Special Relativity
93 Relativistic Momentum
Learning Objectives
- Calculate the relativistic momentum of an object moving at speeds comparable to the speed of light.
- Explain why modern physics uses rest mass rather than the older concept of relativistic mass.
- Describe how the conservation of momentum remains valid within Einstein's theory of special relativity.
- Compare classical momentum with relativistic momentum and identify the conditions under which the classical approximation is valid.
In classical mechanics, momentum is defined as the product of an object's mass and its velocity:
This simple relationship accurately describes everything from moving vehicles to sporting events because the speeds involved are much smaller than the speed of light. As we learned in the previous chapter, however, the rules governing motion change when objects travel at velocities approaching [latex]c[/latex].
One of the most important consequences of special relativity is that no object with mass can reach or exceed the speed of light. If velocity is limited in this way, an obvious question arises: How does momentum continue to increase when an object is pushed closer and closer to the speed of light?
The answer is that the classical equation for momentum is no longer sufficient. Instead, momentum must be modified by the same relativistic factor that appears throughout Einstein's theory.
Momentum is one of the most fundamental quantities in all of physics. Newton's second law is most generally expressed in terms of momentum, and the law of conservation of momentum remains one of the most powerful tools for analyzing physical systems. Earlier in this textbook, momentum conservation was used to study collisions between everyday objects. In modern physics, exactly the same conservation law allows scientists to investigate particles that are far too small to observe directly.
For example, nearly everything we know about the internal structure of atomic nuclei and elementary particles comes from analyzing high-energy collisions produced in particle accelerators. By measuring the momentum of particles before and after a collision, physicists can infer the existence and properties of particles that may exist for only tiny fractions of a second.
Healthcare Connection
Relativistic momentum is not just important in particle physics laboratories—it also plays a vital role in medicine. Modern radiation therapy systems accelerate electrons and protons to extremely high speeds before directing them toward cancerous tissue. At these velocities, the classical momentum equation no longer predicts particle motion accurately. Treatment planning software therefore relies on relativistic momentum to calculate how charged particles travel through magnetic fields and deposit energy inside the patient's body.
Einstein's first postulate states that the laws of physics must be identical in every inertial reference frame. Therefore, if momentum is conserved in one inertial frame, it must also be conserved in every other inertial frame. This requirement is satisfied only if momentum is defined using the relativistic expression given below.
Relativistic Momentum
The momentum of an object moving at relativistic speeds is
where
- [latex]m[/latex] is the object's rest mass,
- [latex]u[/latex] is its speed relative to the observer, and
- [latex]\gamma[/latex] is the Lorentz factor,
[latex]\gamma=\frac{1}{\sqrt{1-\frac{u^2}{c^2}}}.[/latex]
Notice that the symbol [latex]u[/latex] is used here for the object's velocity. This distinguishes it from the symbol [latex]v[/latex], which was used in the previous chapter to represent the relative velocity between two observers.
At everyday speeds, the Lorentz factor is extremely close to 1, so relativistic momentum becomes
which is simply the familiar classical expression. Consequently, classical mechanics remains an excellent approximation whenever an object's speed is much less than the speed of light.
Example 93.1: Relativistic Momentum of an Electron in a Medical Linear Accelerator
Problem
A medical linear accelerator (LINAC) used for radiation therapy accelerates electrons to a kinetic energy of 6.00 MeV before they strike a target to produce high-energy X-rays. Calculate the electron's relativistic momentum, and compare it with the momentum predicted by classical mechanics at the same speed.
Strategy
At this energy, the electron is moving at a substantial fraction of the speed of light, so we must use the relativistic expressions for total energy and momentum rather than the classical formula [latex]p=mu[/latex]. We first find the Lorentz factor [latex]\gamma[/latex] from the electron's total energy, then use it to find the electron's speed and its relativistic momentum.
Solution
- Identify the known quantities.
- Kinetic energy: [latex]KE=6.00\ \text{MeV}[/latex]
- Electron rest energy: [latex]mc^2=0.511\ \text{MeV}[/latex]
- Electron rest mass: [latex]m=9.11\times10^{-31}\ \text{kg}[/latex]
- Find the total energy and the Lorentz factor.
[latex]E=KE+mc^2=6.00\ \text{MeV}+0.511\ \text{MeV}=6.511\ \text{MeV}[/latex][latex]\gamma=\frac{E}{mc^2}=\frac{6.511\ \text{MeV}}{0.511\ \text{MeV}}=12.7[/latex]
- Find the electron's speed.Solving [latex]\gamma=1/\sqrt{1-u^2/c^2}[/latex] for [latex]u[/latex] gives
[latex]u=c\sqrt{1-\frac{1}{\gamma^2}}=c\sqrt{1-\frac{1}{(12.7)^2}}=0.997c[/latex]
- Calculate the relativistic momentum.
[latex]p=\gamma mu=(12.7)(9.11\times10^{-31}\ \text{kg})(0.997)(3.00\times10^8\ \text{m/s})[/latex][latex]p=3.47\times10^{-21}\ \text{kg}\cdot\text{m/s}[/latex]
- Compare with the classical prediction.Using the same speed in the classical formula [latex]p=mu[/latex] (that is, ignoring the Lorentz factor) gives
[latex]p_{\text{classical}}=mu=(9.11\times10^{-31}\ \text{kg})(0.997)(3.00\times10^8\ \text{m/s})[/latex][latex]p_{\text{classical}}=2.72\times10^{-22}\ \text{kg}\cdot\text{m/s}[/latex]
Discussion
The relativistic momentum is about 12.7 times larger than the classical prediction at the same speed—exactly a factor of [latex]\gamma[/latex], as expected from [latex]p=\gamma mu[/latex]. If a treatment-planning system used the classical formula for an electron beam at this energy, it would substantially underestimate the electron's true momentum and mispredict how the beam bends in the magnetic-steering components of the accelerator. This is why, as noted earlier in this chapter, modern radiation-therapy planning software must use the relativistic expression for momentum rather than the classical one.
Section Summary
- Classical momentum, [latex]p=mu[/latex], is only an approximation that holds when an object's speed is much smaller than the speed of light.
- The relativistic momentum of an object of rest mass [latex]m[/latex] moving at speed [latex]u[/latex] is
[latex]p=\gamma mu,[/latex]
where [latex]\gamma=1/\sqrt{1-u^2/c^2}[/latex] is the Lorentz factor.
- As an object's speed approaches the speed of light, [latex]\gamma[/latex] grows without bound, so its momentum can increase indefinitely even though its speed can never reach or exceed [latex]c[/latex].
- Conservation of momentum remains valid in special relativity, provided the relativistic expression for momentum is used consistently in every inertial reference frame.
Glossary
- Momentum
- A measure of an object's motion that depends on both its mass and its velocity. Momentum is conserved whenever the net external force on a system is zero.
- Rest mass
- The mass of an object measured in a reference frame where the object is at rest. In modern physics, this is simply called the object's mass.
- Relativistic momentum
- The momentum of an object moving at relativistic speeds, given by
[latex]p=\gamma mu.[/latex]
- Lorentz factor
- The dimensionless factor
[latex]\gamma=\frac{1}{\sqrt{1-\frac{u^2}{c^2}}},[/latex]
which describes how relativistic effects increase as an object's speed approaches the speed of light.
- Conservation of momentum
- The principle that the total momentum of an isolated system remains constant if no net external force acts on the system.
A measure of an object's motion that depends on both its mass and its velocity. Momentum is conserved whenever the net external force on a system is zero.
The mass of an object measured in a reference frame where the object is at rest. In modern physics, this is simply called the object's mass.
The momentum of an object moving at relativistic speeds, given by
[latex]p=\gamma mu.[/latex]
The dimensionless quantity that determines the magnitude of relativistic effects:
[latex]\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}.[/latex]
The principle that the total momentum of an isolated system remains constant if no net external force acts on the system.
