Relativistic Mass and Momentum (HSC SSCE Physics): Revision Notes

Relativistic Mass and Momentum

Introduction

According to Einstein's theory of special relativity, mass is not a constant property—it depends on the velocity of an object relative to an observer. Just as we've seen with length contraction and time dilation, mass also changes at high speeds approaching the speed of light. This has profound implications for understanding motion at relativistic velocities and explains why nothing with mass can reach the speed of light.

Rest mass and relativistic mass

When we measure the mass of an object that is stationary in our reference frame, we obtain what is called the rest mass or proper mass, denoted as $m_0$. This is a fundamental property of the object that never changes—it's the same regardless of who measures it, as long as the object is at rest relative to them.

However, when an object moves at high speeds relative to an observer, its mass appears to increase. This increased mass is called the relativistic mass or relativistically corrected mass, denoted simply as $m$. The relationship between rest mass and relativistic mass is given by:

$m = \gamma m_0 = \frac{m_0}{\sqrt{1 - \frac{v^2}{c^2}}}$

where:

• $m$ is the relativistic mass
• $m_0$ is the rest mass
• $v$ is the velocity of the object relative to the observer
• $c$ is the speed of light ($3.0 \times 10^8 \text{ m s}^{-1}$)
• $\gamma$ is the Lorentz factor

At everyday speeds, this effect is negligible, but at speeds close to $c$, the mass increase becomes very significant.

Calculating relativistic mass

Let's look at how to calculate the relativistic mass of a particle moving at high speed.

Relativistic momentum

In classical physics, momentum is defined as $p = mv$, where $m$ is the mass and $v$ is the velocity. Special relativity uses the same basic equation, but now we must use the relativistic mass instead of the rest mass.

The magnitude of an object's relativistic momentum, $p$, is given by:

$p = mv = \frac{m_0 v}{\sqrt{1 - \frac{v^2}{c^2}}}$

We can also express this in terms of the rest momentum $p_0 = m_0 v$:

$p = \frac{p_0}{\sqrt{1 - \frac{v^2}{c^2}}} = \gamma p_0$

Why nothing can reach the speed of light

The increase in relativistic mass has a crucial consequence: it becomes progressively harder to accelerate an object as its speed increases. Consider a rocket engine that provides a constant force $F_R$. Using Newton's second law:

$F = ma$

At low speeds, we can write:

$$m_{\text{rocket}} a_{\text{rocket}} = F_R$$ $$a = \frac{F_R}{m_{\text{rocket}}}$$

However, at relativistic speeds, we must use the relativistic mass:

$$\gamma m_{\text{rocket}} a = F_R$$ $$a = \frac{F_R}{\gamma m_{\text{rocket}}}$$

Since $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$ increases dramatically as $v$ approaches $c$, the acceleration $a$ becomes smaller and smaller. This means:

$a \propto \frac{1}{\gamma} = \sqrt{1 - \frac{v^2}{c^2}}$

Particle accelerators and synchrotrons

Scientists observe the effects of relativistic mass in particle accelerators, where subatomic particles are accelerated to speeds very close to the speed of light. Two important types of particle accelerators are:

• Synchrotrons: Circular accelerators that use electric and magnetic fields to control and accelerate particles
• Particle colliders: Accelerators where particles are smashed together at high energies

As particles approach the speed of light in these machines, they behave as if they have much more mass than their rest mass. This makes them increasingly difficult to accelerate further. Additionally, forcing high-speed particles to follow a curved path requires enormous centripetal force:

$F = \frac{mv^2}{r}$

As both the effective mass $m$ and velocity $v$ increase, either the force must increase or the radius $r$ of the curve must be larger. This is why very high-energy particle accelerators, such as the Large Hadron Collider (LHC) and the Tevatron, need to be built as large rings.

The Australian Synchrotron

The Australian Synchrotron is an example of a facility that accelerates electrons to approximately $299792 \text{ km s}^{-1}$—extremely close to the speed of light ($299792.458 \text{ km s}^{-1}$). At this speed, relativistic effects are highly significant.

Synchrotrons like the Australian Synchrotron are used not only for particle physics experiments but also to generate powerful beams of electromagnetic radiation, including X-rays. These beams have applications in many fields of science, engineering, and medicine, from studying the structure of proteins to developing new materials.