Physics

Examples

Decompose , to decompose 4D tensorial expressions (free and/or contracted indices) into the space and time parts.

gamma_ , representing the three-dimensional metric tensor, with which the element of spatial distance is defined as  .

Redefine , to redefine the coordinates and the spacetime metric according to changes in the signature from any of the four possible signatures(− + + +), (+ − − −), (+ + + −) and ((− + + +) to any of the other ones.

EnergyMomentum , is a computational representation for the energy-momentum tensor entering Einstein's equations as well as their 3+1 form, the ADMEquations .

Examples

Note the different color for , now a 4D tensor representing the metric of a generic 3-dimensional hypersurface induced by the 4D spacetime metric . All the ThreePlusOne tensors are displayed in black to distinguish them of the corresponding 4D or 3D tensors. The particular hypersurface  operates is parameterized by the Lapse   and the Shift  .

The induced metric is defined in terms of the UnitNormalVector   and the 4D metric  as

where  is defined in terms of the Lapse   and the derivative of a scalar function t that can be interpreted as a global time function

The TimeVector  is defined in terms of the Lapse   and the Shift   and this vector   as

The ExtrinsicCurvature  is defined in terms of the LieDerivative  of  

The metric is also a projection tensor in that it projects 4D tensors into the 3D hypersurface Σ. The definition for any 4D tensor that is also a 3D tensor in Σ, can thus be written directly by contracting their indices with . In the case of Christoffel3 , Ricci3  and Riemann3,  these tensors can be defined by replacing the 4D metric  by  and the 4D Christoffel symbols  by the ThreePlusOne  in the definitions of the corresponding 4D tensors. So, for instance

When working with the ADM formalism, the line element of an arbitrary spacetime metric can be expressed in terms of the differentials of the coordinates , the Lapse , the Shift  and the spatial components of the 3D metric gamma3_ . From this line element one can derive the relation between the Lapse , the spatial part of the Shift , the spatial part of the gamma3_  metric and the  components of the 4D spacetime metric.

For this purpose, define a tensor representing the differentials of the coordinates and an alias  

The expression for the line element in terms of the Lapse  and Shift   is (see [2], eq.(2.123))

Compare this expression with the 3+1 decomposition of the line element in an arbitrary system. To avoid the automatic evaluation of the metric components, work with the inert form of the metric %g_

The second and third terms on the right-hand side are equal

Taking the difference between this expression and the one in terms of the Lapse  and Shift  we get

Taking coefficients, we get equations for the Shift , the Lapse  and the spatial components of the metric gamma3_

Using these equations, these quantities can all be expressed in terms of the time and space components of the 4D metric  and

Examples: Decompose, gamma_

Define  now an arbitrary tensor

So  is a 4D tensor with only one free index, where the position of the time-like component is the position of the different sign in the signature, that you can query about via

To perform a decomposition into space and time, set - for instance - the lowercase latin letters from i to s to represent spaceindices and

Accordingly, the 3+1 decomposition of  is

The 3+1 decomposition of the inert representation %g_[mu,nu] of the 4D spacetime metric; use the inert representation when you do not want the actual components of the metric appearing in the output

Note the position of the component %g_[0, 0], related to the trailing position of the time-like component in the signature .

Compare the decomposition of the 4D inert with the decomposition of the 4D active spacetime metric

Note that in general the 3D space part of  is not equal to the 3D metric  whose definition includes another term (see [1] Landau & Lifshitz, eq.(84.7)).

The 3D space part of  is actually equal to the 3D metric

To derive the formula  for the covariant components of the 3D metric, Decompose into 3+1 the identity

To the side, for illustration purposes, these are the 3 + 1 decompositions, first excluding the repeated indices, then excluding the free indices

Compare with a full decomposition

 is a symmetric matrix of equations involving non-contracted occurrences of ,  and . Isolate, in , , that you input as %g_[~j, ~0], and substitute into

Collect , that you input as %g_[~j, ~i]

Since the right-hand side is the identity matrix and, from , , the expression between parenthesis, multiplied by -1, is the reciprocal of the contravariant 3D metric , that is the covariant 3D metric , in accordance to its definition for the signature

Example: Redefine

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