Consider two events with coordinates and in an inertial frame of reference S. The coordinates of these two events in an inertial frame S' moving to the right with speed relative to frame S are and .
If we assume that the y and z coordinates are constant in the two frames, then is the space interval, more commonly known as distance, between the events as measured in frame S', and is the space interval in the frame S. These two intervals are more commonly written and , respectively. Similarly, is called the time interval between the events in frame S', and is called the time interval in frame S. These two intervals are more commonly written and , respectively.
Part A  

Find the value of . 
Express your answer in terms of , , , and .
ANSWER: 


In classical physics, the space interval and time interval are each separately conserved. In special relativity, however, the conserved quantity is a mixture of space and time called the spacetime interval . To avoid possibly having to take the square root of a negative number, we usually talk about the square of the spacetime interval, which is what you actually showed to be conserved in this part: .
Part B  

A pair of events are observed to have coordinates and in a frame S. What is the proper time interval between the two events? 
Express your answer in seconds to three significant figures.
ANSWER: 


Just as conservation of energy allowed you to solve problems where Newton's laws would have been difficult to work with, conservation of the spacetime interval can simplify many relativity problems. In any situation, the proper time between two events will equal the spacetime interval divided by , since the space interval is, by definition, equal to zero in the frame where the proper time is measured.
You may have heard that in the general theory of relativity (the relativistic description of gravity), gravity is described by a "curvature of spacetime" or that matter curves spacetime. This essentially means that, in the presence of matter, the spacetime interval is no longer conserved. The difficult mathematics required to describe how the spacetime interval changes, as well as the conceptual leaps needed to understand how the shape of space and time can be changed, led to the widely quoted figure that five years after the theory had been put forward, only three people on earth understood it.
You may have heard that in the general theory of relativity (the relativistic description of gravity), gravity is described by a "curvature of spacetime" or that matter curves spacetime. This essentially means that, in the presence of matter, the spacetime interval is no longer conserved. The difficult mathematics required to describe how the spacetime interval changes, as well as the conceptual leaps needed to understand how the shape of space and time can be changed, led to the widely quoted figure that five years after the theory had been put forward, only three people on earth understood it.
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ReplyDeletenp. I'd be adding more to the blog, but my professor stopped giving us Mastering Physics homework a few weeks back.
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