Riemann Tensor

  1. Find an expression for
    ∇c∇dT
    a
    b − ∇d∇cT
    a
    b
    (1)
    in terms of the Riemann tensor.
  2. Suppose we lived in a universe that was not isotropic, but instead had
    the cylindrical metric
    ds2 = −dt2 + dz2 + f
    2
    (z)(dr2 + r
    2
    dφ2
    ) (2)
    for some function f(z).
    a) Find the nonzero Ricci tensors in terms of f(z) and its derivatives.
    b) Find the Ricci scalar in terms of f(z) and its derivatives.
    c) Find the nonzero components of the energy momentum tensor in terms of
    f(z) and its derivatives.
  3. Consider a Riemann tensor on the form
    Rµνσκ = λ(gµσgνκ − gµκgνσ) (3)
    where gµσ is the metric and λ is a positive constant.
    a) Show that Rµνσκ has the proper symmetry properties when interchanging the indices.
    b) Find the Ricci tensor, Ricci scalar and Einstein tensor in terms of the metric.
    c) Show that the Einstein tensor satisfies the Bianchi identity(i.e. ∇µGµν = 0?).
    (Note, every space with Rµν ∝ gµν is called an Einstein space.)
    4.(CH 4. Lecture notes) Using the weak field approximation, find the following equation.
    a) Γµ
    νρ =
    1
    2
    (h
    µ
    ν,ρ + h
    µ
    ρ,ν − h

    νρ )
    b) Rµ
    νρσ =
    1
    2
    (h
    µ
    ν,σρ + h
    µ
    σ,νρ + h

    νρ σ − h
    µ
    ν,ρσ − h
    µ
    ρ,νσ − h

    νσ ρ)
    c) Rµν = −
    1
    2
    (h
    σ
    µν,σ + h
    σ
    σ,µν − h
    σ
    µ,νσ − h
    σ
    ν,µσ)
  4. Using the following equation,
    T
    µν =
    1
    µ0
    
    F
    µ
    σF
    νσ −
    1
    4
    g
    µνF
    ρσFρσ
    , (4)
    calculate the each components of the Electromagnetic stress−energy tensor T µν in a matrix form.
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Gender Issue

The Questions

  1. Describe the gender issue that both organizations/groups address. Discuss the history of that gender issue, need to do additional research for this portion of the assignment.
  2. Connect the gender issue to contemporary social topics and explain how this issue effects our society and/or our globalized world, need to do additional research for this portion of the assignment.
  3. Describe in detail how the issue is related to one of the chapter’s topic in the book of “Gendered lives: communication, gender, and culture.”( written by J. Wood)
  4. Describe the origins, history, membership, and goals of each organization/group.
  5. Describe the stance and beliefs of each organization/group on that same gender issue. How are the organizations similar and/or different from each other and how is each organization trying to address that specific gender issue.
  6. Describe the cultural impact of each organization/group.
  7. Explain what you learned about gender by studying this issue and the 2 different groups/organizations your chose.

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