# PHI321 Spacetime problems

 Due handed at the beginning of class on Thursday March 3. A particle moves back and forth along a line, increasing in speed. Graph. How many equivalence classes in Galilean spacetime are there for a particle that is at rest? A particle that is moving at a constant speed? Why are the previous two questions trick questions? In Galilean spacetime, there is no such thing as absolute velocity. Is there such a thing as absolute acceleration? If not, why not? If so, describe a spacetime in which there is no notion of absolute acceleration. Hint: to move from Aristotelian spacetime to Galilean spacetime, we got rid of the notion of absolute velocity by counting two graphs as equivalent (picturing the same spacetime) if they differed by a shear transformation. Perhaps we can get rid of absolute acceleration with an analogous move? Draw a two-dimensional Cartesian grid. Label the axes x and t, and mark a scale on these axes. Make the x axis the horizontal axis, and the t axis the vertical one. Pick two points that are not on the same vertical line. Name them Ann and Bob. Label each point with its x and t coordinates. Definition: The interval between a=(x1,t1) and b=(x2, t2) is defined by I(a,b) = (x1-x2)^2 - (t1-t2)^2. It's convenient to write this as Delta{x}^2 - Delta{t}^2. Note also that the definition is reminiscent of--though different than--the definition of squared distance. Calculate the interval between Ann and Bob. If it's negative or zero, move Bob so that the interval between Ann and Bob is positive. Draw a new, empty grid with the same scale. Redraw Ann and Bob so that the following conditions are met: (a) Ann is at the origin (b) Bob has the same t-coordinate as Ann (c) The interval between Ann and Bob is the same on the new grid as on the old grid. Draw a new, empty grid with the same scale. Mark a point on the grid, and name it Fred. Shade the set of points that are at a positive interval from Fred. Draw stripes over the set of points that are at a negative interval from Fred. Mark in dark black the set of points that are at zero interval from Fred. Mark with a squiggly line the set of points that are at an interval of exactly 9 units from the origin. Mark with a sawtooth line the set of points that are at an interval of exactly -9 units from the origin. Draw the path of a particle that moves rather slowly. Estimate the total time elapsed along the path by drawing a series of connected line segments that approximate the path, and adding up the "length" of each line segment. (In this context, the "length" of a segment is the square root of the absolute value of the interval from one end of it to the other. Think of this on analogy with the following fact: the length of a line segment is the square root of the squared distance from one end of it to the other.) Do the same for a particle that moves back and forth at speeds approaching the speed of light. Draw a spacetime diagram of the following story. At time 0, Tweedle and Beedle are both at the origin (O), and both of their watches read: 0 time elapsed. Tweedle just sits there (draw his path through spacetime as a vertical line), but Beedle launches to the right at 3/4 of the speed of light. When Beedle's watch reads "4 units of time elapsed", he turns around (at spacetime point B) and returns at 3/4 of the speed of light. When Beedle gets back to Tweedle, what does Tweedle's watch read? (In other words, what is the time elapsed along the path that Tweedle traverses from A to C?) Draw a spacetime diagram of the following story. At spacetime point L (x=-3, t=-3) a photon is shot to the right. At spacetime point R (x=3, t=-3) a photon is shot to the left. The photons intersect at the origin (O). A (point-sized) train moving to the right at a constant velocity of 1/2 the speed of light passes through x=0 just as the two photons intersect there. Meanwhile, an observer just sits stationary at the origin, with a clock that reads "0 time elapsed" when the two photons hit him. Mark as "A" the spacetime point at which the observer's clock reads "4 units of time elapsed". Draw new axes, and mark point "O" at the origin. Redraw the story above in such a way that the path of the train on the new diagram is a vertical line, and in such a way that all intervals are preserved. In other words, make the interval between any two points in the new diagram be the same as the interval between those two points in the old diagram. Draw a spacetime diagram of a bar (call it "Ben") at rest with its left edge at x=0. Calculate the length of Ben (i.e., choose a time, and calculate the difference in x-coordinate between the position of the left edge of Ben at that time, and the position of the right edge of Ben at that time). Add another bar (call it "Bessy") that moves at 4/5 the speed of light to the right, and whose left side passes through x=0 at t=0. Calculate the length of Bessy. Draw a new diagram that represents the same situation from the reference frame of Bessy. I.e., redraw the diagram so that Bessy stays at rest at the origin, and so that intervals are preserved. Calculate the length of Bessy in this new frame (i.e., choose a time, and calculate the difference in x-coordinate between the position of the left edge of Bessie at that time, and the position of the right edge of Bessy at that time). Calculate the length of Ben in this new frame. Adam Elga | adame@princeton.edu | Princeton University Last modified: Wed Mar 2 21:40:57 EST 2005