lecture14

lecture14 - Lecture 14 Notes: 07 / 25 4-vectors Interval...

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Lecture 14 Notes: 07 / 25 4-vectors Interval between events Last time, we implicitly made use of the concept of an event , which is described by the coordinates ( x, y, z ) and the time t . The coordinates and the time vary depending on the observer, and can be translated from one observer's frame to another via the Lorentz transformation. The coordinates and the time together make up a 4-vector X = (ct, x, y, z ). A 4-vector is defined as a 4-component object that takes on different values for different observers according to the Lorentz transformation. Suppose we have two events at X 1 = ( ct 1 , x 1 , y 1 , z 1 ) and X 2 = ( ct 2 , x 2 , y 2 , z 2 ). The separation between these two events is the difference between their coordinates in time and space: Since X 1 and X 2 are 4-vectors, X is a 4-vector as well, and transforms according to the Lorentz transformation. Moreover, it has an invariant length, which is a 4-scalar , meaning that it is the same according to all observers. The invariant length of the interval is known as the invariant interval . Let us consider the meaning of this invariant interval. Suppose that there exists an observer B who is moving in such a way that he is at position ( x 1 , y 1 , z 1 ) at t 1 and at position ( x 2 , y 2 , z 2 ) at t 2 . In this case, according to this observer, both events occur at his own position, that is, at the origin: x 1B = x 2B = y 1B = y 2B = z 1B = z 2B = 0. The observer measures the invariant interval to be Thus s is the proper time between the two events, multiplied by a factor of c . Recall from last lecture that the proper time for a process was the time elapsed for that process according to the observer who is at rest with respect to it. In this case, the positions of the two events are the same according to the observer, so if the events were caused by the same process, then the observer would be at rest with respect to it.
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Thus, the invariant interval s is equal to t 0 , the proper time that elapses between the two events. This is the time from one event to another according to an observer who is present at one event and moves uniformly in a straight line, in such a way as to be present at the next. Defined this way, this quantity is clearly a 4-scalar: different observers might measure different times between these events, but they all agree that an observer moving from one event to another would measure a length of time equal to s . Note that s 2 can be negative, implying an imaginary “proper time” between the two events. This happens when Thus if is s 2 negative, the observer would have to move faster than the speed of light to get from event 1 to event 2. This is not possible; thus there is no proper time between the two events, and s 2 does not have this physical interpretation. However, it can be shown that in this case, there exists an observer for which the two events are simultaneous. - s 2 then gives the square of the distance between the two events
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lecture14 - Lecture 14 Notes: 07 / 25 4-vectors Interval...

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