An electron is accelerated to 99% of the speed of light (v = 0.99c, γ ≈ 7). A student says 'the electron's mass has increased by a factor of 7.' What is the most precise modern assessment of this statement?
ACorrect — relativistic mass increases with velocity, so the electron is effectively about 7 times heavier.
BIncorrect — the electron's rest mass m is a Lorentz invariant unchanged by its speed. What increases by γ is the electron's momentum, not its mass.
CIncorrect — at 0.99c, quantum effects dominate and the classical concept of mass no longer applies.
DPartially correct — the electron's inertia increases, meaning it behaves like it has 7 times more mass for practical purposes.
Rest mass m is a Lorentz invariant — it has the same value in every reference frame and does not change with velocity. The notion of 'relativistic mass' γm is an outdated framing. What does increase with γ is the momentum: p = γmv grows dramatically as v approaches c. The practical consequence (the electron is increasingly hard to accelerate further) is real, but it arises because momentum and energy grow, not because mass increases. Modern physics reserves 'mass' for the invariant rest mass.
Question 2 Multiple Choice
A photon has zero rest mass. According to the energy-momentum relation E² = (pc)² + (mc²)², what does this imply about a photon's energy and momentum?
AA photon has zero energy since E² = (pc)² gives E = pc = 0 for a massless particle.
BA photon has energy E = pc, so it carries momentum proportional to its energy despite having no rest mass.
CThe energy-momentum relation does not apply to photons, which must be treated using quantum mechanics instead.
DA photon has only rest energy mc² = 0, confirming it has no energy at all.
Setting m = 0 gives E² = (pc)², so E = pc. Photons carry momentum p = E/c despite having no rest mass — a purely relativistic result with no Newtonian analogue, and the reason light exerts radiation pressure. The energy-momentum relation is universally valid for all particles, massive or massless, in any reference frame. This frame-independence is precisely why it is so useful: you never need to pick a specific frame to apply it.
Question 3 True / False
At low velocities (v << c), the relativistic kinetic energy formula K = (γ−1)mc² reduces to the familiar Newtonian expression ½mv².
TTrue
FFalse
Answer: True
At low speeds, γ = 1/√(1 − v²/c²) ≈ 1 + v²/2c² (first-order Taylor expansion). Therefore K = (γ − 1)mc² ≈ (v²/2c²)mc² = ½mv². This is the required correspondence principle: any correct relativistic formula must reduce to the Newtonian result at everyday speeds. Relativistic mechanics is not a replacement for Newtonian mechanics — it is a generalization that contains Newton's laws as a special case in the low-velocity limit.
Question 4 True / False
In special relativity, the rest mass of a particle increases as it moves faster, which is why it becomes very difficult to accelerate a massive particle to the speed of light.
TTrue
FFalse
Answer: False
Rest mass is a Lorentz invariant — it does not change with velocity. The reason a massive particle cannot reach the speed of light is that its momentum p = γmv diverges as v → c (because γ → ∞), requiring infinite energy to continue accelerating. The barrier is not growing mass but growing momentum: each additional increment of speed toward c requires exponentially more energy. 'Mass increases' is an old pedagogical shorthand that conflates invariant rest mass with the γ-factor in momentum.
Question 5 Short Answer
Why must relativistic momentum be defined as p = γmv rather than the classical p = mv?
Think about your answer, then reveal below.
Model answer: Because classical momentum p = mv is not conserved in all inertial frames. When two observers apply conservation of momentum to a collision using p = mv, they disagree on whether momentum was conserved. Defining momentum using proper time — p = m(dx/dτ) = γmv — produces a quantity that is conserved in every inertial frame, as required by the principle of relativity.
The motivation is physical necessity, not mathematical convenience. Proper time τ is the time measured in the particle's own rest frame — a Lorentz-invariant quantity. Differentiating position with respect to proper time (rather than coordinate time) automatically builds in the γ factor. The result is a momentum that all inertial observers agree is conserved in collisions, which is the foundational requirement for the concept of momentum to be physically meaningful.