General Relativity Assignment #3

 

(1)        Download the GRTensor Package for MAPLE and use it to calculate all of the geometrical objects that you calculated by hand in the previous assignment in problem # 7. Hand in all of the input and output data from MAPLE. (NOTE: DO THIS QUESTION ONLY IF EXPLICITLY INSTRUCTED TO DO SO.)

 

(2)        Show that adjusting the pressure in a perfect fluid solution of the Einstein Equations by a constant amount is equivalent to introducing the cosmological constant into the field equations when a perfect fluid is present.

 

(3)        Calculate the equations that are analogous to the equation of continuity and the Navier-Stokes equations of motion for the full General Relativity case when a perfect fluid is present.

 

(4)        Show explicitly that the 4^th differential equation arising from the vacuum field equations is made redundant by the existence of the Bianchi Identity constraint in the Schwarzschild solution.

 

(5)        Explicitly verify all of the details that were left out in the calculation of the isotropic coordinate line element from identification with the Schwarzschild line element.

 

(6)        Transform the Schwarzschild line element using the transformations

 

             

 

            Find the coefficient of . .

 

 

(7)        What is the character of the coordinates of 

(a)  in the line element

 

 

            (b)  in the line element