dt2d2xλ+Γμνλdtdxμdtdxν=0 is geodesic equation.
For R2 we have g11=g22=1, g12=g21=0. So since Γμνλ=21gλm(∂xν∂gmμ+∂xμ∂gmν−∂xm∂gμν), we have Γμνλ=0 for every (λ,μ,ν)∈{1,2}3.
Then geodesic equation for R2 is dt2d2x1=0,dt2d2x2=0. We have x1=At+B,x2=Ct+D, that is ACt=C(x1−B)=A(x2−D), so Ax2+Ex1+F=0, where E=−C,F=−AD+BC.
It is equation of straight line.
Answer: straight line between two given points.
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