1-1 Expressions and Formulas s2

The scalar product and the metric of flat spacetime

PROBLEM SET

PROBLEM 1. Show that the following relationships hold between the basis four-vectors. Hint: Look at Eq. 5 of the lecture.

et × et = -1,

ex × ex = 1,

ey × ey = 1,

ez × ez = 1,

ea × eb = 0 for a ¹ b.

PROBLEM 2. Cite evidence that the basis four vectors are orthogonal.

PROBLEM 3. Cite evidence that the basis four-vectors are of unit length.

PROBLEM 4. If we digress, for a moment, and look back at three-space, Eq. 5 of the lecture would reduce to

hmn = .

If we define the scalar product in Euclidean space as a × b = hmnambn and add the further restriction that

a2 = a × a = (ax)2 + (ay)2 + (az)2,

find the values for all of the hmn, i.e.: find the metric for Euclidean space. Then express the Euclidean line element in terms of hmn.

PROBLEM 5. Let a = (cat, ax, ay, az) and b = (cbt, bx, by, bz). Find the scalar product of a and b using a × b = habaabb.

PROBLEM 6. Is the matrix for hab symmetric about its main diagonal, as required? Why did it turn out to be symmetric? Hint: Look at the line element for Minkowski space.

PROBLEM 7. Suppose there is a space having a line element given by

ds2 = -(cdt)2 + dx2 + dxdy + dydx + dy2 + dz2.

What do you predict the matrix xab will look like? We use xab instead of hab because they will not be the same.

PROBLEM 8. Suppose there is a space having a line element given by

ds2 = -(cdt)2 + dx2 + dxdy + dy2 + dz2.

What do you predict the matrix xab will look like? Hint: Remember we require symmetry about the main diagonal. Thus, dxdy can be written dxdy/2 + dydx/2 to maintain symmetry.

PROBLEM 9. Suppose there is a space having a line element given by

ds2 = -(cdt)2 + dr2 + r2dq2 + r2sin2q df2.

What do you predict the matrix gab will look like? Hint: We are just using t, r, q, and f instead of t, x, y, and z. Then use ds2 = gabdxadxb,

where we define dx0 º cdt, dx1 º dr, dx2 º dq, dx3 º df.