The relativity of simultaneity is one of the most subtle and profound consequences of Special Relativity.
Events that are simultaneous in one inertial frame are generally not simultaneous in another frame moving relative to the first.
This has deep implications for how we understand the nature of time and causality — particularly in astrophysical contexts where relativistic motion is involved (e.g. AGN jets, relativistic shocks).
Consider two inertial frames:
- Frame S ("lab frame")
- Frame S′ moving at velocity v along the x-axis relative to S
Suppose two events occur at different spatial positions x1 and x2, but are simultaneous in S:
t1=t2
We ask: are these events still simultaneous in S′?
The Lorentz transformation for time is:
t′=γ(t−c2vx).
Consider two events with spacetime coordinates:
(x1,t),(x2,t)
Their corresponding times in S′ are:
t1′=γ(t−c2vx1)
t2′=γ(t−c2vx2)
Now compute the absolute value of the time difference:
∣Δt′∣=∣t2′−t1′∣=γ∣∣∣∣−c2v(x2−x1)∣∣∣∣
Thus:
∣Δt′∣=γc2v∣x2−x1∣
Or, equivalently:
Δt′=γc2vΔx
where we now define:
Δx=∣x2−x1∣≥0
Thus, for any pair of events simultaneous in S, an observer in S′ will see the events with a time delay of:
Δt′=γc2vΔx
where:
- Δx=∣x2−x1∣ is the spatial separation of the events in S.
- Δt′ is the time separation between the events in S′.
- The time difference in S′ is proportional to the spatial separation Δx between the events in S.
- The greater the velocity v, the larger the time difference Δt′.
- Events simultaneous in S are generally not simultaneous in S′.
- The notion of "simultaneous emission" in the rest frame of the AGN does not carry over to the observer frame.
- The ordering of events can change between frames for spacelike-separated events.
- This affects:
- Observed light curves
- Time delays in multi-wavelength AGN observations
- Interpretation of variability in relativistic jets
- Care must be taken when reconstructing AGN geometry from observed timing.
The relativity of simultaneity is often visualized using spacetime diagrams:
- In S: lines of constant t are horizontal.
- In S′: lines of constant t′ are tilted.
Thus, what is simultaneous in S is not simultaneous in S′.
- The Lorentz transformation mixes space and time.
- Simultaneity is relative — not absolute.
- The time separation between events depends on their spatial separation and the relative velocity of the frames.
- This effect is quantitatively described by:
Δt′=γc2vΔx
- This principle is essential for properly interpreting relativistic astrophysical phenomena such as AGN jets.
Example: AGN Jet Emission
Suppose an AGN jet emits two light pulses at positions separated by:
Δx=1016 cm
The jet is moving at:
v=0.99c⇒γ≈7.09
In the AGN rest frame (S), the pulses are emitted simultaneously.
In the observer's frame (S′):
Δt′=(7.09)×c20.99c×1016 cm
Δt′≈7.09×0.99×3×1010 cm/s1016 cm≈2.34×106 s≈27 days
Interpretation: In the observer frame, the two "simultaneous" emissions are observed as separated by about 27 days!
- Rindler, W. Introduction to Special Relativity
- Jackson, J.D. Classical Electrodynamics, Chapter 11
- Urry & Padovani (1995), Unified Schemes for Radio-Loud AGN
- Blandford & Königl (1979), Relativistic Jets as Compact Radio Sources