless points with motion that could be defined as "pure and total." More specifically, he explains that this motion does not have force, direction or speed, but has the potential for force, direction and speed everywhere within it.

"Birds of a feather flock together" during the 19th and 20th Centuries

Further down the line we find the greats such as J. J. Thomson (1881), Oliver Heaviside (1888), George Frederick Charles Searle (1897), Wilhelm Wien (1900), Max Abraham (1902), and Hendrik Antoon Lorentz (1904) who tried to understand how and electrostatic field affected the mass of a charged object. This idea became known as "electromagnetic mass". Our last gentleman, Lorentz, presented these two expressions in defining electromagnetic and longitudinal mass:

Suppose we defined $m_0$ as

$m_{0}=\frac{4}{3}\cdot\frac{E_{em}}{c^{2}}$

then we can find that

$m_{L}=\frac{m_{0}}{\left(\sqrt{1-\frac{v^{2}}{c^{2}}}\right)^{3}},\quad m_{T}=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$

It seems now that the ideas start to take shape as equations and evolve!

The Man Who's Conjecture became a Millenium Problem and the Gentleman Who Fixed His Problem

Going back a slight bit to the year of 1900, we find Henri Poincaré who attempted to save the "center of mass" within Lorentz's theory by defining electromagnetic radiation energy based off of an imaginary fluid which had momentum and mass.

$m_{em}=\frac{E_{em}}{c^2}$

However, this led to complications in the concept of radiation. Going foward to 1904, we find Friedrich Hasenöhrl who shows how Lorentz's expression themselves solve Poincaré's problem. How ironic!

Alright, we now have one more journey through time to go through!

The Definition and its Derivation

BEWARE: THE DERIVATION OF THE DEFINITION OF MASS-ENERGY EQUIVALENCE INVOLVES SOME CALCULUS! THOSE WHO WISH TO NOT GET THEMSELVES INVOLVED IN CALCULUS SHOULD SKIP AHEAD TO THE DEFINITION AT THE BOTTOM OF THIS SECTION!

Alright! We made it! Now we arrive in the year of 1905 where Albert Einstein presents his paper and he shows the world his great work! We find that Einstein propses the following definition for mass-energy equivalence. In his paper, einstein presents the correct derivation for the kinetic energy of particles.

$E_{k}=mc^{2}\left(\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}-1\right)$

(This is where we get our famous equation $E=mc^2$!) However, question remains: which formulation applies to bodies at rest? Einstein soon took up the challenge in his paper Does the inertia of a body depend upon its energy content?.

To solve the problem, he took two pulses of light being emitted in opposite directions which had an energy of $E_0$ before and $E_1$ after emission respectively. We see that from a moving frame, this becomes $H_0$ and $H_1$ respectively. From this, Einstein finds:

$\left(H_{0}-E_{0}\right)-\left(H_{1}-E_{1}\right)=E\left(\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}-1\right)$

He further argued that $H ? E$ only differ from a kinetic energy $K$ due to an additive constant, which gives

$K_{0}-K_{1}=E\left(\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}-1\right)$

We perform a taylor series expansion on the right hand side that affects $\frac{v}{c}$. However, we ignore anything that is higher than the 3rd order. From this, Einstein then found that:

$K_{0}-K_{1}=\frac{E}{c^{2}}\cdot\frac{v^{2}}{2}$

With this, Einstein concluded that the emission reduces the body's mass by $\frac{E}{c^2}$, and thus we find mass-energy equivalence.

Thus the formal definition: Mass–energy equivalence is the concept that the mass of an object or system is a measure of its energy content.

Concluding Remarks

Now due to Einstein's work, we have been able to understand why when particles move at extremely high speeds, they seem to "lose" mass. This idea has been consistently used in special relativity and general relativity often in the modern physics of today (it even gets itself involved in quantum mechanics!). For example, this allowed us to understand parts of the behavior of elementary particles such as neutrinos! I hope that you understood what was going on (as this involved a lot of physics and math), and I hope that you weren't bored by the history of mass-energy equivalence!