Sophie Zabar, clairvoyante, cried out in pain at precisely...

Sophie Zabar, clairvoyante, cried out in pain at precisely the instant her twin brother, $500 \mathrm{km}$ away, hit his thumb with a hammer. A skeptical scientist observed both events (brother's accident, Sophie's cry) from an airplane traveling at $\frac{12}{13} c$ to the right (see Fig. 12.19 ). Which event occurred first, according to the scientist? How much earlier was it, in seconds?

Key Concepts

Inertial Reference Frames

Inertial reference frames are coordinate systems in which objects either remain at rest or move at constant velocity unless acted upon by external forces. In the context of special relativity, analyzing events from different inertial frames can lead to different measurements of time and space intervals, which is a key aspect of understanding how observers in relative motion perceive events differently.

Lorentz Transformation

Lorentz transformations are the set of equations that relate the space and time coordinates of events in different inertial frames moving relative to each other at a constant velocity. This mathematical framework is essential for translating event timings and positions between observers, thereby accounting for effects like time dilation and the relativity of simultaneity.

Relativity of Simultaneity

The relativity of simultaneity is the principle that the concept of two events occurring at the same time is not absolute but depends on the observer's frame of reference. In special relativity, events that are simultaneous in one frame may not be simultaneous in another moving frame, which is a direct result of the finite speed of light and the structure of spacetime encapsulated by the Lorentz transformations.

Special Relativity

Special relativity is the theory developed by Einstein that transformed our understanding of space and time. It postulates that the laws of physics are the same in all inertial frames and that the speed of light is constant in all these frames, leading to phenomena such as time dilation, length contraction, and the relativity of simultaneity.

Transcript

00:01 In this problem we're looking at a relativistic airplane with a scientist on it.

00:06 We have a brother and a sister.

00:08 The sister, sophie, is supposed to be a clairvoyant.

00:12 The brother has an accident and sophie claims to react to it instantaneously from 500 kilometers away.

00:21 The problem wants to know if the scientist agrees that they happened at the same time or if one happened before the other from the scientist's frame of reference.

00:31 So the brother and sophie are in frame s and the scientist is in frame s bar because the plane is moving at 12 thirds the speed of light.

00:53 So we take the brother's accident to be time t equals zero in both frames.

01:20 So that means that in frame s since they happen at the same time in frame s, sophie's prize t equals zero.

01:41 X equals 500 kilometers.

01:58 So to find the time in the airplane's frame of reference moving at relativistic speed we're finding t bar and that is the lorentz transform.

02:23 So it's gamma times t minus v over c squared x.

02:42 That's equation 12 .18 in the textbook.

02:52 So that's equal to zero by definition or by choice of the coordinates which be used.

03:10 So that means we have minus t bar is equal to minus gamma v x over c squared.

03:51 So this is negative.

03:53 So this that means sophie cried in the scientist's frame of reference.

04:28 So then the question is how much.

04:31 So to calculate that we're finding t with a bar.

04:39 We have to calculate the gamma term first...

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