Total Energy and Rest Energy
Total Energy and Rest Energy
The first postulate of relativity states that the laws of physics are the same in all inertial frames. Einstein showed that the law of conservation of energy is valid relativistically, if we define energy to include a relativistic factor.
Total Energy
Total energy \(E\) is defined to be
\(E={\mathrm{\gamma mc}}^{2},\)
where \(m\) is mass, \(c\) is the speed of light, \(\gamma =\cfrac{1}{\sqrt{1-\cfrac{{v}^{2}}{{c}^{2}}}}\), and \(v\) is the velocity of the mass relative to an observer. There are many aspects of the total energy \(E\) that we will discuss—among them are how kinetic and potential energies are included in \(E\), and how \(E\) is related to relativistic momentum. But first, note that at rest, total energy is not zero. Rather, when \(v=0\), we have \(\gamma =1\), and an object has rest energy.
Rest Energy
Rest energy is
\({E}_{0}={\mathrm{mc}}^{2}.\)
This is the correct form of Einstein’s most famous equation, which for the first time showed that energy is related to the mass of an object at rest. For example, if energy is stored in the object, its rest mass increases. This also implies that mass can be destroyed to release energy. The implications of these first two equations regarding relativistic energy are so broad that they were not completely recognized for some years after Einstein published them in 1907, nor was the experimental proof that they are correct widely recognized at first. Einstein, it should be noted, did understand and describe the meanings and implications of his theory.
Example: Calculating Rest Energy: Rest Energy is Very Large
Calculate the rest energy of a 1.00-g mass.
Strategy
One gram is a small mass—less than half the mass of a penny. We can multiply this mass, in SI units, by the speed of light squared to find the equivalent rest energy.
Solution
- Identify the knowns. \(m=1\text{.00}×{\text{10}}^{-3}\phantom{\rule{0.25em}{0ex}}\text{kg}\); \(c=3\text{.}\text{00}×{\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{m/s}\)
- Identify the unknown. \({E}_{0}\)
- Choose the appropriate equation. \({E}_{0}={\mathrm{mc}}^{2}\)
- Plug the knowns into the equation.
\(\begin{array}{lll}{E}_{0}& =& {\mathrm{mc}}^{2}=(1.00×{\text{10}}^{-3}\phantom{\rule{0.25em}{0ex}}\text{kg})(3.00×{\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{m/s}{)}^{2}\\ & =& 9.00×{\text{10}}^{\text{13}}\phantom{\rule{0.25em}{0ex}}\text{kg}\cdot {\text{m}}^{2}{\text{/s}}^{2}\end{array}\)
- Convert units.
Noting that \(1\phantom{\rule{0.25em}{0ex}}\text{kg}\cdot {\text{m}}^{2}{\text{/s}}^{2}=1 J\), we see the rest mass energy is
\({E}_{0}=9\text{.}\text{00}×{\text{10}}^{\text{13}}\phantom{\rule{0.25em}{0ex}}\text{J}.\)
Discussion
This is an enormous amount of energy for a 1.00-g mass. We do not notice this energy, because it is generally not available. Rest energy is large because the speed of light \(c\) is a large number and \({c}^{2}\) is a very large number, so that \({\mathrm{mc}}^{2}\) is huge for any macroscopic mass. The \(9\text{.}\text{00}×{\text{10}}^{\text{13}}\phantom{\rule{0.25em}{0ex}}\text{J}\) rest mass energy for 1.00 g is about twice the energy released by the Hiroshima atomic bomb and about 10,000 times the kinetic energy of a large aircraft carrier. If a way can be found to convert rest mass energy into some other form (and all forms of energy can be converted into one another), then huge amounts of energy can be obtained from the destruction of mass.
Today, the practical applications of the conversion of mass into another form of energy, such as in nuclear weapons and nuclear power plants, are well known. But examples also existed when Einstein first proposed the correct form of relativistic energy, and he did describe some of them. Nuclear radiation had been discovered in the previous decade, and it had been a mystery as to where its energy originated. The explanation was that, in certain nuclear processes, a small amount of mass is destroyed and energy is released and carried by nuclear radiation. But the amount of mass destroyed is so small that it is difficult to detect that any is missing. Although Einstein proposed this as the source of energy in the radioactive salts then being studied, it was many years before there was broad recognition that mass could be and, in fact, commonly is converted to energy. (See this figure.)
The Sun (a) and the Susquehanna Steam Electric Station (b) both convert mass into energy—the Sun via nuclear fusion, the electric station via nuclear fission. (credits: (a) NASA/Goddard Space Flight Center, Scientific Visualization Studio; (b) U.S. government)
Because of the relationship of rest energy to mass, we now consider mass to be a form of energy rather than something separate. There had not even been a hint of this prior to Einstein’s work. Such conversion is now known to be the source of the Sun’s energy, the energy of nuclear decay, and even the source of energy keeping Earth’s interior hot.
This lesson is part of:
Special Relativity