Relativity of Simultaneity

The relativity of simultaneity is a fundamental consequence of special relativity stating that whether two spatially separated events occur at the same time is not absolute, but depends on the observer's inertial frame of reference.

Mathematical Formulation

Consider two events $A$ and $B$ in an inertial frame $S$, with coordinates $(t_A,x_A)$ and $(t_B,x_B)$. If these events are simultaneous in $S$, then $\mathrm{\Delta }t=t_B-t_A=0$.

According to the Lorentz boost, the time interval $\mathrm{\Delta }{t}^{\prime }$ in a frame $S^{\prime }$ moving with relative velocity $v$ along the $x$-axis is given by:

where $\gamma =\frac{1}{\sqrt{1-v^2/c^2}}$ is the Lorentz factor and $\mathrm{\Delta }x=x_B-x_A$. Substituting $\mathrm{\Delta }t=0$:

Since $v\ne 0$ and $\mathrm{\Delta }x\ne 0$ for spatially separated events, $\mathrm{\Delta }{t}^{\prime }\ne 0$. Thus, the events are not simultaneous in frame $S^{\prime }$. The sign of $\mathrm{\Delta }{t}^{\prime }$ depends on the sign of $\mathrm{\Delta }x$ and the direction of $v$, leading to the "leading clocks lag" effect.

Geometric Interpretation in Minkowski Space

In the framework of Minkowski space, an observer's definition of "now" corresponds to a three-dimensional hyperplane of simultaneity that is orthogonal (in the sense of the Minkowski metric) to their worldline (four-velocity vector).

Because different inertial observers have non-parallel worldlines, their respective hyperplanes of simultaneity intersect at different angles. This implies that:

• Events that lie on one observer's hyperplane of simultaneity do not, in general, lie on another's.
• There is no unique, universal slicing of spacetime into "time-slices" (foliation) that all observers agree upon.

Physical Implications

1. Lack of Absolute Chronology: There is no objective way to order events that are spacelike separated. For any two spacelike separated events $A$ and $B$, one can find an inertial frame where $A$ precedes $B$, one where $B$ precedes $A$, and one where they are simultaneous.
2. Causality: The relativity of simultaneity does not violate causality because it only applies to events with a spacelike invariant interval. Events that are causally connected (timelike or lightlike separation) have a temporal order that is invariant across all inertial frames.

See Also

• Andromeda paradox