3. Energy in Relativity

Energy Changes at Relativistic Speeds

In classical mechanics, the kinetic energy of a moving object is:

KE = ½mv²

and for ordinary speeds this equation works extremely well.

But Special Relativity changes our understanding of energy when objects travel at speeds approaching:

the speed of light.

At relativistic speeds, we need to distinguish three important quantities:

rest energy

kinetic energy

and:

total relativistic energy.

These quantities are closely related, but they do not mean the same thing.

---

Three Types of Energy

For a massive particle:

Rest energy

E₀ = mc²

Total relativistic energy

E = γmc²

Relativistic kinetic energy

KE = E − E₀

Therefore:

KE = (γ − 1)mc²

where:

γ = 1 / √(1 − v²/c²).

These equations form the foundation of:

relativistic energy calculations.

---

Rest Energy

Every object with rest mass possesses energy even when it is:

not moving.

This is its rest energy:

E₀ = mc²

where:

• E₀ = rest energy
• m = rest mass
• c = speed of light

Rest energy depends only on the object's:

rest mass.

It does not depend on its velocity.

---

Example: Rest Energy

Suppose a particle has mass:

m = 2.0 × 10⁻²⁷ kg.

Its rest energy is:

E₀ = mc²

E₀ = (2.0 × 10⁻²⁷)(3.00 × 10⁸)²

E₀ = (2.0 × 10⁻²⁷)(9.00 × 10¹⁶)

E₀ = 1.8 × 10⁻¹⁰ J.

Even while stationary, the particle possesses:

1.8 × 10⁻¹⁰ J of rest energy.

---

Total Relativistic Energy

When an object moves, its total energy is greater than its:

rest energy.

The total relativistic energy is:

E = γmc²

where:

γ = 1 / √(1 − v²/c²).

Because γ depends on velocity, the total energy increases as:

velocity increases.

---

The Lorentz Factor Again

The Lorentz factor appears throughout Special Relativity:

γ = 1 / √(1 − v²/c²).

At rest:

v = 0

so:

γ = 1.

Therefore:

E = mc².

At higher speeds:

γ > 1,

so:

E > mc².

The extra energy is the particle's:

kinetic energy.

---

Total Energy = Rest Energy + Kinetic Energy

A useful relationship is:

E = E₀ + KE

Therefore:

KE = E − E₀.

Since:

E = γmc²

and:

E₀ = mc²,

we obtain:

KE = γmc² − mc²

which gives:

KE = (γ − 1)mc².

---

The Energy Picture

Think of the total energy as:

Total energy = Rest energy + Kinetic energy

or:

E = mc² + (γ − 1)mc².

The object's rest energy exists even when:

v = 0.

Its kinetic energy appears because the object is:

moving relative to the observer.

---

Classical Kinetic Energy

At ordinary speeds, we use:

This equation remains an excellent approximation when:

v ≪ c.

However, it becomes increasingly inaccurate when:

v approaches c.

At relativistic speeds we instead use:

KE = (γ − 1)mc².

---

Classical vs Relativistic Kinetic Energy

At low speeds:

KE_rel ≈ ½mv².

At high speeds:

KE_rel > ½mv².

The difference becomes dramatic as:

v → c.

The classical curve continues smoothly beyond c mathematically.

The relativistic curve instead rises extremely steeply as velocity approaches:

c.

---

Why Classical Kinetic Energy Still Works

Relativity does not make classical physics useless.

When:

v ≪ c,

the relativistic kinetic-energy equation reduces approximately to:

KE ≈ ½mv².

This is the:

classical limit.

For everyday objects, relativistic corrections are so small that classical mechanics is usually more convenient.

---

Worked Example 1: Total Energy at 0.60c

A particle has rest mass:

m = 2.0 × 10⁻²⁷ kg

and moves at:

0.60c.

First calculate γ:

γ = 1/√(1 − 0.60²)

γ = 1/√0.64

γ = 1.25.

Now calculate:

E = γmc²

E = (1.25)(2.0 × 10⁻²⁷)(9.00 × 10¹⁶)

E = 2.25 × 10⁻¹⁰ J.

The particle's total relativistic energy is:

2.25 × 10⁻¹⁰ J.

---

Finding the Kinetic Energy

The particle's rest energy is:

E₀ = mc²

E₀ = 1.80 × 10⁻¹⁰ J.

Therefore:

KE = E − E₀

KE = 2.25 × 10⁻¹⁰ − 1.80 × 10⁻¹⁰

KE = 4.5 × 10⁻¹¹ J.

We could also calculate this directly:

KE = (γ − 1)mc².

---

Worked Example 2: Particle at 0.80c

Suppose:

m = 3.0 × 10⁻²⁷ kg

and:

v = 0.80c.

First:

γ = 1.667.

Rest energy:

E₀ = mc²

E₀ = (3.0 × 10⁻²⁷)(9.00 × 10¹⁶)

E₀ = 2.70 × 10⁻¹⁰ J.

Total energy:

E = γmc²

E = (1.667)(2.70 × 10⁻¹⁰)

E ≈ 4.50 × 10⁻¹⁰ J.

Kinetic energy:

KE = E − E₀

KE = 4.50 × 10⁻¹⁰ − 2.70 × 10⁻¹⁰

KE = 1.80 × 10⁻¹⁰ J.

---

A Useful Calculation Strategy

For most relativistic energy problems:

Step 1: Identify m and v

Write the mass in:

kilograms.

Step 2: Calculate γ

γ = 1/√(1 − v²/c²).

Step 3: Calculate rest energy if needed

E₀ = mc².

Step 4: Calculate total energy

E = γmc².

Step 5: Calculate kinetic energy if needed

KE = (γ − 1)mc².

Step 6: Check the relationship

Your answers should satisfy:

E = E₀ + KE.

---

Comparing the Three Energies

Suppose:

mc² = 100 units of energy.

If:

γ = 1.5,

then:

rest energy = 100

total energy = 150

and:

kinetic energy = 50.

Notice:

150 = 100 + 50.

This simple relationship is useful for checking calculations.

---

How Energy Changes with Speed

Consider the same particle moving at different speeds.

The pattern becomes striking near:

c.

---

At 0.80c

At:

v = 0.80c

we have:

γ ≈ 1.667.

Therefore:

E ≈ 1.667E₀

and:

KE ≈ 0.667E₀.

The kinetic energy is already about:

two-thirds of the rest energy.

---

At 0.90c

At:

v = 0.90c

we have:

γ ≈ 2.294.

Therefore:

E ≈ 2.294E₀

and:

KE ≈ 1.294E₀.

The kinetic energy is now greater than the particle's:

rest energy.

---

At 0.99c

At:

v = 0.99c

we have:

γ ≈ 7.09.

Therefore:

E ≈ 7.09E₀

and:

KE ≈ 6.09E₀.

The particle's kinetic energy is more than six times its:

rest energy.

---

Why Kinetic Energy Rises So Rapidly

The key is again the:

Lorentz factor.

As:

v → c,

then:

v²/c² → 1.

Therefore:

1 − v²/c² → 0.

So:

γ → ∞.

Since:

KE = (γ − 1)mc²,

the kinetic energy increases without bound as a massive particle's speed approaches:

c.

---

Why Massive Objects Cannot Reach c

Suppose we continue supplying energy to a particle.

At relatively low speeds, additional energy produces noticeable increases in:

velocity.

Near c, additional energy still increases the particle's:

kinetic energy and momentum,

but produces progressively smaller increases in speed.

To reach exactly:

v = c

would require:

γ → ∞.

Therefore, the required energy would also approach:

infinity.

A massive particle cannot be accelerated to:

c.

---

A Useful Way to Think About It

Near the speed of light:

more energy does not mean proportionally more speed.

Instead:

enormous increases in energy produce increasingly tiny increases in velocity.

This is very different from the prediction of:

classical mechanics.

---

Worked Example 3: Kinetic Energy at 0.90c

A particle has rest mass:

m = 1.0 × 10⁻²⁷ kg.

It travels at:

0.90c.

First:

γ ≈ 2.294.

Rest energy:

E₀ = mc²

E₀ = (1.0 × 10⁻²⁷)(9.00 × 10¹⁶)

E₀ = 9.00 × 10⁻¹¹ J.

Kinetic energy:

KE = (γ − 1)mc²

KE = (2.294 − 1)(9.00 × 10⁻¹¹)

KE ≈ 1.16 × 10⁻¹⁰ J.

Total energy:

E = γmc²

E ≈ 2.06 × 10⁻¹⁰ J.

Check:

E₀ + KE ≈ E

9.00 × 10⁻¹¹ + 1.16 × 10⁻¹⁰ ≈ 2.06 × 10⁻¹⁰ J.

Correct.

---

Finding γ from Energy

Sometimes total energy is given instead of:

velocity.

Since:

E = γmc²

and:

E₀ = mc²,

we can write:

γ = E/E₀.

For example, if:

E = 3E₀,

then:

γ = 3.

We can then use γ to determine the particle's:

velocity.

---

Finding Velocity from γ

Start with:

γ = 1/√(1 − v²/c²).

Rearranging gives:

v = c√(1 − 1/γ²).

This allows us to calculate velocity when:

γ is known.

---

Worked Example 4: Finding Speed from Energy

Suppose a particle's total energy is:

2mc².

Therefore:

γ = 2.

Use:

v = c√(1 − 1/γ²).

Substitute:

v = c√(1 − 1/4)

v = c√(3/4)

v ≈ 0.866c.

So a particle whose total energy is twice its rest energy travels at approximately:

86.6% of the speed of light.

---

Finding Speed from Kinetic Energy

Suppose:

KE = mc².

Use:

KE = (γ − 1)mc².

Therefore:

1 = γ − 1

so:

γ = 2.

Therefore:

v ≈ 0.866c.

This means that when a particle's kinetic energy equals its rest energy, its speed is approximately:

0.866c.

---

Energy and Momentum

Energy and momentum are deeply connected in Special Relativity.

The fundamental relationship is:

E² = p²c² + m²c⁴.

where:

• E = total relativistic energy
• p = relativistic momentum
• m = rest mass
• c = speed of light

This equation is one of the most important relationships in:

relativistic mechanics.

---

Connection to E = mc²

Suppose the particle is at rest.

Then:

p = 0.

The energy-momentum equation becomes:

E² = m²c⁴.

Therefore:

E = mc².

So Einstein's famous equation is the special case for a particle:

at rest.

---

Connection to Relativistic Momentum

For a moving massive particle:

p = γmv

and:

E = γmc².

Dividing:

p/E = v/c².

Therefore:

v = pc²/E.

This connects three important quantities:

velocity, momentum, and energy.

---

Photons

Photons have:

zero rest mass.

Therefore:

m = 0.

The energy-momentum relationship becomes:

E² = p²c².

So:

E = pc.

Photons therefore possess both:

energy

and:

momentum

despite having zero rest mass.

---

Why Photons Are Different

For a massive particle:

E = γmc².

For a photon, this form is not used because:

m = 0

and:

v = c,

which would make γ undefined.

Instead, photons are described using:

E = pc.

Their energy can also be written:

E = hf,

where f is the photon's:

frequency.

---

Energy in Particle Physics

Relativistic energy becomes essential in:

particle accelerators.

Accelerators add enormous amounts of kinetic energy to particles.

As the particles approach c:

their speed changes very little

while:

their energy and momentum continue increasing dramatically.

This is why particle accelerators are often described by their:

beam energy

rather than simply their particle speed.

---

Example: Two High-Energy Particles

Suppose two identical particles approach one another with:

equal and opposite momenta.

Their total momentum is:

zero.

However, their total energy is:

not zero.

Each particle has:

rest energy + kinetic energy.

When they collide, some of the available energy can appear in:

• new particles
• kinetic energy of products
• radiation

while total energy and momentum remain:

conserved.

---

Energy Can Produce Massive Particles

High-energy collisions can produce new particles if sufficient energy is:

available.

For example:

collision energy → new particle rest energy + kinetic energy

This is an important application of:

mass-energy equivalence.

The energy does not simply disappear.

It is redistributed among the:

products of the interaction.

---

Conservation of Relativistic Energy

For an isolated system:

total energy before = total energy after.

This includes all relevant forms of energy:

• rest energy
• kinetic energy
• radiation
• internal energy
• other field or interaction energy as appropriate

In relativistic collisions, both:

energy

and:

momentum

must be conserved.

---

Worked Example 5: Total Energy from Momentum

Suppose a particle has:

pc = 3.0 GeV

and:

mc² = 4.0 GeV.

Use:

E² = p²c² + m²c⁴.

Therefore:

E² = (3.0 GeV)² + (4.0 GeV)²

E² = 9.0 + 16.0

E² = 25.0 GeV².

Therefore:

E = 5.0 GeV.

This resembles the familiar:

3-4-5 right triangle relationship.

---

Finding Kinetic Energy from Total Energy

For the previous particle:

E = 5.0 GeV

and:

E₀ = mc² = 4.0 GeV.

Therefore:

KE = E − E₀

KE = 5.0 − 4.0

KE = 1.0 GeV.

---

Energy Units in Particle Physics

The SI unit of energy is:

joule (J).

Particle physics often uses:

electronvolts (eV).

Common units include:

• keV = 10³ eV
• MeV = 10⁶ eV
• GeV = 10⁹ eV
• TeV = 10¹² eV

One electronvolt is approximately:

1.602 × 10⁻¹⁹ J.

---

Rest Mass in Energy Units

Particle physicists often describe rest mass using its energy equivalent.

For example, instead of expressing a particle mass only in kilograms, they may give:

mc² in MeV or GeV.

Mass itself may also be quoted in:

MeV/c²

or:

GeV/c².

This makes relativistic calculations much more convenient.

---

Example: Proton Rest Energy

A proton's rest energy is approximately:

938 MeV.

Therefore:

E₀ ≈ 938 MeV.

If the proton has:

500 MeV

of kinetic energy, its total energy is:

E = E₀ + KE

E ≈ 938 + 500

E ≈ 1438 MeV.

---

Example: Electron Rest Energy

An electron has a rest energy of approximately:

0.511 MeV.

If an electron has:

1.00 MeV

of kinetic energy, its total energy is:

E = 0.511 + 1.00

E = 1.511 MeV.

The electron is then moving at a strongly:

relativistic speed.

---

Relativistic vs Classical Example

Consider a particle moving at:

0.80c.

Relativistically:

γ = 1.667.

Therefore:

KE_rel = 0.667mc².

Classically:

KE_class = ½mv²

KE_class = ½m(0.80c)²

KE_class = 0.32mc².

Compare:

relativistic KE = 0.667mc²

classical KE = 0.32mc².

The classical prediction is less than half the correct relativistic value.

---

At 0.99c

At:

0.99c

γ ≈ 7.09.

Therefore:

KE_rel ≈ 6.09mc².

Classically:

KE_class ≈ 0.490mc².

The difference is enormous.

This demonstrates why classical kinetic energy cannot be used for:

ultrarelativistic particles.

---

Energy Does Not Mean "Relativistic Mass"

Older treatments sometimes describe:

γm

as "relativistic mass."

Modern physics generally avoids this terminology.

Instead, mass is normally treated as the invariant:

rest mass.

As velocity increases:

energy increases

and:

momentum increases,

while the invariant rest mass remains:

unchanged.

---

Energy Depends on Reference Frame

Kinetic energy depends on:

relative motion.

Suppose an astronaut is sitting inside a spacecraft.

In the astronaut's frame, the astronaut is:

at rest.

Therefore:

KE = 0.

An observer on Earth may see the astronaut moving rapidly.

That observer measures:

KE > 0.

Kinetic energy is therefore:

frame-dependent.

---

Rest Energy Is Different

Rest energy is based on the invariant:

rest mass.

All inertial observers agree on:

m.

Therefore, they agree on:

E₀ = mc².

They may disagree about the particle's:

kinetic energy and total energy

because those depend on the observer's reference frame.

---

Energy-Momentum as a Unified Idea

Space and time combine into:

spacetime.

Similarly, energy and momentum combine into:

four-momentum.

This reveals a deep structural relationship within Special Relativity.

Different observers may measure different:

energies and momenta,

but the quantity:

E² − p²c²

remains related to the invariant rest mass:

m²c⁴.

---

Energy and the Speed Limit

The relativistic energy equation explains why c is an:

unreachable speed for massive particles.

As:

v → c

we have:

γ → ∞.

Therefore:

E = γmc² → ∞.

No finite amount of energy can accelerate an object with nonzero rest mass to exactly:

c.

---

Real-World Application: Particle Accelerators

Particle accelerators provide direct practical applications of relativistic energy.

Particles can travel at speeds extremely close to:

c.

Adding more energy does not significantly increase their speed.

Instead, it dramatically increases their:

energy and momentum.

Without Special Relativity, the behaviour of these particles could not be described correctly.

---

Real-World Application: Cosmic Rays

High-energy cosmic rays produce particles travelling extremely close to:

c.

Two particles can have almost identical speeds but very different:

energies.

At ultrarelativistic speeds, velocity becomes a poor indicator of:

particle energy.

Momentum and energy provide much more useful information.

---

Real-World Application: Stars

Relativistic energy relationships are also important in:

• nuclear reactions
• stellar interiors
• supernovae
• neutron stars
• high-energy radiation
• cosmic particle interactions

Special Relativity therefore connects directly with:

astrophysics.

---

Common Misconception: E = mc² Is the Total Energy of Every Moving Object

For an object at rest:

E₀ = mc².

For a moving massive particle:

E = γmc².

So mc² is the rest energy, not generally the total energy of a moving particle.

---

Common Misconception: Classical Kinetic Energy Is Always Valid

The equation:

KE = ½mv²

is an approximation.

It works extremely well at:

low speeds.

Near c, however, we must use:

KE = (γ − 1)mc².

---

Common Misconception: An Object's Rest Mass Increases as It Speeds Up

In modern terminology:

rest mass remains invariant.

What increases with speed is the object's:

energy and momentum.

This language avoids confusion and connects naturally with modern particle physics.

---

Common Misconception: A Photon Has No Energy Because It Has No Mass

Photons have zero:

rest mass.

But they have both:

energy and momentum.

For photons:

E = pc

and:

E = hf.

Therefore, zero rest mass does not mean:

zero energy.

---

Common Misconception: A Particle Reaches c If We Give It Enough Energy

No finite amount of energy is sufficient.

For a massive particle:

γ → ∞ as v → c.

Therefore:

E → ∞.

The particle can approach c increasingly closely but cannot:

reach it.

---

Putting the Equations Together

The major equations for this topic are:

Rest energy

E₀ = mc²

Total relativistic energy

E = γmc²

Kinetic energy

KE = (γ − 1)mc²

Lorentz factor

γ = 1/√(1 − v²/c²)

Energy-momentum relationship

E² = p²c² + m²c⁴

Photon relationship

E = pc

These equations form a connected system rather than:

separate facts to memorize.

---

Choosing the Correct Equation

If the question asks for:

rest energy → use E₀ = mc²

If it asks for:

total energy at a known velocity → use E = γmc²

If it asks for:

kinetic energy at a known velocity → use KE = (γ − 1)mc²

If momentum is given:

use E² = p²c² + m²c⁴

If the particle is a photon:

use E = pc.

Choosing the correct equation is often the most important first step.

---

Check Your Understanding

1. Define rest energy.

2. State the equation for rest energy.

3. State the equation for total relativistic energy.

4. State the equation for relativistic kinetic energy.

5. Explain the difference between rest energy and kinetic energy.

6. Explain the difference between kinetic energy and total energy.

7. Calculate γ for a particle travelling at 0.60c.

8. A particle has mass 2.0 × 10⁻²⁷ kg. Calculate its rest energy.

9. The same particle travels at 0.60c. Calculate its total energy.

10. Calculate its kinetic energy.

11. Why does classical kinetic energy work at low speeds?

12. Why does relativistic kinetic energy increase rapidly near c?

13. What happens to γ as v approaches c?

14. Explain why a massive particle cannot reach c.

15. A particle's total energy equals twice its rest energy. Determine γ.

16. What speed corresponds to γ = 2?

17. State the relativistic energy-momentum relationship.

18. Show how E = mc² follows from the energy-momentum equation for a stationary particle.

19. What is the energy-momentum relationship for a photon?

20. Explain why relativistic energy is important in particle accelerators.

---

Key Terms

• Rest energy: Energy associated with an object's invariant rest mass.
• Rest mass: Invariant mass measured in an object's rest frame.
• Total relativistic energy: Total energy of a moving massive particle, E = γmc².
• Kinetic energy: Energy associated with relative motion.
• Relativistic kinetic energy: Kinetic energy given by KE = (γ − 1)mc².
• Lorentz factor: Factor γ describing how relativistic effects depend on speed.
• Classical limit: Low-speed condition under which relativistic equations reduce approximately to classical equations.
• Momentum: Vector quantity describing motion.
• Relativistic momentum: Momentum given by p = γmv.
• Energy-momentum relation: Equation connecting total energy, momentum, and rest mass.
• Photon: Massless quantum of electromagnetic radiation.
• Electronvolt: Unit of energy commonly used in atomic and particle physics.
• Particle accelerator: Device that gives charged particles large amounts of kinetic energy.
• Four-momentum: Relativistic quantity combining energy and three-dimensional momentum.
• Invariant: Quantity that has the same value for all inertial observers.

---

Key Takeaways

• Relativity distinguishes between rest energy, kinetic energy, and total energy.
• Rest energy is given by E₀ = mc².
• Rest energy exists even when an object is stationary.
• Total relativistic energy is given by E = γmc².
• Kinetic energy is the difference between total energy and rest energy.
• Therefore, KE = (γ − 1)mc².
• At low speeds, relativistic kinetic energy approaches the classical expression ½mv².
• Classical kinetic energy becomes increasingly inaccurate as velocity approaches c.
• The Lorentz factor γ increases rapidly at relativistic speeds.
• As v approaches c, γ increases without bound.
• Therefore, the kinetic energy required to accelerate a massive particle toward c also increases without bound.
• No finite amount of energy can accelerate an object with nonzero rest mass to exactly c.
• Near c, enormous increases in energy produce only very small increases in speed.
• This behaviour is directly observed in particle accelerators.
• Rest mass remains invariant; modern physics generally avoids describing mass as increasing with speed.
• Kinetic energy and total energy depend on the observer's reference frame.
• Energy and momentum are related by E² = p²c² + m²c⁴.
• For a stationary particle, this reduces to E = mc².
• Photons have zero rest mass but still possess energy and momentum.
• For photons, E = pc.
• High-energy collisions can redistribute energy into rest energy, kinetic energy, radiation, and new particles, subject to conservation laws.
• Relativistic energy is fundamental to particle physics, nuclear physics, astrophysics, and modern accelerator science.
• Energy and momentum together form one of the central unified structures of Special Relativity.