(a) Show that the coordinate transformation (24.60a) brings the metric ds 2 = dx ...

(a) Show that the coordinate transformation (24.60a) brings the metric ds² = η_αβdx^αdx^β into the form of Eq. (24.60b), accurate to linear order in separation x_ĵ from the origin of coordinates.

(b) Compute the connection coefficients for the coordinate basis of Eq. (24.60b) at an arbitrary event on the observer’s world line. Do so first by hand calculations, and then verify your results using symbolic-manipulation software on a computer.

(c) Using the connection coefficients from part (b), show that the rate of change of the basis vectors e_α̂ along the observer’s world line is given by Eq. (24.61).

(d) Using the connection coefficients from part (b), show that the low-velocity limit of the geodesic equation [Eq. (24.67)] is given by Eq. (24.68).

Equations.

Transcribed Image Text:

x² = x³ + {a^{x®² + ²x²x²x®, xº = x²(1+ajx¹). (24.60a)