Physics[Ricci]  The Ricci tensor

Calling Sequence


Ricci[]
convert(expression, Ricci)


Parameters


mu, nu



the indices, as names representing integer numbers between 0 and the spacetime dimension, they can also be the numbers themselves





Description


•

The Ricci[mu, nu], displayed as , is a computational representation for the Ricci tensor, defined in terms of the Riemann tensor as

•

From this definition, and because of the symmetries of the Riemann tensor with respect to interchanging the positions of is indices the Ricci tensor is symmetric with respect to interchanging the position of its indices.

•

Expressing the Riemann tensor in terms of the Christoffel symbols and its derivatives, an equivalent definition is

•

When the indices of Ricci assume integer values they are expected to be between 0 and the spacetime dimension, prefixed by ~ when they are contravariant, and the corresponding value of Ricci is returned. When the indices have symbolic values Ricci returns unevaluated after normalizing its indices taking into account their symmetry property.

•

Computations performed with the Physics package commands take into account Einstein's sum rule for repeated indices  see `.` and Simplify. The distinction between covariant and contravariant indices in the input of tensors is done by prefixing contravariant ones with ~, say as in ~mu; in the output, contravariant indices are displayed as superscripts. For contracted indices, you can enter them one covariant and one contravariant. Note however that  provided that the spacetime metric is galilean (Euclidean or Minkowski), or the object is a tensor also in curvilinear coordinates  this distinction in the input is not relevant, and so contracted indices can be entered as both covariant or both contravariant, in which case they will be automatically rewritten as one covariant and one contravariant. Tensors can have spacetime and space indices at the same time. To change the type of letter used to represent spacetime or space indices see Setup.

•

When Physics is loaded, the default spacetime is galilean, of Minkowski type. During a Maple session, the value of any component of is automatically determined by the value of the spacetime metric and so all the elements of Ricci are automatically zero. To set the spacetime metric to something different use Setup. Also, at least one system of coordinates must be set in order to compute the derivatives entering the definition of the Christoffel symbols, used to construct the Ricci tensor. For that purpose see Coordinates or Setup.

•

By default, when Physics is loaded, the spacetime is set to Minkowski type, and so all the elements of Ricci are automatically zero. To set the spacetime metric to something different use Setup. Also, at least one coordinates system must be set in order to compute the derivatives entering the definition of the Christoffel symbols, used to construct the Ricci tensor and so Ricci. For that purpose see Coordinates or Setup.

•

Besides being indexed with two indices, Ricci accepts three keywords:

–

matrix: (synonym: Matrix, array, Array, or no indices whatsoever, as in Ricci[]) returns a Matrix that when indexed with numerical values from 1 to the dimension of spacetime it returns the value of each of the components of Ricci. If this keyword is passed together with indices, that can be covariant or contravariant, the resulting array takes into account the character of the indices.

–

nonzero: returns a set of equations, with the lefthandside as a sequence of two positive numbers identifying the element of and the corresponding value in the righthandside. Note that this set is actually the output of the ArrayElems command when passing to it the Array obtained with the keyword array.

–

scalar: returns , the trace of the Ricci tensor, also known as the Ricci scalar, an invariant.

•

Some automatic checking and normalization are carried out each time you enter Ricci[...]. The checking is concerned with possible syntax errors. The automatic normalization takes into account the symmetry of Ricci[mu,nu] with respect to interchanging the positions of the indices mu and nu.

•

The %Ricci command is the inert form of Ricci, so it represents the same mathematical operation but without performing it. To perform the operation, use value.

•

Sometimes it is convenient to rewrite tensorial expressions involving the Einstein tensor in terms of the Ricci tensor. For this purpose use convert(expression, Ricci)  see the Examples section.



Compatibility


•

The Physics[Ricci] command was introduced in Maple 16.



Examples


>


>


 (1) 
Set up a coordinate system to work with  the first one to be set is automatically taken as the differentiation variables for d_, the covariant derivative D_ and the dAlembertian
>


 (2) 
When Physics is initialized, the default spacetime metric is of Minkowski type. You can see the metric querying Setup, as in Setup(metric);, or directly entering the metric as g_[], with no indices
>


 (3) 
Check the nonzero components of Christoffel, used to construct the Ricci tensor entering the definition of Ricci: because the default spacetime is of Minkowski type, there are none
>


 (4) 
Hence
>


 (5) 
and the same is valid for all the general relativity tensors defined in terms of Christoffel and derivatives of the metric g_.
To set the scenario as a curved spacetime set the metric using Setup, for instance indicating the square of the spacetime interval. In this example, we also choose to work in spherical coordinates, so consider for instance the metric defined by
>


 (6) 
>


 (7) 
>


 (8) 
Now when the indices are not numerical, Ricci returns itself after normalizing its second and third indices taking advantage of the symmetry of its indices
>


 (9) 
>


 (10) 
>


 (11) 
To express Ricci in terms of the Christoffel symbols and its derivatives convert to Christoffel
>


 (12) 
Sometimes it is convenient to rewrite tensorial expressions involving the Einstein tensor in terms of the Ricci tensor
>


 (13) 
>


 (14) 
Check the value of for = = 1
>


 (15) 
Now for with = = 1 (note you enter the value of the contravariant index prefixed by ~)
>


 (16) 
To compute with a representation for Ricci without actually performing the operation, use the inert form %Ricci. To afterwards perform the operation use value
>


 (17) 
>


 (18) 
To have a more compact display in the following examples, suppress the display of the dependency of and have the display of derivatives in jet notation, indexed (see Typesetting)
>


>


 (19) 
The Ricci scalar can be computed entering Ricci[scalar] or Ricci[mu, mu]
>


 (20) 
The nonzero components of the Ricci tensor
>


 (21) 
The Matrix form of
>


 (22) 
As with all the general relativity tensors of the Physics package, you can obtain the same result for the all covariant case by entering the tensor without indices, as in Ricci[]. This is the matrix form of (note the contravariant index , prefixed by ~)
>


 (23) 
To use this matrix, because its components were computed already taking into account the (covariant/contravariant) character of its indices, you do not need to indicate furthermore that character. So for we have
>


>


 (24) 
Compare with the output obtained entering Ricci[~1, 1] in eq (14)
>


 (25) 
>




See Also


`.`, Array, ArrayElems, Christoffel, Coordinates, d_, DifferentialGeometry[Tensor][Ricci], Einstein, Matrix, Physics, Physics conventions, Physics examples, Riemann, Setup, Typesetting, value


References



Landau, L.D., and Lifshitz, E.M. The Classical Theory of Fields, Course of Theoretical Physics Volume 2, fourth revised English edition. Elsevier, 1975.


