Special objects

In general relativity calculations a few objects play an important rôle. They are the metric tensor, the Kronecker delta, the Christoffel symbol, the Levi-Civita symbol and tensor, the Riemann tensor and the Weyl tensor. All of them may be introduced to the system by separate declarations with arbitrary identifiers. If the used identifier has not been defined as tensor (or symbol) yet then it is done automatically at each command. The names of the indices are arbitrary in the declaration. The appropriate symmetry properties of the objects are also declared:

(



The metric tensor is symmetric and its covariant derivative is equal to zero. The inverse of the metric tensor is denoted by $gij$ (G(I,J)) and $gi$_{\>\> j} (G(I,-J)) is equal to the identity map (Kronecker delta).

• Kronecker delta or identity map

  

• Christoffel symbol

  

  The Christoffel symbol is symmetric with respect to its last two indices.

• Levi-Civita tensor and symbol

  

  The Levi-Civita symbol and tensor are fully antisymmetric objects and are permitted to have two or more indices. The number of their indices fixes the dimension of space-time and if both of them are defined then the number of their indices must be consistent. The covariant derivative of the Levi-Civita tensor is equal to zero.

• Riemann tensor (as well as Ricci tensor and curvature scalar)

  

  The syntax of the declaration command RIEMANN TENSOR differs from the others because

  • it introduces the Ricci tensor and the curvature scalar as well
  • there are different definitions of the Riemann and Ricci tensors.
  PLUS means that the Misner-Thorne-Wheeler[2] definition is used; MINUS would mean that the other definition - differing in a minus sign - is used. The first PLUS (or MINUS) relates to the definiton of the Riemann tensor, the second one to the definition of the Ricci tensor. For example, in the case of PLUS,PLUS the Riemann tensor, the Ricci tensor and the curvature scalar are defined as follows

  $A$_i;jk - A_i;kj = R^l_{\>\> ijk} A_l R_ij = R^l_{\>\> ilj} R = R^l_{\>\> l}.

  The Riemann tensor is antisymmetric in its first two and last two indices and symmetric for exchanging of the first and last pairs of indices:

  $R$_ijkl=R_[ij]kl=R_ij[kl] R_ijkl=R_klij.

  The first and second Bianchy identities

  $Ri$_{\>\> [jkl]} = 0 R^ij_{\>\> \>\> [kl;m]} = 0

  can be activated by special commands (see page ).
  The Ricci tensor is symmetric.

• Weyl tensor

  

  The Weyl tensor has all the symmetry properties of the Riemann tensor, it satisfies the first Bianchy identity and is trace-free:

  $Wi$_{\>\> [jkl]} = 0 Wⁱ_{\>\> jik} = 0.

  It can be introduced after declaring the dimension of space-time which must be greater than three (or may be an identifier).

In the case of any special object only one can be defined at the same time: in other words a new declaration always overwrites the previous one. For example the command

defines the tensor G4(-i,-j) as a symmetric tensor and a metric tensor as well. If later in the calculation the command

appears, it has the following effect: it defines G3(-i,-j) as a symmetric tensor and as a metric tensor instead of G4(-i,-j). All the properties of G4(-i,-j) - except that it is a metric tensor - will be kept. The same holds for the other special object definitions too.

The special objects according to the description above can be removed by the commands

namely the tensors and the symbols together with all of their properties that will remain, only they will not play the rôle of the deleted special object any longer.