Relativistic Energy
The National Spherical Torus Experiment (NSTX) has a fusion reactor in which hydrogen isotopes undergo fusion to produce helium. In this process, a relatively small mass of fuel is converted into a large amount of energy. ...
The National Spherical Torus Experiment (NSTX) has a fusion reactor in which hydrogen isotopes undergo fusion to produce helium. In this process, a relatively small mass of fuel is converted into a large amount of energy. (credit: Princeton Plasma Physics Laboratory)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.
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 E size 12{E} {} is defined to be
where m size 12{m} {} is mass, c size 12{c} {} is the speed of light, γ=11−v2c2 size 12{γ= { {1} over { sqrt {1 - { {v rSup { size 8{2} } } over {c rSup { size 8{2} } } } } } } } {}, and v size 12{v} {} is the velocity of the mass relative to an observer. There are many aspects of the total energy E size 12{E} {} that we will discuss—among them are how kinetic and potential energies are included in E size 12{E} {}, and how E size 12{E} {} is related to relativistic momentum. But first, note that at rest, total energy is not zero. Rather, when v=0 size 12{v=0} {}, we have γ=1 size 12{γ=1} {}, and an object has rest energy.
Rest energy is
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.
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=1.00×10−3kg size 12{m=1 "." "00" times "10" rSup { size 8{ - 3} } `"kg"} {}; c=3.00×108m/s size 12{c=3 "." "00" times "10" rSup { size 8{8} } `"m/s"} {}
- Identify the unknown. E0 size 12{E rSub { size 8{0} } } {}
- Choose the appropriate equation. E0=mc2 size 12{E rSub { size 8{0} } = ital "mc" rSup { size 8{2} } } {}
- Plug the knowns into the equation.
E 0 = mc 2 = ( 1.00 × 10 − 3 kg ) ( 3.00 × 10 8 m/s ) 2 = 9.00 × 10 13 kg ⋅ m 2 /s 2
- Convert units.
Noting that 1kg⋅m2/s2=1 J size 12{1`"kg" cdot m rSup { size 8{2} } "/s" rSup { size 8{2} } =1`J} {}, we see the rest mass energy is
E0=9.00×1013J. size 12{E rSub { size 8{0} } =9 "." "00" times "10" rSup { size 8{"13"} } `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 size 12{c} {} is a large number and c2 size 12{c rSup { size 8{2} } } {} is a very large number, so that mc2 size 12{ ital "mc" rSup { size 8{2} } } {} is huge for any macroscopic mass. The 9.00×1013J size 12{9 "." "00" times "10" rSup { size 8{"13"} } `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 [link].)
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.
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.
A car battery is rated to be able to move 600 ampere-hours (A·h) of charge at 12.0 V. (a) Calculate the increase in rest mass of such a battery when it is taken from being fully depleted to being fully charged. (b) What percent increase is this, given the battery’s mass is 20.0 kg?
Strategy
In part (a), we first must find the energy stored in the battery, which equals what the battery can supply in the form of electrical potential energy. Since PEelec=qV size 12{"PE" rSub { size 8{"elec"} } = ital "qV"} {}, we have to calculate the charge q size 12{q} {} in 600A·h, which is the product of the current I and the time t size 12{t} {}. We then multiply the result by 12.0 V. We can then calculate the battery’s increase in mass using ΔE=PEelec=(Δm)c2 size 12{ΔE="PE" rSub { size 8{"elec"} } = ( Δm ) c rSup { size 8{2} } } {}. Part (b) is a simple ratio converted to a percentage.
Solution for (a)
- Identify the knowns. I⋅t=600 A⋅h; V=12.0V size 12{V="12" "." 0`V} {}; c=3.00×108m/s
- Identify the unknown. Δm size 12{m} {}
- Choose the appropriate equation. PEelec=(Δm)c2
- Rearrange the equation to solve for the unknown. Δm=PEelecc2 size 12{m= { {"PE" rSub { size 8{"elec"} } } over {c rSup { size 8{2} } } } } {}
- Plug the knowns into the equation.
Δm = PE elec c 2 = qV c 2 = ( I t ) V c 2 = ( 600 A ⋅ h ) ( 12.0 V ) ( 3.00 × 10 8 ) 2 .
Write amperes A as coulombs per second (C/s), and convert hours to seconds.
Δm = ( 600 C/s ⋅ h 3600 s 1 h ( 12.0 J/C ) ( 3.00 × 10 8 m/s ) 2 = ( 2.16 × 10 6 C ) ( 12.0 J/C ) ( 3.00 × 10 8 m/s ) 2Using the conversion 1kg⋅m2/s2=1J, we can write the mass as
Δm = 2.88 × 10 − 10 kg .
Solution for (b)
- Identify the knowns. Δm=2.88×10−10kg; m=20.0 kg
- Identify the unknown. % change
- Choose the appropriate equation. % increase=Δmm×100% size 12{%" increase"= { {Δm} over {m} } times "100"%} {}
- Plug the knowns into the equation.
% increase = Δm m × 100% = 2.88 × 10 − 10 kg 20.0 kg × 100% = 1.44 × 10 − 9 % .
Discussion
Both the actual increase in mass and the percent increase are very small, since energy is divided by c2 size 12{c rSup { size 8{2} } } {}, a very large number. We would have to be able to measure the mass of the battery to a precision of a billionth of a percent, or 1 part in 1011, to notice this increase. It is no wonder that the mass variation is not readily observed. In fact, this change in mass is so small that we may question how you could verify it is real. The answer is found in nuclear processes in which the percentage of mass destroyed is large enough to be measured. The mass of the fuel of a nuclear reactor, for example, is measurably smaller when its energy has been used. In that case, stored energy has been released (converted mostly to heat and electricity) and the rest mass has decreased. This is also the case when you use the energy stored in a battery, except that the stored energy is much greater in nuclear processes, making the change in mass measurable in practice as well as in theory.
Kinetic energy is energy of motion. Classically, kinetic energy has the familiar expression 12mv2 size 12{ { {1} over {2} } ital "mv" rSup { size 8{2} } } {}. 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 E0=mc2. The work increases this to the total energy E=γmc2. Thus,
Relativistically, we have Wnet=KErel size 12{W="KE" rSub { size 8{"rel"} } } {}.
Relativistic kinetic energy is
When motionless, we have v=0 size 12{v=0} {} and
so that KErel=0 size 12{"KE" rSub { size 8{"rel"} } =0} {} 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 12mv2 size 12{ { {1} over {2} } ital "mv" rSup { size 8{2} } } {}. To show that the classical expression for kinetic energy is obtained at low velocities, we note that the binomial expansion for γ size 12{γ} {} 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 γ size 12{γ} {} 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 v<<c size 12{v"<<"c} {}.
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 γ size 12{γ} {} becomes infinite as v size 12{v} {} approaches c size 12{c} {}, so that KErel also becomes infinite as the velocity approaches the speed of light. (See [link].) An infinite amount of work (and, hence, an infinite amount of energy input) is required to accelerate a mass to the speed of light.
No object with mass can attain 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 c size 12{c} {} always add to less than c size 12{c} {}. Both the relativistic form for kinetic energy and the ultimate speed limit being c size 12{c} {} 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.
This graph of KErel size 12{"KE" rSub { size 8{"rel"} } } {} versus velocity shows how kinetic energy approaches infinity as velocity approaches the speed of light. It is thus not possible for an object having mass to reach the speed of light. Also shown is KEclass size 12{"KE" rSub { size 8{"class"} } } {}, the classical kinetic energy, which is similar to relativistic kinetic energy at low velocities. Note that much more energy is required to reach high velocities than predicted classically.
An electron has a velocity v=0.990c size 12{v=0 "." "990"c} {}. (a) Calculate the kinetic energy in MeV of the electron. (b) Compare this with the classical value for kinetic energy at this velocity. (The mass of an electron is 9.11×10−31 kg size 12{9 "." "11" times "10" rSup { size 8{ - "31"} } " kg"} {}.)
Strategy
The expression for relativistic kinetic energy is always correct, but for (a) it must be used since the velocity is highly relativistic (close to c size 12{c} {}). First, we will calculate the relativistic factor γ size 12{γ} {}, and then use it to determine the relativistic kinetic energy. For (b), we will calculate the classical kinetic energy (which would be close to the relativistic value if v size 12{v} {} were less than a few percent of c size 12{c} {}) and see that it is not the same.
Solution for (a)
- Identify the knowns. v=0.990c size 12{v=0 "." "990"c} {}; m=9.11×10−31kg size 12{m=9 "." "11" times "10" rSup { size 8{ - "31"} } `"kg"} {}
- Identify the unknown. KErel size 12{"KE" rSub { size 8{"rel"} } } {}
- Choose the appropriate equation. KErel=γ−1mc2 size 12{"KE" rSub { size 8{"rel"} } = left (γ - 1 right ) ital "mc" rSup { size 8{2} } } {}
- Plug the knowns into the equation.
First calculate γ size 12{γ} {}. We will carry extra digits because this is an intermediate calculation.
γ = 1 1 − v 2 c 2 = 1 1 − ( 0 . 990 c ) 2 c 2 = 1 1 − ( 0 . 990 ) 2 = 7 . 0888 alignl { stack { size 12{γ= { {1} over { sqrt {1 - { {v rSup { size 8{2} } } over {c rSup { size 8{2} } } } } } } } {} # = { {1} over { sqrt {1 - { { ( 0 "." "990" ital " c" ) rSup { size 8{2} } } over {c rSup { size 8{2} } } } } } } {} # - { {1} over { sqrt {1 - ( 0 "." "990" ) rSup { size 8{2} } } } } {} # =7 "." "0888" {} } } {}Next, we use this value to calculate the kinetic energy.
KE rel = ( γ − 1 ) mc 2 = ( 7.0888 − 1 ) ( 9.11 × 10 – 31 kg ) ( 3.00 × 10 8 m/s ) 2 = 4.99 × 10 –13 J - Convert units.
KE rel = ( 4.99 × 10 –13 J ) 1 MeV 1.60 × 10 – 13 J = 3.12 MeV alignl { stack { size 12{"KE" rSub { size 8{"rel"} } = ( 4 "." "99" times "10" rSup { size 8{"-13"} } " J" ) left ( { {"1 MeV"} over {1 "." "60" times "10" rSup { size 8{ - "13"} } " J"} } right )} {} # =3 "." "12"" MeV" {} } } {}
Solution for (b)
- List the knowns. v=0.990c size 12{v=0 "." "990"c} {}; m=9.11×10−31kg
- List the unknown. KEclass
- Choose the appropriate equation. KEclass=12mv2
- Plug the knowns into the equation.
KE class = 1 2 mv 2 = 1 2 ( 9.00 × 10 – 31 kg ) ( 0.990 ) 2 ( 3.00 × 10 8 m/s ) 2 = 4.02 × 10 – 14 J
- Convert units.
KE class = 4.02 × 10 – 14 J 1 MeV 1.60 × 10 – 13 J = 0.251 MeV
Discussion
As might be expected, since the velocity is 99.0% of the speed of light, the classical kinetic energy is significantly off from the correct relativistic value. Note also that the classical value is much smaller than the relativistic value. In fact, KErel/KEclass=12.4 size 12{"KE" rSub { size 8{"rel"} } "/KE" rSub { size 8{"class"} } ="12" "." 4} {} here. This is some indication of how difficult it is to get a mass moving close to the speed of light. Much more energy is required than predicted classically. Some people interpret this extra energy as going into increasing the mass of the system, but, as discussed in Relativistic Momentum, this cannot be verified unambiguously. What is certain is that ever-increasing amounts of energy are needed to get the velocity of a mass a little closer to that of light. An energy of 3 MeV is a very small amount for an electron, and it can be achieved with present-day particle accelerators. SLAC, for example, can accelerate electrons to over 50×109eV=50,000 MeV size 12{"50" times "10" rSup { size 8{9} } "eV"="50,000"`"MeV"} {}.
Is there any point in getting v size 12{v} {} a little closer to c than 99.0% or 99.9%? The answer is yes. We learn a great deal by doing this. The energy that goes into a high-velocity mass can be converted to any other form, including into entirely new masses. (See [link].) Most of what we know about the substructure of matter and the collection of exotic short-lived particles in nature has been learned this way. Particles are accelerated to extremely relativistic energies and made to collide with other particles, producing totally new species of particles. Patterns in the characteristics of these previously unknown particles hint at a basic substructure for all matter. These particles and some of their characteristics will be covered in Particle Physics.
The Fermi National Accelerator Laboratory, near Batavia, Illinois, was a subatomic particle collider that accelerated protons and antiprotons to attain energies up to 1 Tev (a trillion electronvolts). The circular ponds near the rings were built to dissipate waste heat. This accelerator was shut down in September 2011. (credit: Fermilab, Reidar Hahn)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 E size 12{E} {} is the relativistic total energy and p size 12{p} {} 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 mc2 (so this equation is consistent with the discussion of rest energy above). However, as the mass is accelerated, its momentum p increases, thus increasing the total energy. At sufficiently high velocities, the rest energy term (mc2)2 becomes negligible compared with the momentum term (pc)2; thus, E=pc at extremely relativistic velocities.
If we consider momentum p size 12{p} {} to be distinct from mass, we can determine the implications of the equation E2=(pc)2+(mc2)2, size 12{E rSup { size 8{2} } = ( ital "pc" ) rSup { size 8{2} } + ( ital "mc" ) rSup { size 8{2} } } {} for a particle that has no mass. If we take m size 12{m} {} to be zero in this equation, then E=pc size 12{E= ital "pc"} {}, or p=E/c size 12{p=E/c} {}. 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 c size 12{c} {} and only at speed c size 12{c} {}. While it is beyond the scope of this text to examine the relationship in the equation E2=(pc)2+(mc2)2, size 12{E rSup { size 8{2} } = ( ital "pc" ) rSup { size 8{2} } + ( ital "mc" ) rSup { size 8{2} } } {} in detail, we can see that the relationship has important implications in special relativity.
- Examine the situation to determine that it is necessary to use relativity. Relativistic effects are related to γ=11−v2c2 size 12{γ= { {1} over { sqrt {1 - { {v rSup { size 8{2} } } over {c rSup { size 8{2} } } } } } } } {}, the quantitative relativistic factor. If γ size 12{γ} {} is very close to 1, then relativistic effects are small and differ very little from the usually easier classical calculations.
- Identify exactly what needs to be determined in the problem (identify the unknowns).
- Make a list of what is given or can be inferred from the problem as stated (identify the knowns). Look in particular for information on relative velocity v size 12{v} {}.
- Make certain you understand the conceptual aspects of the problem before making any calculations. Decide, for example, which observer sees time dilated or length contracted before plugging into equations. If you have thought about who sees what, who is moving with the event being observed, who sees proper time, and so on, you will find it much easier to determine if your calculation is reasonable.
- Determine the primary type of calculation to be done to find the unknowns identified above. You will find the section summary helpful in determining whether a length contraction, relativistic kinetic energy, or some other concept is involved.
- Do not round off during the calculation. As noted in the text, you must often perform your calculations to many digits to see the desired effect. You may round off at the very end of the problem, but do not use a rounded number in a subsequent calculation.
- Check the answer to see if it is reasonable: Does it make sense? This may be more difficult for relativity, since we do not encounter it directly. But you can look for velocities greater than c size 12{c} {} or relativistic effects that are in the wrong direction (such as a time contraction where a dilation was expected).
A photon decays into an electron-positron pair. What is the kinetic energy of the electron if its speed is 0.992c size 12{c} {}?
- Relativistic energy is conserved as long as we define it to include the possibility of mass changing to energy.
- Total Energy is defined as: E=γmc2, where γ=11−v2c2.
- Rest energy is E0=mc2, 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.
- 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.
- The relativistic work-energy theorem is Wnet=E−E0=γmc2−mc2=γ−1mc2.
- Relativistically, Wnet=KErel , where KErel is the relativistic kinetic energy.
- Relativistic kinetic energy is KErel=γ−1mc2, where γ=11−v2c2. At low velocities, relativistic kinetic energy reduces to classical kinetic energy.
- 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.
- The equation E2=(pc)2+(mc2)2 relates the relativistic total energy E and the relativistic momentum p. At extremely high velocities, the rest energy mc2 becomes negligible, and E=pc.
How are the classical laws of conservation of energy and conservation of mass modified by modern relativity?
What happens to the mass of water in a pot when it cools, assuming no molecules escape or are added? Is this observable in practice? Explain.
Consider a thought experiment. You place an expanded balloon of air on weighing scales outside in the early morning. The balloon stays on the scales and you are able to measure changes in its mass. Does the mass of the balloon change as the day progresses? Discuss the difficulties in carrying out this experiment.
The mass of the fuel in a nuclear reactor decreases by an observable amount as it puts out energy. Is the same true for the coal and oxygen combined in a conventional power plant? If so, is this observable in practice for the coal and oxygen? Explain.
We know that the velocity of an object with mass has an upper limit of c size 12{c} {}. Is there an upper limit on its momentum? Its energy? Explain.
Given the fact that light travels at c size 12{c} {}, can it have mass? Explain.
If you use an Earth-based telescope to project a laser beam onto the Moon, you can move the spot across the Moon’s surface at a velocity greater than the speed of light. Does this violate modern relativity? (Note that light is being sent from the Earth to the Moon, not across the surface of the Moon.)
What is the rest energy of an electron, given its mass is 9.11×10−31 kg? Give your answer in joules and MeV.
8.20 × 10 − 14 J
0.512 MeV
Find the rest energy in joules and MeV of a proton, given its mass is 1.67×10−27 kg.
If the rest energies of a proton and a neutron (the two constituents of nuclei) are 938.3 and 939.6 MeV respectively, what is the difference in their masses in kilograms?
2 . 3 × 10 − 30 kg size 12{2 "." 3 times "10" rSup { size 8{ - "30"} } `"kg"} {}
The Big Bang that began the universe is estimated to have released 1068 J size 12{10 rSup { size 8{"68"} } " J"} {} of energy. How many stars could half this energy create, assuming the average star’s mass is 4.00×1030 kg size 12{4 "." "00" times "10" rSup { size 8{"30"} } " kg"} {}?
A supernova explosion of a 2.00×1031 kg size 12{2 "." "00" times "10" rSup { size 8{"31"} } " kg"} {} star produces 1.00×1044 kg size 12{1 "." "00" times "10" rSup { size 8{"44"} } " kg"} {} of energy. (a) How many kilograms of mass are converted to energy in the explosion? (b) What is the ratio Δm/m size 12{Δm/m} {} of mass destroyed to the original mass of the star?
(a) 1 . 11 ×
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