Sunday, January 30, 2011

Mastering Physics: ± The Space-Time Interval

For this problem, assume that any two frames have the same origin.
Consider two events with coordinates (t_1,x_1) and (t_2,x_2) in an inertial frame of reference S. The coordinates of these two events in an inertial frame S' moving to the right with speed u relative to frame S are (t_1',x_1') and (t_2',x_2').
If we assume that the y and z coordinates are constant in the two frames, then x_2'-x_1' is the space interval, more commonly known as distance, between the events as measured in frame S', and x_2-x_1 is the space interval in the frame S. These two intervals are more commonly written Deltax' and Deltax, respectively. Similarly, t_2'-t_1' is called the time interval between the events in frame S', and t_2-t_1 is called the time interval in frame S. These two intervals are more commonly written Deltat' and Deltat, respectively.

Part A
Find the value of c^2 (\Delta t')^2 - (\Delta x')^2.
Express your answer in terms of gamma, Deltax, Deltat, and c.

  c^2 (\Delta t')^2 - (\Delta x')^2  = c^{2}\left({\Delta}t\right)^{2}-\left({\Delta}x\right)^{2}

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 space-time interval s. To avoid possibly having to take the square root of a negative number, we usually talk about the square of the space-time interval, which is what you actually showed to be conserved in this part: s^2=(c \Delta t)^2 - (\Delta x)^2.
Part B
A pair of events are observed to have coordinates (0\; \rm s,0\; m) and (50.0\; \rm s, 9.00\times 10^9 \; m) in a frame S. What is the proper time interval Deltatau between the two events?
Express your answer in seconds to three significant figures.

  Deltatau  = 40.0
  \rm s
Just as conservation of energy allowed you to solve problems where Newton's laws would have been difficult to work with, conservation of the space-time interval can simplify many relativity problems. In any situation, the proper time between two events will equal the space-time interval divided by c, since the space interval Deltax 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 space-time" or that matter curves space-time. This essentially means that, in the presence of matter, the space-time interval is no longer conserved. The difficult mathematics required to describe how the space-time 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.


  1. np. I'd be adding more to the blog, but my professor stopped giving us Mastering Physics homework a few weeks back.