Question 2 Lorentz Transformation of the Field Strength Tensor (10 points) Note: Use the Sympy simplify() function for t

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Question 2 Lorentz Transformation Of The Field Strength Tensor 10 Points Note Use The Sympy Simplify Function For T 1 (50.51 KiB) Viewed 231 times

Question 2 Lorentz Transformation of the Field Strength Tensor (10 points) Note: Use the Sympy simplify() function for this question when printing the final output of a computation!! In this exercise we shall implement the Lorentz Transformation of the tensor F. We can start by recalling the general Lorentz transformation for a (p, q)- tensor. We define a (p, q)-tensor as a tensor with p contravariant indicies and q covariant indicies an example of this would be, T- Now we say that that a (p, q)-tensor that depends on the 4 vector x transforms under a Lorentz transformation A as follows, VV2-V₂ The (T)¹2-₁2-√(x) = Aª¹₁^²³, (^¯¹)¹₁... (1-¹) TPIP2-PP 01020 (X). ...V и Using this transformation Law answer the following questions. (a) Define the field strength tensor as a sympy matrix using the symbols E₁ = Ex, E₂ = Ey and so on. (2 points) In [4]: # your solution to 2. (a) here!! (b) Using the definition of the field strength tensor above calculate using sympy the Lorentz transformation of FV under a boost in the (i) x direction and (ii) for a boost in the y direction. Hint: The matrix for a boost in the y direction can be obtained from question 1 (a). (4 points) B₂ sinh ($) - Ey cosh (d) -B₂ cosh (p) + Ey sinh (4) -By sinh ($) - E₂ cosh (0) Bycosh()+ Ez sinh () - Bx 0 Bx 0 In [44]: #your solution to 2. (b) (i) here!! Out[44]: 0 Ex –B, sinh() + E, cosh(p) By sinh ($) + E₂ cosh (p) - Ex 0 Bz cosh (¢) – Ey sinh (p) - By cosh (p) - E₂ sinh (4)
(c) Calculate the transformed field strength tensor F, after a boost in the z direction. (4 points) Hint: First transform the tensor F and then lower the indicies using the metric n = diag(+1,-1,-1,-1). You can find the matrix for a boost in the z direction from question 1 (a). In [6]: #your solution to 2. (c) here !!