# Special Relativity

## Relativistic Energy


### Learning Objectives

By the end of this section, you will be able to:
1. Compute total energy of a relativistic object.
2. Compute the kinetic energy of a relativistic object.
3. Describe rest energy, and explain how it can be converted to other forms.
4. Explain why massive particles cannot reach C.
A tokamak is a form of experimental fusion reactor, which can change mass to energy. Accomplishing this requires an understanding of relativistic energy. Nuclear reactors are proof of the conservation of relativistic energy.
Conservation of energy is one of the most important laws in physics. Not only does energy have many important forms, but each form can be converted to any other. We know that classically the total amount of energy in a system remains constant. Relativistically, energy is still conserved, provided its definition is altered to include the possibility of mass changing to energy, as in the reactions that occur within a nuclear reactor. Relativistic energy is intentionally defined so that it will be conserved in all inertial frames, just as is the case for relativistic momentum. As a consequence, we learn that several fundamental quantities are related in ways not known in classical physics. All of these relationships are verified by experiment and have fundamental consequences. The altered definition of energy contains some of the most fundamental and spectacular new insights into nature found in recent history.

### 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.
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.
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 .)
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.

### Stored Energy and Potential Energy

What happens to energy stored in an object at rest, such as the energy put into a battery by charging it, or the energy stored in a toy gun’s compressed spring? The energy input becomes part of the total energy of the object and, thus, increases its rest mass. All stored and potential energy becomes mass in a system. Why is it we don’t ordinarily notice this? In fact, conservation of mass (meaning total mass is constant) was one of the great laws verified by 19th-century science. Why was it not noticed to be incorrect? The following example helps answer these questions.

### Kinetic Energy and the Ultimate Speed Limit

Kinetic energy is energy of motion. Classically, kinetic energy has the familiar expression . The relativistic expression for kinetic energy is obtained from the work-energy theorem. This theorem states that the net work on a system goes into kinetic energy. If our system starts from rest, then the work-energy theorem is
Relativistically, at rest we have rest energy . The work increases this to the total energy . Thus,
Relativistically, we have .
When motionless, we have  and
so that  at rest, as expected. But the expression for relativistic kinetic energy (such as total energy and rest energy) does not look much like the classical . To show that the classical expression for kinetic energy is obtained at low velocities, we note that the binomial expansion for  at low velocities gives
A binomial expansion is a way of expressing an algebraic quantity as a sum of an infinite series of terms. In some cases, as in the limit of small velocity here, most terms are very small. Thus the expression derived for  here is not exact, but it is a very accurate approximation. Thus, at low velocities,
Entering this into the expression for relativistic kinetic energy gives
So, in fact, relativistic kinetic energy does become the same as classical kinetic energy when .
It is even more interesting to investigate what happens to kinetic energy when the velocity of an object approaches the speed of light. We know that  becomes infinite as  approaches , so that KErel also becomes infinite as the velocity approaches the speed of light. (See .) An infinite amount of work (and, hence, an infinite amount of energy input) is required to accelerate a mass to the speed of light.
So the speed of light is the ultimate speed limit for any particle having mass. All of this is consistent with the fact that velocities less than  always add to less than . Both the relativistic form for kinetic energy and the ultimate speed limit being  have been confirmed in detail in numerous experiments. No matter how much energy is put into accelerating a mass, its velocity can only approach—not reach—the speed of light.

### Relativistic Energy and Momentum

We know classically that kinetic energy and momentum are related to each other, since
Relativistically, we can obtain a relationship between energy and momentum by algebraically manipulating their definitions. This produces
where  is the relativistic total energy and  is the relativistic momentum. This relationship between relativistic energy and relativistic momentum is more complicated than the classical, but we can gain some interesting new insights by examining it. First, total energy is related to momentum and rest mass. At rest, momentum is zero, and the equation gives the total energy to be the rest energy  (so this equation is consistent with the discussion of rest energy above). However, as the mass is accelerated, its momentum  increases, thus increasing the total energy. At sufficiently high velocities, the rest energy term  becomes negligible compared with the momentum term

; thus,

 at extremely relativistic velocities.
If we consider momentum  to be distinct from mass, we can determine the implications of the equation   for a particle that has no mass. If we take  to be zero in this equation, then , or . Massless particles have this momentum. There are several massless particles found in nature, including photons (these are quanta of electromagnetic radiation). Another implication is that a massless particle must travel at speed  and only at speed . While it is beyond the scope of this text to examine the relationship in the equation  in detail, we can see that the relationship has important implications in special relativity.

### Test Prep for AP Courses


### Section Summary

1. Relativistic energy is conserved as long as we define it to include the possibility of mass changing to energy.
2. Total Energy is defined as:
, where

.
3. Rest energy is
,  meaning that mass is a form of energy. If energy is stored in an object, its mass increases. Mass can be destroyed to release energy.
4. We do not ordinarily notice the increase or decrease in mass of an object because the change in mass is so small for a large increase in energy.
5. The relativistic work-energy theorem is .
6. Relativistically,
 ,
where   is the relativistic kinetic energy.
7. Relativistic kinetic energy is , where . At low velocities, relativistic kinetic energy reduces to classical kinetic energy.
8. No object with mass can attain the speed of light because an infinite amount of work and an infinite amount of energy input is required to accelerate a mass to the speed of light.
9. The equation  relates the relativistic total energy

 and the relativistic momentum
. At extremely high velocities, the rest energy
 becomes negligible, and
.

### Conceptual Questions


### Problems & Exercises
