Question

Consider a two-dimensional geometry, known as the hyperbolic plane, with coordinates (z, y) and line element: ds^2 = g(dx^2 + dy^2), where g = 2(d^2 + dy^2),0. This space is Euclidean, as it has no timelike dimensions. a) What is the distance (as measured with this metric) from a point (z, y) = (0, y) on the y-axis to the point (z, y) = (z, oy) at a finite value of z = zo? Interpret your result. b) Write out the Lagrangian for this metric, using the Euclidean action S = âˆ« ds, and hence obtain the geodesic equations. Hint: The coordinate "time" parameter in this case - call it X - is purely a label along the trajectory, but geodesics still have a meaning as the "straight line" paths in this geometry. You can write ds = LdX, so that S = âˆ« LdX with L = âˆš(dx/dX). c) By comparing the geodesic equations you obtain to the general form below: d^2x show that the nonzero Christoffel symbols are Î“^x_yy = Î“^y_xy = Î“^y_yx = -1/z and Î“^yy_y = 1/z. d) Using the form of the equations, show that the possible geodesics are: (i) semi-circles centered on the y-axis, and (ii) straight lines at constant y.

          Consider a two-dimensional geometry, known as the hyperbolic plane, with coordinates (z, y) and line element:
ds^2 = g(dx^2 + dy^2),
where g = 2(d^2 + dy^2),0.
This space is Euclidean, as it has no timelike dimensions.

a) What is the distance (as measured with this metric) from a point (z, y) = (0, y) on the y-axis to the point (z, y) = (z, oy) at a finite value of z = zo? Interpret your result.

b) Write out the Lagrangian for this metric, using the Euclidean action S = âˆ« ds, and hence obtain the geodesic equations. Hint: The coordinate "time" parameter in this case - call it X - is purely a label along the trajectory, but geodesics still have a meaning as the "straight line" paths in this geometry. You can write ds = LdX, so that S = âˆ« LdX with L = âˆš(dx/dX).

c) By comparing the geodesic equations you obtain to the general form below:
d^2x
show that the nonzero Christoffel symbols are Î“^x_yy = Î“^y_xy = Î“^y_yx = -1/z and Î“^yy_y = 1/z.

d) Using the form of the equations, show that the possible geodesics are: (i) semi-circles centered on the y-axis, and (ii) straight lines at constant y.

10 marks consider a two dimensional geometry known as the hyperbolic plane with coordinates zy and line element ds2gdxdx 2d22dy20 this space is euclideanas it has no timelike dimensions a wh 54859

Added by Kathy S.

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