ds²=- Consider the following spacetime: dr² dt2 · + r² (də² + sin² 0 dø²), where > 0 is a constant. (a) Let u t-tanh(r/) for r 0, where the dot denotes the derivative with respect to the affine parameter along the geodesics? Sketch the radial null geodesics in the (u, r) plane for 0
ds²=- Consider the following spacetime: dr² dt2 · + r² (də² + sin² 0 dø²), where > 0 is a constant. (a) Let u t-tanh(r/) for r 0, where the dot denotes the derivative with respect to the affine parameter along the geodesics? Sketch the radial null geodesics in the (u, r) plane for 0
Related questions
Question
Transcribed Image Text:ds²=-
Consider the following spacetime:
dr²
dt2
· + r² (də² + sin² 0 dø²),
where > 0 is a constant.
(a) Let u t-tanh(r/) for r<l. Use the coordinates (u, r, 0,6) to show that
the surface of r = is non-singular. (Hint: Recall that tanh (x)=—-—.)
(b) Show that the vector field gabu is null.
(c) Show that the radial null geodesics obey either
2
du
du
0 or
dr
dr
1-
For r<, which of these families of geodesics is outgoing, i.e., => 0, where
the dot denotes the derivative with respect to the affine parameter along the
geodesics? Sketch the radial null geodesics in the (u, r) plane for 0<r<l, where
the r-axis is horizontal and the lines of constant u are inclined at 45° with respect
to the horizontal.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 3 images
