Mathematics 138
Problems due Wednesday, April 13, 2005
Euclid is online at http://aleph0.clarku.edu/~djoyce/java/elements/toc.html
1. Space traveler Buck leaves Earth (position y = 0) at time t = 0.  This is event A.  He travels for
25 years at a speed of vA = 3/5 to reach a star that is 15 years from earth.  This is event B.  He now returns to earth on a slower spaceship whose speed is a mere vC =5/13.  His return is event C. His twin brother Bill stays on earth.
a. Draw a Minkowski triangle showing the triangle made from these three events.  Calculate the length of each side and the value of sinh A, sinh B, and sinh C.  Confirm the Law of Sines numerically.
b. During the trip, Buck ages by c+a, while Bill ages by b. The ratio (c+a)/b is less than 1, and it
decreases if Buck travels faster.  Using the Law of Sines, find an expression for (c+a)/b in terms of
sinh A and sinh C only, then find one in terms of vA and vC only.
c. Write down the 3x3 matrix for the Lorentz transformation that makes line AB horizontal, and use it to calculate the transformed coordinates for events A, B, and C.
2. Determine the bisector of the angle between the line y =  t and the line y = - t. Find the matrix for a Lorentz transformation that carries y = - t into the bisector and that carries the bisector into y =  t.
3a. Show that the vectors v = (13, 5) and w = (20, 16) are both forward timelike vectors with the
same Minkowski length. This means that you can use them to create a "Minkowski rhombus" whose sides are all timelike.
b. Prove that the diagonals of a rhombus are orthogonal.  If you use vectors and the fact that the inner product is distributive, the proof will be valid for both Euclidean and Minkowski geometry, even though a Minkowski rhombus is not a Euclidean rhombus!
c. Draw a diagram to illustrate your theorem for the case of the rhombus made from v and w of part a. Remember that in Minkowski geometry, lines are orthogonal if the product of their slopes is 1, which means that one line must be timelike and the other spacelike.
d. Another special type of paralleolgram in Minkowski geometry is one that has a lightlike diagonal. Prove that for such a parallelogram, the timelike diagonal has a squared length equal to twice the sum of the squares of the lengths of the two sides.  State this theorem in physical terms. The plot begins, "Buck and Bill set off from Earth at constant velocities v1 and v2 respectively.  After a while, Buck sends a light signal to Bill and changes his velocity to v2. When Bill receives the signal, he changes his velocity to v1. Eventually Buck catches up to Bill and discovers to his surprise that sister Betty, who left Earth at the same time but never changed her velocity, has also just caught Bill. When the three siblings compare their ships' logs, they discover that..."
e. Confirm this theorem numerically for the simple case where v1 = 0, v2 = 3/5, and Bill waits two years befoere sending the light signal to Buck.

4. Suppose that you possess a straightedge and a "Minkowski compass" that permits you to draw a circle (a Euclidean hyperbola) with a timelike radius. Tracing the hyperbolas in the lecture notes will work in practice. Draw diagrams to show how you would use this to carry out the Minkowski versions of two classic constructions of Euclid, as follows:
a. Constructing the perpendicular bisector of a timelike segment.  Draw this segment with slope 1/2, and the bisector should have slope 2.
b. Constructing the bisector of the angle between two timelike lines.  Choose slopes of 0 and 4/5 for the lines, and the angle bisector should have slope 1/2.
5. A good way to parameterize one branch of a type 2 hyperbola with radius 1 is 
t = sinh s   y = cosh s. Then t2 - y2 = -1: the radius is spacelike and has length 1.
a. Use this parameterization to prove that the tangent to the hyperbola at any point is Minkowski-orthogonal to the diameter.
b. Prove that the Euclidean area (which is also the Minkowski area) of the region bounded by the
hyperbola , the line t = 0, and the line joining the origin to the point t = sinh S, y = cosh S) is S/2.
Hint: use polar coordinates, and make the change of variable tan q = tanh s.  The derivative of tanh s is sech2 s.
c. The Minkowski length of an arc of the hyperbola is given by  . Show that this antiderivative equals s. This means that the parameter s is what physicists call "proper time," the time that elapses on a clock whose "world line" is the hyperbola.
6. Another of Euclid's wonderful theorems about circles is Book III, Proposition 35,
"If in a circle two straight lines cut one another, then the rectangle contained by the segments of the one equals the rectangle contained by the segments of the other."
a. Prove this theorem.  You could look up Euclid's proof and use it, but you can invent a shorter proof based on the theorem that the two lines joining a point on a circle to the ends of a chord make a constant directed angle with one another.
b. State and prove a Minkowski version of this theorem, involving two intersecting chords of one branch of a type 2 circle (hyperbola).
7a. Find the type 2 Minkowski circle (hyperbola) that passes through the three events
A: t = 0, y = 0
B: t =10, y = 2
C: t = 32, y = 16
Of course, the center of this circle is the intersection of the perpendicular bisector of two of its sides. See Euclid, Book IV, Proposition 5. This construction is illustrated on the Web site.
b. How is the radius of this circle related to the value of for the triangle determined by the three events?

8. Find the type 2 Minkowski circle (hyperbola, with two branches) that is tangent to the three sides of the isosceles triangle formed by events
A: t = 0, y = 0
B: t = 20, y = 0
C: t = 10, y = 6
Determine the radius and center of the circle, find the points of tangency, and draw a diagram showing the circle and the triangle.  Hint: Euclid solves the Euclidean version of this problem in Book IV, Proposition 4. There is a diagram on the Web site that illustrates the construction.
9a. Suppose that A and B are two points on one branch of a type 1 circle (hyperbola with timelike radius) and C is a point on the other branch.  Prove that the (Minkowski) angle between the timelike lines AC and BC is independent of C.  This is one Minkowski version of Yaglom's alternative definition of a circle in terms of segment and angle measure.

b. A Minkowski theorem, related to Euclid's Book III, Proposition 36, that involves only timelike
lines, is the following:
Take the two branches of a type 1 circle (hyperbola) with timelike radius.  Consider two timelike lines, one of which intersects the left branch at A and the right branch at B, and the other of which intersects the left branch at C and the right branch at D. Suppose the two chords intersect at a point M between the branches of the hyperbola.
Then (AM)(MB) = (CM)(MD).

Prove it. Hint: Draw AD and BC, then use part a and the Law of Sines