Simultaneity and Relativity of Time

Simultaneity is the key “mind bend” of special relativity: two events can be simultaneous in one inertial frame but not in another.

This maintained page sets up every later derivation, so keep frame labels and simultaneity conditions explicit before moving on.

1. Definitions (Must Know)

• Event: something that happens at a specific place and time, described by (x,t) in a chosen frame.
• Simultaneous (in a given frame): two events have the same time coordinate in that frame (t₁ = t₂).
• Inertial frame: a frame moving at constant velocity where the laws of physics take their standard form.

2. Key Ideas (What Earns Marks)

• “Simultaneous” is not just “I saw them at the same time.” You must account for signal travel time using a procedure consistent with the postulates.
• If all inertial observers measure the same speed of light in vacuum (c), then different inertial frames disagree on what events are simultaneous.
• Relativity of simultaneity is not an “illusion”; it is a consequence of how space and time coordinates relate between frames.

Quick comparison:

3. Detailed Explanations

A. Why “seeing” is not the same as “happening”

Light takes time to reach you. If you see two flashes simultaneously, you can only conclude the events were simultaneous in your frame if you also know:

• you are equidistant from the event locations (in your frame), or
• you have a synchronized clock network and have corrected for signal delays.

B. The train/boxcar lightning thought experiment (core idea)

Two lightning bolts strike the two ends of a moving boxcar. Consider two observers:

• O on the ground, midway between the ground strike marks.
• O' on the boxcar, midway between the boxcar ends (in the boxcar’s frame).

In the ground frame, if the light reaches O at the same time and the distances are equal, O concludes the strikes were simultaneous in the ground frame.

But O' is moving toward one flash and away from the other (in the ground frame). Since both observers must measure light moving at speed c, O' concludes the strike at the front end happened first in the boxcar frame.

The conclusion: Events simultaneous in one inertial frame are generally not simultaneous in another inertial frame moving relative to it.

C. Connection to the postulates

This is a direct consequence of:

• Postulate 1: laws of physics same in all inertial frames, and
• Postulate 2: c is the same in all inertial frames.

4. Common Mistakes

• Saying “they saw different times because of signal delay” and stopping there (each observer corrects for signal travel in their own frame).
• Assuming simultaneity is absolute, then trying to force Galilean time (t' = t) onto special relativity.
• Forgetting that “midpoint” depends on frame (lengths and simultaneity are linked).

5. Exam Tips

• When asked “are they simultaneous?”, always include “in which frame?”
• Use the safe phrasing: “Simultaneity is frame-dependent because all inertial observers measure the same light speed.”
• If you use the train thought experiment, state explicitly who is moving toward which flash (this anchors the reasoning).

6. Worked Examples

Example 1: Interpreting “I saw both flashes together”Core

You are standing still on the ground exactly midway between two lampposts. Both flash and you see the light at the same time. Are the flashes simultaneous in the ground frame?

Answer

Yes (in the ground frame), because you are equidistant from both flash locations in that frame, and light speed is the same from both. Equal travel time implies equal emission time in the same frame.

Example 2: Train lightning (concept)Core

In the train/boxcar scenario, why can both observers be “right” even though they disagree?

Answer

Because “simultaneous” means “same time coordinate” in a specific inertial frame. Different inertial frames assign different time coordinates to the same pair of events while still using the same physical laws and the same value of c.

Example 3: Events simultaneous in S, not in S' (numerical)Core

In frame S, two events are simultaneous (Δ t = 0) and separated by Δ x = 240 m along + x. Frame S' moves at v = 0.60c along + x relative to S.

Find Δ t'.

Answer

Use the Lorentz difference form: Δ t' = γ(Δ t-vΔ x/c²) Here Δ t = 0 and γ = 1/sqrt1-0.60² = 1.25: Δ t' = -gammavΔ x/c²

Δ t' = -1.25(0.60c)(240)/c² = -6.00 × 10⁻⁷ s So the events are not simultaneous in S'.

Example 4: “Saw simultaneously” but not at the midpointCore

Two flashes occur at lampposts A and B separated by 200 m (in the ground frame). You are standing 50 m from A and 150 m from B (in the ground frame). You see both flashes at the same time.

Are the flashes simultaneous in the ground frame?

Answer

No. In the ground frame you are not equidistant from the flash locations. Seeing both at the same time means the nearer flash must have occurred later than the farther flash to compensate for the shorter travel time.

Example 5: What do you need to claim simultaneity in your frame?Core

State one valid procedure to determine whether two distant events are simultaneous in your inertial frame.

Answer

Use synchronized clocks at the two event locations (synchronized in that frame), record the event times on those local clocks, and compare the time readings. (Equivalently, correct for signal travel time using a consistent synchronization convention in that frame.)

7. Mind Stretchers

Mind stretcher 1: What becomes impossible without frame-dependent simultaneity?Extension

If simultaneity were absolute (same for all inertial frames), what would that imply about the speed of light measurements in different inertial frames?

Answer

It would force a Galilean-style velocity addition for light (some observers would measure c± u), contradicting the postulate that all inertial observers measure c in vacuum.

Mind stretcher 2: Why simultaneity and length measurement are linkedExtension

Explain briefly why a “length of a moving object” measurement in your frame requires simultaneity in your frame.

Answer

To measure length you need the positions of both ends at the same time in your frame. If you use different times for the two ends, the object has moved in between and you are not measuring a single-frame length; this is why relativity of simultaneity is tied to length contraction.

8. Optional/Enrichment: Lorentz Transformations and a Formula

Using Lorentz transformations, the time difference between two events in different frames depends on both Δ t and Δ x. This is why spatial separation matters for simultaneity.

Next steps:

• Lorentz Transformations
• Time Dilation
• Length Contraction

Next in the maintained sequence: Lorentz Transformations.

8. Practice, Quiz and Next Step

Close your notes and use Simultaneity and Relativity of Time in the supplied context below. This requires a constructed explanation or working, not recognition of an option.

Fresh context: An unfamiliar data set or physical system requires you to apply Simultaneity and Relativity of Time while stating the model, regime and assumptions.

1. Retrieve: define simultaneity and relativity of time in your own words, including units, sign or conditions where relevant.
2. Represent: Choose and label an appropriate diagram, graph, table or symbolic model; derive or justify the relationship used.
3. Apply: Reach a conclusion, then evaluate it using units, uncertainty, a limiting case and one practical or modelling limitation.

Check the response before looking back

• The model, regime, coordinates and assumptions are explicit.
• The derivation or multi-step reasoning is visible rather than implied.
• The conclusion is tested against units, data quality and a limiting case.
• A practical control, uncertainty or model limitation is evaluated where applicable.

If one check fails, name that exact gap, revisit the matching explanation or worked example, and redo the task with different values or a different situation. Then use theH3 Physics course hub orpractice browser for an independent re-test.