page_title: Relativity of simultaneity
section_title: Lorentz transformation
hierarchical_section_title: Relativity of simultaneity / Lorentz transformation
context_page_description: In physics, the relativity of simultaneity is the concept that distant simultaneity – whether two spatially separated events occur at the same time – is not absolute, but depends on the observer's reference frame.
context_section_description: The relativity of simultaneity can be demonstrated using the Lorentz transformation, which relates the coordinates used by one observer to coordinates used by another in uniform relative motion with respect to the first.
  Assume that the first observer uses coordinates labeled t, x, y, and z, while the second observer uses coordinates labeled t', x', y', and z'. Now suppose that the first observer sees the second moving in the x-direction at a velocity v. And suppose that the observers' coordinate axes are parallel and that they have the same origin. Then the Lorentz transformation expresses how the coordinates are related:
  where c is the speed of light.  If two events happen at the same time in the frame of the first observer, they will have identical values of the t-coordinate.  However, if they have different values of the x-coordinate (different positions in the x-direction), they will have different values of the t' coordinate, so they will happen at different times in that frame. The term that accounts for the failure of absolute simultaneity is the v x/c².
  The equation t' = constant defines a "line of simultaneity" in the (x', t' ) coordinate system for the second (moving) observer, just as the equation t = constant defines the "line of simultaneity" for the first (stationary) observer in the (x, t) coordinate system. From the above equations for the Lorentz transform it can be seen that t' is constant if and only if t – v x/c² = constant. Thus the set of points that make t constant are different from the set of points that makes t' constant. That is, the set of events which are regarded as simultaneous depends on the frame of reference used to make the comparison.
  Graphically, this can be represented on a space-time diagram by the fact that a plot of the set of points regarded as simultaneous generates a line which depends on the observer.  In the space-time diagram, the dashed line represents a set of points considered to be simultaneous with the origin by an observer moving with a velocity v of one-quarter of the speed of light.  The dotted horizontal line represents the set of points regarded as simultaneous with the origin by a stationary observer.  This diagram is drawn using the (x, t) coordinates of the stationary observer, and is scaled so that the speed of light is one, i.e., so that a ray of light would be represented by a line with a 45° angle from the x axis.  From our previous analysis, given that v = 0.25 and c = 1, the equation of the dashed line of simultaneity is t – 0.25x = 0 and with v = 0, the equation of the dotted line of simultaneity is t = 0.
  In general the second observer traces out a worldline in the spacetime of the first observer described by t = x/v, and the set of simultaneous events for the second observer (at the origin) is described by the line t = vx. Note the multiplicative inverse relation of the slopes of the worldline and simultaneous events, in accord with the principle of hyperbolic orthogonality.
caption: A spacetime diagram showing the set of points regarded as simultaneous by a stationary observer (horizontal dotted line) and the set of points regarded as simultaneous by an observer moving at v = 0.25c (dashed line)