Mass Energy Equivalence

Albert Einstein may well be one of the most influential and certainly most famous physicists that the world has ever seen. Whilst his work on the photoelectric effect is what he earned the Nobel prize for in 1921, it's his work on Special Relativity, with its mind-bending ideas about space and time that have lingered on in popular culture. If Einstein is one of the most famous physicists of all time, his mass-energy equivalence equation \(E=mc^2\) is surely the most famous equation in all of science. Walk into any physics student's dorm room and you're bound to find a poster of Einstein and his famous equation! So what exactly was Einstein's great discovery about the nature of mass and energy, and why is a simple equation like \(E=mc^2\) so profound? Well, read on to find out exactly that...

Einstein's Mass-Energy Equivalence Postulates

Before we can get into the core of mass-energy equivalence and its consequences, we'll first need to cover some basic principles of special relativity. The special theory of relativity, special relativity for short, is one of the core theories of modern physics developed by Albert Einstein in 1905, building on the earlier work of Hendrik Lorentz and Henri Poincaré. Special relativity is largely concerned with the relationship between space and time, and the physics of objects moving at speeds close to the speed of light \(c=3\times10^8\,\mathrm{m}\,\mathrm{s}^{-1}\). Many strange and ground-breaking phenomena were predicted and later confirmed, by special relativity. For example, time dilation states that a stationary clock ticks faster than a clock that is moving relative to it. The principle of mass-energy equivalence is also one of the central ideas within special relativity.

Fig. 1 - Special relativity, and the mass-energy equivalence principle, are regarded as the crowning jewels in the work of Albert Einstein, revolutionizing how we understand space and time.

Special relativity is based upon two fundamental postulates, from which all other effects can be derived. These postulates, and by extension all of special relativity, rely on the idea of an inertial reference frame.

The postulates can be summarised as:

• Principle of relativity- The laws of physics are equivalent in all inertial reference frames. This means that, in an inertial frame, no experiment can be done to determine the absolute velocity of the frame, only the velocity relative to some other object.

• Principle of invariant light speed- Light is always propagated through a vacuum at the same finite speed \(c\) independent of the speed of the body which emits it.

Principle of Mass-Energy Equivalence

Special relativity states that the mass of a body is dependent on its velocity relative to the observer measuring it, however, we can define the mass of the body in its rest frame as the intrinsic mass of the body. Special relativity states that whilst a body has no kinetic energy in its rest frame, it still has intrinsic rest energy. For example, when two (approximately) stationary particle-anti-particle pairs annihilate, the light energy emitted is equal to the combined rest energies of the particles. The principle of mass-energy equivalence shows that this rest energy is entirely equivalent to the rest mass of the body, up to a constant.

The idea of mass and energy being equivalent is wholly new to special relativity and is absent in any form of Newtonian physics. This is an astonishingly profound piece of physics, and its simple statement belies some extraordinary consequences.

Mass-Energy Equivalence Formula

Mass energy equivalence is mathematically defined by Einstein's famous formula

\[E=mc^2,\]

with \(E\) and \(m\) being the rest energy and mass of the body respectively. From this equation, we see that the conversion constant between mass and energy is the speed of light squared \(c^2=9\times10^{16}\,\mathrm{m/s^2}\).

This is a huge number and so even very small rest masses are equivalent to huge amounts of energy. For example, nuclear weapons with only a few kilograms of fissionable material produce as much energy as tens of thousands of tonnes of regular TNT.

Fig . 2 - Huge amounts of energy can be released through the conversion of rest mass to energy such as in a nuclear fission bomb

When calculating quantities using this formula it's important to take note of which units are being used for mass and energy. In nuclear and particle physics it is more common to use the unit of mega-electron volts \(\mathrm{MeV}\) rather than Joules \(\mathrm J\).

It is also common to use units of \(\mathrm{MeV}\,\mathrm{c}^{-2}\) for mass when performing calculations with Einstein's equation. These units are equivalent to setting the speed of light \(c=1\) making energy and mass exactly equal in these units.

The mass-energy equivalence formula can be seen as a special case of Einstein's more general energy-momentum equation

\[E_r^2=(pc)^2+(m_0c^2)^2.\]

This equation gives the relativistic energy \(E_r\) of a particle with intrinsic rest mass \(m_0\), moving with a momentum \(p\) relative to the reference frame it is being observed in. So we can see the total relativistic energy of a particle is composed of a kinetic component associated with its momentum relative to an observer, as well as a component stemming from the intrinsic rest mass of the particle which is constant in all frames. If we are in the rest frame of the particle, \(p=0\) and the equation becomes Einstein's mass-energy equivalence relation,\[E_r^2=(m_0c^2)^2\Rightarrow E_r=m_0c^2.\]

Conversely, for massless particles such as photons, the particle's total energy is entirely dependent on the momentum\[E_r=pc\]

This demonstrates that there is no possible rest frame for photons, as they always travel at a velocity \(c\) no matter the reference frame.

Mass-Energy Equivalence Meaning

To get a better grasp of what this mass-energy equivalence means let's take a look at some direct physical consequences of this fact.

Nuclear Reactions and the Mass-Defect

The splitting of the atom in the 1930s may well be one of the single most influential experiments in human history. The discovery that huge amounts of energy can be generated from the process of nuclear fission kickstarted the nuclear age, with both nuclear power plants and nuclear weapons proliferating across the globe throughout the 20th century. The source of this awesome power is entirely dependent on mass-energy equivalence.

Nuclei within atoms are composed of protons and neutrons bound together by the strong nuclear force. The binding force of the strong force means that when bonded within a nucleus, the protons and neutrons have lower energy than they would do as separate particles. Hence, when protons and neutrons first form into nuclei during nuclear fusion, large amounts of energy are released. Due to mass-energy equivalence, this difference in energy, known as the nuclear binding energy, must be accounted for in the mass of the nuclei meaning the mass of a nucleus is lower than the combined masses of the constituent particles. This mass difference is known as the mass defect and can be experimentally measured.

For example, the atomic mass of a Helium atom, in atomic mass units \(\mathrm{u}\) is \(4.0015084\,\mathrm{u}\), whereas two protons and two neutrons have a total mass of \[\begin{align}2m_p+2m_n&=2\cdot1.000728\mathrm{u}+2\cdot1.008665\,\mathrm{u}\\&=4.03190\,\mathrm{u}\end{align}\]

So we see the mass defect of helium is

\[\begin{align}\Delta m&=4.03190\,\mathrm{u}-4.00150\,\mathrm{u}\\&=0.0303\,\mathrm{u}=5.03\times10^{-29}\,\mathrm{kg}\end{align}\]

This difference in mass is due to the binding energy released when the nucleus was first formed, which by the mass-energy equivalence corresponds to a loss of mass. We can calculate the binding energy of the nuclei \(\Delta E\) using the formula for mass-energy equivalence

\[\Delta E=\Delta m c^2.\]

Using the mass defect for helium we find that the binding energy for a helium atom is

\[\begin{align}\Delta E&=5.03\times10^{-29}\cdot\left(3.00\times10^8\right)^2\\&=45.3\times10^{-20}\,\mathrm{J}.\end{align}\]

It is this mass defect/binding energy that is behind the awesome power of nuclear fission reaction. Whenever fission occurs a heavy unstable nuclei splits into two smaller nuclei. The combined mass of these product nuclei is smaller than the mass of the original nucleus as the binding energy of the product nuclei is greater. The excess binding energy is released as a by-product of the reaction, this energy release is harnessed in nuclear power stations and nuclear weapons.

Fig. 3 - A diagram showing the fission of Uranium-235 into Krypton-89 and Barium-144 due to the addition of a neutron. The combined mass of the product nuclei is lower than the mass of the original nucleus.

Particle-anti particle annihilation

The existence of anti-particles was first postulated by Dirac in 1932 and experimentally confirmed shortly after. These anti-particles have almost identical properties to that of 'regular' matter particles except their charge is opposite. For example, a positron or anti-electron has the same mass and spin as an electron but a positive charge. When a particle and anti-particle pair collide they annihilate releasing energy often in the form of photons, though other particles can be produced. If the kinetic energy of the particle and anti-particle is negligible compared with their rest energies, we can think of annihilation as a process by which the total rest mass of the two particles is converted into energy given off as two photons. Hence the energy of the photons given off can be calculated from the rest masses of the particles by using Einstein's mass-energy equivalence relation.

Fig. 4 - A Feynman diagram showing an electron-positron pair annihilating one another. The positron is shown moving backward in time, with the photons produced shown as wavy lines labelled \(\gamma\).

Mass Energy Equivalence Examples

To recap let's take a look at some example calculations associated with mass-energy equivalence.