Chapter 6: Relativity of Simultaneity
Perhaps the most mind-bending consequence of special relativity: events that are simultaneous in one reference frame are NOT simultaneous in another. There is no universal "now" that all observers share. The very concept of "at the same time" depends on your state of motion.
Einstein's Train Thought Experiment
Setup
- • A train travels at high speed along a track
- • Lightning strikes both ends of the train simultaneously (in the platform frame)
- • An observer M stands at the midpoint of the platform
- • An observer M' rides at the midpoint of the train
Platform Observer (M)
M sees the light from both strikes arrive at the same time. Since M is equidistant from both strike locations and light travels at c both ways, M concludes: the strikes were simultaneous.
Train Observer (M')
M' is also at the midpoint, but M' is moving toward the front strike and away from the rear strike. The light from the front arrives first! M' concludes: the front strike happened first.
Who is right? Both are! There is no experiment either can do to prove the other wrong. The speed of light is c in both frames—this is the second postulate. Simultaneity itself is relative.
The Mathematics of Simultaneity
The Lorentz transformation for time shows how simultaneity is frame-dependent:
\( t' = \gamma \left( t - \frac{vx}{c^2} \right) \)
Time in frame S' depends on both t AND x from frame S
Consider two events that are simultaneous in S (t₁ = t₂ = 0) but at different locations (x₁ ≠ x₂):
\( t_1' = \gamma \left( 0 - \frac{vx_1}{c^2} \right) = -\frac{\gamma v x_1}{c^2} \)
\( t_2' = \gamma \left( 0 - \frac{vx_2}{c^2} \right) = -\frac{\gamma v x_2}{c^2} \)
\( \Delta t' = t_2' - t_1' = -\frac{\gamma v}{c^2}(x_2 - x_1) \neq 0 \) if x₁ ≠ x₂
Events that are simultaneous and separated in space in one frame are NOT simultaneous in another frame. The time difference depends on the spatial separation and the relative velocity.
Profound Implications
1. No Universal "Now"
When you look at a distant star, there is no universal answer to "what is happening there right now?" The answer depends on your state of motion. An observer moving toward the star and one moving away would have completely different "now" slices through spacetime.
2. The Block Universe
Relativity suggests that past, present, and future all exist equally—a view called "eternalism" or the "block universe." Since different observers disagree on which events are "now," it's hard to maintain that only the present moment exists.
3. Length Contraction Explained
Length contraction is fundamentally about simultaneity! To measure length, you must mark both ends "at the same time." But different observers disagree on what "same time" means, so they measure different lengths.
4. Causality is Preserved
Despite the relativity of simultaneity, cause always precedes effect in ALL frames. This is because only events with spacelike separation can have their order reversed—and such events cannot be causally connected (nothing can travel faster than light).
Worked Example
Problem
Two events occur simultaneously in frame S, separated by Δx = 300 m. Frame S' moves at v = 0.8c relative to S. What is the time difference between the events in S'?
Solution
Step 1: Calculate γ = 1/√(1 - 0.8²) = 1/√(0.36) = 1/0.6 = 5/3 ≈ 1.667
Step 2: Use \( \Delta t' = -\frac{\gamma v \Delta x}{c^2} \)
Step 3: \( \Delta t' = -\frac{1.667 \times 0.8c \times 300\text{ m}}{c^2} \)
Step 4: \( \Delta t' = -\frac{1.667 \times 0.8 \times 300}{3 \times 10^8} \) s
Step 5: \( \Delta t' = -1.33 \times 10^{-6} \) s = −1.33 μs
Result: In frame S', the event at larger x happens 1.33 μs BEFORE the other event! The negative sign indicates the order is reversed compared to what simultaneity would suggest.
The Andromeda Paradox
A thought-provoking illustration of relativity of simultaneity, proposed by Roger Penrose:
The Andromeda galaxy is about 2.5 million light-years away. Consider two people passing each other on Earth—one walking toward Andromeda, one walking away, at just 3 mph relative to each other.
Due to the relativity of simultaneity, their "now" slices through Andromeda differ by about one day!
If an alien invasion fleet is launching from Andromeda, one person's "now" has the fleet already on its way, while the other's "now" has the launch decision not yet made.
The philosophical puzzle: How can the future be "undecided" if different observers can disagree about whether a distant event has "already happened"? This challenges our intuitions about free will and the nature of time.
Common Misconceptions
❌ "Simultaneity depends on when you see the events"
No! Relativity of simultaneity is NOT about light travel time. We always correct for light travel time when determining when an event "really" happened. Even after this correction, observers still disagree on simultaneity.
❌ "One frame is right, the other is wrong"
No! All inertial frames are equally valid. Neither observer is "really" moving or "really" at rest. There is no privileged frame that tells us the "true" simultaneity.
❌ "Causality can be violated"
No! Only events with spacelike separation can have their order reversed. Events with timelike separation (where one could causally affect the other) maintain their order in ALL frames. You cannot kill your grandfather before you're born!