# Simultaneity in Special Relativity and lorentz transformation

• kevin86
In summary, this relativistic space-time diagram displays in true magnitudes the readings (date times) of two inertial reference frames clocks. Two events are simultaneous in one reference frame if they have the same t coordinate, but not in the other reference frame.
kevin86
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.

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

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.

## 1. What is simultaneity in special relativity?

Simultaneity in special relativity refers to the concept that the notion of "simultaneous events" is relative and depends on the observer's frame of reference. In other words, two events that may appear simultaneous to one observer may not be simultaneous to another observer who is moving at a different velocity.

## 2. What is the Lorentz transformation?

The Lorentz transformation is a mathematical formula used in special relativity to describe how measurements of space and time change between different inertial frames of reference. It takes into account the effects of time dilation and length contraction, and it is essential for understanding the concept of simultaneity in special relativity.

## 3. How does the Lorentz transformation affect the measurement of time?

The Lorentz transformation shows that time is relative and can appear to pass at different rates for different observers, depending on their relative velocities. This is known as time dilation and is a fundamental aspect of special relativity. It means that two simultaneous events for one observer may appear to occur at different times for another observer.

## 4. How does the Lorentz transformation affect the measurement of space?

The Lorentz transformation also shows that space is relative and can appear differently to different observers depending on their relative velocities. This is known as length contraction and is another fundamental aspect of special relativity. It means that objects in motion may appear shorter in the direction of their motion to an observer compared to a stationary observer.

## 5. Can two events be simultaneous in all inertial frames of reference?

No, according to special relativity, two events cannot be simultaneous in all inertial frames of reference. This is because the concept of simultaneity is relative and depends on the observer's frame of reference. A simultaneous event for one observer may not be simultaneous for another observer who is moving at a different velocity. The only exception is when the two observers are at rest relative to each other.

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