The command "restart" is useful to clear memory from Maple. It is good practice to initialize a worksheet with it.

Transcription

1 The command "restart" is useful to clear memory from Maple. It is good practice to initialize a worksheet with it. Notes: To generate a command line use "ctrl+j" or "ctrl+k" To generate a text line hit "ctrl+t" on an empty command line. To use Maple we will use the following packages. To find information on a certain package type? before the name to learn more about it. For example, F The next command is needed to specify the coordinates and the name of the manifold, I typically just call the manifold M Now to introduce the coframe basis, this is done using the "evaldg" command (1) (2) M M (3) (4)

2 With the orthonormal basis we can construct the metric using "evaldg" Here "&s" denotes the symmetric tensor product. M (5) We are still working with a coordinate basis here, so we can compute the Christoffel symbols (beware that this displays as a tensor in Maple, it is NOT a tensor) M (6) Similarly we can compute the curvature tensor relative to the coordinate basis I have suppressed the output by ending the command with a " : " to see the output, replace this with a " ; ". M I want to work with my frame basis, because this is less painful for my laptop due to the smaller set of components to compute To do this, we use the " FrameData " command, which computes the exterior derivatives of the frame basis elements and represents it in terms of the coframe basis (think: Cartan's 1st structure equation) M (7)

3 (7) Using this, we can pass the output to DGsetup and define the frame, here I have specified that the frame basis is labelled "e" and the coframe basis is labelled "w" Adding ", verbose" causes the output to display this info, you can remove ", verbose" to reduce the output. M The following coordinates have been protected: The following vector fields have been defined and protected: The following differential 1-forms have been defined and protected: frame name: F (8) Since we are working in a frame basis now we have to reintroduce the metric, which is now simplify "eta_{ab}", the metric for Minkowski space (well, it looks like that, we've hidden the metric "under the hood" ) F Now we can compute the connection coefficients relative to the frame basis: F (9) (10) To compute the curvature tensor we use the same command with the metric defined relative to the

4 frame basis "gframe". I have labelled it "Riem" Since the output is pretty big, I have taken the tensor and converted it to an array called "RiemA" This is to make it easier to handle with for-loops and if statements F The code I have written below will range through the four-indices, however I have chosen the "j" and "l" indices so that "i <j" and "k <l " This will reduce the number of repeated components, although there will be repetition when (ij) = (kl). Note: In Maple to make the not-equal sign, type "<" Additionally, I have introduced an if statement that ignores the components of RiemA that are zero. F

5 (11) Let's look at the Vaidya metric I will define the multivariable function "f" to be a specific function: F (12) Here "m(t)" is an arbitrary function of the t coordinate I can now compute the Ricci Tensor, the Weyl tensor and the Riemann tensor (I will suppress the output for Weyl and Riemann because it is rather large). F (13)

6 F F So, Vaidya is a non-vacuum metric, let's now look at Schwarzschild by requiring that "m(t)" is constant F (14) F F (15) (16) F (17) F