# Problem deriving failure of simultaneity at a distance

• Roo2
In summary, to derive the failure of simultaneity at a distance, one must use the second equation of the Lorentz transformation and recognize that events that are perceived as simultaneous in one frame of reference may not be perceived as simultaneous in another frame of reference due to time dilation. This is evident in the Michelson-Morley experiment where the null result can be explained solely by spatial Lorentz contraction, but time dilation is necessary to account for the difference in perceived time and position between the observer in the stationary and moving reference frames.
Roo2
Problem deriving "failure of simultaneity at a distance"

## Homework Statement

I'm trying to derive the failure of simultaneity at a distance from the spatial part of the Lorentz transformation. In his treatment of the subject, Feynman glosses over the algebra/derivation and produces the formula:

I tried to start with the first equation of the Lorentz transformation and arrive at the difference in simultaneous time. As seen from my work below, I'm not having much luck. Where is my math incorrect?

On a related note, in reading about the Michelson-Morley experiment, it appears that spatial Lorentz contraction completely accounts for the null result. Why, then, is the time dilation necessary? Wouldn't that double-count the difference in perceived time and position between the observer in the stationary and moving reference frames?

## Homework Equations

The Lorentz contraction, shown above

## The Attempt at a Solution

You need to use the second equation of the Lorentz transformation, and recognize that in the S frame of reference, t1 = t2. That is, as reckoned from the S frame of reference, two events occur simultaneously (i.e., at t1=t2=t) at x1 and x2. As reckoned from the S' frame of reference (i.e., using a set of clocks synchronized in the S' frame of reference), the two events are observed to not occur simultaneously.

## What is the problem of deriving failure of simultaneity at a distance?

The problem of deriving failure of simultaneity at a distance stems from the fact that the concept of simultaneity is relative and can be influenced by the observer's frame of reference. This means that different observers may experience events as simultaneous or not simultaneous depending on their relative positions and velocities.

## How does this problem affect our understanding of time and space?

This problem challenges the traditional understanding of time and space as absolute and fixed. It suggests that time and space are relative concepts that are dependent on the observer's frame of reference.

## What experiments have been conducted to investigate this problem?

One famous experiment that was conducted to investigate this problem is the Michelson-Morley experiment, which aimed to detect the presence of an ether that was believed to be the medium through which light traveled. The results of this experiment showed that the speed of light is constant regardless of the observer's frame of reference.

## How is this problem relevant in modern physics?

The problem of deriving failure of simultaneity at a distance is relevant in modern physics as it is one of the fundamental concepts that led to the development of Einstein's theory of relativity. This theory revolutionized our understanding of space and time and has been confirmed by numerous experiments and observations.

## What are some potential implications of this problem?

This problem has significant implications for our understanding of the universe and its fundamental laws. It challenges our traditional notions of causality and the concept of a universal time frame. It also has practical implications for technologies that rely on precise measurements of time and distance, such as GPS systems.

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