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Tensors and Relativity: Assignment 5

# Problem

### Peter Dunsby Room 310a Applied Mathematics

• (1) A rocket of height h undergoes a constant acceleration g. Light rays emitted from the top at t=0 and reach the bottom at times and . Prove that, so long as (i.e. the rocket moves non- relativistically), one has

• (2) In 2D Cartesian coordinates

is a scalar and

are two vectors.

(2a) Compute f as a function of polar coordinates and find the components of and on the polar basis, expressing them as functions of r and .

(2b) Find the components of in Cartesians and obtain them in polars by (i) direct calculation in polars, and (ii) transforming from Cartesian coordinates.

(2c) Use the metric tensor in polar coordinates to find the polar components of the one- forms and associated with and . Obtain these components by transformation of the Cartesian coordinates of and .

• (3) For the vector field of question (2) above, compute;

(3a) in Cartesian coordinates;

(3b) the transformation to polars;

(3c) the components using the Christoffel symbols given in the lecture notes [ Why is this the same as (3b)?];

(3d) the divergence in Cartesian coordinates;

(3e) the divergence in polars using part (3c);

(3f) and the divergence using the formula given in the lecture notes.

• (4) Compute for all possible indices in polar coordinates for the tensor with polar components (,, , )

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