Tensors and Relativity: Assignment 3

Problem


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Peter Dunsby Room 310a Applied Mathematics


  • (1) Prove, by writing out all the terms, the validity of the following

    Let the components of be , those of be and those of be . Find (i) ; (ii) ; (iii) ; (iv) .

    Hint!

     

  • (2) Given the following vectors in :

    (2a) show that they are linearly independent.

    (2b) Find the components of if

    (2c) Find the value of for

    (2d) Determine whether the one- forms , , and are linearly independent if , , , , , , , .

    Hint!

     

  • (3) Let and be two one forms. Prove by trying two vectors and as arguments, that . Then find the components of .

    Hint!

     

  • (4a) Given the components of a tensor as the matrix

    find (i) the components of the symmetric tensor and the antisymmetric tensor ; (ii) the components of ; (iii) the components of ; and (iv) the components of .

    (4b) For the tensor whose components are , does it make sense to speak of its symmetric and antisymmetric parts? If so, define them. If not, say why.

    (4c) Raise an index of the metric tensor to prove that

     

    Hint!

     

  • (5) Suppose A is an antisymmetric tensor, B is a symmetric tensor, C is an arbitrary tensor, and D is an arbitrary tensor. Prove:

    (5a) ;

    (5b) ;

    (5c) .

    Hint!

     

  • (6) In some frame , the four- vector fields and have the components [ with units where c=1 ]:

     

    and the scalar has the value

     

    (6a) Find , , . Is suitable as a 4- velocity field? Is ?

    (6b) Find the spatial velocity of a particle whose four- velocity is , for arbitrary t. What happens to it in the limits , ?

    (6c) Find for all .

    (6d) Find for all and .

    (6e) Show that for all . Show that for all .

    (6f) Find .

    (6g) Find for all .

    (6h) Find and compare with (f) above. Why are the two answers similar?

    (6i) Find for all . Find for all . What are the numbers the components of?

    (6j) Find , . and .

    Hint!

     

 

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