Appendix A Riemann Curvature Tensor - Springer 978-3-319-01842-3/1.pdf244 Appendix A: Riemann...

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Transcript of Appendix A Riemann Curvature Tensor - Springer 978-3-319-01842-3/1.pdf244 Appendix A: Riemann...

  • Appendix ARiemann Curvature Tensor

    We will often use the notation (...), and (...); for a partial and covariant derivativerespectively, and the [anti]-symmetrization brackets defined here by

    T[ab] := 12

    (Tab Tba) and T(ab) := 12 (Tab + Tba) , (A.1)

    for a tensor of second rank Tab, but easily generalized for tensors of arbitrary rank.We can define the Riemannian curvature tensor in coordinate representation by

    the action of the commutator of two covariant derivatives on a vector field v

    [,] v = R v, (A.2)

    with the explicit formula in terms of the symmetric Christoffel symbols

    R = + . (A.3)

    From this definition it is obvious that R possesses the following symmetries

    R = R = R = R . (A.4)

    In addition, there is an identity for the cyclic permutation of the last three indices

    R[] = 13 (R + R + R) = 0. (A.5)

    An arbitrary tensor of fourth rank in d dimension has d4 independent components.Since a tensor is built from tensor products, we can think of R as being composedof two (d d) matrices A1 and A2 . From (A.4), it follows that A1 and A2 areantisymmetric, each having n(n 1)/2 independent components. Writing R RA1 A2 , where we have collected the index pairs () A1 and () A2,this corresponds to a matrix, in which each index A1 and A2 labels d(d 1)/2

    C. F. Steinwachs, Non-minimal Higgs Inflation and Frame Dependence 243in Cosmology, Springer Theses, DOI: 10.1007/978-3-319-01842-3, Springer International Publishing Switzerland 2014

  • 244 Appendix A: Riemann Curvature Tensor

    components. Taking into account the symmetry under pairwise exchange of ()and (), we can consider RA1 A2 as a symmetric matrix in A

    1 and A2, havingA1(A1 + 1)/2 independent components. Altogether, we find

    1

    2

    [1

    2d (d 1)

    ] {[1

    2d (d 1)

    ]+ 1

    }= 1

    8d4 1

    4d3 + 3

    8d2 1

    4d (A.6)

    independent components. We still need to figure out how many components arerelated by the cyclic identity (A.5) in d dimensions. We can artificially write

    R = 18[R R + R R + () ()

    ](A.7)

    and similar expressions for R and R . Inserting these expressions into (A.5)leads to the condition R[] = 0. This totally antisymmetric object is identicalzero for two identical indices but gives one additional constraint for each choice offour distinct, orderless indices, reducing the number of independent components byone respectively. In d dimensions, the number of additional constraints correspondsto choose 4 out of d. For integers d, n there exists a product formula to calculatethe binomial (

    dn

    )=

    nk=1

    d n + kk

    . (A.8)

    Inserting n = 4, we obtain(

    d4

    )= 1

    24[d (d 1) (d 2) (d 3)] = 1

    24d4 1

    4d3 + 11

    24d2 1

    4d. (A.9)

    Thus, the number of independent components IRiem(d) of R is given by

    IRiem(d) := 14

    d (d 1)[

    1

    2d (d 1) + 1

    ]

    (d4

    )= 1

    12d2 (d2 1). (A.10)

    We consider IRiem(d) for different dimensions d:

    IRiem(1) = 0, IRiem(2) = 1, IRiem(3) = 6, IRiem(4) = 20. (A.11)

    This shows that gravity in one dimension is trivial, since there is no dynamical degreeof freedom. It further shows that gravity in two dimensions can be described by theRicci scalar R and in three dimensions by R(3) = R(3) . In general, the d-dimensionalRicci tensor R := gR has IRic(d) := 12 d (d + 1) independent components.In particular IRic(4) = 10. Thus, ten components of R not contained in R stillremain. They are contained in the Weyl tensor

  • Appendix A: Riemann Curvature Tensor 245

    C := R 2d 2

    (g[R] g[R]

    ) + 2(d 1) (d 2) g[g] R.

    (A.12)It follows that the Weyl tensor has (for d > 3)

    IWeyl(d) = IRiem(d) IRic(d) = 112

    d (d + 1) (d + 2) (d 3) (A.13)

    independent components, and in particular IWeyl(4) = 10.

  • Appendix BVariations and Derivatives

    B.1 Covariant Differentiation in General Relativity

    The action of the metric compatible covariant derivative g = 0 with respect tothe Christoffel symbol on a general tensor is defined as

    T ...... = T ...... + T ...... + ... + T ...... T ...... ... T ......, (B.1)

    with the symmetric Christoffel symbol

    (g) :=1

    2g

    ( g + g g

    )= (g). (B.2)

    For scalar functions = . For vector fields v, a useful formula is

    v = 1g

    (g v

    )or

    g v =

    (g v

    ). (B.3)

    B.2 Functional Derivative

    We use again the condensed DeWitt notation for a generalized field i = A(x),introduced in Sect. 4.4. We want to emphasize that an object with two DeWitt indiceslike i j corresponds to a two-point function or a generalized bi-tensor AB(x, x )in conventional notation. The primed indices like j refer to the space-time point x .This means we can construct objects which behave as tensors of different rank atdifferent space-time points. A particular case of a generalized bi-tensor is a bi-scalarwhich has no indices A, B, .. at all. A special bi-scalar, in turn, is the Dirac Delta-Distribution (x, x ) which is defined in a 2-dimensional curved space-time withmetric g(x) for a general test field A(x) by the equation

    C. F. Steinwachs, Non-minimal Higgs Inflation and Frame Dependence 247in Cosmology, Springer Theses, DOI: 10.1007/978-3-319-01842-3, Springer International Publishing Switzerland 2014

    http://dx.doi.org/10.1007/978-3-319-01842-3_4

  • 248 Appendix B: Variations and Derivatives

    A(x) =

    d2x |g(x )|1/2 (x, x ) A(x ), (B.4)

    For simplicity, we specify 2 = 4. In the condensed notation (B.4) takes the form i = i j j

    with

    ij = |g(x )|1/2 AB (x, x ) = AB (x, x ) . (B.5)

    The quantity (x, x ) := |g(x )|1/2(x, x ) is no longer a bi-scalar, since it transformsas a scalar at the point x , but as a scalar density at the point x . The expansion ofZ [] up to linear order in condensed notation is defined by

    Z [ + ] =: Z [] + Z, i [] i , (B.6)

    with i := i and

    Z, i [] i =

    d4x Z [(x)] A(x )

    A(x ) (B.7)

    in conventional notation. Therefore, it seems obvious to define the functional deriv-ative Z, i = Z [(x)]/ A(x ) by the variation [3]

    lim0

    1

    (Z [ + ] Z []) =:

    d4x Z [(x)]

    A(x ) A(x ). (B.8)

    For the special case Z = Id, we obtain from (B.8)

    A =

    d4x A(x)

    B(x )B(x ) or

    A(x)

    B(x )= AB (x, x ). (B.9)

    B.3 Lie Derivative

    The Lie derivative L of an arbitrary tensor T ...... in the direction of the vector isa measure of the difference between T

    ...... (x

    ) dragged along the integral curve of compared to the tensor T

    ...... (x

    ) at this point. Calculating the infinitesimalflow of T

    ...... (x

    ) along back from x to x with the infinitesimal coordinatetransformation matrix x

    /x = + , yields

    T ...

    ... (x) = T ...... +

    (,T

    ...... + ... + ,T

    ......

    , T

    ...... ...

    , T

    ......

    ).

    (B.10)

    Taylor expansion of T ...... (x) around x yields

  • Appendix B: Variations and Derivatives 249

    T ...... (x + ) = T ...... (x) + T ......, . (B.11)

    Since the right-hand sides of (B.10) and (B.11) involve only quantities at x , we cancompare these two objects at x and define the Lie derivative as the difference

    (L T )...... (x) := lim0T ...... (x

    ) T ...... (x )

    = T ......, T ...... , ... T ...... ,+ T ...... , + ... + T ...... , . (B.12)

    A special case is the Lie derivative of the metric tensor. The vanishing of the Liederivative

    (L g) = 0 in direction signalizes a symmetry of the space-timemanifold and means that the vector is a generator of an isometry

    (L g) = g, + g , + g , = ; + ; = 0. (B.13)The vector fields obeying (B.13) are called Killing vector fields.

    B.4 Variation of Metric Quantities

    In the case of pure gravity, i = g(x) and i = h(x), we find from (B.9)g(x)

    g(x )= (x, x ) with := () =

    1

    2

    (

    +

    ). (B.14)

    By using (B.8) with F() = F(g) = (g)1 and g := (g)1, it follows

    g g = g( g) h . (B.15)

    By using (B.8) with F(g) = g1/2 and the identity det(g) = exp { tr[log(g)] }, wefind

    g g1/2 = 1

    2g1/2 g h . (B.16)

    By using (B.4) with an arbitrary test function t (x) = 0 that does not depend on gand differentiating both sides, we find

    d4x

    [

    g(y)(x, x )

    ]t (x ) = 0 or

    g(y)(x, x ) = 0. (B.17)

    Combining (B.14) with (B.17) leads to

  • 250 Appendix B: Variations and Derivatives

    2g(x)

    g(y) g(z)= 0 or 2g g = g h = 0. (B.18)

    With the basic results (B.15), (B.16) and (B.18), we can construct the variation ofmore complicated objects. Using the fact that the operations of variation and partialdifferentiation commute, we calculate the variation of the Christoffel symbol

    g =

    1

    2gg

    ( g + g g

    )h

    + 12g

    ( h + h h

    ). (B.19)

    Using (B.1), we rewrite the s acting on the hs in terms of the s

    g = g h +

    1

    2g

    ( h + h h+ h + h + h + h h h

    )

    = g h +1

    2g

    ( h + h h + 2h

    )

    = 12g

    ( h + h h) = 12 g(h; + h; h;

    ).

    (B.20)

    Then, the variation of the Riemann tensor yields

    gR = g

    (

    +

    )= (g), (g), + (g) + (g)

    (g) (g). (B.21)

    Using (B.1), we can again rewrite the partial derivative in terms of a covariant deriv-ative plus terms proportional to the connection

    (g), = (g); (g) + (g) + (g). (B.22)

    Substituting this expression (and the same term interchanging and ) in (B.21),we find that all terms pr