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Curvature and the Einstein Equation

This is theMathematica notebookCurvature and the

Einstein Equation available from thebook website. From a given metricg ,

it computes thecomponents of thefollowing : the inverse metric, g

, theChristoffel symbols or affineconnection,

1

2g g g g,

stands for the partial derivative x, the Riemann tensor,

R

,

the Ricci tensor

R R,

the scalar curvature,

R gR,

and the Einstein tensor,

G R 1

2g R.

Youmust input thecovariantcomponents of themetric tensor g by editing the relevant input line in thisMathematica

notebook. You mayalso wish to change thenames of thecoordinates. Only thenonzero components of theabove quantities

aredisplayed as theoutput. All thecomponents computed are in thecoordinate basis in which themetric wasspecified.

Clearing the values of symbols :

First clear any values that may already have been assigned to the names of the various objects to be calculated. The names of the

coordinates that you will use are also cleared.

Clearcoord, metric, inversemetric, affine, riemann, ricci, scalar, einstein, r, , , t

Setting the dimension

The dimension n of the spacetime (or space) must be set :

n 4

4

Defining a list of coordinates :

The example given here is the Schwarzschild metric. The coordinates choice of Schwarzschild is appropriate for this spherically

symmetric spacetime.

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coord r, , , t

r, , , t

Youcanchange thenames of thecoordinates by simplyediting thedefinition ofcoord, for example,

to coord x, y, z, t, when another setof coordinates names is more appropriate. In this program

indices ranges over 1 to n. Thus for spacetime they range from 1 to 4 and x4

isthesame asx0

used in the text.

Defining the metric:

Input the metric as a list of list, i.e., as a matrix. You can input the components of any metric here, but you must specify them as

explicit functions of the coordinates.

wormhole 1, 0, 0, 0, 0, b2 r2, 0, 0, 0, 0, b2 r2 Sin2, 0, 0, 0, 0, 1;

schwarzschild

1 2 m r^1, 0, 0, 0, 0, r^2, 0, 0, 0, 0, r ^2 Sin, 0, 0, 0, 0, 1 2 m r;

metric wormhole

1, 0, 0, 0, 0, b2 r2, 0, 0, 0, 0, b2 r2 Sin2, 0, 0, 0, 0, 1You can also display this in matrix form.

metric MatrixForm

1 0 0 0

0 b2 r2 0 0

0 0 b2 r2 Sin2 00 0 0 1

Note :

It is important not to use the symbols, i, j, k, l, s, or n as constants or coordinates in the metric that you specify above. The reason

is that the first five of those symbols are used as summation or table indices in the calculations done below, and n is the dimen-

sion of space. For example, ifm were used as a summation or table index below, then you would get the wrong answer for the

present metric because m in the metric would be treated as an index, rather than as the mass.

Calculating the inverse metric :

The inverse metric is obtained through matrix inversion.

inversemetric SimplifyInversemetric

1, 0, 0, 0, 0,1

b2 r2

, 0, 0, 0, 0,Csc2

b2 r2

, 0, 0, 0, 0, 1

This can also be displayed in matrix form.

2 Curvature and the Einstein Equation.nb

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inversemetric MatrixForm

1 0 0 0

01

b2r20 0

0 0Csc2

b2r20

0 0 0 1

Calculating the Christoffel symbols :

The calculation of the components of the Christoffel symbols is done by transcribing the definition given earlier into the notation

of Mathematica and using the Mathematica functions for taking partial derivatives, Sum for summing over repeated indices,

Table for forming a list of components, and Simplify for simplifying the result.

affine :

affine SimplifyTable1 2 Suminversemetrici, s Dmetrics, j, coordk Dmetrics, k, coordj Dmetricj, k, coords,

s, 1, n, i, 1, n, j, 1, n, k, 1, n

Displaying the Christoffel symbols :

The nonzero Christoffel symbols are displayed below. You need not follow the details of constructing the functions that we use

for that purpose. In the output the symbol [1, 2, 3] stands for 123. Because the Christoffel symbols are symmetric under

interchange of the last two indices, only the independent components are displayed.

listaffine : TableIfUnsameQaffinei, j, k, 0,ToStringi, j, k, affinei, j, k, i, 1, n, j, 1, n, k, 1, j

TableFormPartitionDeleteCasesFlattenlistaffine, Null, 2, TableSpacing 2, 2

1, 2, 2 r1, 3, 3 r Sin2

2, 2, 1r

b2r2

2, 3, 3 Cos Sin3, 3, 1 r

b2r2

3, 3, 2 Cot

Calculating and displaying the Riemann tensor :

The components of the Riemann tensor,R, are calculated using the definition given above.

riemann :

riemann SimplifyTableDaffinei, j, l, coordk Daffinei, j, k, coordl

Sumaffines, j, l affinei, k, s affines, j, k affinei, l, s, s, 1, n,i, 1, n, j, 1, n, k, 1, n, l, 1, n

The nonzero components are displayed by the following functions. In the output, the symbol R[1, 2, 1, 3] stands for R1213, and

similarly for the other components. You can obtain R[1, 2, 3, 1] from R[1, 2, 1, 3] using the antisymmetry of the Riemann tensor

under exchange of the last two indices. The antisymmetry under exchange of the first two indices ofR is not evident in the

output because the components ofR are the displayed.

Curvature and the Einstein Equation.nb 3

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listriemann :

TableIfUnsameQriemanni, j, k, l, 0, ToStringRi, j, k, l, riemanni, j, k, l,i, 1, n, j, 1, n, k, 1, n, l, 1, k 1

TableFormPartitionDeleteCasesFlattenlistriemann, Null, 2, TableSpacing 2, 2

R1, 2, 2, 1 mr

R1, 3, 3, 1 m Sin2r

R1, 4, 4, 1 2 m 2 mrr4

R2, 1, 2, 1 m2 mr r2

R2, 3, 3, 2 2 m Sin2r

R2, 4, 4, 2 m 2 mrr4

R3, 1, 3, 1 m2 mr r2

R3, 2, 3, 2 2 mr

R3, 4, 4, 3m

2 mr

r4

R4, 1, 4, 1 2 m2 mr r2

R4, 2, 4, 2 mr

R4, 3, 4, 3 m Sin2r

Calculating and displaying the Ricci tensor :

The Ricci tensor R was defined by summing the first and third indices of the Riemann tensor (which has the first index already

raised).

ricci : ricci SimplifyTableSumriemanni, j, i, l, i, 1, n, j, 1, n, l, 1, n

Next we display thenonzero components. In theoutputR1, 2 denotes R12, andsimilarly for theother components.

listricci :

TableIfUnsameQriccij, l, 0, ToStringRj, l, riccij, l, j, 1, n, l, 1, n

TableFormPartitionDeleteCasesFlattenlistricci, Null, 2, TableSpacing 2, 2

A vanishing table (as with the Schwarzschild metric example) means that the vacuum Einstein equation is satisfied.

Calculating the scalar curvature :

The scalar curvature R is calculated using the inverse mtric and the Ricci tensor. The result is displayed in the output line.

scalar SimplifySuminversemetrici, j riccii, j, i, 1, n, j, 1, n

0

4 Curvature and the Einstein Equation.nb

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Calculating the Einstein tensor :

The Einstein tensor, G = R -1

2gR, is found from the tensors already calculated.

einstein : einstein Simplifyricci 1 2 scalar metric

listeinstein : TableIfUnsameQeinsteinj, l, 0, ToStringGj, l, einsteinj, l, j, 1, n, l, 1, j

TableFormPartitionDeleteCasesFlattenlisteinstein, Null, 2, TableSpacing 2, 2

A vanishing table means that the vacuum Einstein equation is satisfied!

Curvature and the Einstein Equation.nb 5

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