# Simultaneity in Special Relativity and lorentz transformation

## Main Question or Discussion Point

How do I derive the equation for Simultaneityfrom one of the lorentz transformation.

The textbook focused mainly on the mathematically derivation without using the lorentz transformations, and I cannot find any answers online.

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robphy
Homework Helper
Gold Member
Take two spatially-separated events A and B that are simultaneous in one frame: So, x_A=/=x_B and t_A=t_B. After a Lorentz Transformation, you'll find that in another frame t'_A=/=t'_B.

simultaneity

kevin86 said:
How do I derive the equation for Simultaneityfrom one of the lorentz transformation.

The textbook focused mainly on the mathematically derivation without using the lorentz transformations, and I cannot find any answers online.
Please have a critical look at

arXiv.org > physics > physics/0511062
Physics, abstract
physics/0511062

Illustrating the relativity of simultaneity

Subj-class: Physics Education

We present a relativistic space-time diagram that displays in true magnitudes the readings (date times) of two inertial reference frames clocks. One reference frame is the rest frame for one clock. This diagram shows that two events simultaneous in one reference frames are not compulsory simultaneous in the other frame. This approach has a bi-dimensional character.

Full-text: PDF only

pervect
Staff Emeritus
kevin86 said:
How do I derive the equation for Simultaneityfrom one of the lorentz transformation.

The textbook focused mainly on the mathematically derivation without using the lorentz transformations, and I cannot find any answers online.
Try the wikipedia article on the relativity of simultaneity

http://en.wikipedia.org/wiki/Relativity_of_simultaneity

Basically, events are simultaneous if they have the same t coordinate.

Suppose we have two frames: S1, with coordinates x and t, and S2, with coordinates x' and t'.

Then two events are simultaneous in frame S1 if they have the same t coordinate, i.e. t(event1) = t(event2).

Two events are simultaneous in frame S2 if they have the same t' coordinate, i.e. t'(event1) = t'(event2).

The two sets above are not the same.

Because the Lorentz transform gives t' and x' in terms of t and x, one can determine the equation of a 'line of simultaneity' in S2 in terms of x and t.

Let us find the set of events simultaneous with the origin. Then we have

t' = gamma * (t - v*x/c^2) = 0

This means that the equation of t'=0 is t = vx/c^2 , which defines the equation of the "line of simultaneity" of events in S2 in terms of the coordinates (t,x) of S1.

If you work out a more general example, you'll find that all lines of simultaneity have the same slope on the space-time diagram, which by the example above is slope = dx/dt = c^2 / v.