M3 The Twin Paradox

Moving Clocks Run Slow

A clock that is moving relative to you measures less time between the same two events than a clock at rest with you.

Lorentz Factor

(γ) Quantifies how time intervals and lengths differ between two frames in relative motion

v = speed of moving object

c = speed of light

γ = 1 / √ [1- (v² / c²)]

Lorentz Factor and Speed

As things get closer to the speed of light, the bigger the difference in the rate if time

• At speed 0c, Lorenz Factor is 1

• At speed 0.8c, Lorenz Factor is 1.667

Twin Paradox

Imagine two twins, Alice and Bob.

• Eve stays on Earth while Sam travels on a spaceship at a very high speed (close to the speed of light) to a distant star and then returns.

From Eve’s perspective on Earth, Sam’s clock is ticking slower

From Sam’s perspective on the spaceship, Eve’s clock is ticking slower

• When Sam comes back, they both think the other is younger than them

Spacetime

Space and time are linked into one four-dimensional fabric

• Spacetime interval equation:

• s² = spacetime (quantity all observers agree on)

• (cΔt)² = amount you move in time

• x,y,z amount you move in directions of space

s² = (cΔt)² − Δx² − Δy² −Δz²

Moving in Spacetime

Moving in space means motion is subtracted from the amount you move in time

• With a conversion rate of the speed of light

Moving in Spacetime Example

Scenario 1: Wait outside your door all day

• Moving through time but not space

Scenario 2: (At light speed) Walk from your door, to the park, to the gym, and come back

• Moving through time AND space

• Space is subtracted from time

Acceleration

When Sam accelerates and moves close to the speed of light, his path through spacetime is different from the twin who stays on Earth

• The “distance” through space time affects how much time passes → Bob ages slower

• 4 ‘accelerations’

  • Starting from Earth

  • Slowing down

  • Starting from the Destination

  • Slowing down

Signals

The Earth twin sends a light signal once every year (according to Earth clocks).

• Signals travel at speed c

• The traveling twin moves away at high speed, turns around, and comes back.

Going Away Signals

While the traveler is moving away from Earth:

• Each new light signal is sent from farther behind

• The traveler is running away from the light

• Each signal has to catch up

Turnaround

At the turnaround:

• The traveler changes reference frames

Nothing magical happens to the light at that instant, but:

• The traveler’s idea of which Earth events are ‘now’ jumps forward

• In spacetime terms: their line of simultaneity tilts

Return Signals

Now the traveler is moving toward Earth:

• They are moving into the incoming light

• Each signal meets them faster

• Signals arrive more frequently than once per year

Spaceship twin observes Earth’s clock running faster than their own

Earth twin observes Spaceship’s clock running slower than their own

Longer Spacetime

In spacetime:

• Everyone moves through spacetime at the same “speed”

• How much time you experience depends on your path through spacetime

• Fritz’s path:

  • Straight line (no frame switching)

  • Maximizes proper time → ages more

• Vera’s path:

  • Bent path (because of turnaround / acceleration)

  • Shorter proper time → ages less

So even though Vera traveled farther in space, she traveled less in time.