I have recently started a course on general relativity, following some classical textbooks, where I am learning how to describe the curvature of spacetime in a nutshell. I came across an interesting exercise proposed by a professor, where one must compute the geodesics for a given 2×2 metric. In this document, I will solve this exercise as part of a new series of posts where I document my progress in general relativity.
Geodesics equations
We consider the coordinates:
x1=xx2=y
The metric tensor is given by:
gij=(−1−yb001+yb1)
Our goal is to derive the geodesic equations for these coordinates using variational principles:
dsdum+21gmi(∂kgip+∂pgik−∂igkp)ukup=0
Where:
u1u2=dsdx1=dsdx=dsdx2=dsdy
Since gmi is the inverse of gij, it satisfies:
gijgmi=δjm
The inverse metric tensor is computed as:
gmi=(−1+yb1001+yb)
Geodesic equation for x
For the coordinate x1=x, the only relevant term in the geodesic equation is g11:
ds2d2x+21g11(∂kg1p+∂pg1k−∂1gkp)ukup=0.
The only nonzero Christoffel term arises when k=2 and p=1:
∂2g11+∂1g12−∂1g21=y2b
Thus, the geodesic equation for x simplifies to:
ds2d2x−21(1+yb)y2bdsdydsdx=0
Geodesic equation for y
Similarly, for the coordinate y, only the g22 term contributes, and the relevant partial derivatives are:
p=1,k=1p=2,k=2⇒−∂2g11=−y2b,⇒∂2g22+∂2g22=2∂2g22=(1+yb)2y22b.
Thus, the geodesic equation for y takes the form:
ds2d2y−21y2b(1+yb)(dsdx)2+(1+yb)y2b(dsdy)2=0